1,1,109,131,0.1393317,"\int x^3 \sqrt{b x+c x^2} \, dx","Integrate[x^3*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\frac{105 b^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-105 b^4+70 b^3 c x-56 b^2 c^2 x^2+48 b c^3 x^3+384 c^4 x^4\right)\right)}{1920 c^{9/2}}","\frac{7 b^5 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{128 c^{9/2}}-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left(b x+c x^2\right)^{3/2}}{48 c^3}-\frac{7 b x \left(b x+c x^2\right)^{3/2}}{40 c^2}+\frac{x^2 \left(b x+c x^2\right)^{3/2}}{5 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^4 + 70*b^3*c*x - 56*b^2*c^2*x^2 + 48*b*c^3*x^3 + 384*c^4*x^4) + (105*b^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(1920*c^(9/2))","A",1
2,1,98,105,0.1204374,"\int x^2 \sqrt{b x+c x^2} \, dx","Integrate[x^2*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^3-10 b^2 c x+8 b c^2 x^2+48 c^3 x^3\right)-\frac{15 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{192 c^{7/2}}","-\frac{5 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{7/2}}+\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left(b x+c x^2\right)^{3/2}}{24 c^2}+\frac{x \left(b x+c x^2\right)^{3/2}}{4 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3 - 10*b^2*c*x + 8*b*c^2*x^2 + 48*c^3*x^3) - (15*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(192*c^(7/2))","A",1
3,1,87,81,0.1091925,"\int x \sqrt{b x+c x^2} \, dx","Integrate[x*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 b^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-3 b^2+2 b c x+8 c^2 x^2\right)\right)}{24 c^{5/2}}","\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{\left(b x+c x^2\right)^{3/2}}{3 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^2 + 2*b*c*x + 8*c^2*x^2) + (3*b^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(24*c^(5/2))","A",1
4,1,74,60,0.0958506,"\int \sqrt{b x+c x^2} \, dx","Integrate[Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} (b+2 c x)-\frac{b^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{4 c^{3/2}}","\frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 c^{3/2}}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(b + 2*c*x) - (b^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(4*c^(3/2))","A",1
5,1,66,42,0.0923924,"\int \frac{\sqrt{b x+c x^2}}{x} \, dx","Integrate[Sqrt[b*x + c*x^2]/x,x]","\sqrt{x (b+c x)} \left(\frac{b^{3/2} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{c} \sqrt{x} (b+c x)}+1\right)","\sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{\sqrt{c}}",1,"Sqrt[x*(b + c*x)]*(1 + (b^(3/2)*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[c]*Sqrt[x]*(b + c*x)))","A",1
6,1,63,47,0.0815987,"\int \frac{\sqrt{b x+c x^2}}{x^2} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^2,x]","\frac{2 \sqrt{x (b+c x)} \left(\frac{\sqrt{c} \sqrt{x} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}-1\right)}{x}","2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)-\frac{2 \sqrt{b x+c x^2}}{x}",1,"(2*Sqrt[x*(b + c*x)]*(-1 + (Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/x","A",1
7,1,21,23,0.0096839,"\int \frac{\sqrt{b x+c x^2}}{x^3} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^3,x]","-\frac{2 (x (b+c x))^{3/2}}{3 b x^3}","-\frac{2 \left(b x+c x^2\right)^{3/2}}{3 b x^3}",1,"(-2*(x*(b + c*x))^(3/2))/(3*b*x^3)","A",1
8,1,29,48,0.010591,"\int \frac{\sqrt{b x+c x^2}}{x^4} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^4,x]","\frac{2 (x (b+c x))^{3/2} (2 c x-3 b)}{15 b^2 x^4}","\frac{4 c \left(b x+c x^2\right)^{3/2}}{15 b^2 x^3}-\frac{2 \left(b x+c x^2\right)^{3/2}}{5 b x^4}",1,"(2*(x*(b + c*x))^(3/2)*(-3*b + 2*c*x))/(15*b^2*x^4)","A",1
9,1,51,74,0.0108397,"\int \frac{\sqrt{b x+c x^2}}{x^5} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^5,x]","-\frac{2 \sqrt{x (b+c x)} \left(15 b^3+3 b^2 c x-4 b c^2 x^2+8 c^3 x^3\right)}{105 b^3 x^4}","-\frac{16 c^2 \left(b x+c x^2\right)^{3/2}}{105 b^3 x^3}+\frac{8 c \left(b x+c x^2\right)^{3/2}}{35 b^2 x^4}-\frac{2 \left(b x+c x^2\right)^{3/2}}{7 b x^5}",1,"(-2*Sqrt[x*(b + c*x)]*(15*b^3 + 3*b^2*c*x - 4*b*c^2*x^2 + 8*c^3*x^3))/(105*b^3*x^4)","A",1
10,1,62,100,0.0126814,"\int \frac{\sqrt{b x+c x^2}}{x^6} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^6,x]","\frac{2 \sqrt{x (b+c x)} \left(-35 b^4-5 b^3 c x+6 b^2 c^2 x^2-8 b c^3 x^3+16 c^4 x^4\right)}{315 b^4 x^5}","\frac{32 c^3 \left(b x+c x^2\right)^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left(b x+c x^2\right)^{3/2}}{105 b^3 x^4}+\frac{4 c \left(b x+c x^2\right)^{3/2}}{21 b^2 x^5}-\frac{2 \left(b x+c x^2\right)^{3/2}}{9 b x^6}",1,"(2*Sqrt[x*(b + c*x)]*(-35*b^4 - 5*b^3*c*x + 6*b^2*c^2*x^2 - 8*b*c^3*x^3 + 16*c^4*x^4))/(315*b^4*x^5)","A",1
11,1,73,126,0.0131627,"\int \frac{\sqrt{b x+c x^2}}{x^7} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^7,x]","-\frac{2 \sqrt{x (b+c x)} \left(315 b^5+35 b^4 c x-40 b^3 c^2 x^2+48 b^2 c^3 x^3-64 b c^4 x^4+128 c^5 x^5\right)}{3465 b^5 x^6}","-\frac{256 c^4 \left(b x+c x^2\right)^{3/2}}{3465 b^5 x^3}+\frac{128 c^3 \left(b x+c x^2\right)^{3/2}}{1155 b^4 x^4}-\frac{32 c^2 \left(b x+c x^2\right)^{3/2}}{231 b^3 x^5}+\frac{16 c \left(b x+c x^2\right)^{3/2}}{99 b^2 x^6}-\frac{2 \left(b x+c x^2\right)^{3/2}}{11 b x^7}",1,"(-2*Sqrt[x*(b + c*x)]*(315*b^5 + 35*b^4*c*x - 40*b^3*c^2*x^2 + 48*b^2*c^3*x^3 - 64*b*c^4*x^4 + 128*c^5*x^5))/(3465*b^5*x^6)","A",1
12,1,120,134,0.1515586,"\int x^2 \left(b x+c x^2\right)^{3/2} \, dx","Integrate[x^2*(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\frac{105 b^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-105 b^5+70 b^4 c x-56 b^3 c^2 x^2+48 b^2 c^3 x^3+1664 b c^4 x^4+1280 c^5 x^5\right)\right)}{7680 c^{9/2}}","\frac{7 b^6 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{512 c^{9/2}}-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left(b x+c x^2\right)^{3/2}}{192 c^3}-\frac{7 b \left(b x+c x^2\right)^{5/2}}{60 c^2}+\frac{x \left(b x+c x^2\right)^{5/2}}{6 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^5 + 70*b^4*c*x - 56*b^3*c^2*x^2 + 48*b^2*c^3*x^3 + 1664*b*c^4*x^4 + 1280*c^5*x^5) + (105*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(7680*c^(9/2))","A",1
13,1,109,110,0.1349963,"\int x \left(b x+c x^2\right)^{3/2} \, dx","Integrate[x*(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^4-10 b^3 c x+8 b^2 c^2 x^2+176 b c^3 x^3+128 c^4 x^4\right)-\frac{15 b^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{640 c^{7/2}}","-\frac{3 b^5 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{128 c^{7/2}}+\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left(b x+c x^2\right)^{3/2}}{16 c^2}+\frac{\left(b x+c x^2\right)^{5/2}}{5 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^4 - 10*b^3*c*x + 8*b^2*c^2*x^2 + 176*b*c^3*x^3 + 128*c^4*x^4) - (15*b^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(640*c^(7/2))","A",1
14,1,98,89,0.1218301,"\int \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right)\right)}{64 c^{5/2}}","\frac{3 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left(b x+c x^2\right)^{3/2}}{8 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x^3) + (3*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(64*c^(5/2))","A",1
15,1,87,78,0.1096589,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x,x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(3 b^2+14 b c x+8 c^2 x^2\right)-\frac{3 b^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{24 c^{3/2}}","-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{3/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left(b x+c x^2\right)^{3/2}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(3*b^2 + 14*b*c*x + 8*c^2*x^2) - (3*b^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(24*c^(3/2))","A",1
16,1,69,72,0.095879,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^2} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^2,x]","\frac{1}{4} \sqrt{x (b+c x)} \left(\frac{3 b^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1}}+5 b+2 c x\right)","\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left(b x+c x^2\right)^{3/2}}{2 x}",1,"(Sqrt[x*(b + c*x)]*(5*b + 2*c*x + (3*b^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b])))/4","A",1
17,1,46,64,0.0117213,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^3} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^3,x]","-\frac{2 b \sqrt{x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x}{b}\right)}{x \sqrt{\frac{c x}{b}+1}}","-\frac{2 \left(b x+c x^2\right)^{3/2}}{x^2}+3 c \sqrt{b x+c x^2}+3 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)",1,"(-2*b*Sqrt[x*(b + c*x)]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x)/b)])/(x*Sqrt[1 + (c*x)/b])","C",1
18,1,48,68,0.0120992,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^4} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^4,x]","-\frac{2 b \sqrt{x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x}{b}\right)}{3 x^2 \sqrt{\frac{c x}{b}+1}}","2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left(b x+c x^2\right)^{3/2}}{3 x^3}",1,"(-2*b*Sqrt[x*(b + c*x)]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((c*x)/b)])/(3*x^2*Sqrt[1 + (c*x)/b])","C",1
19,1,21,23,0.0109322,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^5} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^5,x]","-\frac{2 (x (b+c x))^{5/2}}{5 b x^5}","-\frac{2 \left(b x+c x^2\right)^{5/2}}{5 b x^5}",1,"(-2*(x*(b + c*x))^(5/2))/(5*b*x^5)","A",1
20,1,29,48,0.0137304,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^6} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^6,x]","\frac{2 (x (b+c x))^{5/2} (2 c x-5 b)}{35 b^2 x^6}","\frac{4 c \left(b x+c x^2\right)^{5/2}}{35 b^2 x^5}-\frac{2 \left(b x+c x^2\right)^{5/2}}{7 b x^6}",1,"(2*(x*(b + c*x))^(5/2)*(-5*b + 2*c*x))/(35*b^2*x^6)","A",1
21,1,40,74,0.0145114,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^7} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^7,x]","-\frac{2 (x (b+c x))^{5/2} \left(35 b^2-20 b c x+8 c^2 x^2\right)}{315 b^3 x^7}","-\frac{16 c^2 \left(b x+c x^2\right)^{5/2}}{315 b^3 x^5}+\frac{8 c \left(b x+c x^2\right)^{5/2}}{63 b^2 x^6}-\frac{2 \left(b x+c x^2\right)^{5/2}}{9 b x^7}",1,"(-2*(x*(b + c*x))^(5/2)*(35*b^2 - 20*b*c*x + 8*c^2*x^2))/(315*b^3*x^7)","A",1
22,1,51,100,0.0167161,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^8} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^8,x]","\frac{2 (x (b+c x))^{5/2} \left(-105 b^3+70 b^2 c x-40 b c^2 x^2+16 c^3 x^3\right)}{1155 b^4 x^8}","\frac{32 c^3 \left(b x+c x^2\right)^{5/2}}{1155 b^4 x^5}-\frac{16 c^2 \left(b x+c x^2\right)^{5/2}}{231 b^3 x^6}+\frac{4 c \left(b x+c x^2\right)^{5/2}}{33 b^2 x^7}-\frac{2 \left(b x+c x^2\right)^{5/2}}{11 b x^8}",1,"(2*(x*(b + c*x))^(5/2)*(-105*b^3 + 70*b^2*c*x - 40*b*c^2*x^2 + 16*c^3*x^3))/(1155*b^4*x^8)","A",1
23,1,62,126,0.0182047,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^9} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^9,x]","-\frac{2 (x (b+c x))^{5/2} \left(1155 b^4-840 b^3 c x+560 b^2 c^2 x^2-320 b c^3 x^3+128 c^4 x^4\right)}{15015 b^5 x^9}","-\frac{256 c^4 \left(b x+c x^2\right)^{5/2}}{15015 b^5 x^5}+\frac{128 c^3 \left(b x+c x^2\right)^{5/2}}{3003 b^4 x^6}-\frac{32 c^2 \left(b x+c x^2\right)^{5/2}}{429 b^3 x^7}+\frac{16 c \left(b x+c x^2\right)^{5/2}}{143 b^2 x^8}-\frac{2 \left(b x+c x^2\right)^{5/2}}{13 b x^9}",1,"(-2*(x*(b + c*x))^(5/2)*(1155*b^4 - 840*b^3*c*x + 560*b^2*c^2*x^2 - 320*b*c^3*x^3 + 128*c^4*x^4))/(15015*b^5*x^9)","A",1
24,1,142,163,0.1923999,"\int x^2 \left(a x+b x^2\right)^{5/2} \, dx","Integrate[x^2*(a*x + b*x^2)^(5/2),x]","\frac{\sqrt{x (a+b x)} \left(\sqrt{b} \left(315 a^7-210 a^6 b x+168 a^5 b^2 x^2-144 a^4 b^3 x^3+128 a^3 b^4 x^4+20736 a^2 b^5 x^5+33792 a b^6 x^6+14336 b^7 x^7\right)-\frac{315 a^{15/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right)}{114688 b^{11/2}}","-\frac{45 a^8 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{16384 b^{11/2}}+\frac{45 a^6 (a+2 b x) \sqrt{a x+b x^2}}{16384 b^5}-\frac{15 a^4 (a+2 b x) \left(a x+b x^2\right)^{3/2}}{2048 b^4}+\frac{3 a^2 (a+2 b x) \left(a x+b x^2\right)^{5/2}}{128 b^3}-\frac{9 a \left(a x+b x^2\right)^{7/2}}{112 b^2}+\frac{x \left(a x+b x^2\right)^{7/2}}{8 b}",1,"(Sqrt[x*(a + b*x)]*(Sqrt[b]*(315*a^7 - 210*a^6*b*x + 168*a^5*b^2*x^2 - 144*a^4*b^3*x^3 + 128*a^3*b^4*x^4 + 20736*a^2*b^5*x^5 + 33792*a*b^6*x^6 + 14336*b^7*x^7) - (315*a^(15/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(114688*b^(11/2))","A",1
25,1,131,139,0.1684116,"\int x \left(a x+b x^2\right)^{5/2} \, dx","Integrate[x*(a*x + b*x^2)^(5/2),x]","\frac{\sqrt{x (a+b x)} \left(\frac{105 a^{13/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}+\sqrt{b} \left(-105 a^6+70 a^5 b x-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+7424 a b^5 x^5+3072 b^6 x^6\right)\right)}{21504 b^{9/2}}","\frac{5 a^7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{1024 b^{9/2}}-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left(a x+b x^2\right)^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left(a x+b x^2\right)^{5/2}}{24 b^2}+\frac{\left(a x+b x^2\right)^{7/2}}{7 b}",1,"(Sqrt[x*(a + b*x)]*(Sqrt[b]*(-105*a^6 + 70*a^5*b*x - 56*a^4*b^2*x^2 + 48*a^3*b^3*x^3 + 4736*a^2*b^4*x^4 + 7424*a*b^5*x^5 + 3072*b^6*x^6) + (105*a^(13/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(21504*b^(9/2))","A",1
26,1,120,118,0.170802,"\int \left(a x+b x^2\right)^{5/2} \, dx","Integrate[(a*x + b*x^2)^(5/2),x]","\frac{\sqrt{x (a+b x)} \left(\sqrt{b} \left(15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right)-\frac{15 a^{11/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right)}{1536 b^{7/2}}","-\frac{5 a^6 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{512 b^{7/2}}+\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left(a x+b x^2\right)^{3/2}}{192 b^2}+\frac{(a+2 b x) \left(a x+b x^2\right)^{5/2}}{12 b}",1,"(Sqrt[x*(a + b*x)]*(Sqrt[b]*(15*a^5 - 10*a^4*b*x + 8*a^3*b^2*x^2 + 432*a^2*b^3*x^3 + 640*a*b^4*x^4 + 256*b^5*x^5) - (15*a^(11/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(1536*b^(7/2))","A",1
27,1,109,107,0.1408348,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x,x]","\frac{\sqrt{x (a+b x)} \left(\frac{15 a^{9/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}+\sqrt{b} \left(-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right)\right)}{640 b^{5/2}}","\frac{3 a^5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{128 b^{5/2}}-\frac{3 a^3 (a+2 b x) \sqrt{a x+b x^2}}{128 b^2}+\frac{a (a+2 b x) \left(a x+b x^2\right)^{3/2}}{16 b}+\frac{1}{5} \left(a x+b x^2\right)^{5/2}",1,"(Sqrt[x*(a + b*x)]*(Sqrt[b]*(-15*a^4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 336*a*b^3*x^3 + 128*b^4*x^4) + (15*a^(9/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(640*b^(5/2))","A",1
28,1,98,101,0.1248031,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^2} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^2,x]","\frac{\sqrt{x (a+b x)} \left(\sqrt{b} \left(15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right)-\frac{15 a^{7/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right)}{192 b^{3/2}}","-\frac{5 a^4 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left(a x+b x^2\right)^{3/2}+\frac{\left(a x+b x^2\right)^{5/2}}{4 x}",1,"(Sqrt[x*(a + b*x)]*(Sqrt[b]*(15*a^3 + 118*a^2*b*x + 136*a*b^2*x^2 + 48*b^3*x^3) - (15*a^(7/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(192*b^(3/2))","A",1
29,1,80,94,0.1174561,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^3} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^3,x]","\frac{1}{24} \sqrt{x (a+b x)} \left(\frac{15 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x} \sqrt{\frac{b x}{a}+1}}+33 a^2+26 a b x+8 b^2 x^2\right)","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left(a x+b x^2\right)^{5/2}}{x^2}-\frac{5}{3} b \left(a x+b x^2\right)^{3/2}",1,"(Sqrt[x*(a + b*x)]*(33*a^2 + 26*a*b*x + 8*b^2*x^2 + (15*a^(5/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x]*Sqrt[1 + (b*x)/a])))/24","A",1
30,1,48,92,0.0118702,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^4} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^4,x]","-\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x}{a}\right)}{x \sqrt{\frac{b x}{a}+1}}","\frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)+\frac{5 b \left(a x+b x^2\right)^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2}-\frac{2 \left(a x+b x^2\right)^{5/2}}{x^3}",1,"(-2*a^2*Sqrt[x*(a + b*x)]*Hypergeometric2F1[-5/2, -1/2, 1/2, -((b*x)/a)])/(x*Sqrt[1 + (b*x)/a])","C",1
31,1,50,89,0.0125139,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^5} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^5,x]","-\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{a}\right)}{3 x^2 \sqrt{\frac{b x}{a}+1}}","5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)+5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left(a x+b x^2\right)^{3/2}}{3 x^2}-\frac{2 \left(a x+b x^2\right)^{5/2}}{3 x^4}",1,"(-2*a^2*Sqrt[x*(a + b*x)]*Hypergeometric2F1[-5/2, -3/2, -1/2, -((b*x)/a)])/(3*x^2*Sqrt[1 + (b*x)/a])","C",1
32,1,50,91,0.0126298,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^6} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^6,x]","-\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x}{a}\right)}{5 x^3 \sqrt{\frac{b x}{a}+1}}","2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)-\frac{2 b^2 \sqrt{a x+b x^2}}{x}-\frac{2 \left(a x+b x^2\right)^{5/2}}{5 x^5}-\frac{2 b \left(a x+b x^2\right)^{3/2}}{3 x^3}",1,"(-2*a^2*Sqrt[x*(a + b*x)]*Hypergeometric2F1[-5/2, -5/2, -3/2, -((b*x)/a)])/(5*x^3*Sqrt[1 + (b*x)/a])","C",1
33,1,21,23,0.0145901,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^7} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^7,x]","-\frac{2 (x (a+b x))^{7/2}}{7 a x^7}","-\frac{2 \left(a x+b x^2\right)^{7/2}}{7 a x^7}",1,"(-2*(x*(a + b*x))^(7/2))/(7*a*x^7)","A",1
34,1,36,48,0.0116265,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^8} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^8,x]","\frac{2 (a+b x)^3 \sqrt{x (a+b x)} (2 b x-7 a)}{63 a^2 x^5}","\frac{4 b \left(a x+b x^2\right)^{7/2}}{63 a^2 x^7}-\frac{2 \left(a x+b x^2\right)^{7/2}}{9 a x^8}",1,"(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-7*a + 2*b*x))/(63*a^2*x^5)","A",1
35,1,47,74,0.0126814,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^9} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^9,x]","-\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left(63 a^2-28 a b x+8 b^2 x^2\right)}{693 a^3 x^6}","-\frac{16 b^2 \left(a x+b x^2\right)^{7/2}}{693 a^3 x^7}+\frac{8 b \left(a x+b x^2\right)^{7/2}}{99 a^2 x^8}-\frac{2 \left(a x+b x^2\right)^{7/2}}{11 a x^9}",1,"(-2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(63*a^2 - 28*a*b*x + 8*b^2*x^2))/(693*a^3*x^6)","A",1
36,1,58,100,0.0151739,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^{10}} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^10,x]","\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left(-231 a^3+126 a^2 b x-56 a b^2 x^2+16 b^3 x^3\right)}{3003 a^4 x^7}","\frac{32 b^3 \left(a x+b x^2\right)^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left(a x+b x^2\right)^{7/2}}{429 a^3 x^8}+\frac{12 b \left(a x+b x^2\right)^{7/2}}{143 a^2 x^9}-\frac{2 \left(a x+b x^2\right)^{7/2}}{13 a x^{10}}",1,"(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-231*a^3 + 126*a^2*b*x - 56*a*b^2*x^2 + 16*b^3*x^3))/(3003*a^4*x^7)","A",1
37,1,69,126,0.0154519,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^{11}} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^11,x]","-\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left(3003 a^4-1848 a^3 b x+1008 a^2 b^2 x^2-448 a b^3 x^3+128 b^4 x^4\right)}{45045 a^5 x^8}","-\frac{256 b^4 \left(a x+b x^2\right)^{7/2}}{45045 a^5 x^7}+\frac{128 b^3 \left(a x+b x^2\right)^{7/2}}{6435 a^4 x^8}-\frac{32 b^2 \left(a x+b x^2\right)^{7/2}}{715 a^3 x^9}+\frac{16 b \left(a x+b x^2\right)^{7/2}}{195 a^2 x^{10}}-\frac{2 \left(a x+b x^2\right)^{7/2}}{15 a x^{11}}",1,"(-2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(3003*a^4 - 1848*a^3*b*x + 1008*a^2*b^2*x^2 - 448*a*b^3*x^3 + 128*b^4*x^4))/(45045*a^5*x^8)","A",1
38,1,80,152,0.0173076,"\int \frac{\left(a x+b x^2\right)^{5/2}}{x^{12}} \, dx","Integrate[(a*x + b*x^2)^(5/2)/x^12,x]","\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left(-9009 a^5+6006 a^4 b x-3696 a^3 b^2 x^2+2016 a^2 b^3 x^3-896 a b^4 x^4+256 b^5 x^5\right)}{153153 a^6 x^9}","\frac{512 b^5 \left(a x+b x^2\right)^{7/2}}{153153 a^6 x^7}-\frac{256 b^4 \left(a x+b x^2\right)^{7/2}}{21879 a^5 x^8}+\frac{64 b^3 \left(a x+b x^2\right)^{7/2}}{2431 a^4 x^9}-\frac{32 b^2 \left(a x+b x^2\right)^{7/2}}{663 a^3 x^{10}}+\frac{4 b \left(a x+b x^2\right)^{7/2}}{51 a^2 x^{11}}-\frac{2 \left(a x+b x^2\right)^{7/2}}{17 a x^{12}}",1,"(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-9009*a^5 + 6006*a^4*b*x - 3696*a^3*b^2*x^2 + 2016*a^2*b^3*x^3 - 896*a*b^4*x^4 + 256*b^5*x^5))/(153153*a^6*x^9)","A",1
39,1,39,50,0.0466382,"\int x \sqrt{2 x-x^2} \, dx","Integrate[x*Sqrt[2*x - x^2],x]","\frac{1}{6} \sqrt{-((x-2) x)} \left(2 x^2-x-3\right)-\sin ^{-1}\left(\sqrt{1-\frac{x}{2}}\right)","-\frac{1}{3} \left(2 x-x^2\right)^{3/2}-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{2} \sin ^{-1}(1-x)",1,"(Sqrt[-((-2 + x)*x)]*(-3 - x + 2*x^2))/6 - ArcSin[Sqrt[1 - x/2]]","A",1
40,1,63,52,0.0392001,"\int x \sqrt{3 x-4 x^2} \, dx","Integrate[x*Sqrt[3*x - 4*x^2],x]","\frac{2 x \left(-512 x^3+480 x^2+36 x-81\right)-81 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left(\sqrt{1-\frac{4 x}{3}}\right)}{768 \sqrt{-x (4 x-3)}}","-\frac{1}{12} \left(3 x-4 x^2\right)^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left(1-\frac{8 x}{3}\right)",1,"(2*x*(-81 + 36*x + 480*x^2 - 512*x^3) - 81*Sqrt[3 - 4*x]*Sqrt[x]*ArcSin[Sqrt[1 - (4*x)/3]])/(768*Sqrt[-(x*(-3 + 4*x))])","A",1
41,1,43,48,0.031091,"\int x \sqrt{x+x^2} \, dx","Integrate[x*Sqrt[x + x^2],x]","\frac{1}{24} \sqrt{x (x+1)} \left(8 x^2+2 x+\frac{3 \sinh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x+1} \sqrt{x}}-3\right)","\frac{1}{3} \left(x^2+x\right)^{3/2}-\frac{1}{8} (2 x+1) \sqrt{x^2+x}+\frac{1}{8} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2+x}}\right)",1,"(Sqrt[x*(1 + x)]*(-3 + 2*x + 8*x^2 + (3*ArcSinh[Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x])))/24","A",1
42,1,98,128,0.1891381,"\int \frac{x^4}{\sqrt{b x+c x^2}} \, dx","Integrate[x^4/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\frac{105 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-105 b^3+70 b^2 c x-56 b c^2 x^2+48 c^3 x^3\right)\right)}{192 c^{9/2}}","\frac{35 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{9/2}}-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^3 + 70*b^2*c*x - 56*b*c^2*x^2 + 48*c^3*x^3) + (105*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(192*c^(9/2))","A",1
43,1,87,102,0.1636785,"\int \frac{x^3}{\sqrt{b x+c x^2}} \, dx","Integrate[x^3/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^2-10 b c x+8 c^2 x^2\right)-\frac{15 b^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{24 c^{7/2}}","-\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{7/2}}+\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^2 - 10*b*c*x + 8*c^2*x^2) - (15*b^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(24*c^(7/2))","A",1
44,1,88,76,0.0490766,"\int \frac{x^2}{\sqrt{b x+c x^2}} \, dx","Integrate[x^2/Sqrt[b*x + c*x^2],x]","\frac{3 b^{5/2} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)+\sqrt{c} x \left(-3 b^2-b c x+2 c^2 x^2\right)}{4 c^{5/2} \sqrt{x (b+c x)}}","\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 c^{5/2}}-\frac{3 b \sqrt{b x+c x^2}}{4 c^2}+\frac{x \sqrt{b x+c x^2}}{2 c}",1,"(Sqrt[c]*x*(-3*b^2 - b*c*x + 2*c^2*x^2) + 3*b^(5/2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*c^(5/2)*Sqrt[x*(b + c*x)])","A",1
45,1,71,47,0.040515,"\int \frac{x}{\sqrt{b x+c x^2}} \, dx","Integrate[x/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{c} x (b+c x)-b^{3/2} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{c^{3/2} \sqrt{x (b+c x)}}","\frac{\sqrt{b x+c x^2}}{c}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{3/2}}",1,"(Sqrt[c]*x*(b + c*x) - b^(3/2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(c^(3/2)*Sqrt[x*(b + c*x)])","A",1
46,1,57,28,0.0161892,"\int \frac{1}{\sqrt{b x+c x^2}} \, dx","Integrate[1/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{c} \sqrt{x (b+c x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{\sqrt{c}}",1,"(2*Sqrt[b]*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[c]*Sqrt[x*(b + c*x)])","B",1
47,1,21,21,0.0062134,"\int \frac{1}{x \sqrt{b x+c x^2}} \, dx","Integrate[1/(x*Sqrt[b*x + c*x^2]),x]","-\frac{2 (b+c x)}{b \sqrt{x (b+c x)}}","-\frac{2 \sqrt{b x+c x^2}}{b x}",1,"(-2*(b + c*x))/(b*Sqrt[x*(b + c*x)])","A",1
48,1,29,48,0.0127764,"\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^2*Sqrt[b*x + c*x^2]),x]","\frac{2 \sqrt{x (b+c x)} (2 c x-b)}{3 b^2 x^2}","\frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2}",1,"(2*Sqrt[x*(b + c*x)]*(-b + 2*c*x))/(3*b^2*x^2)","A",1
49,1,40,74,0.0116996,"\int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^3*Sqrt[b*x + c*x^2]),x]","-\frac{2 \sqrt{x (b+c x)} \left(3 b^2-4 b c x+8 c^2 x^2\right)}{15 b^3 x^3}","-\frac{16 c^2 \sqrt{b x+c x^2}}{15 b^3 x}+\frac{8 c \sqrt{b x+c x^2}}{15 b^2 x^2}-\frac{2 \sqrt{b x+c x^2}}{5 b x^3}",1,"(-2*Sqrt[x*(b + c*x)]*(3*b^2 - 4*b*c*x + 8*c^2*x^2))/(15*b^3*x^3)","A",1
50,1,51,100,0.0135319,"\int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^4*Sqrt[b*x + c*x^2]),x]","\frac{2 \sqrt{x (b+c x)} \left(-5 b^3+6 b^2 c x-8 b c^2 x^2+16 c^3 x^3\right)}{35 b^4 x^4}","\frac{32 c^3 \sqrt{b x+c x^2}}{35 b^4 x}-\frac{16 c^2 \sqrt{b x+c x^2}}{35 b^3 x^2}+\frac{12 c \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{2 \sqrt{b x+c x^2}}{7 b x^4}",1,"(2*Sqrt[x*(b + c*x)]*(-5*b^3 + 6*b^2*c*x - 8*b*c^2*x^2 + 16*c^3*x^3))/(35*b^4*x^4)","A",1
51,1,62,126,0.0162146,"\int \frac{1}{x^5 \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^5*Sqrt[b*x + c*x^2]),x]","-\frac{2 \sqrt{x (b+c x)} \left(35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right)}{315 b^5 x^5}","-\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}",1,"(-2*Sqrt[x*(b + c*x)]*(35*b^4 - 40*b^3*c*x + 48*b^2*c^2*x^2 - 64*b*c^3*x^3 + 128*c^4*x^4))/(315*b^5*x^5)","A",1
52,1,50,97,0.0126637,"\int \frac{x^4}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^4/(b*x + c*x^2)^(3/2),x]","\frac{2 x^4 \sqrt{\frac{c x}{b}+1} \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{c x}{b}\right)}{7 b \sqrt{x (b+c x)}}","\frac{15 b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 c^{7/2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{2 x^3}{c \sqrt{b x+c x^2}}",1,"(2*x^4*Sqrt[1 + (c*x)/b]*Hypergeometric2F1[3/2, 7/2, 9/2, -((c*x)/b)])/(7*b*Sqrt[x*(b + c*x)])","C",1
53,1,50,69,0.0115241,"\int \frac{x^3}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^3/(b*x + c*x^2)^(3/2),x]","\frac{2 x^3 \sqrt{\frac{c x}{b}+1} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{c x}{b}\right)}{5 b \sqrt{x (b+c x)}}","-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{5/2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{2 x^2}{c \sqrt{b x+c x^2}}",1,"(2*x^3*Sqrt[1 + (c*x)/b]*Hypergeometric2F1[3/2, 5/2, 7/2, -((c*x)/b)])/(5*b*Sqrt[x*(b + c*x)])","C",1
54,1,67,48,0.0740457,"\int \frac{x^2}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^2/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)-2 \sqrt{c} x}{c^{3/2} \sqrt{x (b+c x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{3/2}}-\frac{2 x}{c \sqrt{b x+c x^2}}",1,"(-2*Sqrt[c]*x + 2*Sqrt[b]*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(c^(3/2)*Sqrt[x*(b + c*x)])","A",1
55,1,17,19,0.0046239,"\int \frac{x}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x/(b*x + c*x^2)^(3/2),x]","\frac{2 x}{b \sqrt{x (b+c x)}}","\frac{2 x}{b \sqrt{b x+c x^2}}",1,"(2*x)/(b*Sqrt[x*(b + c*x)])","A",1
56,1,22,24,0.0070869,"\int \frac{1}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^(-3/2),x]","-\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}}","-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}",1,"(-2*(b + 2*c*x))/(b^2*Sqrt[x*(b + c*x)])","A",1
57,1,40,51,0.0108676,"\int \frac{1}{x \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(x*(b*x + c*x^2)^(3/2)),x]","\frac{2 \left(-b^2+4 b c x+8 c^2 x^2\right)}{3 b^3 x \sqrt{x (b+c x)}}","\frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}}",1,"(2*(-b^2 + 4*b*c*x + 8*c^2*x^2))/(3*b^3*x*Sqrt[x*(b + c*x)])","A",1
58,1,49,77,0.0121079,"\int \frac{1}{x^2 \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(b*x + c*x^2)^(3/2)),x]","-\frac{2 \left(b^3-2 b^2 c x+8 b c^2 x^2+16 c^3 x^3\right)}{5 b^4 x^2 \sqrt{x (b+c x)}}","-\frac{16 c^2 (b+2 c x)}{5 b^4 \sqrt{b x+c x^2}}+\frac{4 c}{5 b^2 x \sqrt{b x+c x^2}}-\frac{2}{5 b x^2 \sqrt{b x+c x^2}}",1,"(-2*(b^3 - 2*b^2*c*x + 8*b*c^2*x^2 + 16*c^3*x^3))/(5*b^4*x^2*Sqrt[x*(b + c*x)])","A",1
59,1,62,103,0.0151044,"\int \frac{1}{x^3 \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(b*x + c*x^2)^(3/2)),x]","\frac{2 \left(-5 b^4+8 b^3 c x-16 b^2 c^2 x^2+64 b c^3 x^3+128 c^4 x^4\right)}{35 b^5 x^3 \sqrt{x (b+c x)}}","\frac{128 c^3 (b+2 c x)}{35 b^5 \sqrt{b x+c x^2}}-\frac{32 c^2}{35 b^3 x \sqrt{b x+c x^2}}+\frac{16 c}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2}{7 b x^3 \sqrt{b x+c x^2}}",1,"(2*(-5*b^4 + 8*b^3*c*x - 16*b^2*c^2*x^2 + 64*b*c^3*x^3 + 128*c^4*x^4))/(35*b^5*x^3*Sqrt[x*(b + c*x)])","A",1
60,1,50,122,0.013332,"\int \frac{x^6}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x^6/(a*x + b*x^2)^(5/2),x]","\frac{2 x^5 \sqrt{\frac{b x}{a}+1} \, _2F_1\left(\frac{5}{2},\frac{9}{2};\frac{11}{2};-\frac{b x}{a}\right)}{9 a^2 \sqrt{x (a+b x)}}","\frac{35 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{4 b^{9/2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^5}{3 b \left(a x+b x^2\right)^{3/2}}",1,"(2*x^5*Sqrt[1 + (b*x)/a]*Hypergeometric2F1[5/2, 9/2, 11/2, -((b*x)/a)])/(9*a^2*Sqrt[x*(a + b*x)])","C",1
61,1,50,94,0.0125648,"\int \frac{x^5}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x^5/(a*x + b*x^2)^(5/2),x]","\frac{2 x^4 \sqrt{\frac{b x}{a}+1} \, _2F_1\left(\frac{5}{2},\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right)}{7 a^2 \sqrt{x (a+b x)}}","-\frac{5 a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{b^{7/2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^4}{3 b \left(a x+b x^2\right)^{3/2}}",1,"(2*x^4*Sqrt[1 + (b*x)/a]*Hypergeometric2F1[5/2, 7/2, 9/2, -((b*x)/a)])/(7*a^2*Sqrt[x*(a + b*x)])","C",1
62,1,84,71,0.1507663,"\int \frac{x^4}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x^4/(a*x + b*x^2)^(5/2),x]","\frac{x \left(6 \sqrt{a} \sqrt{x} (a+b x) \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)-2 \sqrt{b} x (3 a+4 b x)\right)}{3 b^{5/2} (x (a+b x))^{3/2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{b^{5/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}-\frac{2 x^3}{3 b \left(a x+b x^2\right)^{3/2}}",1,"(x*(-2*Sqrt[b]*x*(3*a + 4*b*x) + 6*Sqrt[a]*Sqrt[x]*(a + b*x)*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(3*b^(5/2)*(x*(a + b*x))^(3/2))","A",1
63,1,21,23,0.0082903,"\int \frac{x^3}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x^3/(a*x + b*x^2)^(5/2),x]","\frac{2 x^3}{3 a (x (a+b x))^{3/2}}","\frac{2 x^3}{3 a \left(a x+b x^2\right)^{3/2}}",1,"(2*x^3)/(3*a*(x*(a + b*x))^(3/2))","A",1
64,1,29,51,0.011301,"\int \frac{x^2}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x^2/(a*x + b*x^2)^(5/2),x]","\frac{2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}}","\frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left(a x+b x^2\right)^{3/2}}",1,"(2*x^2*(3*a + 2*b*x))/(3*a^2*(x*(a + b*x))^(3/2))","A",1
65,1,38,48,0.013199,"\int \frac{x}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[x/(a*x + b*x^2)^(5/2),x]","-\frac{2 x \left(3 a^2+12 a b x+8 b^2 x^2\right)}{3 a^3 (x (a+b x))^{3/2}}","\frac{2 x}{3 a \left(a x+b x^2\right)^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}}",1,"(-2*x*(3*a^2 + 12*a*b*x + 8*b^2*x^2))/(3*a^3*(x*(a + b*x))^(3/2))","A",1
66,1,48,54,0.0148318,"\int \frac{1}{\left(a x+b x^2\right)^{5/2}} \, dx","Integrate[(a*x + b*x^2)^(-5/2),x]","\frac{-2 a^3+12 a^2 b x+48 a b^2 x^2+32 b^3 x^3}{3 a^4 (x (a+b x))^{3/2}}","\frac{16 b (a+2 b x)}{3 a^4 \sqrt{a x+b x^2}}-\frac{2 (a+2 b x)}{3 a^2 \left(a x+b x^2\right)^{3/2}}",1,"(-2*a^3 + 12*a^2*b*x + 48*a*b^2*x^2 + 32*b^3*x^3)/(3*a^4*(x*(a + b*x))^(3/2))","A",1
67,1,62,80,0.0166628,"\int \frac{1}{x \left(a x+b x^2\right)^{5/2}} \, dx","Integrate[1/(x*(a*x + b*x^2)^(5/2)),x]","-\frac{2 \left(3 a^4-8 a^3 b x+48 a^2 b^2 x^2+192 a b^3 x^3+128 b^4 x^4\right)}{15 a^5 x (x (a+b x))^{3/2}}","-\frac{128 b^2 (a+2 b x)}{15 a^5 \sqrt{a x+b x^2}}+\frac{16 b (a+2 b x)}{15 a^3 \left(a x+b x^2\right)^{3/2}}-\frac{2}{5 a x \left(a x+b x^2\right)^{3/2}}",1,"(-2*(3*a^4 - 8*a^3*b*x + 48*a^2*b^2*x^2 + 192*a*b^3*x^3 + 128*b^4*x^4))/(15*a^5*x*(x*(a + b*x))^(3/2))","A",1
68,1,73,106,0.0193688,"\int \frac{1}{x^2 \left(a x+b x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(a*x + b*x^2)^(5/2)),x]","\frac{2 \left(-3 a^5+6 a^4 b x-16 a^3 b^2 x^2+96 a^2 b^3 x^3+384 a b^4 x^4+256 b^5 x^5\right)}{21 a^6 x^2 (x (a+b x))^{3/2}}","\frac{256 b^3 (a+2 b x)}{21 a^6 \sqrt{a x+b x^2}}-\frac{32 b^2 (a+2 b x)}{21 a^4 \left(a x+b x^2\right)^{3/2}}+\frac{4 b}{7 a^2 x \left(a x+b x^2\right)^{3/2}}-\frac{2}{7 a x^2 \left(a x+b x^2\right)^{3/2}}",1,"(2*(-3*a^5 + 6*a^4*b*x - 16*a^3*b^2*x^2 + 96*a^2*b^3*x^3 + 384*a*b^4*x^4 + 256*b^5*x^5))/(21*a^6*x^2*(x*(a + b*x))^(3/2))","A",1
69,1,27,26,0.0299258,"\int \frac{x}{\sqrt{4 x-x^2}} \, dx","Integrate[x/Sqrt[4*x - x^2],x]","-\sqrt{-((x-4) x)}-4 \sin ^{-1}\left(\sqrt{1-\frac{x}{4}}\right)","-\sqrt{4 x-x^2}-2 \sin ^{-1}\left(1-\frac{x}{2}\right)",1,"-Sqrt[-((-4 + x)*x)] - 4*ArcSin[Sqrt[1 - x/4]]","A",1
70,1,40,28,0.0133163,"\int \frac{x}{\sqrt{-4 x+x^2}} \, dx","Integrate[x/Sqrt[-4*x + x^2],x]","\frac{(x-4) x-4 \sqrt{-((x-4) x)} \sin ^{-1}\left(\sqrt{1-\frac{x}{4}}\right)}{\sqrt{(x-4) x}}","\sqrt{x^2-4 x}+4 \tanh ^{-1}\left(\frac{x}{\sqrt{x^2-4 x}}\right)",1,"((-4 + x)*x - 4*Sqrt[-((-4 + x)*x)]*ArcSin[Sqrt[1 - x/4]])/Sqrt[(-4 + x)*x]","A",0
71,1,47,46,0.0408605,"\int \frac{x^2}{\sqrt{2 x-x^2}} \, dx","Integrate[x^2/Sqrt[2*x - x^2],x]","\frac{1}{2} \left(-\sqrt{2-x} x^{3/2}-3 \sqrt{-((x-2) x)}-6 \sin ^{-1}\left(\sqrt{1-\frac{x}{2}}\right)\right)","-\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x)",1,"(-(Sqrt[2 - x]*x^(3/2)) - 3*Sqrt[-((-2 + x)*x)] - 6*ArcSin[Sqrt[1 - x/2]])/2","A",1
72,1,64,136,0.0333953,"\int x^{7/2} \sqrt{b x+c x^2} \, dx","Integrate[x^(7/2)*Sqrt[b*x + c*x^2],x]","\frac{2 (x (b+c x))^{3/2} \left(128 b^4-192 b^3 c x+240 b^2 c^2 x^2-280 b c^3 x^3+315 c^4 x^4\right)}{3465 c^5 x^{3/2}}","\frac{256 b^4 \left(b x+c x^2\right)^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left(b x+c x^2\right)^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left(b x+c x^2\right)^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left(b x+c x^2\right)^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left(b x+c x^2\right)^{3/2}}{11 c}",1,"(2*(x*(b + c*x))^(3/2)*(128*b^4 - 192*b^3*c*x + 240*b^2*c^2*x^2 - 280*b*c^3*x^3 + 315*c^4*x^4))/(3465*c^5*x^(3/2))","A",1
73,1,53,108,0.0267094,"\int x^{5/2} \sqrt{b x+c x^2} \, dx","Integrate[x^(5/2)*Sqrt[b*x + c*x^2],x]","\frac{2 (x (b+c x))^{3/2} \left(-16 b^3+24 b^2 c x-30 b c^2 x^2+35 c^3 x^3\right)}{315 c^4 x^{3/2}}","-\frac{32 b^3 \left(b x+c x^2\right)^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left(b x+c x^2\right)^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left(b x+c x^2\right)^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left(b x+c x^2\right)^{3/2}}{9 c}",1,"(2*(x*(b + c*x))^(3/2)*(-16*b^3 + 24*b^2*c*x - 30*b*c^2*x^2 + 35*c^3*x^3))/(315*c^4*x^(3/2))","A",1
74,1,42,80,0.0214629,"\int x^{3/2} \sqrt{b x+c x^2} \, dx","Integrate[x^(3/2)*Sqrt[b*x + c*x^2],x]","\frac{2 (x (b+c x))^{3/2} \left(8 b^2-12 b c x+15 c^2 x^2\right)}{105 c^3 x^{3/2}}","\frac{16 b^2 \left(b x+c x^2\right)^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left(b x+c x^2\right)^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left(b x+c x^2\right)^{3/2}}{7 c}",1,"(2*(x*(b + c*x))^(3/2)*(8*b^2 - 12*b*c*x + 15*c^2*x^2))/(105*c^3*x^(3/2))","A",1
75,1,31,52,0.016752,"\int \sqrt{x} \sqrt{b x+c x^2} \, dx","Integrate[Sqrt[x]*Sqrt[b*x + c*x^2],x]","\frac{2 (x (b+c x))^{3/2} (3 c x-2 b)}{15 c^2 x^{3/2}}","\frac{2 \left(b x+c x^2\right)^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left(b x+c x^2\right)^{3/2}}{15 c^2 x^{3/2}}",1,"(2*(x*(b + c*x))^(3/2)*(-2*b + 3*c*x))/(15*c^2*x^(3/2))","A",1
76,1,23,25,0.0112041,"\int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx","Integrate[Sqrt[b*x + c*x^2]/Sqrt[x],x]","\frac{2 (x (b+c x))^{3/2}}{3 c x^{3/2}}","\frac{2 \left(b x+c x^2\right)^{3/2}}{3 c x^{3/2}}",1,"(2*(x*(b + c*x))^(3/2))/(3*c*x^(3/2))","A",1
77,1,60,53,0.0249885,"\int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^(3/2),x]","\frac{2 \sqrt{x} \sqrt{b+c x} \left(\sqrt{b+c x}-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b+c x}}{\sqrt{b}}\right)\right)}{\sqrt{x (b+c x)}}","\frac{2 \sqrt{b x+c x^2}}{\sqrt{x}}-2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)",1,"(2*Sqrt[x]*Sqrt[b + c*x]*(Sqrt[b + c*x] - Sqrt[b]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/Sqrt[x*(b + c*x)]","A",1
78,1,51,54,0.0386811,"\int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^(5/2),x]","-\frac{c x \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left(\sqrt{\frac{c x}{b}+1}\right)+b+c x}{\sqrt{x} \sqrt{x (b+c x)}}","-\frac{c \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{\sqrt{b}}-\frac{\sqrt{b x+c x^2}}{x^{3/2}}",1,"-((b + c*x + c*x*Sqrt[1 + (c*x)/b]*ArcTanh[Sqrt[1 + (c*x)/b]])/(Sqrt[x]*Sqrt[x*(b + c*x)]))","A",1
79,1,42,86,0.0137915,"\int \frac{\sqrt{b x+c x^2}}{x^{7/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^(7/2),x]","-\frac{2 c^2 (x (b+c x))^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x}{b}+1\right)}{3 b^3 x^{3/2}}","\frac{c^2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}}",1,"(-2*c^2*(x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x)/b])/(3*b^3*x^(3/2))","C",1
80,1,42,114,0.0136753,"\int \frac{\sqrt{b x+c x^2}}{x^{9/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^(9/2),x]","\frac{2 c^3 (x (b+c x))^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{c x}{b}+1\right)}{3 b^4 x^{3/2}}","-\frac{c^3 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{8 b^{5/2}}+\frac{c^2 \sqrt{b x+c x^2}}{8 b^2 x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{12 b x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 x^{7/2}}",1,"(2*c^3*(x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, 1 + (c*x)/b])/(3*b^4*x^(3/2))","C",1
81,1,42,142,0.0143966,"\int \frac{\sqrt{b x+c x^2}}{x^{11/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/x^(11/2),x]","-\frac{2 c^4 (x (b+c x))^{3/2} \, _2F_1\left(\frac{3}{2},5;\frac{5}{2};\frac{c x}{b}+1\right)}{3 b^5 x^{3/2}}","\frac{5 c^4 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{64 b^{7/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}",1,"(-2*c^4*(x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 5, 5/2, 1 + (c*x)/b])/(3*b^5*x^(3/2))","C",1
82,1,75,164,0.0423486,"\int x^{7/2} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[x^(7/2)*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{5/2} \left(-256 b^5+640 b^4 c x-1120 b^3 c^2 x^2+1680 b^2 c^3 x^3-2310 b c^4 x^4+3003 c^5 x^5\right)}{45045 c^6 x^{5/2}}","-\frac{512 b^5 \left(b x+c x^2\right)^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left(b x+c x^2\right)^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left(b x+c x^2\right)^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left(b x+c x^2\right)^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left(b x+c x^2\right)^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left(b x+c x^2\right)^{5/2}}{15 c}",1,"(2*(x*(b + c*x))^(5/2)*(-256*b^5 + 640*b^4*c*x - 1120*b^3*c^2*x^2 + 1680*b^2*c^3*x^3 - 2310*b*c^4*x^4 + 3003*c^5*x^5))/(45045*c^6*x^(5/2))","A",1
83,1,64,136,0.0334647,"\int x^{5/2} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[x^(5/2)*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{5/2} \left(128 b^4-320 b^3 c x+560 b^2 c^2 x^2-840 b c^3 x^3+1155 c^4 x^4\right)}{15015 c^5 x^{5/2}}","\frac{256 b^4 \left(b x+c x^2\right)^{5/2}}{15015 c^5 x^{5/2}}-\frac{128 b^3 \left(b x+c x^2\right)^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left(b x+c x^2\right)^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left(b x+c x^2\right)^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left(b x+c x^2\right)^{5/2}}{13 c}",1,"(2*(x*(b + c*x))^(5/2)*(128*b^4 - 320*b^3*c*x + 560*b^2*c^2*x^2 - 840*b*c^3*x^3 + 1155*c^4*x^4))/(15015*c^5*x^(5/2))","A",1
84,1,53,108,0.0294658,"\int x^{3/2} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[x^(3/2)*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{5/2} \left(-16 b^3+40 b^2 c x-70 b c^2 x^2+105 c^3 x^3\right)}{1155 c^4 x^{5/2}}","-\frac{32 b^3 \left(b x+c x^2\right)^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left(b x+c x^2\right)^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left(b x+c x^2\right)^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left(b x+c x^2\right)^{5/2}}{11 c}",1,"(2*(x*(b + c*x))^(5/2)*(-16*b^3 + 40*b^2*c*x - 70*b*c^2*x^2 + 105*c^3*x^3))/(1155*c^4*x^(5/2))","A",1
85,1,42,80,0.0236041,"\int \sqrt{x} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[Sqrt[x]*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{5/2} \left(8 b^2-20 b c x+35 c^2 x^2\right)}{315 c^3 x^{5/2}}","\frac{16 b^2 \left(b x+c x^2\right)^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left(b x+c x^2\right)^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left(b x+c x^2\right)^{5/2}}{9 c \sqrt{x}}",1,"(2*(x*(b + c*x))^(5/2)*(8*b^2 - 20*b*c*x + 35*c^2*x^2))/(315*c^3*x^(5/2))","A",1
86,1,31,52,0.0187531,"\int \frac{\left(b x+c x^2\right)^{3/2}}{\sqrt{x}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/Sqrt[x],x]","\frac{2 (x (b+c x))^{5/2} (5 c x-2 b)}{35 c^2 x^{5/2}}","\frac{2 \left(b x+c x^2\right)^{5/2}}{7 c x^{3/2}}-\frac{4 b \left(b x+c x^2\right)^{5/2}}{35 c^2 x^{5/2}}",1,"(2*(x*(b + c*x))^(5/2)*(-2*b + 5*c*x))/(35*c^2*x^(5/2))","A",1
87,1,23,25,0.0136033,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{3/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(3/2),x]","\frac{2 (x (b+c x))^{5/2}}{5 c x^{5/2}}","\frac{2 \left(b x+c x^2\right)^{5/2}}{5 c x^{5/2}}",1,"(2*(x*(b + c*x))^(5/2))/(5*c*x^(5/2))","A",1
88,1,70,76,0.0439494,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{5/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(5/2),x]","\frac{2 \sqrt{x} \sqrt{b+c x} \left(\sqrt{b+c x} (4 b+c x)-3 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+c x}}{\sqrt{b}}\right)\right)}{3 \sqrt{x (b+c x)}}","-2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)+\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left(b x+c x^2\right)^{3/2}}{3 x^{3/2}}",1,"(2*Sqrt[x]*Sqrt[b + c*x]*(Sqrt[b + c*x]*(4*b + c*x) - 3*b^(3/2)*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(3*Sqrt[x*(b + c*x)])","A",1
89,1,40,75,0.0160909,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{7/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(7/2),x]","\frac{2 c (x (b+c x))^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{c x}{b}+1\right)}{5 b^2 x^{5/2}}","\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-3 \sqrt{b} c \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)-\frac{\left(b x+c x^2\right)^{3/2}}{x^{5/2}}",1,"(2*c*(x*(b + c*x))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x)/b])/(5*b^2*x^(5/2))","C",1
90,1,72,83,0.0474253,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{9/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(9/2),x]","-\frac{2 b^2+3 c^2 x^2 \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left(\sqrt{\frac{c x}{b}+1}\right)+7 b c x+5 c^2 x^2}{4 x^{3/2} \sqrt{x (b+c x)}}","-\frac{3 c^2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left(b x+c x^2\right)^{3/2}}{2 x^{7/2}}",1,"-1/4*(2*b^2 + 7*b*c*x + 5*c^2*x^2 + 3*c^2*x^2*Sqrt[1 + (c*x)/b]*ArcTanh[Sqrt[1 + (c*x)/b]])/(x^(3/2)*Sqrt[x*(b + c*x)])","A",1
91,1,42,111,0.0178486,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{11/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(11/2),x]","\frac{2 c^3 (x (b+c x))^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{c x}{b}+1\right)}{5 b^4 x^{5/2}}","\frac{c^3 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left(b x+c x^2\right)^{3/2}}{3 x^{9/2}}",1,"(2*c^3*(x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, 1 + (c*x)/b])/(5*b^4*x^(5/2))","C",1
92,1,42,139,0.0167769,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{13/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(13/2),x]","-\frac{2 c^4 (x (b+c x))^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{c x}{b}+1\right)}{5 b^5 x^{5/2}}","-\frac{3 c^4 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{64 b^{5/2}}+\frac{3 c^3 \sqrt{b x+c x^2}}{64 b^2 x^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{32 b x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{8 x^{7/2}}-\frac{\left(b x+c x^2\right)^{3/2}}{4 x^{11/2}}",1,"(-2*c^4*(x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, 1 + (c*x)/b])/(5*b^5*x^(5/2))","C",1
93,1,42,167,0.0169452,"\int \frac{\left(b x+c x^2\right)^{3/2}}{x^{15/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/x^(15/2),x]","\frac{2 c^5 (x (b+c x))^{5/2} \, _2F_1\left(\frac{5}{2},6;\frac{7}{2};\frac{c x}{b}+1\right)}{5 b^6 x^{5/2}}","\frac{3 c^5 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{128 b^{7/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left(b x+c x^2\right)^{3/2}}{5 x^{13/2}}",1,"(2*c^5*(x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 6, 7/2, 1 + (c*x)/b])/(5*b^6*x^(5/2))","C",1
94,1,53,108,0.0317377,"\int \frac{x^{7/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[x^(7/2)/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{x (b+c x)} \left(-16 b^3+8 b^2 c x-6 b c^2 x^2+5 c^3 x^3\right)}{35 c^4 \sqrt{x}}","-\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c}",1,"(2*Sqrt[x*(b + c*x)]*(-16*b^3 + 8*b^2*c*x - 6*b*c^2*x^2 + 5*c^3*x^3))/(35*c^4*Sqrt[x])","A",1
95,1,42,80,0.0200877,"\int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[x^(5/2)/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{x (b+c x)} \left(8 b^2-4 b c x+3 c^2 x^2\right)}{15 c^3 \sqrt{x}}","\frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c}",1,"(2*Sqrt[x*(b + c*x)]*(8*b^2 - 4*b*c*x + 3*c^2*x^2))/(15*c^3*Sqrt[x])","A",1
96,1,30,52,0.0156016,"\int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[x^(3/2)/Sqrt[b*x + c*x^2],x]","\frac{2 (c x-2 b) \sqrt{x (b+c x)}}{3 c^2 \sqrt{x}}","\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}",1,"(2*(-2*b + c*x)*Sqrt[x*(b + c*x)])/(3*c^2*Sqrt[x])","A",1
97,1,21,23,0.0090899,"\int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx","Integrate[Sqrt[x]/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{x (b+c x)}}{c \sqrt{x}}","\frac{2 \sqrt{b x+c x^2}}{c \sqrt{x}}",1,"(2*Sqrt[x*(b + c*x)])/(c*Sqrt[x])","A",1
98,1,48,32,0.0110571,"\int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[b*x + c*x^2]),x]","-\frac{2 \sqrt{x} \sqrt{b+c x} \tanh ^{-1}\left(\frac{\sqrt{b+c x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{x (b+c x)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{\sqrt{b}}",1,"(-2*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(Sqrt[b]*Sqrt[x*(b + c*x)])","A",1
99,1,63,56,0.0544549,"\int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^(3/2)*Sqrt[b*x + c*x^2]),x]","\frac{2 c \sqrt{x (b+c x)} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{c x}{b}+1}\right)}{2 \sqrt{\frac{c x}{b}+1}}-\frac{b}{2 c x}\right)}{b^2 \sqrt{x}}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{b^{3/2}}-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}",1,"(2*c*Sqrt[x*(b + c*x)]*(-1/2*b/(c*x) + ArcTanh[Sqrt[1 + (c*x)/b]]/(2*Sqrt[1 + (c*x)/b])))/(b^2*Sqrt[x])","A",1
100,1,40,89,0.0117576,"\int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^(5/2)*Sqrt[b*x + c*x^2]),x]","-\frac{2 c^2 \sqrt{x (b+c x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{c x}{b}+1\right)}{b^3 \sqrt{x}}","-\frac{3 c^2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{4 b^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}}",1,"(-2*c^2*Sqrt[x*(b + c*x)]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (c*x)/b])/(b^3*Sqrt[x])","C",1
101,1,40,117,0.0115059,"\int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/(x^(7/2)*Sqrt[b*x + c*x^2]),x]","\frac{2 c^3 \sqrt{x (b+c x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};\frac{c x}{b}+1\right)}{b^4 \sqrt{x}}","\frac{5 c^3 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{8 b^{7/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}",1,"(2*c^3*Sqrt[x*(b + c*x)]*Hypergeometric2F1[1/2, 4, 3/2, 1 + (c*x)/b])/(b^4*Sqrt[x])","C",1
102,1,75,164,0.0335948,"\int \frac{x^{13/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(13/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} \left(256 b^5+128 b^4 c x-32 b^3 c^2 x^2+16 b^2 c^3 x^3-10 b c^4 x^4+7 c^5 x^5\right)}{63 c^6 \sqrt{x (b+c x)}}","\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}",1,"(2*Sqrt[x]*(256*b^5 + 128*b^4*c*x - 32*b^3*c^2*x^2 + 16*b^2*c^3*x^3 - 10*b*c^4*x^4 + 7*c^5*x^5))/(63*c^6*Sqrt[x*(b + c*x)])","A",1
103,1,64,136,0.028769,"\int \frac{x^{11/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(11/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} \left(-128 b^4-64 b^3 c x+16 b^2 c^2 x^2-8 b c^3 x^3+5 c^4 x^4\right)}{35 c^5 \sqrt{x (b+c x)}}","-\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}}",1,"(2*Sqrt[x]*(-128*b^4 - 64*b^3*c*x + 16*b^2*c^2*x^2 - 8*b*c^3*x^3 + 5*c^4*x^4))/(35*c^5*Sqrt[x*(b + c*x)])","A",1
104,1,52,108,0.0225825,"\int \frac{x^{9/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(9/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} \left(16 b^3+8 b^2 c x-2 b c^2 x^2+c^3 x^3\right)}{5 c^4 \sqrt{x (b+c x)}}","\frac{32 b^3 \sqrt{x}}{5 c^4 \sqrt{b x+c x^2}}+\frac{16 b^2 x^{3/2}}{5 c^3 \sqrt{b x+c x^2}}-\frac{4 b x^{5/2}}{5 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{7/2}}{5 c \sqrt{b x+c x^2}}",1,"(2*Sqrt[x]*(16*b^3 + 8*b^2*c*x - 2*b*c^2*x^2 + c^3*x^3))/(5*c^4*Sqrt[x*(b + c*x)])","A",1
105,1,41,80,0.0177034,"\int \frac{x^{7/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(7/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} \left(-8 b^2-4 b c x+c^2 x^2\right)}{3 c^3 \sqrt{x (b+c x)}}","-\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}}",1,"(2*Sqrt[x]*(-8*b^2 - 4*b*c*x + c^2*x^2))/(3*c^3*Sqrt[x*(b + c*x)])","A",1
106,1,28,48,0.0133231,"\int \frac{x^{5/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(5/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} (2 b+c x)}{c^2 \sqrt{x (b+c x)}}","\frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}}",1,"(2*Sqrt[x]*(2*b + c*x))/(c^2*Sqrt[x*(b + c*x)])","A",1
107,1,21,23,0.0069504,"\int \frac{x^{3/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[x^(3/2)/(b*x + c*x^2)^(3/2),x]","-\frac{2 \sqrt{x}}{c \sqrt{x (b+c x)}}","-\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}}",1,"(-2*Sqrt[x])/(c*Sqrt[x*(b + c*x)])","A",1
108,1,37,56,0.0076088,"\int \frac{\sqrt{x}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[x]/(b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{x} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x}{b}+1\right)}{b \sqrt{x (b+c x)}}","\frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{b^{3/2}}",1,"(2*Sqrt[x]*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x)/b])/(b*Sqrt[x*(b + c*x)])","C",1
109,1,38,81,0.0082946,"\int \frac{1}{\sqrt{x} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[x]*(b*x + c*x^2)^(3/2)),x]","-\frac{2 c \sqrt{x} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{c x}{b}+1\right)}{b^2 \sqrt{x (b+c x)}}","\frac{3 c \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{b^{5/2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}}",1,"(-2*c*Sqrt[x]*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x)/b])/(b^2*Sqrt[x*(b + c*x)])","C",1
110,1,40,117,0.0099123,"\int \frac{1}{x^{3/2} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^(3/2)*(b*x + c*x^2)^(3/2)),x]","\frac{2 c^2 \sqrt{x} \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{c x}{b}+1\right)}{b^3 \sqrt{x (b+c x)}}","-\frac{15 c^2 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{4 b^{7/2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}",1,"(2*c^2*Sqrt[x]*Hypergeometric2F1[-1/2, 3, 1/2, 1 + (c*x)/b])/(b^3*Sqrt[x*(b + c*x)])","C",1
111,1,40,145,0.0101835,"\int \frac{1}{x^{5/2} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^(5/2)*(b*x + c*x^2)^(3/2)),x]","-\frac{2 c^3 \sqrt{x} \, _2F_1\left(-\frac{1}{2},4;\frac{1}{2};\frac{c x}{b}+1\right)}{b^4 \sqrt{x (b+c x)}}","\frac{35 c^3 \tanh ^{-1}\left(\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right)}{8 b^{9/2}}-\frac{35 c^3 \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}-\frac{35 c^2}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{7 c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{1}{3 b x^{5/2} \sqrt{b x+c x^2}}",1,"(-2*c^3*Sqrt[x]*Hypergeometric2F1[-1/2, 4, 1/2, 1 + (c*x)/b])/(b^4*Sqrt[x*(b + c*x)])","C",1
112,1,57,81,0.037236,"\int (d x)^m \left(b x+c x^2\right)^3 \, dx","Integrate[(d*x)^m*(b*x + c*x^2)^3,x]","x^4 (d x)^m \left(\frac{b^3}{m+4}+\frac{3 b^2 c x}{m+5}+\frac{3 b c^2 x^2}{m+6}+\frac{c^3 x^3}{m+7}\right)","\frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)}",1,"x^4*(d*x)^m*(b^3/(4 + m) + (3*b^2*c*x)/(5 + m) + (3*b*c^2*x^2)/(6 + m) + (c^3*x^3)/(7 + m))","A",1
113,1,41,58,0.0334633,"\int (d x)^m \left(b x+c x^2\right)^2 \, dx","Integrate[(d*x)^m*(b*x + c*x^2)^2,x]","x^3 (d x)^m \left(\frac{b^2}{m+3}+\frac{2 b c x}{m+4}+\frac{c^2 x^2}{m+5}\right)","\frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)}",1,"x^3*(d*x)^m*(b^2/(3 + m) + (2*b*c*x)/(4 + m) + (c^2*x^2)/(5 + m))","A",1
114,1,25,35,0.0143164,"\int (d x)^m \left(b x+c x^2\right) \, dx","Integrate[(d*x)^m*(b*x + c*x^2),x]","x^2 (d x)^m \left(\frac{b}{m+2}+\frac{c x}{m+3}\right)","\frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)}",1,"x^2*(d*x)^m*(b/(2 + m) + (c*x)/(3 + m))","A",1
115,1,25,25,0.0066753,"\int \frac{(d x)^m}{b x+c x^2} \, dx","Integrate[(d*x)^m/(b*x + c*x^2),x]","\frac{(d x)^m \, _2F_1\left(1,m;m+1;-\frac{c x}{b}\right)}{b m}","\frac{(d x)^m \, _2F_1\left(1,m;m+1;-\frac{c x}{b}\right)}{b m}",1,"((d*x)^m*Hypergeometric2F1[1, m, 1 + m, -((c*x)/b)])/(b*m)","A",1
116,1,30,33,0.0082393,"\int \frac{(d x)^m}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d*x)^m/(b*x + c*x^2)^2,x]","\frac{(d x)^m \, _2F_1\left(2,m-1;m;-\frac{c x}{b}\right)}{b^2 (m-1) x}","-\frac{d (d x)^{m-1} \, _2F_1\left(2,m-1;m;-\frac{c x}{b}\right)}{b^2 (1-m)}",1,"((d*x)^m*Hypergeometric2F1[2, -1 + m, m, -((c*x)/b)])/(b^2*(-1 + m)*x)","A",1
117,1,32,37,0.0087151,"\int \frac{(d x)^m}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d*x)^m/(b*x + c*x^2)^3,x]","\frac{(d x)^m \, _2F_1\left(3,m-2;m-1;-\frac{c x}{b}\right)}{b^3 (m-2) x^2}","-\frac{d^2 (d x)^{m-2} \, _2F_1\left(3,m-2;m-1;-\frac{c x}{b}\right)}{b^3 (2-m)}",1,"((d*x)^m*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*x)/b)])/(b^3*(-2 + m)*x^2)","A",1
118,1,70,73,0.1184258,"\int (d x)^m \left(b x+c x^2\right)^{5/2} \, dx","Integrate[(d*x)^m*(b*x + c*x^2)^(5/2),x]","\frac{2 b^2 (b+c x)^3 \sqrt{x (b+c x)} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right)}{7 c^3}","\frac{2 b^2 (b+c x) \left(b x+c x^2\right)^{5/2} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c x}{b}+1\right)}{7 c^3 x^2}",1,"(2*b^2*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)^3*Sqrt[x*(b + c*x)]*Hypergeometric2F1[7/2, -5/2 - m, 9/2, 1 + (c*x)/b])/(7*c^3)","A",1
119,1,60,71,0.1107675,"\int (d x)^m \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d*x)^m*(b*x + c*x^2)^(3/2),x]","-\frac{2 (x (b+c x))^{5/2} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{5}{2}} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right)}{5 b}","-\frac{2 b (b+c x) \left(b x+c x^2\right)^{3/2} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right)}{5 c^2 x}",1,"(-2*(-((c*x)/b))^(-5/2 - m)*(d*x)^m*(x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, -3/2 - m, 7/2, 1 + (c*x)/b])/(5*b)","A",1
120,1,60,67,0.067998,"\int (d x)^m \sqrt{b x+c x^2} \, dx","Integrate[(d*x)^m*Sqrt[b*x + c*x^2],x]","-\frac{2 (x (b+c x))^{3/2} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{3}{2}} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right)}{3 b}","\frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right)}{3 c}",1,"(-2*(-((c*x)/b))^(-3/2 - m)*(d*x)^m*(x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, -1/2 - m, 5/2, 1 + (c*x)/b])/(3*b)","A",1
121,1,58,65,0.057079,"\int \frac{(d x)^m}{\sqrt{b x+c x^2}} \, dx","Integrate[(d*x)^m/Sqrt[b*x + c*x^2],x]","-\frac{2 \sqrt{x (b+c x)} (d x)^m \left(-\frac{c x}{b}\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right)}{b}","\frac{2 (b+c x) (d x)^m \left(-\frac{c x}{b}\right)^{\frac{1}{2}-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right)}{c \sqrt{b x+c x^2}}",1,"(-2*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*Sqrt[x*(b + c*x)]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, 1 + (c*x)/b])/b","A",1
122,1,58,66,0.0901812,"\int \frac{(d x)^m}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d*x)^m/(b*x + c*x^2)^(3/2),x]","\frac{2 (d x)^m \left(-\frac{c x}{b}\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right)}{b \sqrt{x (b+c x)}}","\frac{2 x (b+c x) (d x)^m \left(-\frac{c x}{b}\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right)}{b \left(b x+c x^2\right)^{3/2}}",1,"(2*(-((c*x)/b))^(1/2 - m)*(d*x)^m*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, 1 + (c*x)/b])/(b*Sqrt[x*(b + c*x)])","A",1
123,1,60,71,0.1083481,"\int \frac{(d x)^m}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d*x)^m/(b*x + c*x^2)^(5/2),x]","\frac{2 (d x)^m \left(-\frac{c x}{b}\right)^{\frac{3}{2}-m} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right)}{3 b (x (b+c x))^{3/2}}","-\frac{2 c x^2 (b+c x) (d x)^m \left(-\frac{c x}{b}\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right)}{3 b^2 \left(b x+c x^2\right)^{5/2}}",1,"(2*(-((c*x)/b))^(3/2 - m)*(d*x)^m*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, 1 + (c*x)/b])/(3*b*(x*(b + c*x))^(3/2))","A",1
124,1,53,55,0.0127819,"\int (d x)^m \left(b x+c x^2\right)^p \, dx","Integrate[(d*x)^m*(b*x + c*x^2)^p,x]","\frac{x (d x)^m (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,m+p+1;m+p+2;-\frac{c x}{b}\right)}{m+p+1}","\frac{x (d x)^m \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,m+p+1;m+p+2;-\frac{c x}{b}\right)}{m+p+1}",1,"(x*(d*x)^m*(x*(b + c*x))^p*Hypergeometric2F1[-p, 1 + m + p, 2 + m + p, -((c*x)/b)])/((1 + m + p)*(1 + (c*x)/b)^p)","A",1
125,1,47,49,0.0074638,"\int x^3 \left(b x+c x^2\right)^p \, dx","Integrate[x^3*(b*x + c*x^2)^p,x]","\frac{x^4 (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+4;p+5;-\frac{c x}{b}\right)}{p+4}","\frac{x^4 \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+4;p+5;-\frac{c x}{b}\right)}{p+4}",1,"(x^4*(x*(b + c*x))^p*Hypergeometric2F1[-p, 4 + p, 5 + p, -((c*x)/b)])/((4 + p)*(1 + (c*x)/b)^p)","A",1
126,1,47,49,0.0077973,"\int x^2 \left(b x+c x^2\right)^p \, dx","Integrate[x^2*(b*x + c*x^2)^p,x]","\frac{x^3 (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+3;p+4;-\frac{c x}{b}\right)}{p+3}","\frac{x^3 \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+3;p+4;-\frac{c x}{b}\right)}{p+3}",1,"(x^3*(x*(b + c*x))^p*Hypergeometric2F1[-p, 3 + p, 4 + p, -((c*x)/b)])/((3 + p)*(1 + (c*x)/b)^p)","A",1
127,1,47,49,0.0077583,"\int x \left(b x+c x^2\right)^p \, dx","Integrate[x*(b*x + c*x^2)^p,x]","\frac{x^2 (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+2;p+3;-\frac{c x}{b}\right)}{p+2}","\frac{x^2 \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+2;p+3;-\frac{c x}{b}\right)}{p+2}",1,"(x^2*(x*(b + c*x))^p*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x)/b)])/((2 + p)*(1 + (c*x)/b)^p)","A",1
128,1,40,42,0.0064316,"\int \frac{\left(b x+c x^2\right)^p}{x} \, dx","Integrate[(b*x + c*x^2)^p/x,x]","\frac{(x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p;p+1;-\frac{c x}{b}\right)}{p}","\frac{\left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p;p+1;-\frac{c x}{b}\right)}{p}",1,"((x*(b + c*x))^p*Hypergeometric2F1[-p, p, 1 + p, -((c*x)/b)])/(p*(1 + (c*x)/b)^p)","A",1
129,1,45,50,0.0080695,"\int \frac{\left(b x+c x^2\right)^p}{x^2} \, dx","Integrate[(b*x + c*x^2)^p/x^2,x]","\frac{(x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(p-1,-p;p;-\frac{c x}{b}\right)}{(p-1) x}","-\frac{\left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(p-1,-p;p;-\frac{c x}{b}\right)}{(1-p) x}",1,"((x*(b + c*x))^p*Hypergeometric2F1[-1 + p, -p, p, -((c*x)/b)])/((-1 + p)*x*(1 + (c*x)/b)^p)","A",1
130,1,47,52,0.0081352,"\int \frac{\left(b x+c x^2\right)^p}{x^3} \, dx","Integrate[(b*x + c*x^2)^p/x^3,x]","\frac{(x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(p-2,-p;p-1;-\frac{c x}{b}\right)}{(p-2) x^2}","-\frac{\left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(p-2,-p;p-1;-\frac{c x}{b}\right)}{(2-p) x^2}",1,"((x*(b + c*x))^p*Hypergeometric2F1[-2 + p, -p, -1 + p, -((c*x)/b)])/((-2 + p)*x^2*(1 + (c*x)/b)^p)","A",1
131,1,58,61,0.0140414,"\int (d x)^{5/2} \left(b x+c x^2\right)^p \, dx","Integrate[(d*x)^(5/2)*(b*x + c*x^2)^p,x]","\frac{x (d x)^{5/2} (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+\frac{7}{2};p+\frac{9}{2};-\frac{c x}{b}\right)}{p+\frac{7}{2}}","\frac{2 x (d x)^{5/2} \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+\frac{7}{2};p+\frac{9}{2};-\frac{c x}{b}\right)}{2 p+7}",1,"(x*(d*x)^(5/2)*(x*(b + c*x))^p*Hypergeometric2F1[-p, 7/2 + p, 9/2 + p, -((c*x)/b)])/((7/2 + p)*(1 + (c*x)/b)^p)","A",1
132,1,58,61,0.0121616,"\int (d x)^{3/2} \left(b x+c x^2\right)^p \, dx","Integrate[(d*x)^(3/2)*(b*x + c*x^2)^p,x]","\frac{x (d x)^{3/2} (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+\frac{5}{2};p+\frac{7}{2};-\frac{c x}{b}\right)}{p+\frac{5}{2}}","\frac{2 x (d x)^{3/2} \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+\frac{5}{2};p+\frac{7}{2};-\frac{c x}{b}\right)}{2 p+5}",1,"(x*(d*x)^(3/2)*(x*(b + c*x))^p*Hypergeometric2F1[-p, 5/2 + p, 7/2 + p, -((c*x)/b)])/((5/2 + p)*(1 + (c*x)/b)^p)","A",1
133,1,58,61,0.0108881,"\int \sqrt{d x} \left(b x+c x^2\right)^p \, dx","Integrate[Sqrt[d*x]*(b*x + c*x^2)^p,x]","\frac{x \sqrt{d x} (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right)}{p+\frac{3}{2}}","\frac{2 x \sqrt{d x} \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right)}{2 p+3}",1,"(x*Sqrt[d*x]*(x*(b + c*x))^p*Hypergeometric2F1[-p, 3/2 + p, 5/2 + p, -((c*x)/b)])/((3/2 + p)*(1 + (c*x)/b)^p)","A",1
134,1,58,61,0.0121244,"\int \frac{\left(b x+c x^2\right)^p}{\sqrt{d x}} \, dx","Integrate[(b*x + c*x^2)^p/Sqrt[d*x],x]","\frac{x (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right)}{\left(p+\frac{1}{2}\right) \sqrt{d x}}","\frac{2 x \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right)}{(2 p+1) \sqrt{d x}}",1,"(x*(x*(b + c*x))^p*Hypergeometric2F1[-p, 1/2 + p, 3/2 + p, -((c*x)/b)])/((1/2 + p)*Sqrt[d*x]*(1 + (c*x)/b)^p)","A",1
135,1,58,61,0.0130679,"\int \frac{\left(b x+c x^2\right)^p}{(d x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^p/(d*x)^(3/2),x]","\frac{x (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right)}{\left(p-\frac{1}{2}\right) (d x)^{3/2}}","-\frac{2 x \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right)}{(1-2 p) (d x)^{3/2}}",1,"(x*(x*(b + c*x))^p*Hypergeometric2F1[-1/2 + p, -p, 1/2 + p, -((c*x)/b)])/((-1/2 + p)*(d*x)^(3/2)*(1 + (c*x)/b)^p)","A",1
136,1,58,61,0.0140063,"\int \frac{\left(b x+c x^2\right)^p}{(d x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^p/(d*x)^(5/2),x]","\frac{x (x (b+c x))^p \left(\frac{c x}{b}+1\right)^{-p} \, _2F_1\left(p-\frac{3}{2},-p;p-\frac{1}{2};-\frac{c x}{b}\right)}{\left(p-\frac{3}{2}\right) (d x)^{5/2}}","-\frac{2 x \left(\frac{c x}{b}+1\right)^{-p} \left(b x+c x^2\right)^p \, _2F_1\left(p-\frac{3}{2},-p;p-\frac{1}{2};-\frac{c x}{b}\right)}{(3-2 p) (d x)^{5/2}}",1,"(x*(x*(b + c*x))^p*Hypergeometric2F1[-3/2 + p, -p, -1/2 + p, -((c*x)/b)])/((-3/2 + p)*(d*x)^(5/2)*(1 + (c*x)/b)^p)","A",1
137,1,33,71,0.0133369,"\int x^4 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x^5 \sqrt{(a+b x)^2} (6 a+5 b x)}{30 (a+b x)}","\frac{b x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{a x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}",1,"(x^5*Sqrt[(a + b*x)^2]*(6*a + 5*b*x))/(30*(a + b*x))","A",1
138,1,33,71,0.0092156,"\int x^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x^4 \sqrt{(a+b x)^2} (5 a+4 b x)}{20 (a+b x)}","\frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}",1,"(x^4*Sqrt[(a + b*x)^2]*(5*a + 4*b*x))/(20*(a + b*x))","A",1
139,1,33,71,0.009132,"\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x^3 \sqrt{(a+b x)^2} (4 a+3 b x)}{12 (a+b x)}","\frac{b x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}",1,"(x^3*Sqrt[(a + b*x)^2]*(4*a + 3*b*x))/(12*(a + b*x))","A",1
140,1,33,61,0.0080487,"\int x \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x^2 \sqrt{(a+b x)^2} (3 a+2 b x)}{6 (a+b x)}","\frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{3 b^2}-\frac{a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2}",1,"(x^2*Sqrt[(a + b*x)^2]*(3*a + 2*b*x))/(6*(a + b*x))","A",1
141,1,30,32,0.0077357,"\int \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} (2 a+b x)}{2 (a+b x)}","\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b}",1,"(x*Sqrt[(a + b*x)^2]*(2*a + b*x))/(2*(a + b*x))","A",1
142,1,27,62,0.0091786,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x,x]","\frac{\sqrt{(a+b x)^2} (a \log (x)+b x)}{a+b x}","\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}",1,"(Sqrt[(a + b*x)^2]*(b*x + a*Log[x]))/(a + b*x)","A",1
143,1,31,65,0.0088912,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^2,x]","\frac{\sqrt{(a+b x)^2} (b x \log (x)-a)}{x (a+b x)}","\frac{b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(-a + b*x*Log[x]))/(x*(a + b*x))","A",1
144,1,31,35,0.006368,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^3} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]","-\frac{\sqrt{(a+b x)^2} (a+2 b x)}{2 x^2 (a+b x)}","-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2}",1,"-1/2*(Sqrt[(a + b*x)^2]*(a + 2*b*x))/(x^2*(a + b*x))","A",1
145,1,33,71,0.0073658,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^4} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^4,x]","-\frac{\sqrt{(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)}","-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}",1,"-1/6*(Sqrt[(a + b*x)^2]*(2*a + 3*b*x))/(x^3*(a + b*x))","A",1
146,1,33,71,0.0076042,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^5} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^5,x]","-\frac{\sqrt{(a+b x)^2} (3 a+4 b x)}{12 x^4 (a+b x)}","-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}",1,"-1/12*(Sqrt[(a + b*x)^2]*(3*a + 4*b*x))/(x^4*(a + b*x))","A",1
147,1,33,71,0.0070964,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^6} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^6,x]","-\frac{\sqrt{(a+b x)^2} (4 a+5 b x)}{20 x^5 (a+b x)}","-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}",1,"-1/20*(Sqrt[(a + b*x)^2]*(4*a + 5*b*x))/(x^5*(a + b*x))","A",1
148,1,55,151,0.0169221,"\int x^5 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x^6 \sqrt{(a+b x)^2} \left(84 a^3+216 a^2 b x+189 a b^2 x^2+56 b^3 x^3\right)}{504 (a+b x)}","\frac{3 a b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^3 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}",1,"(x^6*Sqrt[(a + b*x)^2]*(84*a^3 + 216*a^2*b*x + 189*a*b^2*x^2 + 56*b^3*x^3))/(504*(a + b*x))","A",1
149,1,55,151,0.0141795,"\int x^4 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x^5 \sqrt{(a+b x)^2} \left(56 a^3+140 a^2 b x+120 a b^2 x^2+35 b^3 x^3\right)}{280 (a+b x)}","\frac{3 a b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^2 b x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{a^3 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}",1,"(x^5*Sqrt[(a + b*x)^2]*(56*a^3 + 140*a^2*b*x + 120*a*b^2*x^2 + 35*b^3*x^3))/(280*(a + b*x))","A",1
150,1,55,151,0.01453,"\int x^3 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x^4 \sqrt{(a+b x)^2} \left(35 a^3+84 a^2 b x+70 a b^2 x^2+20 b^3 x^3\right)}{140 (a+b x)}","\frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}",1,"(x^4*Sqrt[(a + b*x)^2]*(35*a^3 + 84*a^2*b*x + 70*a*b^2*x^2 + 20*b^3*x^3))/(140*(a + b*x))","A",1
151,1,55,96,0.0145237,"\int x^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x^3 \sqrt{(a+b x)^2} \left(20 a^3+45 a^2 b x+36 a b^2 x^2+10 b^3 x^3\right)}{60 (a+b x)}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{6 b^3}-\frac{2 a \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{5 b^3}+\frac{a^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{4 b^3}",1,"(x^3*Sqrt[(a + b*x)^2]*(20*a^3 + 45*a^2*b*x + 36*a*b^2*x^2 + 10*b^3*x^3))/(60*(a + b*x))","A",1
152,1,55,61,0.0136604,"\int x \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[x*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x^2 \sqrt{(a+b x)^2} \left(10 a^3+20 a^2 b x+15 a b^2 x^2+4 b^3 x^3\right)}{20 (a+b x)}","\frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{5 b^2}-\frac{a (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{4 b^2}",1,"(x^2*Sqrt[(a + b*x)^2]*(10*a^3 + 20*a^2*b*x + 15*a*b^2*x^2 + 4*b^3*x^3))/(20*(a + b*x))","A",1
153,1,23,32,0.00878,"\int \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{(a+b x) \left((a+b x)^2\right)^{3/2}}{4 b}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{4 b}",1,"((a + b*x)*((a + b*x)^2)^(3/2))/(4*b)","A",1
154,1,52,143,0.0167491,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x,x]","\frac{\sqrt{(a+b x)^2} \left(6 a^3 \log (x)+b x \left(18 a^2+9 a b x+2 b^2 x^2\right)\right)}{6 (a+b x)}","\frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}",1,"(Sqrt[(a + b*x)^2]*(b*x*(18*a^2 + 9*a*b*x + 2*b^2*x^2) + 6*a^3*Log[x]))/(6*(a + b*x))","A",1
155,1,56,142,0.0183558,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^2,x]","\frac{\sqrt{(a+b x)^2} \left(-2 a^3+6 a^2 b x \log (x)+6 a b^2 x^2+b^3 x^3\right)}{2 x (a+b x)}","\frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(-2*a^3 + 6*a*b^2*x^2 + b^3*x^3 + 6*a^2*b*x*Log[x]))/(2*x*(a + b*x))","A",1
156,1,55,141,0.0154761,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^3,x]","-\frac{\sqrt{(a+b x)^2} \left(a^3+6 a^2 b x-6 a b^2 x^2 \log (x)-2 b^3 x^3\right)}{2 x^2 (a+b x)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}",1,"-1/2*(Sqrt[(a + b*x)^2]*(a^3 + 6*a^2*b*x - 2*b^3*x^3 - 6*a*b^2*x^2*Log[x]))/(x^2*(a + b*x))","A",1
157,1,57,145,0.0189884,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^4,x]","-\frac{\sqrt{(a+b x)^2} \left(a \left(2 a^2+9 a b x+18 b^2 x^2\right)-6 b^3 x^3 \log (x)\right)}{6 x^3 (a+b x)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}",1,"-1/6*(Sqrt[(a + b*x)^2]*(a*(2*a^2 + 9*a*b*x + 18*b^2*x^2) - 6*b^3*x^3*Log[x]))/(x^3*(a + b*x))","A",1
158,1,53,37,0.0116815,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^5,x]","-\frac{\sqrt{(a+b x)^2} \left(a^3+4 a^2 b x+6 a b^2 x^2+4 b^3 x^3\right)}{4 x^4 (a+b x)}","-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 a x^4}",1,"-1/4*(Sqrt[(a + b*x)^2]*(a^3 + 4*a^2*b*x + 6*a*b^2*x^2 + 4*b^3*x^3))/(x^4*(a + b*x))","A",1
159,1,55,76,0.0153166,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^6,x]","-\frac{\sqrt{(a+b x)^2} \left(4 a^3+15 a^2 b x+20 a b^2 x^2+10 b^3 x^3\right)}{20 x^5 (a+b x)}","\frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5}",1,"-1/20*(Sqrt[(a + b*x)^2]*(4*a^3 + 15*a^2*b*x + 20*a*b^2*x^2 + 10*b^3*x^3))/(x^5*(a + b*x))","A",1
160,1,55,151,0.012222,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]","-\frac{\sqrt{(a+b x)^2} \left(10 a^3+36 a^2 b x+45 a b^2 x^2+20 b^3 x^3\right)}{60 x^6 (a+b x)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}",1,"-1/60*(Sqrt[(a + b*x)^2]*(10*a^3 + 36*a^2*b*x + 45*a*b^2*x^2 + 20*b^3*x^3))/(x^6*(a + b*x))","A",1
161,1,55,151,0.0126187,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^8,x]","-\frac{\sqrt{(a+b x)^2} \left(20 a^3+70 a^2 b x+84 a b^2 x^2+35 b^3 x^3\right)}{140 x^7 (a+b x)}","-\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}",1,"-1/140*(Sqrt[(a + b*x)^2]*(20*a^3 + 70*a^2*b*x + 84*a*b^2*x^2 + 35*b^3*x^3))/(x^7*(a + b*x))","A",1
162,1,55,151,0.0155885,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^9,x]","-\frac{\sqrt{(a+b x)^2} \left(35 a^3+120 a^2 b x+140 a b^2 x^2+56 b^3 x^3\right)}{280 x^8 (a+b x)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}",1,"-1/280*(Sqrt[(a + b*x)^2]*(35*a^3 + 120*a^2*b*x + 140*a*b^2*x^2 + 56*b^3*x^3))/(x^8*(a + b*x))","A",1
163,1,77,231,0.0224039,"\int x^5 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x^6 \sqrt{(a+b x)^2} \left(462 a^5+1980 a^4 b x+3465 a^3 b^2 x^2+3080 a^2 b^3 x^3+1386 a b^4 x^4+252 b^5 x^5\right)}{2772 (a+b x)}","\frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}",1,"(x^6*Sqrt[(a + b*x)^2]*(462*a^5 + 1980*a^4*b*x + 3465*a^3*b^2*x^2 + 3080*a^2*b^3*x^3 + 1386*a*b^4*x^4 + 252*b^5*x^5))/(2772*(a + b*x))","A",1
164,1,77,181,0.0188986,"\int x^4 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x^5 \sqrt{(a+b x)^2} \left(252 a^5+1050 a^4 b x+1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+700 a b^4 x^4+126 b^5 x^5\right)}{1260 (a+b x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5}",1,"(x^5*Sqrt[(a + b*x)^2]*(252*a^5 + 1050*a^4*b*x + 1800*a^3*b^2*x^2 + 1575*a^2*b^3*x^3 + 700*a*b^4*x^4 + 126*b^5*x^5))/(1260*(a + b*x))","A",1
165,1,77,144,0.018464,"\int x^3 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x^4 \sqrt{(a+b x)^2} \left(126 a^5+504 a^4 b x+840 a^3 b^2 x^2+720 a^2 b^3 x^3+315 a b^4 x^4+56 b^5 x^5\right)}{504 (a+b x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac{3 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4}",1,"(x^4*Sqrt[(a + b*x)^2]*(126*a^5 + 504*a^4*b*x + 840*a^3*b^2*x^2 + 720*a^2*b^3*x^3 + 315*a*b^4*x^4 + 56*b^5*x^5))/(504*(a + b*x))","A",1
166,1,77,107,0.0196139,"\int x^2 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x^3 \sqrt{(a+b x)^2} \left(56 a^5+210 a^4 b x+336 a^3 b^2 x^2+280 a^2 b^3 x^3+120 a b^4 x^4+21 b^5 x^5\right)}{168 (a+b x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3}",1,"(x^3*Sqrt[(a + b*x)^2]*(56*a^5 + 210*a^4*b*x + 336*a^3*b^2*x^2 + 280*a^2*b^3*x^3 + 120*a*b^4*x^4 + 21*b^5*x^5))/(168*(a + b*x))","A",1
167,1,77,61,0.0185743,"\int x \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[x*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x^2 \sqrt{(a+b x)^2} \left(21 a^5+70 a^4 b x+105 a^3 b^2 x^2+84 a^2 b^3 x^3+35 a b^4 x^4+6 b^5 x^5\right)}{42 (a+b x)}","\frac{\left(a^2+2 a b x+b^2 x^2\right)^{7/2}}{7 b^2}-\frac{a (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{6 b^2}",1,"(x^2*Sqrt[(a + b*x)^2]*(21*a^5 + 70*a^4*b*x + 105*a^3*b^2*x^2 + 84*a^2*b^3*x^3 + 35*a*b^4*x^4 + 6*b^5*x^5))/(42*(a + b*x))","A",1
168,1,23,32,0.0098427,"\int \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{(a+b x) \left((a+b x)^2\right)^{5/2}}{6 b}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{6 b}",1,"((a + b*x)*((a + b*x)^2)^(5/2))/(6*b)","A",1
169,1,74,221,0.0213914,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x,x]","\frac{\sqrt{(a+b x)^2} \left(60 a^5 \log (x)+b x \left(300 a^4+300 a^3 b x+200 a^2 b^2 x^2+75 a b^3 x^3+12 b^4 x^4\right)\right)}{60 (a+b x)}","\frac{b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^4 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}",1,"(Sqrt[(a + b*x)^2]*(b*x*(300*a^4 + 300*a^3*b*x + 200*a^2*b^2*x^2 + 75*a*b^3*x^3 + 12*b^4*x^4) + 60*a^5*Log[x]))/(60*(a + b*x))","A",1
170,1,79,220,0.0227207,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^2,x]","\frac{\sqrt{(a+b x)^2} \left(-12 a^5+60 a^4 b x \log (x)+120 a^3 b^2 x^2+60 a^2 b^3 x^3+20 a b^4 x^4+3 b^5 x^5\right)}{12 x (a+b x)}","\frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}",1,"(Sqrt[(a + b*x)^2]*(-12*a^5 + 120*a^3*b^2*x^2 + 60*a^2*b^3*x^3 + 20*a*b^4*x^4 + 3*b^5*x^5 + 60*a^4*b*x*Log[x]))/(12*x*(a + b*x))","A",1
171,1,79,222,0.022395,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^3,x]","\frac{\sqrt{(a+b x)^2} \left(-3 a^5-30 a^4 b x+60 a^3 b^2 x^2 \log (x)+60 a^2 b^3 x^3+15 a b^4 x^4+2 b^5 x^5\right)}{6 x^2 (a+b x)}","\frac{b^5 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}",1,"(Sqrt[(a + b*x)^2]*(-3*a^5 - 30*a^4*b*x + 60*a^2*b^3*x^3 + 15*a*b^4*x^4 + 2*b^5*x^5 + 60*a^3*b^2*x^2*Log[x]))/(6*x^2*(a + b*x))","A",1
172,1,79,222,0.0221314,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^4,x]","\frac{\sqrt{(a+b x)^2} \left(-2 a^5-15 a^4 b x-60 a^3 b^2 x^2+60 a^2 b^3 x^3 \log (x)+30 a b^4 x^4+3 b^5 x^5\right)}{6 x^3 (a+b x)}","\frac{b^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{5 a b^4 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^2 b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(-2*a^5 - 15*a^4*b*x - 60*a^3*b^2*x^2 + 30*a*b^4*x^4 + 3*b^5*x^5 + 60*a^2*b^3*x^3*Log[x]))/(6*x^3*(a + b*x))","A",1
173,1,79,219,0.0196904,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^5,x]","-\frac{\sqrt{(a+b x)^2} \left(3 a^5+20 a^4 b x+60 a^3 b^2 x^2+120 a^2 b^3 x^3-60 a b^4 x^4 \log (x)-12 b^5 x^5\right)}{12 x^4 (a+b x)}","\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}",1,"-1/12*(Sqrt[(a + b*x)^2]*(3*a^5 + 20*a^4*b*x + 60*a^3*b^2*x^2 + 120*a^2*b^3*x^3 - 12*b^5*x^5 - 60*a*b^4*x^4*Log[x]))/(x^4*(a + b*x))","A",1
174,1,79,223,0.0249572,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^6,x]","-\frac{\sqrt{(a+b x)^2} \left(a \left(12 a^4+75 a^3 b x+200 a^2 b^2 x^2+300 a b^3 x^3+300 b^4 x^4\right)-60 b^5 x^5 \log (x)\right)}{60 x^5 (a+b x)}","\frac{b^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}",1,"-1/60*(Sqrt[(a + b*x)^2]*(a*(12*a^4 + 75*a^3*b*x + 200*a^2*b^2*x^2 + 300*a*b^3*x^3 + 300*b^4*x^4) - 60*b^5*x^5*Log[x]))/(x^5*(a + b*x))","A",1
175,1,75,37,0.0152159,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]","-\frac{\sqrt{(a+b x)^2} \left(a^5+6 a^4 b x+15 a^3 b^2 x^2+20 a^2 b^3 x^3+15 a b^4 x^4+6 b^5 x^5\right)}{6 x^6 (a+b x)}","-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6}",1,"-1/6*(Sqrt[(a + b*x)^2]*(a^5 + 6*a^4*b*x + 15*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 15*a*b^4*x^4 + 6*b^5*x^5))/(x^6*(a + b*x))","B",1
176,1,77,76,0.0161266,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]","-\frac{\sqrt{(a+b x)^2} \left(6 a^5+35 a^4 b x+84 a^3 b^2 x^2+105 a^2 b^3 x^3+70 a b^4 x^4+21 b^5 x^5\right)}{42 x^7 (a+b x)}","\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 a x^7}",1,"-1/42*(Sqrt[(a + b*x)^2]*(6*a^5 + 35*a^4*b*x + 84*a^3*b^2*x^2 + 105*a^2*b^3*x^3 + 70*a*b^4*x^4 + 21*b^5*x^5))/(x^7*(a + b*x))","A",1
177,1,77,116,0.0162331,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]","-\frac{\sqrt{(a+b x)^2} \left(21 a^5+120 a^4 b x+280 a^3 b^2 x^2+336 a^2 b^3 x^3+210 a b^4 x^4+56 b^5 x^5\right)}{168 x^8 (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6}",1,"-1/168*(Sqrt[(a + b*x)^2]*(21*a^5 + 120*a^4*b*x + 280*a^3*b^2*x^2 + 336*a^2*b^3*x^3 + 210*a*b^4*x^4 + 56*b^5*x^5))/(x^8*(a + b*x))","A",1
178,1,77,229,0.0185733,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]","-\frac{\sqrt{(a+b x)^2} \left(56 a^5+315 a^4 b x+720 a^3 b^2 x^2+840 a^2 b^3 x^3+504 a b^4 x^4+126 b^5 x^5\right)}{504 x^9 (a+b x)}","-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}",1,"-1/504*(Sqrt[(a + b*x)^2]*(56*a^5 + 315*a^4*b*x + 720*a^3*b^2*x^2 + 840*a^2*b^3*x^3 + 504*a*b^4*x^4 + 126*b^5*x^5))/(x^9*(a + b*x))","A",1
179,1,77,231,0.017127,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^11,x]","-\frac{\sqrt{(a+b x)^2} \left(126 a^5+700 a^4 b x+1575 a^3 b^2 x^2+1800 a^2 b^3 x^3+1050 a b^4 x^4+252 b^5 x^5\right)}{1260 x^{10} (a+b x)}","-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}",1,"-1/1260*(Sqrt[(a + b*x)^2]*(126*a^5 + 700*a^4*b*x + 1575*a^3*b^2*x^2 + 1800*a^2*b^3*x^3 + 1050*a*b^4*x^4 + 252*b^5*x^5))/(x^10*(a + b*x))","A",1
180,1,77,231,0.0168922,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]","-\frac{\sqrt{(a+b x)^2} \left(252 a^5+1386 a^4 b x+3080 a^3 b^2 x^2+3465 a^2 b^3 x^3+1980 a b^4 x^4+462 b^5 x^5\right)}{2772 x^{11} (a+b x)}","-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}",1,"-1/2772*(Sqrt[(a + b*x)^2]*(252*a^5 + 1386*a^4*b*x + 3080*a^3*b^2*x^2 + 3465*a^2*b^3*x^3 + 1980*a*b^4*x^4 + 462*b^5*x^5))/(x^11*(a + b*x))","A",1
181,1,68,182,0.0248299,"\int \frac{x^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[x^4/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(12 a^4 \log (a+b x)+b x \left(-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right)\right)}{12 b^5 \sqrt{(a+b x)^2}}","\frac{x^4 (a+b x)}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^3 (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^4 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*x*(-12*a^3 + 6*a^2*b*x - 4*a*b^2*x^2 + 3*b^3*x^3) + 12*a^4*Log[a + b*x]))/(12*b^5*Sqrt[(a + b*x)^2])","A",1
182,1,57,144,0.0185911,"\int \frac{x^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[x^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(b x \left(6 a^2-3 a b x+2 b^2 x^2\right)-6 a^3 \log (a+b x)\right)}{6 b^4 \sqrt{(a+b x)^2}}","-\frac{a x^2 (a+b x)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^3 (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*x*(6*a^2 - 3*a*b*x + 2*b^2*x^2) - 6*a^3*Log[a + b*x]))/(6*b^4*Sqrt[(a + b*x)^2])","A",1
183,1,45,106,0.0148231,"\int \frac{x^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[x^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(2 a^2 \log (a+b x)+b x (b x-2 a)\right)}{2 b^3 \sqrt{(a+b x)^2}}","-\frac{a x (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x]))/(2*b^3*Sqrt[(a + b*x)^2])","A",1
184,1,33,62,0.0100152,"\int \frac{x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[x/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) (b x-a \log (a+b x))}{b^2 \sqrt{(a+b x)^2}}","\frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b^2}-\frac{a (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*x - a*Log[a + b*x]))/(b^2*Sqrt[(a + b*x)^2])","A",1
185,1,26,35,0.006062,"\int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \log (a+b x)}{b \sqrt{(a+b x)^2}}","\frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*Log[a + b*x])/(b*Sqrt[(a + b*x)^2])","A",1
186,1,31,68,0.0098785,"\int \frac{1}{x \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{(a+b x)^2}}","\frac{\log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(Log[x] - Log[a + b*x]))/(a*Sqrt[(a + b*x)^2])","A",1
187,1,41,103,0.0133724,"\int \frac{1}{x^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{(a+b x) (-b x \log (a+b x)+a+b x \log (x))}{a^2 x \sqrt{(a+b x)^2}}","-\frac{a+b x}{a x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b \log (x) (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x) \log (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"-(((a + b*x)*(a + b*x*Log[x] - b*x*Log[a + b*x]))/(a^2*x*Sqrt[(a + b*x)^2]))","A",1
188,1,59,145,0.0186041,"\int \frac{1}{x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{(a+b x) \left(2 b^2 x^2 \log (a+b x)+a (a-2 b x)-2 b^2 x^2 \log (x)\right)}{2 a^3 x^2 \sqrt{(a+b x)^2}}","\frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{-a-b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"-1/2*((a + b*x)*(a*(a - 2*b*x) - 2*b^2*x^2*Log[x] + 2*b^2*x^2*Log[a + b*x]))/(a^3*x^2*Sqrt[(a + b*x)^2])","A",1
189,1,72,184,0.0206366,"\int \frac{1}{x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{(a+b x) \left(a \left(2 a^2-3 a b x+6 b^2 x^2\right)-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right)}{6 a^4 x^3 \sqrt{(a+b x)^2}}","\frac{b (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{-a-b x}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}",1,"-1/6*((a + b*x)*(a*(2*a^2 - 3*a*b*x + 6*b^2*x^2) + 6*b^3*x^3*Log[x] - 6*b^3*x^3*Log[a + b*x]))/(a^4*x^3*Sqrt[(a + b*x)^2])","A",1
190,1,83,172,0.0255397,"\int \frac{x^4}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{7 a^4+2 a^3 b x-11 a^2 b^2 x^2+12 a^2 (a+b x)^2 \log (a+b x)-4 a b^3 x^3+b^4 x^4}{2 b^5 (a+b x) \sqrt{(a+b x)^2}}","\frac{6 a^2 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(7*a^4 + 2*a^3*b*x - 11*a^2*b^2*x^2 - 4*a*b^3*x^3 + b^4*x^4 + 12*a^2*(a + b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
191,1,71,133,0.0240617,"\int \frac{x^3}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{-5 a^3-4 a^2 b x+4 a b^2 x^2-6 a (a+b x)^2 \log (a+b x)+2 b^3 x^3}{2 b^4 (a+b x) \sqrt{(a+b x)^2}}","-\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-5*a^3 - 4*a^2*b*x + 4*a*b^2*x^2 + 2*b^3*x^3 - 6*a*(a + b*x)^2*Log[a + b*x])/(2*b^4*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
192,1,51,99,0.0182171,"\int \frac{x^2}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{a (3 a+4 b x)+2 (a+b x)^2 \log (a+b x)}{2 b^3 (a+b x) \sqrt{(a+b x)^2}}","-\frac{a^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(3*a + 4*b*x) + 2*(a + b*x)^2*Log[a + b*x])/(2*b^3*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
193,1,33,61,0.0094464,"\int \frac{x}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{-a-2 b x}{2 b^2 (a+b x) \sqrt{(a+b x)^2}}","\frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-a - 2*b*x)/(2*b^2*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
194,1,23,34,0.0079282,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]","-\frac{a+b x}{2 b \left((a+b x)^2\right)^{3/2}}","-\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"-1/2*(a + b*x)/(b*((a + b*x)^2)^(3/2))","A",1
195,1,62,126,0.0262279,"\int \frac{1}{x \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{a (3 a+2 b x)+2 \log (x) (a+b x)^2-2 (a+b x)^2 \log (a+b x)}{2 a^3 (a+b x) \sqrt{(a+b x)^2}}","\frac{1}{2 a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(3*a + 2*b*x) + 2*(a + b*x)^2*Log[x] - 2*(a + b*x)^2*Log[a + b*x])/(2*a^3*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
196,1,81,165,0.0281024,"\int \frac{1}{x^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{-a \left(2 a^2+9 a b x+6 b^2 x^2\right)-6 b x \log (x) (a+b x)^2+6 b x (a+b x)^2 \log (a+b x)}{2 a^4 x (a+b x) \sqrt{(a+b x)^2}}","-\frac{b}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-(a*(2*a^2 + 9*a*b*x + 6*b^2*x^2)) - 6*b*x*(a + b*x)^2*Log[x] + 6*b*x*(a + b*x)^2*Log[a + b*x])/(2*a^4*x*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
197,1,99,209,0.036747,"\int \frac{1}{x^3 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{a \left(-a^3+4 a^2 b x+18 a b^2 x^2+12 b^3 x^3\right)+12 b^2 x^2 \log (x) (a+b x)^2-12 b^2 x^2 (a+b x)^2 \log (a+b x)}{2 a^5 x^2 (a+b x) \sqrt{(a+b x)^2}}","\frac{6 b^2 \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{6 b^2 (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(-a^3 + 4*a^2*b*x + 18*a*b^2*x^2 + 12*b^3*x^3) + 12*b^2*x^2*(a + b*x)^2*Log[x] - 12*b^2*x^2*(a + b*x)^2*Log[a + b*x])/(2*a^5*x^2*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
198,1,106,244,0.0361069,"\int \frac{x^6}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{57 a^6+168 a^5 b x+132 a^4 b^2 x^2-32 a^3 b^3 x^3-68 a^2 b^4 x^4+60 a^2 (a+b x)^4 \log (a+b x)-12 a b^5 x^5+2 b^6 x^6}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(57*a^6 + 168*a^5*b*x + 132*a^4*b^2*x^2 - 32*a^3*b^3*x^3 - 68*a^2*b^4*x^4 - 12*a*b^5*x^5 + 2*b^6*x^6 + 60*a^2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
199,1,93,205,0.03001,"\int \frac{x^5}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-77 a^5-248 a^4 b x-252 a^3 b^2 x^2-48 a^2 b^3 x^3+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-77*a^5 - 248*a^4*b*x - 252*a^3*b^2*x^2 - 48*a^2*b^3*x^3 + 48*a*b^4*x^4 + 12*b^5*x^5 - 60*a*(a + b*x)^4*Log[a + b*x])/(12*b^6*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
200,1,73,171,0.026274,"\int \frac{x^4}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{a \left(25 a^3+88 a^2 b x+108 a b^2 x^2+48 b^3 x^3\right)+12 (a+b x)^4 \log (a+b x)}{12 b^5 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{3 a^2}{b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^4}{4 b^5 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 a^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(25*a^3 + 88*a^2*b*x + 108*a*b^2*x^2 + 48*b^3*x^3) + 12*(a + b*x)^4*Log[a + b*x])/(12*b^5*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
201,1,55,37,0.0146843,"\int \frac{x^3}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a^3-4 a^2 b x-6 a b^2 x^2-4 b^3 x^3}{4 b^4 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{x^4}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-a^3 - 4*a^2*b*x - 6*a*b^2*x^2 - 4*b^3*x^3)/(4*b^4*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
202,1,44,107,0.0140846,"\int \frac{x^2}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a^2-4 a b x-6 b^2 x^2}{12 b^3 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{a^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-a^2 - 4*a*b*x - 6*b^2*x^2)/(12*b^3*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
203,1,33,63,0.0101849,"\int \frac{x}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a-4 b x}{12 b^2 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{a}{4 b^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}-\frac{1}{3 b^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}",1,"(-a - 4*b*x)/(12*b^2*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
204,1,23,34,0.0090991,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-5/2),x]","-\frac{a+b x}{4 b \left((a+b x)^2\right)^{5/2}}","-\frac{1}{4 b (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}",1,"-1/4*(a + b*x)/(b*((a + b*x)^2)^(5/2))","A",1
205,1,84,194,0.033807,"\int \frac{1}{x \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{a \left(25 a^3+52 a^2 b x+42 a b^2 x^2+12 b^3 x^3\right)+12 \log (x) (a+b x)^4-12 (a+b x)^4 \log (a+b x)}{12 a^5 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(25*a^3 + 52*a^2*b*x + 42*a*b^2*x^2 + 12*b^3*x^3) + 12*(a + b*x)^4*Log[x] - 12*(a + b*x)^4*Log[a + b*x])/(12*a^5*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
206,1,103,235,0.0402726,"\int \frac{1}{x^2 \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{-a \left(12 a^4+125 a^3 b x+260 a^2 b^2 x^2+210 a b^3 x^3+60 b^4 x^4\right)-60 b x \log (x) (a+b x)^4+60 b x (a+b x)^4 \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b \log (x) (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-(a*(12*a^4 + 125*a^3*b*x + 260*a^2*b^2*x^2 + 210*a*b^3*x^3 + 60*b^4*x^4)) - 60*b*x*(a + b*x)^4*Log[x] + 60*b*x*(a + b*x)^4*Log[a + b*x])/(12*a^6*x*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
207,1,121,278,0.0434781,"\int \frac{1}{x^3 \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{a \left(-2 a^5+12 a^4 b x+125 a^3 b^2 x^2+260 a^2 b^3 x^3+210 a b^4 x^4+60 b^5 x^5\right)+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(a*(-2*a^5 + 12*a^4*b*x + 125*a^3*b^2*x^2 + 260*a^2*b^3*x^3 + 210*a*b^4*x^4 + 60*b^5*x^5) + 60*b^2*x^2*(a + b*x)^4*Log[x] - 60*b^2*x^2*(a + b*x)^4*Log[a + b*x])/(4*a^7*x^2*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
208,1,47,42,0.0155243,"\int x \left(9+12 x+4 x^2\right)^{5/2} \, dx","Integrate[x*(9 + 12*x + 4*x^2)^(5/2),x]","\frac{x^2 \sqrt{(2 x+3)^2} \left(64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right)}{28 x+42}","\frac{1}{28} \left(4 x^2+12 x+9\right)^{7/2}-\frac{1}{8} (2 x+3) \left(4 x^2+12 x+9\right)^{5/2}",1,"(x^2*Sqrt[(3 + 2*x)^2]*(1701 + 3780*x + 3780*x^2 + 2016*x^3 + 560*x^4 + 64*x^5))/(42 + 28*x)","A",1
209,1,37,42,0.0108108,"\int x \left(9+12 x+4 x^2\right)^{3/2} \, dx","Integrate[x*(9 + 12*x + 4*x^2)^(3/2),x]","\frac{x^2 \sqrt{(2 x+3)^2} \left(16 x^3+90 x^2+180 x+135\right)}{20 x+30}","\frac{1}{20} \left(4 x^2+12 x+9\right)^{5/2}-\frac{3}{16} (2 x+3) \left(4 x^2+12 x+9\right)^{3/2}",1,"(x^2*Sqrt[(3 + 2*x)^2]*(135 + 180*x + 90*x^2 + 16*x^3))/(30 + 20*x)","A",1
210,1,30,42,0.0066654,"\int x \sqrt{9+12 x+4 x^2} \, dx","Integrate[x*Sqrt[9 + 12*x + 4*x^2],x]","\frac{x^2 \sqrt{(2 x+3)^2} (4 x+9)}{6 (2 x+3)}","\frac{1}{12} \left(4 x^2+12 x+9\right)^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9}",1,"(x^2*Sqrt[(3 + 2*x)^2]*(9 + 4*x))/(6*(3 + 2*x))","A",1
211,1,33,48,0.0091772,"\int \frac{x}{\sqrt{9+12 x+4 x^2}} \, dx","Integrate[x/Sqrt[9 + 12*x + 4*x^2],x]","\frac{(2 x+3) (2 x-3 \log (2 x+3)+3)}{4 \sqrt{(2 x+3)^2}}","\frac{1}{4} \sqrt{4 x^2+12 x+9}-\frac{3 (2 x+3) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}",1,"((3 + 2*x)*(3 + 2*x - 3*Log[3 + 2*x]))/(4*Sqrt[(3 + 2*x)^2])","A",1
212,1,27,44,0.0072862,"\int \frac{x}{\left(9+12 x+4 x^2\right)^{3/2}} \, dx","Integrate[x/(9 + 12*x + 4*x^2)^(3/2),x]","\frac{-4 x-3}{8 (2 x+3) \sqrt{(2 x+3)^2}}","\frac{3}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{1}{4 \sqrt{4 x^2+12 x+9}}",1,"(-3 - 4*x)/(8*(3 + 2*x)*Sqrt[(3 + 2*x)^2])","A",1
213,1,27,44,0.0082187,"\int \frac{x}{\left(9+12 x+4 x^2\right)^{5/2}} \, dx","Integrate[x/(9 + 12*x + 4*x^2)^(5/2),x]","\frac{-8 x-3}{48 (2 x+3)^3 \sqrt{(2 x+3)^2}}","\frac{3}{16 (2 x+3) \left(4 x^2+12 x+9\right)^{3/2}}-\frac{1}{12 \left(4 x^2+12 x+9\right)^{3/2}}",1,"(-3 - 8*x)/(48*(3 + 2*x)^3*Sqrt[(3 + 2*x)^2])","A",1
214,1,27,44,0.008267,"\int \frac{x}{\left(9+12 x+4 x^2\right)^{7/2}} \, dx","Integrate[x/(9 + 12*x + 4*x^2)^(7/2),x]","\frac{-4 x-1}{40 (2 x+3)^5 \sqrt{(2 x+3)^2}}","\frac{1}{8 (2 x+3) \left(4 x^2+12 x+9\right)^{5/2}}-\frac{1}{20 \left(4 x^2+12 x+9\right)^{5/2}}",1,"(-1 - 4*x)/(40*(3 + 2*x)^5*Sqrt[(3 + 2*x)^2])","A",1
215,1,33,48,0.0090697,"\int \frac{x}{\sqrt{4+12 x+9 x^2}} \, dx","Integrate[x/Sqrt[4 + 12*x + 9*x^2],x]","\frac{(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt{(3 x+2)^2}}","\frac{1}{9} \sqrt{9 x^2+12 x+4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{9 x^2+12 x+4}}",1,"((2 + 3*x)*(2 + 3*x - 2*Log[2 + 3*x]))/(9*Sqrt[(2 + 3*x)^2])","A",1
216,1,33,48,0.0124016,"\int \frac{x}{\sqrt{4-12 x+9 x^2}} \, dx","Integrate[x/Sqrt[4 - 12*x + 9*x^2],x]","\frac{(3 x-2) (3 x+2 \log (3 x-2)-2)}{9 \sqrt{(2-3 x)^2}}","\frac{1}{9} \sqrt{9 x^2-12 x+4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{9 x^2-12 x+4}}",1,"((-2 + 3*x)*(-2 + 3*x + 2*Log[-2 + 3*x]))/(9*Sqrt[(2 - 3*x)^2])","A",1
217,1,35,48,0.0117609,"\int \frac{x}{\sqrt{-4+12 x-9 x^2}} \, dx","Integrate[x/Sqrt[-4 + 12*x - 9*x^2],x]","\frac{(3 x-2) (3 x+2 \log (2-3 x)-2)}{9 \sqrt{-(2-3 x)^2}}","-\frac{1}{9} \sqrt{-9 x^2+12 x-4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{-9 x^2+12 x-4}}",1,"((-2 + 3*x)*(-2 + 3*x + 2*Log[2 - 3*x]))/(9*Sqrt[-(2 - 3*x)^2])","A",1
218,1,35,48,0.0079643,"\int \frac{x}{\sqrt{-4-12 x-9 x^2}} \, dx","Integrate[x/Sqrt[-4 - 12*x - 9*x^2],x]","\frac{(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt{-(3 x+2)^2}}","-\frac{1}{9} \sqrt{-9 x^2-12 x-4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{-9 x^2-12 x-4}}",1,"((2 + 3*x)*(2 + 3*x - 2*Log[2 + 3*x]))/(9*Sqrt[-(2 + 3*x)^2])","A",1
219,1,15,12,0.0025976,"\int \frac{1+x}{2 x+x^2} \, dx","Integrate[(1 + x)/(2*x + x^2),x]","\frac{\log (x)}{2}+\frac{1}{2} \log (x+2)","\frac{1}{2} \log \left(x^2+2 x\right)",1,"Log[x]/2 + Log[2 + x]/2","A",1
220,1,9,10,0.0042182,"\int \frac{a+2 b x}{a x+b x^2} \, dx","Integrate[(a + 2*b*x)/(a*x + b*x^2),x]","\log (a+b x)+\log (x)","\log \left(a x+b x^2\right)",1,"Log[x] + Log[a + b*x]","A",1
221,1,99,62,0.0151455,"\int (d+e x)^4 \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^4*(b*x + c*x^2),x]","\frac{1}{3} d^3 x^3 (4 b e+c d)+\frac{1}{2} d^2 e x^4 (3 b e+2 c d)+\frac{1}{6} e^3 x^6 (b e+4 c d)+\frac{2}{5} d e^2 x^5 (2 b e+3 c d)+\frac{1}{2} b d^4 x^2+\frac{1}{7} c e^4 x^7","-\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{d (d+e x)^5 (c d-b e)}{5 e^3}+\frac{c (d+e x)^7}{7 e^3}",1,"(b*d^4*x^2)/2 + (d^3*(c*d + 4*b*e)*x^3)/3 + (d^2*e*(2*c*d + 3*b*e)*x^4)/2 + (2*d*e^2*(3*c*d + 2*b*e)*x^5)/5 + (e^3*(4*c*d + b*e)*x^6)/6 + (c*e^4*x^7)/7","A",1
222,1,67,62,0.0162327,"\int (d+e x)^3 \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^3*(b*x + c*x^2),x]","\frac{1}{60} x^2 \left(20 d^2 x (3 b e+c d)+12 e^2 x^3 (b e+3 c d)+45 d e x^2 (b e+c d)+30 b d^3+10 c e^3 x^4\right)","-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}",1,"(x^2*(30*b*d^3 + 20*d^2*(c*d + 3*b*e)*x + 45*d*e*(c*d + b*e)*x^2 + 12*e^2*(3*c*d + b*e)*x^3 + 10*c*e^3*x^4))/60","A",1
223,1,49,55,0.0121174,"\int (d+e x)^2 \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^2*(b*x + c*x^2),x]","\frac{1}{60} x^2 \left(15 e x^2 (b e+2 c d)+20 d x (2 b e+c d)+30 b d^2+12 c e^2 x^3\right)","\frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{3} d x^3 (2 b e+c d)+\frac{1}{2} b d^2 x^2+\frac{1}{5} c e^2 x^5",1,"(x^2*(30*b*d^2 + 20*d*(c*d + 2*b*e)*x + 15*e*(2*c*d + b*e)*x^2 + 12*c*e^2*x^3))/60","A",1
224,1,29,33,0.0049818,"\int (d+e x) \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)*(b*x + c*x^2),x]","\frac{1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x))","\frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4",1,"(x^2*(c*x*(4*d + 3*e*x) + b*(6*d + 4*e*x)))/12","A",1
225,1,17,17,0.0000755,"\int \left(b x+c x^2\right) \, dx","Integrate[b*x + c*x^2,x]","\frac{b x^2}{2}+\frac{c x^3}{3}","\frac{b x^2}{2}+\frac{c x^3}{3}",1,"(b*x^2)/2 + (c*x^3)/3","A",1
226,1,41,45,0.013827,"\int \frac{b x+c x^2}{d+e x} \, dx","Integrate[(b*x + c*x^2)/(d + e*x),x]","\frac{e x (2 b e-2 c d+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3}","\frac{d (c d-b e) \log (d+e x)}{e^3}-\frac{x (c d-b e)}{e^2}+\frac{c x^2}{2 e}",1,"(e*x*(-2*c*d + 2*b*e + c*e*x) + 2*d*(c*d - b*e)*Log[d + e*x])/(2*e^3)","A",1
227,1,41,48,0.0246812,"\int \frac{b x+c x^2}{(d+e x)^2} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^2,x]","\frac{\frac{d (b e-c d)}{d+e x}+(b e-2 c d) \log (d+e x)+c e x}{e^3}","-\frac{d (c d-b e)}{e^3 (d+e x)}-\frac{(2 c d-b e) \log (d+e x)}{e^3}+\frac{c x}{e^2}",1,"(c*e*x + (d*(-(c*d) + b*e))/(d + e*x) + (-2*c*d + b*e)*Log[d + e*x])/e^3","A",1
228,1,52,55,0.0174204,"\int \frac{b x+c x^2}{(d+e x)^3} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^3,x]","\frac{-b e (d+2 e x)+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2}","-\frac{d (c d-b e)}{2 e^3 (d+e x)^2}+\frac{2 c d-b e}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3}",1,"(-(b*e*(d + 2*e*x)) + c*d*(3*d + 4*e*x) + 2*c*(d + e*x)^2*Log[d + e*x])/(2*e^3*(d + e*x)^2)","A",1
229,1,44,60,0.0160655,"\int \frac{b x+c x^2}{(d+e x)^4} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^4,x]","-\frac{b e (d+3 e x)+2 c \left(d^2+3 d e x+3 e^2 x^2\right)}{6 e^3 (d+e x)^3}","\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{d (c d-b e)}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}",1,"-1/6*(b*e*(d + 3*e*x) + 2*c*(d^2 + 3*d*e*x + 3*e^2*x^2))/(e^3*(d + e*x)^3)","A",1
230,1,43,62,0.0156545,"\int \frac{b x+c x^2}{(d+e x)^5} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^5,x]","-\frac{b e (d+4 e x)+c \left(d^2+4 d e x+6 e^2 x^2\right)}{12 e^3 (d+e x)^4}","\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}",1,"-1/12*(b*e*(d + 4*e*x) + c*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)","A",1
231,1,159,137,0.0250386,"\int (d+e x)^4 \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^4*(b*x + c*x^2)^2,x]","\frac{1}{7} e^2 x^7 \left(b^2 e^2+8 b c d e+6 c^2 d^2\right)+\frac{2}{3} d e x^6 \left(b^2 e^2+3 b c d e+c^2 d^2\right)+\frac{1}{5} d^2 x^5 \left(6 b^2 e^2+8 b c d e+c^2 d^2\right)+\frac{1}{3} b^2 d^4 x^3+\frac{1}{2} b d^3 x^4 (2 b e+c d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9","\frac{(d+e x)^7 \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}",1,"(b^2*d^4*x^3)/3 + (b*d^3*(c*d + 2*b*e)*x^4)/2 + (d^2*(c^2*d^2 + 8*b*c*d*e + 6*b^2*e^2)*x^5)/5 + (2*d*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/3 + (e^2*(6*c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (c*e^3*(2*c*d + b*e)*x^8)/4 + (c^2*e^4*x^9)/9","A",1
232,1,127,127,0.0177546,"\int (d+e x)^3 \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^3*(b*x + c*x^2)^2,x]","\frac{1}{6} e x^6 \left(b^2 e^2+6 b c d e+3 c^2 d^2\right)+\frac{1}{5} d x^5 \left(3 b^2 e^2+6 b c d e+c^2 d^2\right)+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8","\frac{1}{6} e x^6 \left(b^2 e^2+6 b c d e+3 c^2 d^2\right)+\frac{1}{5} d x^5 \left(3 b^2 e^2+6 b c d e+c^2 d^2\right)+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8",1,"(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8","A",1
233,1,87,87,0.0139189,"\int (d+e x)^2 \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(b*x + c*x^2)^2,x]","\frac{1}{5} x^5 \left(b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7","\frac{1}{5} x^5 \left(b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{1}{3} b^2 d^2 x^3+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{2} b d x^4 (b e+c d)+\frac{1}{7} c^2 e^2 x^7",1,"(b^2*d^2*x^3)/3 + (b*d*(c*d + b*e)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3 + (c^2*e^2*x^7)/7","A",1
234,1,50,55,0.0101808,"\int (d+e x) \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)*(b*x + c*x^2)^2,x]","\frac{1}{60} x^3 \left(5 b^2 (4 d+3 e x)+6 b c x (5 d+4 e x)+2 c^2 x^2 (6 d+5 e x)\right)","\frac{1}{3} b^2 d x^3+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{4} b x^4 (b e+2 c d)+\frac{1}{6} c^2 e x^6",1,"(x^3*(5*b^2*(4*d + 3*e*x) + 6*b*c*x*(5*d + 4*e*x) + 2*c^2*x^2*(6*d + 5*e*x)))/60","A",1
235,1,30,30,0.0016758,"\int \left(b x+c x^2\right)^2 \, dx","Integrate[(b*x + c*x^2)^2,x]","\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}","\frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}",1,"(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5","A",1
236,1,106,93,0.0413753,"\int \frac{\left(b x+c x^2\right)^2}{d+e x} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x),x]","\frac{\left(b^2 d^2 e^2-2 b c d^3 e+c^2 d^4\right) \log (d+e x)}{e^5}-\frac{d x (c d-b e)^2}{e^4}+\frac{x^2 (b e-c d)^2}{2 e^3}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e}","\frac{d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac{d x (c d-b e)^2}{e^4}+\frac{x^2 (c d-b e)^2}{2 e^3}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e}",1,"-((d*(c*d - b*e)^2*x)/e^4) + ((-(c*d) + b*e)^2*x^2)/(2*e^3) - (c*(c*d - 2*b*e)*x^3)/(3*e^2) + (c^2*x^4)/(4*e) + ((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*Log[d + e*x])/e^5","A",1
237,1,114,107,0.0920633,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^2,x]","\frac{3 e x \left(b^2 e^2-4 b c d e+3 c^2 d^2\right)-6 d \left(b^2 e^2-3 b c d e+2 c^2 d^2\right) \log (d+e x)-\frac{3 d^2 (c d-b e)^2}{d+e x}-3 c e^2 x^2 (c d-b e)+c^2 e^3 x^3}{3 e^5}","-\frac{d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac{2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac{x (c d-b e) (3 c d-b e)}{e^4}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2}",1,"(3*e*(3*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*x - 3*c*e^2*(c*d - b*e)*x^2 + c^2*e^3*x^3 - (3*d^2*(c*d - b*e)^2)/(d + e*x) - 6*d*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*Log[d + e*x])/(3*e^5)","A",1
238,1,116,119,0.065271,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^3,x]","\frac{\frac{4 d \left(b^2 e^2-3 b c d e+2 c^2 d^2\right)}{d+e x}+2 \left(b^2 e^2-6 b c d e+6 c^2 d^2\right) \log (d+e x)-\frac{d^2 (c d-b e)^2}{(d+e x)^2}-2 c e x (3 c d-2 b e)+c^2 e^2 x^2}{2 e^5}","\frac{\left(b^2 e^2-6 b c d e+6 c^2 d^2\right) \log (d+e x)}{e^5}-\frac{d^2 (c d-b e)^2}{2 e^5 (d+e x)^2}+\frac{2 d (2 c d-b e) (c d-b e)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3}",1,"(-2*c*e*(3*c*d - 2*b*e)*x + c^2*e^2*x^2 - (d^2*(c*d - b*e)^2)/(d + e*x)^2 + (4*d*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2))/(d + e*x) + 2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*Log[d + e*x])/(2*e^5)","A",1
239,1,134,120,0.0536803,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^4,x]","\frac{-b^2 e^2 \left(d^2+3 d e x+3 e^2 x^2\right)+b c d e \left(11 d^2+27 d e x+18 e^2 x^2\right)-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left(-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right)}{3 e^5 (d+e x)^3}","-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{e^5 (d+e x)}-\frac{d^2 (c d-b e)^2}{3 e^5 (d+e x)^3}+\frac{d (c d-b e) (2 c d-b e)}{e^5 (d+e x)^2}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4}",1,"(-(b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) + b*c*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + c^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - 6*c*(2*c*d - b*e)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)","A",1
240,1,126,131,0.0441707,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^5,x]","\frac{-b^2 e^2 \left(d^2+4 d e x+6 e^2 x^2\right)-6 b c e \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)+c^2 d \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}","-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{c^2 \log (d+e x)}{e^5}",1,"(-(b^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) - 6*b*c*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)","A",1
241,1,116,132,0.0478629,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^6,x]","-\frac{b^2 e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+3 b c e \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+6 c^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)}{30 e^5 (d+e x)^5}","-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{e^5 (d+e x)}",1,"-1/30*(b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*c*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(e^5*(d + e*x)^5)","A",1
242,1,116,137,0.0401051,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^7,x]","-\frac{b^2 e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+2 b c e \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+2 c^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)}{60 e^5 (d+e x)^6}","-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2}",1,"-1/60*(b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*b*c*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))/(e^5*(d + e*x)^6)","A",1
243,1,117,137,0.0479407,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^8,x]","-\frac{2 b^2 e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+3 b c e \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+2 c^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)}{210 e^5 (d+e x)^7}","-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{c^2}{3 e^5 (d+e x)^3}",1,"-1/210*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(e^5*(d + e*x)^7)","A",1
244,1,225,225,0.0325996,"\int (d+e x)^4 \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^4*(b*x + c*x^2)^3,x]","\frac{1}{4} b^3 d^4 x^4+\frac{1}{3} c e^2 x^9 \left(b^2 e^2+4 b c d e+2 c^2 d^2\right)+\frac{1}{2} b d^2 x^6 \left(2 b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{8} e x^8 \left(b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right)+\frac{1}{7} d x^7 \left(4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right)+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11}","\frac{1}{4} b^3 d^4 x^4+\frac{1}{3} c e^2 x^9 \left(b^2 e^2+4 b c d e+2 c^2 d^2\right)+\frac{1}{2} b d^2 x^6 \left(2 b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{8} e x^8 \left(b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right)+\frac{1}{7} d x^7 \left(4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right)+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11}",1,"(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e + 2*b^2*e^2)*x^6)/2 + (d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3)*x^8)/8 + (c*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/10 + (c^3*e^4*x^11)/11","A",1
245,1,169,162,0.0235809,"\int (d+e x)^3 \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^3*(b*x + c*x^2)^3,x]","\frac{1}{4} b^3 d^3 x^4+\frac{3}{8} c e x^8 \left(b^2 e^2+3 b c d e+c^2 d^2\right)+\frac{1}{2} b d x^6 \left(b^2 e^2+3 b c d e+c^2 d^2\right)+\frac{3}{5} b^2 d^2 x^5 (b e+c d)+\frac{1}{7} x^7 \left(b^3 e^3+9 b^2 c d e^2+9 b c^2 d^2 e+c^3 d^3\right)+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}","\frac{1}{4} b^3 d^3 x^4+\frac{3}{8} c e x^8 \left(b^2 e^2+3 b c d e+c^2 d^2\right)+\frac{1}{7} x^7 (b e+c d) \left(b^2 e^2+8 b c d e+c^2 d^2\right)+\frac{1}{2} b d x^6 \left(b^2 e^2+3 b c d e+c^2 d^2\right)+\frac{3}{5} b^2 d^2 x^5 (b e+c d)+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}",1,"(b^3*d^3*x^4)/4 + (3*b^2*d^2*(c*d + b*e)*x^5)/5 + (b*d*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/2 + ((c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + b^3*e^3)*x^7)/7 + (3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^8)/8 + (c^2*e^2*(c*d + b*e)*x^9)/3 + (c^3*e^3*x^10)/10","A",1
246,1,127,127,0.0222845,"\int (d+e x)^2 \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^2*(b*x + c*x^2)^3,x]","\frac{1}{4} b^3 d^2 x^4+\frac{1}{7} c x^7 \left(3 b^2 e^2+6 b c d e+c^2 d^2\right)+\frac{1}{6} b x^6 \left(b^2 e^2+6 b c d e+3 c^2 d^2\right)+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9","\frac{1}{4} b^3 d^2 x^4+\frac{1}{7} c x^7 \left(3 b^2 e^2+6 b c d e+c^2 d^2\right)+\frac{1}{6} b x^6 \left(b^2 e^2+6 b c d e+3 c^2 d^2\right)+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9",1,"(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9","A",1
247,1,75,75,0.0106382,"\int (d+e x) \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)*(b*x + c*x^2)^3,x]","\frac{1}{4} b^3 d x^4+\frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8","\frac{1}{4} b^3 d x^4+\frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8",1,"(b^3*d*x^4)/4 + (b^2*(3*c*d + b*e)*x^5)/5 + (b*c*(c*d + b*e)*x^6)/2 + (c^2*(c*d + 3*b*e)*x^7)/7 + (c^3*e*x^8)/8","A",1
248,1,43,43,0.0021988,"\int \left(b x+c x^2\right)^3 \, dx","Integrate[(b*x + c*x^2)^3,x]","\frac{b^3 x^4}{4}+\frac{3}{5} b^2 c x^5+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7}","\frac{b^3 x^4}{4}+\frac{3}{5} b^2 c x^5+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7}",1,"(b^3*x^4)/4 + (3*b^2*c*x^5)/5 + (b*c^2*x^6)/2 + (c^3*x^7)/7","A",1
249,1,144,151,0.0802549,"\int \frac{\left(b x+c x^2\right)^3}{d+e x} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x),x]","\frac{15 c e^4 x^4 \left(3 b^2 e^2-3 b c d e+c^2 d^2\right)-12 c^2 e^5 x^5 (c d-3 b e)+60 d^3 (c d-b e)^3 \log (d+e x)-60 d^2 e x (c d-b e)^3+20 e^3 x^3 (b e-c d)^3+30 d e^2 x^2 (c d-b e)^3+10 c^3 e^6 x^6}{60 e^7}","\frac{c x^4 \left(3 b^2 e^2-3 b c d e+c^2 d^2\right)}{4 e^3}-\frac{c^2 x^5 (c d-3 b e)}{5 e^2}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac{d^2 x (c d-b e)^3}{e^6}+\frac{d x^2 (c d-b e)^3}{2 e^5}-\frac{x^3 (c d-b e)^3}{3 e^4}+\frac{c^3 x^6}{6 e}",1,"(-60*d^2*e*(c*d - b*e)^3*x + 30*d*e^2*(c*d - b*e)^3*x^2 + 20*e^3*(-(c*d) + b*e)^3*x^3 + 15*c*e^4*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2)*x^4 - 12*c^2*e^5*(c*d - 3*b*e)*x^5 + 10*c^3*e^6*x^6 + 60*d^3*(c*d - b*e)^3*Log[d + e*x])/(60*e^7)","A",1
250,1,160,166,0.0576376,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^2} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^2,x]","\frac{-5 c^2 e^4 x^4 (2 c d-3 b e)-\frac{20 d^3 (c d-b e)^3}{d+e x}-60 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)+20 c e^3 x^3 (c d-b e)^2+10 e^2 x^2 (c d-b e)^2 (b e-4 c d)+20 d e x (5 c d-2 b e) (c d-b e)^2+4 c^3 e^5 x^5}{20 e^7}","-\frac{c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{d x (5 c d-2 b e) (c d-b e)^2}{e^6}-\frac{x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac{c x^3 (c d-b e)^2}{e^4}+\frac{c^3 x^5}{5 e^2}",1,"(20*d*e*(5*c*d - 2*b*e)*(c*d - b*e)^2*x + 10*e^2*(c*d - b*e)^2*(-4*c*d + b*e)*x^2 + 20*c*e^3*(c*d - b*e)^2*x^3 - 5*c^2*e^4*(2*c*d - 3*b*e)*x^4 + 4*c^3*e^5*x^5 - (20*d^3*(c*d - b*e)^3)/(d + e*x) - 60*d^2*(c*d - b*e)^2*(2*c*d - b*e)*Log[d + e*x])/(20*e^7)","A",1
251,1,207,200,0.127806,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^3} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^3,x]","\frac{6 c e^2 x^2 \left(b^2 e^2-3 b c d e+2 c^2 d^2\right)+4 e x \left(b^3 e^3-9 b^2 c d e^2+18 b c^2 d^2 e-10 c^3 d^3\right)+12 d \left(-b^3 e^3+6 b^2 c d e^2-10 b c^2 d^2 e+5 c^3 d^3\right) \log (d+e x)-4 c^2 e^3 x^3 (c d-b e)-\frac{2 d^3 (c d-b e)^3}{(d+e x)^2}+\frac{12 d^2 (c d-b e)^2 (2 c d-b e)}{d+e x}+c^3 e^4 x^4}{4 e^7}","\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) \log (d+e x)}{e^7}-\frac{x (c d-b e) \left(b^2 e^2-8 b c d e+10 c^2 d^2\right)}{e^6}-\frac{c^2 x^3 (c d-b e)}{e^4}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac{c^3 x^4}{4 e^3}",1,"(4*e*(-10*c^3*d^3 + 18*b*c^2*d^2*e - 9*b^2*c*d*e^2 + b^3*e^3)*x + 6*c*e^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*x^2 - 4*c^2*e^3*(c*d - b*e)*x^3 + c^3*e^4*x^4 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^2 + (12*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(d + e*x) + 12*d*(5*c^3*d^3 - 10*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*Log[d + e*x])/(4*e^7)","A",1
252,1,210,213,0.0984336,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^4} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^4,x]","\frac{6 c e x \left(3 b^2 e^2-12 b c d e+10 c^2 d^2\right)+\frac{18 d \left(b^3 e^3-6 b^2 c d e^2+10 b c^2 d^2 e-5 c^3 d^3\right)}{d+e x}+6 \left(b^3 e^3-12 b^2 c d e^2+30 b c^2 d^2 e-20 c^3 d^3\right) \log (d+e x)-3 c^2 e^2 x^2 (4 c d-3 b e)-\frac{2 d^3 (c d-b e)^3}{(d+e x)^3}+\frac{9 d^2 (2 c d-b e) (c d-b e)^2}{(d+e x)^2}+2 c^3 e^3 x^3}{6 e^7}","-\frac{3 d \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) (c d-b e)}{e^7 (d+e x)}-\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right) \log (d+e x)}{e^7}+\frac{c x \left(3 b^2 e^2-12 b c d e+10 c^2 d^2\right)}{e^6}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac{d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac{3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac{c^3 x^3}{3 e^4}",1,"(6*c*e*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x - 3*c^2*e^2*(4*c*d - 3*b*e)*x^2 + 2*c^3*e^3*x^3 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^3 + (9*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(d + e*x)^2 + (18*d*(-5*c^3*d^3 + 10*b*c^2*d^2*e - 6*b^2*c*d*e^2 + b^3*e^3))/(d + e*x) + 6*(-20*c^3*d^3 + 30*b*c^2*d^2*e - 12*b^2*c*d*e^2 + b^3*e^3)*Log[d + e*x])/(6*e^7)","A",1
253,1,210,213,0.1029734,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^5} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^5,x]","\frac{12 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) \log (d+e x)+\frac{-4 b^3 e^3+48 b^2 c d e^2-120 b c^2 d^2 e+80 c^3 d^3}{d+e x}+\frac{6 d \left(b^3 e^3-6 b^2 c d e^2+10 b c^2 d^2 e-5 c^3 d^3\right)}{(d+e x)^2}-4 c^2 e x (5 c d-3 b e)-\frac{d^3 (c d-b e)^3}{(d+e x)^4}+\frac{4 d^2 (c d-b e)^2 (2 c d-b e)}{(d+e x)^3}+2 c^3 e^2 x^2}{4 e^7}","\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{e^7 (d+e x)}-\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{2 e^7 (d+e x)^2}+\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) \log (d+e x)}{e^7}-\frac{c^2 x (5 c d-3 b e)}{e^6}-\frac{d^3 (c d-b e)^3}{4 e^7 (d+e x)^4}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^3}+\frac{c^3 x^2}{2 e^5}",1,"(-4*c^2*e*(5*c*d - 3*b*e)*x + 2*c^3*e^2*x^2 - (d^3*(c*d - b*e)^3)/(d + e*x)^4 + (4*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(d + e*x)^3 + (6*d*(-5*c^3*d^3 + 10*b*c^2*d^2*e - 6*b^2*c*d*e^2 + b^3*e^3))/(d + e*x)^2 + (80*c^3*d^3 - 120*b*c^2*d^2*e + 48*b^2*c*d*e^2 - 4*b^3*e^3)/(d + e*x) + 12*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Log[d + e*x])/(4*e^7)","A",1
254,1,242,218,0.099441,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^6} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^6,x]","-\frac{b^3 e^3 \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+12 b^2 c e^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)-b c^2 d e \left(137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right)+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left(87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right)}{20 e^7 (d+e x)^5}","-\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7 (d+e x)}+\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{2 e^7 (d+e x)^2}-\frac{d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7 (d+e x)^3}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}-\frac{d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac{c^3 x}{e^6}",1,"-1/20*(b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 12*b^2*c*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - b*c^2*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + 2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)","A",1
255,1,231,228,0.0766246,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^7} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^7,x]","\frac{-b^3 e^3 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)-6 b^2 c e^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)-30 b c^2 e \left(d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right)+c^3 d \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}","-\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac{c^3 \log (d+e x)}{e^7}",1,"(-(b^3*e^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) - 6*b^2*c*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) - 30*b*c^2*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)","A",1
256,1,221,230,0.0671859,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^8} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^8,x]","-\frac{b^3 e^3 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+4 b^2 c e^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+10 b c^2 e \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)+20 c^3 \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)}{140 e^7 (d+e x)^7}","-\frac{c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{4 e^7 (d+e x)^4}-\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{5 e^7 (d+e x)^5}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac{c^3}{e^7 (d+e x)}",1,"-1/140*(b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*b^2*c*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 10*b*c^2*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)","A",1
257,1,221,231,0.0757619,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^9} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^9,x]","-\frac{b^3 e^3 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+3 b^2 c e^2 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+5 b c^2 e \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)+5 c^3 \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)}{280 e^7 (d+e x)^8}","-\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{5 e^7 (d+e x)^5}-\frac{d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{2 e^7 (d+e x)^6}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac{c^3}{2 e^7 (d+e x)^2}",1,"-1/280*(b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*c*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*c^2*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)","A",1
258,1,222,234,0.0711947,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^{10}} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^10,x]","-\frac{5 b^3 e^3 \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+12 b^2 c e^2 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+15 b c^2 e \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)+10 c^3 \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)}{2520 e^7 (d+e x)^9}","-\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{6 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{c^3}{3 e^7 (d+e x)^3}",1,"-1/2520*(5*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*b^2*c*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 15*b*c^2*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)","A",1
259,1,90,99,0.0440231,"\int \frac{(d+e x)^4}{b x+c x^2} \, dx","Integrate[(d + e*x)^4/(b*x + c*x^2),x]","\frac{b c e^2 x \left(6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left(18 d^2+6 d e x+e^2 x^2\right)\right)-6 (c d-b e)^4 \log (b+c x)+6 c^4 d^4 \log (x)}{6 b c^4}","\frac{e^2 x \left(b^2 e^2-4 b c d e+6 c^2 d^2\right)}{c^3}-\frac{(c d-b e)^4 \log (b+c x)}{b c^4}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}+\frac{d^4 \log (x)}{b}+\frac{e^4 x^3}{3 c}",1,"(b*c*e^2*x*(6*b^2*e^2 - 3*b*c*e*(8*d + e*x) + 2*c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 6*c^4*d^4*Log[x] - 6*(c*d - b*e)^4*Log[b + c*x])/(6*b*c^4)","A",1
260,1,59,64,0.0267423,"\int \frac{(d+e x)^3}{b x+c x^2} \, dx","Integrate[(d + e*x)^3/(b*x + c*x^2),x]","\frac{b c e^2 x (-2 b e+6 c d+c e x)-2 (c d-b e)^3 \log (b+c x)+2 c^3 d^3 \log (x)}{2 b c^3}","-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{d^3 \log (x)}{b}+\frac{e^3 x^2}{2 c}",1,"(b*c*e^2*x*(6*c*d - 2*b*e + c*e*x) + 2*c^3*d^3*Log[x] - 2*(c*d - b*e)^3*Log[b + c*x])/(2*b*c^3)","A",1
261,1,42,42,0.0181743,"\int \frac{(d+e x)^2}{b x+c x^2} \, dx","Integrate[(d + e*x)^2/(b*x + c*x^2),x]","\frac{-(c d-b e)^2 \log (b+c x)+b c e^2 x+c^2 d^2 \log (x)}{b c^2}","-\frac{(c d-b e)^2 \log (b+c x)}{b c^2}+\frac{d^2 \log (x)}{b}+\frac{e^2 x}{c}",1,"(b*c*e^2*x + c^2*d^2*Log[x] - (c*d - b*e)^2*Log[b + c*x])/(b*c^2)","A",1
262,1,29,30,0.009761,"\int \frac{d+e x}{b x+c x^2} \, dx","Integrate[(d + e*x)/(b*x + c*x^2),x]","\frac{(b e-c d) \log (b+c x)}{b c}+\frac{d \log (x)}{b}","\frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c}",1,"(d*Log[x])/b + ((-(c*d) + b*e)*Log[b + c*x])/(b*c)","A",1
263,1,18,18,0.0032513,"\int \frac{1}{b x+c x^2} \, dx","Integrate[(b*x + c*x^2)^(-1),x]","\frac{\log (x)}{b}-\frac{\log (b+c x)}{b}","\frac{\log (x)}{b}-\frac{\log (b+c x)}{b}",1,"Log[x]/b - Log[b + c*x]/b","A",1
264,1,48,53,0.0238217,"\int \frac{1}{(d+e x) \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)*(b*x + c*x^2)),x]","\frac{-c d \log (b+c x)+b e \log (d+e x)-b e \log (x)+c d \log (x)}{b c d^2-b^2 d e}","-\frac{c \log (b+c x)}{b (c d-b e)}+\frac{e \log (d+e x)}{d (c d-b e)}+\frac{\log (x)}{b d}",1,"(c*d*Log[x] - b*e*Log[x] - c*d*Log[b + c*x] + b*e*Log[d + e*x])/(b*c*d^2 - b^2*d*e)","A",1
265,1,83,87,0.0942768,"\int \frac{1}{(d+e x)^2 \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(b*x + c*x^2)),x]","\frac{\frac{b e ((d+e x) (2 c d-b e) \log (d+e x)+d (b e-c d))-c^2 d^2 (d+e x) \log (b+c x)}{(d+e x) (c d-b e)^2}+\log (x)}{b d^2}","-\frac{c^2 \log (b+c x)}{b (c d-b e)^2}+\frac{e (2 c d-b e) \log (d+e x)}{d^2 (c d-b e)^2}-\frac{e}{d (d+e x) (c d-b e)}+\frac{\log (x)}{b d^2}",1,"(Log[x] + (-(c^2*d^2*(d + e*x)*Log[b + c*x]) + b*e*(d*(-(c*d) + b*e) + (2*c*d - b*e)*(d + e*x)*Log[d + e*x]))/((c*d - b*e)^2*(d + e*x)))/(b*d^2)","A",1
266,1,116,134,0.2255506,"\int \frac{1}{(d+e x)^3 \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(b*x + c*x^2)),x]","\frac{\frac{e \left(\frac{d (c d-b e) (c d (5 d+4 e x)-b e (3 d+2 e x))}{(d+e x)^2}-2 \left(b^2 e^2-3 b c d e+3 c^2 d^2\right) \log (d+e x)\right)}{d^3}+\frac{2 c^3 \log (b+c x)}{b}}{2 (b e-c d)^3}+\frac{\log (x)}{b d^3}","\frac{e \left(b^2 e^2-3 b c d e+3 c^2 d^2\right) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}-\frac{e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac{e}{2 d (d+e x)^2 (c d-b e)}+\frac{\log (x)}{b d^3}",1,"Log[x]/(b*d^3) + ((2*c^3*Log[b + c*x])/b + (e*((d*(c*d - b*e)*(-(b*e*(3*d + 2*e*x)) + c*d*(5*d + 4*e*x)))/(d + e*x)^2 - 2*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*Log[d + e*x]))/d^3)/(2*(-(c*d) + b*e)^3)","A",1
267,1,116,118,0.0547748,"\int \frac{(d+e x)^5}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(b*x + c*x^2)^2,x]","\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}+\frac{d^4 \log (x) (5 b e-2 c d)}{b^3}+\frac{(b e-c d)^5}{b^2 c^4 (b+c x)}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2}","\frac{(c d-b e)^4 (3 b e+2 c d) \log (b+c x)}{b^3 c^4}-\frac{d^4 \log (x) (2 c d-5 b e)}{b^3}-\frac{(c d-b e)^5}{b^2 c^4 (b+c x)}-\frac{d^5}{b^2 x}+\frac{e^4 x (5 c d-2 b e)}{c^3}+\frac{e^5 x^2}{2 c^2}",1,"-(d^5/(b^2*x)) + (e^4*(5*c*d - 2*b*e)*x)/c^3 + (e^5*x^2)/(2*c^2) + (-(c*d) + b*e)^5/(b^2*c^4*(b + c*x)) + (d^4*(-2*c*d + 5*b*e)*Log[x])/b^3 + ((c*d - b*e)^4*(2*c*d + 3*b*e)*Log[b + c*x])/(b^3*c^4)","A",1
268,1,95,94,0.0881339,"\int \frac{(d+e x)^4}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(b*x + c*x^2)^2,x]","\frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}+\frac{2 d^3 \log (x) (2 b e-c d)}{b^3}-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2}","\frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac{2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2}",1,"-(d^4/(b^2*x)) + (e^4*x)/c^2 - (c*d - b*e)^4/(b^2*c^3*(b + c*x)) + (2*d^3*(-(c*d) + 2*b*e)*Log[x])/b^3 + (2*(c*d - b*e)^3*(c*d + b*e)*Log[b + c*x])/(b^3*c^3)","A",1
269,1,79,87,0.0712696,"\int \frac{(d+e x)^3}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(b*x + c*x^2)^2,x]","\frac{\frac{b (b e-c d)^3}{c^2 (b+c x)}+\frac{(c d-b e)^2 (b e+2 c d) \log (b+c x)}{c^2}+d^2 \log (x) (3 b e-2 c d)-\frac{b d^3}{x}}{b^3}","\frac{(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac{d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac{(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac{d^3}{b^2 x}",1,"(-((b*d^3)/x) + (b*(-(c*d) + b*e)^3)/(c^2*(b + c*x)) + d^2*(-2*c*d + 3*b*e)*Log[x] + ((c*d - b*e)^2*(2*c*d + b*e)*Log[b + c*x])/c^2)/b^3","A",1
270,1,67,73,0.0708784,"\int \frac{(d+e x)^2}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(b*x + c*x^2)^2,x]","\frac{-\frac{b (c d-b e)^2}{c (b+c x)}+2 d \log (x) (b e-c d)+2 d (c d-b e) \log (b+c x)-\frac{b d^2}{x}}{b^3}","-\frac{2 d \log (x) (c d-b e)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}-\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{d^2}{b^2 x}",1,"(-((b*d^2)/x) - (b*(c*d - b*e)^2)/(c*(b + c*x)) + 2*d*(-(c*d) + b*e)*Log[x] + 2*d*(c*d - b*e)*Log[b + c*x])/b^3","A",1
271,1,56,65,0.037328,"\int \frac{d+e x}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)/(b*x + c*x^2)^2,x]","\frac{\frac{b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac{b d}{x}}{b^3}","-\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{c d-b e}{b^2 (b+c x)}-\frac{d}{b^2 x}",1,"(-((b*d)/x) + (b*(-(c*d) + b*e))/(b + c*x) + (-2*c*d + b*e)*Log[x] + (2*c*d - b*e)*Log[b + c*x])/b^3","A",1
272,1,35,43,0.036587,"\int \frac{1}{\left(b x+c x^2\right)^2} \, dx","Integrate[(b*x + c*x^2)^(-2),x]","-\frac{b \left(\frac{c}{b+c x}+\frac{1}{x}\right)-2 c \log (b+c x)+2 c \log (x)}{b^3}","-\frac{2 c \log (x)}{b^3}+\frac{2 c \log (b+c x)}{b^3}-\frac{b+2 c x}{b^2 \left(b x+c x^2\right)}",1,"-((b*(x^(-1) + c/(b + c*x)) + 2*c*Log[x] - 2*c*Log[b + c*x])/b^3)","A",1
273,1,111,110,0.0850417,"\int \frac{1}{(d+e x) \left(b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(b*x + c*x^2)^2),x]","\frac{\left(2 c^3 d-3 b c^2 e\right) \log (b+c x)}{b^3 (b e-c d)^2}+\frac{\log (x) (-b e-2 c d)}{b^3 d^2}+\frac{c^2}{b^2 (b+c x) (b e-c d)}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2}","\frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac{\log (x) (b e+2 c d)}{b^3 d^2}-\frac{c^2}{b^2 (b+c x) (c d-b e)}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2}",1,"-(1/(b^2*d*x)) + c^2/(b^2*(-(c*d) + b*e)*(b + c*x)) + ((-2*c*d - b*e)*Log[x])/(b^3*d^2) + ((2*c^3*d - 3*b*c^2*e)*Log[b + c*x])/(b^3*(-(c*d) + b*e)^2) + (e^3*Log[d + e*x])/(d^2*(c*d - b*e)^2)","A",1
274,1,145,144,0.1562208,"\int \frac{1}{(d+e x)^2 \left(b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(b*x + c*x^2)^2),x]","\frac{2 c^3 (2 b e-c d) \log (b+c x)}{b^3 (b e-c d)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}-\frac{1}{b^2 d^2 x}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}","\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}-\frac{1}{b^2 d^2 x}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}",1,"-(1/(b^2*d^2*x)) - c^3/(b^2*(c*d - b*e)^2*(b + c*x)) - e^3/(d^2*(c*d - b*e)^2*(d + e*x)) - (2*(c*d + b*e)*Log[x])/(b^3*d^3) + (2*c^3*(-(c*d) + 2*b*e)*Log[b + c*x])/(b^3*(-(c*d) + b*e)^3) + (2*e^3*(2*c*d - b*e)*Log[d + e*x])/(d^3*(c*d - b*e)^3)","A",1
275,1,202,203,0.1143684,"\int \frac{(d+e x)^7}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^7/(b*x + c*x^2)^3,x]","\frac{1}{2} \left(\frac{2 (c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{2 d^6 (3 c d-7 b e)}{b^4 x}+\frac{(c d-b e)^7}{b^3 c^5 (b+c x)^2}-\frac{d^7}{b^3 x^2}+\frac{6 d^5 \log (x) \left(7 b^2 e^2-7 b c d e+2 c^2 d^2\right)}{b^5}+\frac{6 (b e-c d)^5 \left(2 b^2 e^2+3 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 c^5}+\frac{2 e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{c^3}\right)","\frac{(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac{d^6 (3 c d-7 b e)}{b^4 x}+\frac{(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}-\frac{d^7}{2 b^3 x^2}+\frac{3 d^5 \log (x) \left(7 b^2 e^2-7 b c d e+2 c^2 d^2\right)}{b^5}-\frac{3 (c d-b e)^5 \left(2 b^2 e^2+3 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 c^5}+\frac{e^6 x (7 c d-3 b e)}{c^4}+\frac{e^7 x^2}{2 c^3}",1,"(-(d^7/(b^3*x^2)) + (2*d^6*(3*c*d - 7*b*e))/(b^4*x) + (2*e^6*(7*c*d - 3*b*e)*x)/c^4 + (e^7*x^2)/c^3 + (c*d - b*e)^7/(b^3*c^5*(b + c*x)^2) + (2*(c*d - b*e)^6*(3*c*d + 4*b*e))/(b^4*c^5*(b + c*x)) + (6*d^5*(2*c^2*d^2 - 7*b*c*d*e + 7*b^2*e^2)*Log[x])/b^5 + (6*(-(c*d) + b*e)^5*(2*c^2*d^2 + 3*b*c*d*e + 2*b^2*e^2)*Log[b + c*x])/(b^5*c^5))/2","A",1
276,1,179,179,0.0813604,"\int \frac{(d+e x)^6}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^6/(b*x + c*x^2)^3,x]","\frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac{d^6}{2 b^3 x^2}+\frac{3 d^4 \log (x) \left(5 b^2 e^2-6 b c d e+2 c^2 d^2\right)}{b^5}-\frac{3 (c d-b e)^4 \left(b^2 e^2+2 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 c^4}+\frac{e^6 x}{c^3}","\frac{3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac{3 d^5 (c d-2 b e)}{b^4 x}+\frac{(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac{d^6}{2 b^3 x^2}+\frac{3 d^4 \log (x) \left(5 b^2 e^2-6 b c d e+2 c^2 d^2\right)}{b^5}-\frac{3 (c d-b e)^4 \left(b^2 e^2+2 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 c^4}+\frac{e^6 x}{c^3}",1,"-1/2*d^6/(b^3*x^2) + (3*d^5*(c*d - 2*b*e))/(b^4*x) + (e^6*x)/c^3 + (c*d - b*e)^6/(2*b^3*c^4*(b + c*x)^2) + (3*(c*d - b*e)^5*(c*d + b*e))/(b^4*c^4*(b + c*x)) + (3*d^4*(2*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*Log[x])/b^5 - (3*(c*d - b*e)^4*(2*c^2*d^2 + 2*b*c*d*e + b^2*e^2)*Log[b + c*x])/(b^5*c^4)","A",1
277,1,165,171,0.0973036,"\int \frac{(d+e x)^5}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(b*x + c*x^2)^3,x]","-\frac{\frac{b^2 (b e-c d)^5}{c^3 (b+c x)^2}-2 d^3 \log (x) \left(10 b^2 e^2-15 b c d e+6 c^2 d^2\right)+\frac{2 (c d-b e)^3 \left(b^2 e^2+3 b c d e+6 c^2 d^2\right) \log (b+c x)}{c^3}+\frac{b^2 d^5}{x^2}-\frac{2 b (c d-b e)^4 (2 b e+3 c d)}{c^3 (b+c x)}+\frac{2 b d^4 (5 b e-3 c d)}{x}}{2 b^5}","\frac{(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac{d^4 (3 c d-5 b e)}{b^4 x}+\frac{(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}-\frac{d^5}{2 b^3 x^2}+\frac{d^3 \log (x) \left(10 b^2 e^2-15 b c d e+6 c^2 d^2\right)}{b^5}-\frac{(c d-b e)^3 \left(b^2 e^2+3 b c d e+6 c^2 d^2\right) \log (b+c x)}{b^5 c^3}",1,"-1/2*((b^2*d^5)/x^2 + (2*b*d^4*(-3*c*d + 5*b*e))/x + (b^2*(-(c*d) + b*e)^5)/(c^3*(b + c*x)^2) - (2*b*(c*d - b*e)^4*(3*c*d + 2*b*e))/(c^3*(b + c*x)) - 2*d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x] + (2*(c*d - b*e)^3*(6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/c^3)/b^5","A",1
278,1,130,136,0.0791888,"\int \frac{(d+e x)^4}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(b*x + c*x^2)^3,x]","-\frac{-\frac{b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac{b^2 d^4}{x^2}+\frac{2 b (b e-c d)^3 (b e+3 c d)}{c^2 (b+c x)}+\frac{2 b d^3 (4 b e-3 c d)}{x}-12 d^2 \log (x) (c d-b e)^2+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5}","\frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}+\frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}-\frac{d^4}{2 b^3 x^2}",1,"-1/2*((b^2*d^4)/x^2 + (2*b*d^3*(-3*c*d + 4*b*e))/x - (b^2*(c*d - b*e)^4)/(c^2*(b + c*x)^2) + (2*b*(-(c*d) + b*e)^3*(3*c*d + b*e))/(c^2*(b + c*x)) - 12*d^2*(c*d - b*e)^2*Log[x] + 12*d^2*(c*d - b*e)^2*Log[b + c*x])/b^5","A",1
279,1,138,137,0.1448033,"\int \frac{(d+e x)^3}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(b*x + c*x^2)^3,x]","-\frac{-6 d \log (x) \left(b^2 e^2-3 b c d e+2 c^2 d^2\right)+6 d \left(b^2 e^2-3 b c d e+2 c^2 d^2\right) \log (b+c x)+\frac{b^2 (b e-c d)^3}{c (b+c x)^2}+\frac{b^2 d^3}{x^2}+\frac{6 b d^2 (b e-c d)}{x}-\frac{6 b d (c d-b e)^2}{b+c x}}{2 b^5}","\frac{3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}+\frac{3 d^2 (c d-b e)}{b^4 x}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}-\frac{d^3}{2 b^3 x^2}",1,"-1/2*((b^2*d^3)/x^2 + (6*b*d^2*(-(c*d) + b*e))/x + (b^2*(-(c*d) + b*e)^3)/(c*(b + c*x)^2) - (6*b*d*(c*d - b*e)^2)/(b + c*x) - 6*d*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*Log[x] + 6*d*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/b^5","A",1
280,1,144,144,0.090787,"\int \frac{(d+e x)^2}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(b*x + c*x^2)^3,x]","\frac{\frac{2 b \left(b^2 e^2-4 b c d e+3 c^2 d^2\right)}{b+c x}+2 \log (x) \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)-2 \left(b^2 e^2-6 b c d e+6 c^2 d^2\right) \log (b+c x)+\frac{b^2 (c d-b e)^2}{(b+c x)^2}-\frac{b^2 d^2}{x^2}-\frac{2 b d (2 b e-3 c d)}{x}}{2 b^5}","\frac{d (3 c d-2 b e)}{b^4 x}+\frac{(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac{(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac{d^2}{2 b^3 x^2}+\frac{\log (x) \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{b^5}-\frac{\left(b^2 e^2-6 b c d e+6 c^2 d^2\right) \log (b+c x)}{b^5}",1,"(-((b^2*d^2)/x^2) - (2*b*d*(-3*c*d + 2*b*e))/x + (b^2*(c*d - b*e)^2)/(b + c*x)^2 + (2*b*(3*c^2*d^2 - 4*b*c*d*e + b^2*e^2))/(b + c*x) + 2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*Log[x] - 2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*Log[b + c*x])/(2*b^5)","A",1
281,1,102,110,0.0781488,"\int \frac{d+e x}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)/(b*x + c*x^2)^3,x]","\frac{-\frac{b \left(b^3 (d+2 e x)+b^2 c x (9 e x-4 d)+6 b c^2 x^2 (e x-3 d)-12 c^3 d x^3\right)}{x^2 (b+c x)^2}+6 c \log (x) (2 c d-b e)+6 c (b e-2 c d) \log (b+c x)}{2 b^5}","\frac{3 c \log (x) (2 c d-b e)}{b^5}-\frac{3 c (2 c d-b e) \log (b+c x)}{b^5}+\frac{3 c d-b e}{b^4 x}+\frac{c (3 c d-2 b e)}{b^4 (b+c x)}+\frac{c (c d-b e)}{2 b^3 (b+c x)^2}-\frac{d}{2 b^3 x^2}",1,"(-((b*(-12*c^3*d*x^3 + 6*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 2*e*x) + b^2*c*x*(-4*d + 9*e*x)))/(x^2*(b + c*x)^2)) + 6*c*(2*c*d - b*e)*Log[x] + 6*c*(-2*c*d + b*e)*Log[b + c*x])/(2*b^5)","A",1
282,1,68,72,0.0453103,"\int \frac{1}{\left(b x+c x^2\right)^3} \, dx","Integrate[(b*x + c*x^2)^(-3),x]","\frac{\frac{b \left(-b^3+4 b^2 c x+18 b c^2 x^2+12 c^3 x^3\right)}{x^2 (b+c x)^2}-12 c^2 \log (b+c x)+12 c^2 \log (x)}{2 b^5}","\frac{6 c^2 \log (x)}{b^5}-\frac{6 c^2 \log (b+c x)}{b^5}+\frac{3 c (b+2 c x)}{b^4 \left(b x+c x^2\right)}-\frac{b+2 c x}{2 b^2 \left(b x+c x^2\right)^2}",1,"((b*(-b^3 + 4*b^2*c*x + 18*b*c^2*x^2 + 12*c^3*x^3))/(x^2*(b + c*x)^2) + 12*c^2*Log[x] - 12*c^2*Log[b + c*x])/(2*b^5)","A",1
283,1,192,193,0.1638444,"\int \frac{1}{(d+e x) \left(b x+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(b*x + c*x^2)^3),x]","\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{c^3}{2 b^3 (b+c x)^2 (b e-c d)}-\frac{1}{2 b^3 d x^2}+\frac{\log (x) \left(b^2 e^2+3 b c d e+6 c^2 d^2\right)}{b^5 d^3}+\frac{c^3 \left(10 b^2 e^2-15 b c d e+6 c^2 d^2\right) \log (b+c x)}{b^5 (b e-c d)^3}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}","\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{b e+3 c d}{b^4 d^2 x}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}-\frac{1}{2 b^3 d x^2}+\frac{\log (x) \left(b^2 e^2+3 b c d e+6 c^2 d^2\right)}{b^5 d^3}-\frac{c^3 \left(10 b^2 e^2-15 b c d e+6 c^2 d^2\right) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}",1,"-1/2*1/(b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) - c^3/(2*b^3*(-(c*d) + b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(b^4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) + (c^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)","A",1
284,1,230,230,0.2489446,"\int \frac{1}{(d+e x)^2 \left(b x+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)^2*(b*x + c*x^2)^3),x]","\frac{c^4 (5 b e-3 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{2 b e+3 c d}{b^4 d^3 x}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{1}{2 b^3 d^2 x^2}+\frac{3 \log (x) \left(b^2 e^2+2 b c d e+2 c^2 d^2\right)}{b^5 d^4}-\frac{3 c^4 \left(5 b^2 e^2-6 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}","\frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{2 b e+3 c d}{b^4 d^3 x}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac{1}{2 b^3 d^2 x^2}+\frac{3 \log (x) \left(b^2 e^2+2 b c d e+2 c^2 d^2\right)}{b^5 d^4}-\frac{3 c^4 \left(5 b^2 e^2-6 b c d e+2 c^2 d^2\right) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}",1,"-1/2*1/(b^3*d^2*x^2) + (3*c*d + 2*b*e)/(b^4*d^3*x) + c^4/(2*b^3*(c*d - b*e)^2*(b + c*x)^2) + (c^4*(-3*c*d + 5*b*e))/(b^4*(-(c*d) + b*e)^3*(b + c*x)) - e^5/(d^3*(c*d - b*e)^3*(d + e*x)) + (3*(2*c^2*d^2 + 2*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^4) - (3*c^4*(2*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^4) + (3*e^5*(2*c*d - b*e)*Log[d + e*x])/(d^4*(c*d - b*e)^4)","A",1
285,1,230,210,0.4935078,"\int (d+e x)^3 \sqrt{b x+c x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\frac{15 b^{3/2} \left(7 b^3 e^3-30 b^2 c d e^2+48 b c^2 d^2 e-32 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)-4 b^2 c^2 e \left(180 d^2+75 d e x+14 e^2 x^2\right)+48 b c^3 \left(10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right)+96 c^4 x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)\right)\right)}{1920 c^{9/2}}","-\frac{b^2 (2 c d-b e) \left(7 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{128 c^{9/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left(7 b^2 e^2-16 b c d e+16 c^2 d^2\right)}{128 c^4}+\frac{e \left(b x+c x^2\right)^{3/2} \left(35 b^2 e^2+42 c e x (2 c d-b e)-150 b c d e+192 c^2 d^2\right)}{240 c^3}+\frac{e \left(b x+c x^2\right)^{3/2} (d+e x)^2}{5 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^4*e^3 + 10*b^3*c*e^2*(45*d + 7*e*x) - 4*b^2*c^2*e*(180*d^2 + 75*d*e*x + 14*e^2*x^2) + 48*b*c^3*(10*d^3 + 10*d^2*e*x + 5*d*e^2*x^2 + e^3*x^3) + 96*c^4*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) + (15*b^(3/2)*(-32*c^3*d^3 + 48*b*c^2*d^2*e - 30*b^2*c*d*e^2 + 7*b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(1920*c^(9/2))","A",1
286,1,164,162,0.2814334,"\int (d+e x)^2 \sqrt{b x+c x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^3 e^2-2 b^2 c e (24 d+5 e x)+8 b c^2 \left(6 d^2+4 d e x+e^2 x^2\right)+16 c^3 x \left(6 d^2+8 d e x+3 e^2 x^2\right)\right)-\frac{3 b^{3/2} \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{192 c^{7/2}}","-\frac{b^2 \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right)}{64 c^3}+\frac{5 e \left(b x+c x^2\right)^{3/2} (2 c d-b e)}{24 c^2}+\frac{e \left(b x+c x^2\right)^{3/2} (d+e x)}{4 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*e^2 - 2*b^2*c*e*(24*d + 5*e*x) + 8*b*c^2*(6*d^2 + 4*d*e*x + e^2*x^2) + 16*c^3*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) - (3*b^(3/2)*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(192*c^(7/2))","A",1
287,1,108,99,0.180619,"\int (d+e x) \sqrt{b x+c x^2} \, dx","Integrate[(d + e*x)*Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 b^{3/2} (b e-2 c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right)\right)}{24 c^{5/2}}","-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left(b x+c x^2\right)^{3/2}}{3 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^2*e + 2*b*c*(3*d + e*x) + 4*c^2*x*(3*d + 2*e*x)) + (3*b^(3/2)*(-2*c*d + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(24*c^(5/2))","A",1
288,1,74,60,0.0739309,"\int \sqrt{b x+c x^2} \, dx","Integrate[Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} (b+2 c x)-\frac{b^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{4 c^{3/2}}","\frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 c^{3/2}}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(b + 2*c*x) - (b^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(4*c^(3/2))","A",1
289,1,137,129,0.3976004,"\int \frac{\sqrt{b x+c x^2}}{d+e x} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x),x]","\frac{\sqrt{x (b+c x)} \left(\frac{(b e-2 c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x}{b}+1}}+\frac{2 \sqrt{d} \sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}+e \sqrt{x}\right)}{e^2 \sqrt{x}}","-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{\sqrt{c} e^2}+\frac{\sqrt{d} \sqrt{c d-b e} \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{e^2}+\frac{\sqrt{b x+c x^2}}{e}",1,"(Sqrt[x*(b + c*x)]*(e*Sqrt[x] + ((-2*c*d + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]) + (2*Sqrt[d]*Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(e^2*Sqrt[x])","A",1
290,1,171,140,0.6525308,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^2,x]","\frac{\sqrt{x (b+c x)} \left(\frac{e \sqrt{x} (c d-b e)}{d+e x}+\frac{2 \sqrt{c} (b e-c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}+\frac{(2 c d-b e) \sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{d} \sqrt{b+c x}}\right)}{e^2 \sqrt{x} (b e-c d)}","-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 \sqrt{d} e^2 \sqrt{c d-b e}}-\frac{\sqrt{b x+c x^2}}{e (d+e x)}+\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{e^2}",1,"(Sqrt[x*(b + c*x)]*((e*(c*d - b*e)*Sqrt[x])/(d + e*x) + (2*Sqrt[c]*(-(c*d) + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + (Sqrt[c*d - b*e]*(2*c*d - b*e)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[b + c*x])))/(e^2*(-(c*d) + b*e)*Sqrt[x])","A",1
291,1,122,127,0.212672,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]","\frac{\sqrt{x (b+c x)} \left(\frac{\sqrt{d} (b (d-e x)+2 c d x)}{(d+e x)^2 (c d-b e)}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{x} \sqrt{b+c x} (c d-b e)^{3/2}}\right)}{4 d^{3/2}}","\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac{b^2 \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{8 d^{3/2} (c d-b e)^{3/2}}",1,"(Sqrt[x*(b + c*x)]*((Sqrt[d]*(2*c*d*x + b*(d - e*x)))/((c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/((c*d - b*e)^(3/2)*Sqrt[x]*Sqrt[b + c*x])))/(4*d^(3/2))","A",1
292,1,191,183,0.2862448,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]","\frac{\sqrt{x (b+c x)} \left(\frac{e x^{3/2} (b+c x)}{(d+e x)^3}-\frac{3 (2 c d-b e) \left(\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{c d-b e} (b (d-e x)+2 c d x)-b^2 (d+e x)^2 \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)\right)}{8 d^{3/2} \sqrt{b+c x} (d+e x)^2 (c d-b e)^{3/2}}\right)}{3 d \sqrt{x} (b e-c d)}","-\frac{b^2 (2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{16 d^{5/2} (c d-b e)^{5/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac{e \left(b x+c x^2\right)^{3/2}}{3 d (d+e x)^3 (c d-b e)}",1,"(Sqrt[x*(b + c*x)]*((e*x^(3/2)*(b + c*x))/(d + e*x)^3 - (3*(2*c*d - b*e)*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(2*c*d*x + b*(d - e*x)) - b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(8*d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[b + c*x]*(d + e*x)^2)))/(3*d*(-(c*d) + b*e)*Sqrt[x])","A",1
293,1,243,258,0.622063,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^5,x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 (d+e x)^2 \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \left(b^2 (d+e x)^2 \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)+\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{c d-b e} (-b d+b e x-2 c d x)\right)}{d^{5/2} \sqrt{b+c x} (c d-b e)^{5/2}}+\frac{40 e x^{3/2} (b+c x) (d+e x) (2 c d-b e)}{d (c d-b e)}+48 e x^{3/2} (b+c x)\right)}{192 d \sqrt{x} (d+e x)^4 (b e-c d)}","-\frac{b^2 \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{128 d^{7/2} (c d-b e)^{7/2}}+\frac{\sqrt{b x+c x^2} \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{5 e \left(b x+c x^2\right)^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac{e \left(b x+c x^2\right)^{3/2}}{4 d (d+e x)^4 (c d-b e)}",1,"(Sqrt[x*(b + c*x)]*(48*e*x^(3/2)*(b + c*x) + (40*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + (3*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(d + e*x)^2*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(5/2)*(c*d - b*e)^(5/2)*Sqrt[b + c*x])))/(192*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^4)","A",1
294,1,308,337,1.0288318,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^6} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]","\frac{\sqrt{x (b+c x)} \left(\frac{8 e x^{3/2} (b+c x) (d+e x)^2 \left(35 b^2 e^2-108 b c d e+108 c^2 d^2\right)}{d^2 (c d-b e)^2}+\frac{15 (d+e x)^3 (2 c d-b e) \left(7 b^2 e^2-16 b c d e+16 c^2 d^2\right) \left(b^2 (d+e x)^2 \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)+\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{c d-b e} (-b d+b e x-2 c d x)\right)}{d^{7/2} \sqrt{b+c x} (c d-b e)^{7/2}}+\frac{336 e x^{3/2} (b+c x) (d+e x) (2 c d-b e)}{d (c d-b e)}+384 e x^{3/2} (b+c x)\right)}{1920 d \sqrt{x} (d+e x)^5 (b e-c d)}","-\frac{b^2 (2 c d-b e) \left(7 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{256 d^{9/2} (c d-b e)^{9/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) \left(7 b^2 e^2-16 b c d e+16 c^2 d^2\right) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{e \left(b x+c x^2\right)^{3/2} \left(35 b^2 e^2-108 b c d e+108 c^2 d^2\right)}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac{7 e \left(b x+c x^2\right)^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac{e \left(b x+c x^2\right)^{3/2}}{5 d (d+e x)^5 (c d-b e)}",1,"(Sqrt[x*(b + c*x)]*(384*e*x^(3/2)*(b + c*x) + (336*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + (8*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*x^(3/2)*(b + c*x)*(d + e*x)^2)/(d^2*(c*d - b*e)^2) + (15*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(d + e*x)^3*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(7/2)*(c*d - b*e)^(7/2)*Sqrt[b + c*x])))/(1920*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^5)","A",1
295,1,312,271,0.6811166,"\int (d+e x)^3 \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left(90 d^2+35 d e x+6 e^2 x^2\right)-16 b^3 c^3 \left(105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right)+32 b^2 c^4 x \left(35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right)+128 b c^5 x^2 \left(105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right)+256 c^6 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)\right)-\frac{105 b^{7/2} \left(3 b^3 e^3-14 b^2 c d e^2+24 b c^2 d^2 e-16 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{35840 c^{11/2}}","-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{1024 c^5}+\frac{(b+2 c x) \left(b x+c x^2\right)^{3/2} (2 c d-b e) \left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{128 c^4}+\frac{e \left(b x+c x^2\right)^{5/2} \left(21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right)}{280 c^3}+\frac{3 b^4 (2 c d-b e) \left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{1024 c^{11/2}}+\frac{e \left(b x+c x^2\right)^{5/2} (d+e x)^2}{7 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(315*b^6*e^3 - 210*b^5*c*e^2*(7*d + e*x) + 28*b^4*c^2*e*(90*d^2 + 35*d*e*x + 6*e^2*x^2) + 32*b^2*c^4*x*(35*d^3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) - 16*b^3*c^3*(105*d^3 + 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 128*b*c^5*x^2*(105*d^3 + 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) - (105*b^(7/2)*(-16*c^3*d^3 + 24*b*c^2*d^2*e - 14*b^2*c*d*e^2 + 3*b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(35840*c^(11/2))","A",1
296,1,197,214,0.4269405,"\int (d+e x)^2 \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(b*x + c*x^2)^(3/2),x]","\frac{(x (b+c x))^{3/2} \left(\frac{\left(7 b^2 e^2-24 b c d e+24 c^2 d^2\right) \left(3 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)+\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left(-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right)\right)}{256 c^{7/2} (b+c x) \sqrt{\frac{c x}{b}+1}}+\frac{7 e x^{5/2} (b+c x) (2 c d-b e)}{10 c}+e x^{5/2} (b+c x) (d+e x)\right)}{6 c x^{3/2}}","-\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left(7 b^2 e^2-24 b c d e+24 c^2 d^2\right)}{512 c^4}+\frac{(b+2 c x) \left(b x+c x^2\right)^{3/2} \left(7 b^2 e^2-24 b c d e+24 c^2 d^2\right)}{192 c^3}+\frac{b^4 \left(7 b^2 e^2-24 b c d e+24 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{512 c^{9/2}}+\frac{7 e \left(b x+c x^2\right)^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left(b x+c x^2\right)^{5/2} (d+e x)}{6 c}",1,"((x*(b + c*x))^(3/2)*((7*e*(2*c*d - b*e)*x^(5/2)*(b + c*x))/(10*c) + e*x^(5/2)*(b + c*x)*(d + e*x) + ((24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x^3) + 3*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(256*c^(7/2)*(b + c*x)*Sqrt[1 + (c*x)/b])))/(6*c*x^(3/2))","A",1
297,1,146,137,0.2519843,"\int (d+e x) \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^4 e-10 b^3 c (3 d+e x)+4 b^2 c^2 x (5 d+2 e x)+16 b c^3 x^2 (15 d+11 e x)+32 c^4 x^3 (5 d+4 e x)\right)-\frac{15 b^{7/2} (b e-2 c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{640 c^{7/2}}","\frac{3 b^4 (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{128 c^{7/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left(b x+c x^2\right)^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left(b x+c x^2\right)^{5/2}}{5 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^4*e - 10*b^3*c*(3*d + e*x) + 4*b^2*c^2*x*(5*d + 2*e*x) + 32*c^4*x^3*(5*d + 4*e*x) + 16*b*c^3*x^2*(15*d + 11*e*x)) - (15*b^(7/2)*(-2*c*d + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(640*c^(7/2))","A",1
298,1,98,89,0.0637739,"\int \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 b^{7/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right)\right)}{64 c^{5/2}}","\frac{3 b^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left(b x+c x^2\right)^{3/2}}{8 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x^3) + (3*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(64*c^(5/2))","A",1
299,1,225,216,0.5521456,"\int \frac{\left(b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} e \sqrt{x} \left(3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-\frac{3 \left(b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}+\frac{48 c^{3/2} d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}\right)}{24 c^{3/2} e^4 \sqrt{x}}","\frac{\sqrt{b x+c x^2} \left(b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right)}{8 c e^3}-\frac{(2 c d-b e) \left(-b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{3/2} e^4}+\frac{d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{e^4}+\frac{\left(b x+c x^2\right)^{3/2}}{3 e}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*e*Sqrt[x]*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - (3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + (48*c^(3/2)*d^(3/2)*(c*d - b*e)^(3/2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(24*c^(3/2)*e^4*Sqrt[x])","A",1
300,1,205,198,0.7419203,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^2,x]","\frac{\sqrt{x (b+c x)} \left(\frac{3 \left(b^2 e^2-8 b c d e+8 c^2 d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x}{b}+1}}+\frac{e \sqrt{x} \left(b e (9 d+5 e x)-2 c \left(6 d^2+3 d e x-e^2 x^2\right)\right)}{d+e x}-\frac{12 \sqrt{d} \sqrt{c d-b e} (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}\right)}{4 e^4 \sqrt{x}}","\frac{3 \left(b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 \sqrt{c} e^4}-\frac{3 \sqrt{d} \sqrt{c d-b e} (2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 e^4}-\frac{3 \sqrt{b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left(b x+c x^2\right)^{3/2}}{e (d+e x)}",1,"(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(b*e*(9*d + 5*e*x) - 2*c*(6*d^2 + 3*d*e*x - e^2*x^2)))/(d + e*x) + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]) - (12*Sqrt[d]*Sqrt[c*d - b*e]*(2*c*d - b*e)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(4*e^4*Sqrt[x])","A",1
301,1,259,205,1.3195781,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^3,x]","\frac{\sqrt{x (b+c x)} \left(\frac{e \sqrt{x} \left(-b^2 e^2 (3 d+5 e x)+b c e \left(15 d^2+23 d e x+4 e^2 x^2\right)-2 c^2 d \left(6 d^2+9 d e x+2 e^2 x^2\right)\right)}{(d+e x)^2}+\frac{12 \sqrt{c} \left(b^2 e^2-3 b c d e+2 c^2 d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}-\frac{3 \sqrt{c d-b e} \left(b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{d} \sqrt{b+c x}}\right)}{4 e^4 \sqrt{x} (b e-c d)}","\frac{3 \left(b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{8 \sqrt{d} e^4 \sqrt{c d-b e}}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{e^4}+\frac{3 \sqrt{b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left(b x+c x^2\right)^{3/2}}{2 e (d+e x)^2}",1,"(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(-(b^2*e^2*(3*d + 5*e*x)) - 2*c^2*d*(6*d^2 + 9*d*e*x + 2*e^2*x^2) + b*c*e*(15*d^2 + 23*d*e*x + 4*e^2*x^2)))/(d + e*x)^2 + (12*Sqrt[c]*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) - (3*Sqrt[c*d - b*e]*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[b + c*x])))/(4*e^4*(-(c*d) + b*e)*Sqrt[x])","A",1
302,1,395,332,0.8333427,"\int (d+e x)^3 \left(b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(b*x + c*x^2)^(5/2),x]","\frac{\sqrt{x (b+c x)} \left(\frac{315 b^{11/2} \left(11 b^3 e^3-54 b^2 c d e^2+96 b c^2 d^2 e-64 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}+\sqrt{c} \left(-3465 b^8 e^3+210 b^7 c e^2 (81 d+11 e x)-84 b^6 c^2 e \left(360 d^2+135 d e x+22 e^2 x^2\right)+144 b^5 c^3 \left(140 d^3+140 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right)-32 b^4 c^4 x \left(420 d^3+504 d^2 e x+243 d e^2 x^2+44 e^3 x^3\right)+256 b^3 c^5 x^2 \left(42 d^3+54 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right)+1536 b^2 c^6 x^3 \left(378 d^3+888 d^2 e x+729 d e^2 x^2+206 e^3 x^3\right)+2048 b c^7 x^4 \left(420 d^3+1044 d^2 e x+891 d e^2 x^2+259 e^3 x^3\right)+4096 c^8 x^5 \left(84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right)\right)\right)}{2064384 c^{13/2}}","-\frac{5 b^2 (b+2 c x) \left(b x+c x^2\right)^{3/2} (2 c d-b e) \left(11 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{12288 c^5}+\frac{(b+2 c x) \left(b x+c x^2\right)^{5/2} (2 c d-b e) \left(11 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{768 c^4}+\frac{e \left(b x+c x^2\right)^{7/2} \left(99 b^2 e^2+154 c e x (2 c d-b e)-486 b c d e+640 c^2 d^2\right)}{2016 c^3}-\frac{5 b^6 (2 c d-b e) \left(11 b^2 e^2-32 b c d e+32 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{32768 c^{13/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left(11 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{32768 c^6}+\frac{e \left(b x+c x^2\right)^{7/2} (d+e x)^2}{9 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3465*b^8*e^3 + 210*b^7*c*e^2*(81*d + 11*e*x) - 84*b^6*c^2*e*(360*d^2 + 135*d*e*x + 22*e^2*x^2) + 256*b^3*c^5*x^2*(42*d^3 + 54*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3) + 144*b^5*c^3*(140*d^3 + 140*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^3) - 32*b^4*c^4*x*(420*d^3 + 504*d^2*e*x + 243*d*e^2*x^2 + 44*e^3*x^3) + 4096*c^8*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 1536*b^2*c^6*x^3*(378*d^3 + 888*d^2*e*x + 729*d*e^2*x^2 + 206*e^3*x^3) + 2048*b*c^7*x^4*(420*d^3 + 1044*d^2*e*x + 891*d*e^2*x^2 + 259*e^3*x^3)) + (315*b^(11/2)*(-64*c^3*d^3 + 96*b*c^2*d^2*e - 54*b^2*c*d*e^2 + 11*b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(2064384*c^(13/2))","A",1
303,1,219,266,0.5577844,"\int (d+e x)^2 \left(b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(b*x + c*x^2)^(5/2),x]","\frac{(x (b+c x))^{5/2} \left(\frac{\left(9 b^2 e^2-32 b c d e+32 c^2 d^2\right) \left(\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left(15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right)-15 b^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)\right)}{6144 c^{9/2} (b+c x)^2 \sqrt{\frac{c x}{b}+1}}+\frac{9 e x^{7/2} (b+c x) (2 c d-b e)}{14 c}+e x^{7/2} (b+c x) (d+e x)\right)}{8 c x^{5/2}}","-\frac{5 b^2 (b+2 c x) \left(b x+c x^2\right)^{3/2} \left(9 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{6144 c^4}+\frac{(b+2 c x) \left(b x+c x^2\right)^{5/2} \left(9 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{384 c^3}-\frac{5 b^6 \left(9 b^2 e^2-32 b c d e+32 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{16384 c^{11/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left(9 b^2 e^2-32 b c d e+32 c^2 d^2\right)}{16384 c^5}+\frac{9 e \left(b x+c x^2\right)^{7/2} (2 c d-b e)}{112 c^2}+\frac{e \left(b x+c x^2\right)^{7/2} (d+e x)}{8 c}",1,"((x*(b + c*x))^(5/2)*((9*e*(2*c*d - b*e)*x^(7/2)*(b + c*x))/(14*c) + e*x^(7/2)*(b + c*x)*(d + e*x) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^2*c^3*x^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - 15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(6144*c^(9/2)*(b + c*x)^2*Sqrt[1 + (c*x)/b])))/(8*c*x^(5/2))","A",1
304,1,171,175,0.2914775,"\int (d+e x) \left(b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(b*x + c*x^2)^(5/2),x]","\frac{(x (b+c x))^{7/2} \left(\frac{49 (2 c d-b e) \left(\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left(15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right)-15 b^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)\right)}{3072 c^{7/2} x^{7/2} \sqrt{\frac{c x}{b}+1}}+7 e (b+c x)^3\right)}{49 c (b+c x)^3}","-\frac{5 b^6 (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{1024 c^{9/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 b^2 (b+2 c x) \left(b x+c x^2\right)^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left(b x+c x^2\right)^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left(b x+c x^2\right)^{7/2}}{7 c}",1,"((x*(b + c*x))^(7/2)*(7*e*(b + c*x)^3 + (49*(2*c*d - b*e)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^2*c^3*x^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - 15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(3072*c^(7/2)*x^(7/2)*Sqrt[1 + (c*x)/b])))/(49*c*(b + c*x)^3)","A",1
305,1,120,118,0.103896,"\int \left(b x+c x^2\right)^{5/2} \, dx","Integrate[(b*x + c*x^2)^(5/2),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} \left(15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right)-\frac{15 b^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{1536 c^{7/2}}","-\frac{5 b^6 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{512 c^{7/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left(b x+c x^2\right)^{3/2}}{192 c^2}+\frac{(b+2 c x) \left(b x+c x^2\right)^{5/2}}{12 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^2*c^3*x^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - (15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(1536*c^(7/2))","A",1
306,1,346,356,0.9725058,"\int \frac{\left(b x+c x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x),x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} e \sqrt{x} \left(-45 b^4 e^4+30 b^3 c e^3 (e x-5 d)+4 b^2 c^2 e^2 \left(660 d^2-295 d e x+186 e^2 x^2\right)+16 b c^3 e \left(-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right)+32 c^4 \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)\right)+\frac{15 \left(3 b^5 e^5+10 b^4 c d e^4+80 b^3 c^2 d^2 e^3-480 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}+\frac{3840 c^{5/2} d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}\right)}{1920 c^{5/2} e^6 \sqrt{x}}","\frac{\left(b x+c x^2\right)^{3/2} \left(3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right)}{48 c e^3}+\frac{\sqrt{b x+c x^2} \left(-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left(-3 b^2 e^2-16 b c d e+16 c^2 d^2\right)+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right)}{128 c^2 e^5}-\frac{(2 c d-b e) \left(3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{e^6}+\frac{\left(b x+c x^2\right)^{5/2}}{5 e}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*e*Sqrt[x]*(-45*b^4*e^4 + 30*b^3*c*e^3*(-5*d + e*x) + 4*b^2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e^2*x^2) + 16*b*c^3*e*(-270*d^3 + 130*d^2*e*x - 85*d*e^2*x^2 + 63*e^3*x^3) + 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + (15*(-256*c^5*d^5 + 640*b*c^4*d^4*e - 480*b^2*c^3*d^3*e^2 + 80*b^3*c^2*d^2*e^3 + 10*b^4*c*d*e^4 + 3*b^5*e^5)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + (3840*c^(5/2)*d^(5/2)*(c*d - b*e)^(5/2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(1920*c^(5/2)*e^6*Sqrt[x])","A",1
307,1,347,314,1.2720125,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^2,x]","\frac{\sqrt{x (b+c x)} \left(-\frac{960 c^{3/2} d^{3/2} \sqrt{c d-b e} \left(b^2 e^2-3 b c d e+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}+\frac{\sqrt{c} e \sqrt{x} \left(15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left(-360 d^2-205 d e x+59 e^2 x^2\right)+8 b c^2 e \left(210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right)-16 c^3 \left(60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right)\right)}{d+e x}-\frac{15 \left(b^4 e^4+16 b^3 c d e^3-144 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}\right)}{192 c^{3/2} e^6 \sqrt{x}}","-\frac{5 \sqrt{b x+c x^2} \left(-b^3 e^3-2 c e x \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right)}{64 c e^5}+\frac{5 \left(-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 e^6}-\frac{5 \left(b x+c x^2\right)^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left(b x+c x^2\right)^{5/2}}{e (d+e x)}",1,"(Sqrt[x*(b + c*x)]*((Sqrt[c]*e*Sqrt[x]*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2*(-360*d^2 - 205*d*e*x + 59*e^2*x^2) + 8*b*c^2*e*(210*d^3 + 110*d^2*e*x - 35*d*e^2*x^2 + 17*e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^2 + 5*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x) - (15*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 144*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + b^4*e^4)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) - (960*c^(3/2)*d^(3/2)*Sqrt[c*d - b*e]*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(192*c^(3/2)*e^6*Sqrt[x])","A",1
308,1,308,282,1.424957,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^3,x]","\frac{\sqrt{x (b+c x)} \left(\frac{30 \sqrt{d} \sqrt{c d-b e} \left(3 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{b+c x}}+\frac{e \sqrt{x} \left(3 b^2 e^2 \left(25 d^2+40 d e x+11 e^2 x^2\right)-2 b c e \left(150 d^3+230 d^2 e x+55 d e^2 x^2-13 e^3 x^3\right)+4 c^2 \left(60 d^4+90 d^3 e x+20 d^2 e^2 x^2-5 d e^3 x^3+2 e^4 x^4\right)\right)}{(d+e x)^2}+\frac{15 \left(b^3 e^3-18 b^2 c d e^2+48 b c^2 d^2 e-32 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x}{b}+1}}\right)}{24 e^6 \sqrt{x}}","-\frac{5 (2 c d-b e) \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 \sqrt{c} e^6}+\frac{5 \sqrt{b x+c x^2} \left(5 b^2 e^2-4 c e x (2 c d-b e)-20 b c d e+16 c^2 d^2\right)}{8 e^5}+\frac{5 \sqrt{d} (4 c d-3 b e) \sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{8 e^6}+\frac{5 \left(b x+c x^2\right)^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left(b x+c x^2\right)^{5/2}}{2 e (d+e x)^2}",1,"(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(3*b^2*e^2*(25*d^2 + 40*d*e*x + 11*e^2*x^2) - 2*b*c*e*(150*d^3 + 230*d^2*e*x + 55*d*e^2*x^2 - 13*e^3*x^3) + 4*c^2*(60*d^4 + 90*d^3*e*x + 20*d^2*e^2*x^2 - 5*d*e^3*x^3 + 2*e^4*x^4)))/(d + e*x)^2 + (15*(-32*c^3*d^3 + 48*b*c^2*d^2*e - 18*b^2*c*d*e^2 + b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]) + (30*Sqrt[d]*Sqrt[c*d - b*e]*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(24*e^6*Sqrt[x])","A",1
309,1,38,26,0.0299474,"\int \frac{\sqrt{2 x+x^2}}{1+x} \, dx","Integrate[Sqrt[2*x + x^2]/(1 + x),x]","\sqrt{x (x+2)} \left(1-\frac{2 \tan ^{-1}\left(\sqrt{\frac{x}{x+2}}\right)}{\sqrt{x} \sqrt{x+2}}\right)","\sqrt{x^2+2 x}-\tan ^{-1}\left(\sqrt{x^2+2 x}\right)",1,"Sqrt[x*(2 + x)]*(1 - (2*ArcTan[Sqrt[x/(2 + x)]])/(Sqrt[x]*Sqrt[2 + x]))","A",0
310,1,48,53,0.0597918,"\int \frac{\left(2 x-x^2\right)^{3/2}}{2-2 x} \, dx","Integrate[(2*x - x^2)^(3/2)/(2 - 2*x),x]","\frac{1}{6} \sqrt{-((x-2) x)} \left(x^2-2 x+\frac{6 \tan ^{-1}\left(\sqrt{\frac{x-2}{x}}\right)}{\sqrt{x-2} \sqrt{x}}-3\right)","-\frac{1}{6} \left(2 x-x^2\right)^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)",1,"(Sqrt[-((-2 + x)*x)]*(-3 - 2*x + x^2 + (6*ArcTan[Sqrt[(-2 + x)/x]])/(Sqrt[-2 + x]*Sqrt[x])))/6","A",0
311,1,46,36,0.0403613,"\int \frac{\sqrt{2 x-x^2}}{2-2 x} \, dx","Integrate[Sqrt[2*x - x^2]/(2 - 2*x),x]","\frac{(x-2) x-2 \sqrt{x-2} \sqrt{x} \tan ^{-1}\left(\sqrt{\frac{x-2}{x}}\right)}{2 \sqrt{-((x-2) x)}}","\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)-\frac{1}{2} \sqrt{2 x-x^2}",1,"((-2 + x)*x - 2*Sqrt[-2 + x]*Sqrt[x]*ArcTan[Sqrt[(-2 + x)/x]])/(2*Sqrt[-((-2 + x)*x)])","A",0
312,1,38,18,0.0105111,"\int \frac{1}{(2-2 x) \sqrt{2 x-x^2}} \, dx","Integrate[1/((2 - 2*x)*Sqrt[2*x - x^2]),x]","-\frac{\sqrt{x-2} \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{x-2}}{\sqrt{x}}\right)}{\sqrt{-((x-2) x)}}","\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)",1,"-((Sqrt[-2 + x]*Sqrt[x]*ArcTan[Sqrt[-2 + x]/Sqrt[x]])/Sqrt[-((-2 + x)*x)])","B",1
313,1,42,36,0.0187362,"\int \frac{1}{(2-2 x) \left(2 x-x^2\right)^{3/2}} \, dx","Integrate[1/((2 - 2*x)*(2*x - x^2)^(3/2)),x]","-\frac{2 \sqrt{x-2} \sqrt{x} \tan ^{-1}\left(\sqrt{\frac{x-2}{x}}\right)+1}{2 \sqrt{-((x-2) x)}}","\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)-\frac{1}{2 \sqrt{2 x-x^2}}",1,"-1/2*(1 + 2*Sqrt[-2 + x]*Sqrt[x]*ArcTan[Sqrt[(-2 + x)/x]])/Sqrt[-((-2 + x)*x)]","A",0
314,1,50,53,0.0439989,"\int \frac{1}{(2-2 x) \left(2 x-x^2\right)^{5/2}} \, dx","Integrate[1/((2 - 2*x)*(2*x - x^2)^(5/2)),x]","\frac{6 (x-2)^{3/2} x^{3/2} \tan ^{-1}\left(\sqrt{\frac{x-2}{x}}\right)+3 x^2-6 x-1}{6 (-((x-2) x))^{3/2}}","-\frac{1}{2 \sqrt{2 x-x^2}}-\frac{1}{6 \left(2 x-x^2\right)^{3/2}}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)",1,"(-1 - 6*x + 3*x^2 + 6*(-2 + x)^(3/2)*x^(3/2)*ArcTan[Sqrt[(-2 + x)/x]])/(6*(-((-2 + x)*x))^(3/2))","A",0
315,1,152,149,0.4482411,"\int \frac{(d+e x)^3}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{x (b+c x)} \left(\sqrt{c} e \left(15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)\right)+\frac{3 \left(-5 b^3 e^3+18 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1}}\right)}{24 c^{7/2}}","\frac{(2 c d-b e) \left(5 b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{8 c^{7/2}}+\frac{e \sqrt{b x+c x^2} \left(15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right)}{24 c^3}+\frac{e \sqrt{b x+c x^2} (d+e x)^2}{3 c}",1,"(Sqrt[x*(b + c*x)]*(Sqrt[c]*e*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*e*x) + 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + (3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^3)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[x]*Sqrt[1 + (c*x)/b])))/(24*c^(7/2))","A",1
316,1,111,110,0.1032679,"\int \frac{(d+e x)^2}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)+\sqrt{c} e x (b+c x) (-3 b e+8 c d+2 c e x)}{4 c^{5/2} \sqrt{x (b+c x)}}","\frac{\left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c}",1,"(Sqrt[c]*e*x*(b + c*x)*(8*c*d - 3*b*e + 2*c*e*x) + Sqrt[b]*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*c^(5/2)*Sqrt[x*(b + c*x)])","A",1
317,1,80,55,0.0593262,"\int \frac{d+e x}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)/Sqrt[b*x + c*x^2],x]","\frac{\sqrt{c} e x (b+c x)-\sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} (b e-2 c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{c^{3/2} \sqrt{x (b+c x)}}","\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{3/2}}+\frac{e \sqrt{b x+c x^2}}{c}",1,"(Sqrt[c]*e*x*(b + c*x) - Sqrt[b]*(-2*c*d + b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(c^(3/2)*Sqrt[x*(b + c*x)])","A",1
318,1,57,28,0.0161329,"\int \frac{1}{\sqrt{b x+c x^2}} \, dx","Integrate[1/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{\sqrt{c} \sqrt{x (b+c x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{\sqrt{c}}",1,"(2*Sqrt[b]*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[c]*Sqrt[x*(b + c*x)])","B",1
319,1,77,68,0.0186075,"\int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[b*x + c*x^2]),x]","\frac{2 \sqrt{x} \sqrt{b+c x} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{d} \sqrt{x (b+c x)} \sqrt{c d-b e}}","\frac{\tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{\sqrt{d} \sqrt{c d-b e}}",1,"(2*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x*(b + c*x)])","A",1
320,1,122,117,0.088959,"\int \frac{1}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[b*x + c*x^2]),x]","\frac{\sqrt{x} \left(\frac{\sqrt{d} e \sqrt{x} (b+c x)}{d+e x}-\frac{\sqrt{b+c x} (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{\sqrt{c d-b e}}\right)}{d^{3/2} \sqrt{x (b+c x)} (b e-c d)}","\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 d^{3/2} (c d-b e)^{3/2}}-\frac{e \sqrt{b x+c x^2}}{d (d+e x) (c d-b e)}",1,"(Sqrt[x]*((Sqrt[d]*e*Sqrt[x]*(b + c*x))/(d + e*x) - ((2*c*d - b*e)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[c*d - b*e]))/(d^(3/2)*(-(c*d) + b*e)*Sqrt[x*(b + c*x)])","A",1
321,1,183,180,0.2328164,"\int \frac{1}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]","\frac{\sqrt{x} \left(-\frac{\sqrt{b+c x} \left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)}{2 d^{3/2} (c d-b e)^{3/2}}+\frac{3 e \sqrt{x} (b+c x) (2 c d-b e)}{2 d (d+e x) (c d-b e)}+\frac{e \sqrt{x} (b+c x)}{(d+e x)^2}\right)}{2 d \sqrt{x (b+c x)} (b e-c d)}","\frac{\left(3 b^2 e^2-8 b c d e+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac{e \sqrt{b x+c x^2}}{2 d (d+e x)^2 (c d-b e)}",1,"(Sqrt[x]*((e*Sqrt[x]*(b + c*x))/(d + e*x)^2 + (3*e*(2*c*d - b*e)*Sqrt[x]*(b + c*x))/(2*d*(c*d - b*e)*(d + e*x)) - ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(2*d^(3/2)*(c*d - b*e)^(3/2))))/(2*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)])","A",1
322,1,129,139,0.104517,"\int \frac{(d+e x)^3}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(b*x + c*x^2)^(3/2),x]","\frac{\sqrt{c} \left(3 b^3 e^3 x+b^2 c e^2 x (e x-6 d)-2 b c^2 d^2 (d-3 e x)-4 c^3 d^3 x\right)-3 b^{5/2} e^2 \sqrt{x} \sqrt{\frac{c x}{b}+1} (b e-2 c d) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)}{b^2 c^{5/2} \sqrt{x (b+c x)}}","\frac{e \sqrt{b x+c x^2} \left(3 b^2 e^2+2 c e x (2 c d-b e)-6 b c d e+8 c^2 d^2\right)}{b^2 c^2}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{5/2}}",1,"(Sqrt[c]*(-4*c^3*d^3*x + 3*b^3*e^3*x - 2*b*c^2*d^2*(d - 3*e*x) + b^2*c*e^2*x*(-6*d + e*x)) - 3*b^(5/2)*e^2*(-2*c*d + b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(b^2*c^(5/2)*Sqrt[x*(b + c*x)])","A",1
323,1,100,101,0.0907089,"\int \frac{(d+e x)^2}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(b*x + c*x^2)^(3/2),x]","\frac{2 b^{5/2} e^2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right)-2 \sqrt{c} \left(b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right)}{b^2 c^{3/2} \sqrt{x (b+c x)}}","\frac{2 e \sqrt{b x+c x^2} (2 c d-b e)}{b^2 c}-\frac{2 (d+e x) (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{3/2}}",1,"(-2*Sqrt[c]*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x)) + 2*b^(5/2)*e^2*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(b^2*c^(3/2)*Sqrt[x*(b + c*x)])","A",1
324,1,30,33,0.0120644,"\int \frac{d+e x}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(b*x + c*x^2)^(3/2),x]","\frac{2 (-b d+b e x-2 c d x)}{b^2 \sqrt{x (b+c x)}}","-\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}",1,"(2*(-(b*d) - 2*c*d*x + b*e*x))/(b^2*Sqrt[x*(b + c*x)])","A",1
325,1,22,24,0.0065695,"\int \frac{1}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^(-3/2),x]","-\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}}","-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}",1,"(-2*(b + 2*c*x))/(b^2*Sqrt[x*(b + c*x)])","A",1
326,1,129,126,0.1110062,"\int \frac{1}{(d+e x) \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(b*x + c*x^2)^(3/2)),x]","\frac{2 \left(\sqrt{d} \sqrt{c d-b e} \left(b^2 e-b c d+b c e x-2 c^2 d x\right)+b^2 e^2 \sqrt{x} \sqrt{b+c x} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)\right)}{b^2 d^{3/2} \sqrt{x (b+c x)} (c d-b e)^{3/2}}","\frac{e^2 \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{d^{3/2} (c d-b e)^{3/2}}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}",1,"(2*(Sqrt[d]*Sqrt[c*d - b*e]*(-(b*c*d) + b^2*e - 2*c^2*d*x + b*c*e*x) + b^2*e^2*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(b^2*d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[x*(b + c*x)])","A",1
327,1,206,207,0.1811096,"\int \frac{1}{(d+e x)^2 \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]","\frac{-3 b^2 e^2 \sqrt{x} \sqrt{b+c x} (d+e x) (b e-2 c d) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)-\sqrt{d} \sqrt{c d-b e} \left(b^3 e^2 (2 d+3 e x)+b^2 c e \left(-4 d^2-2 d e x+3 e^2 x^2\right)+2 b c^2 d \left(d^2-d e x-2 e^2 x^2\right)+4 c^3 d^2 x (d+e x)\right)}{b^2 d^{5/2} \sqrt{x (b+c x)} (d+e x) (c d-b e)^{5/2}}","-\frac{e \sqrt{b x+c x^2} \left(3 b^2 e^2-4 b c d e+4 c^2 d^2\right)}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x) (c d-b e)}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 d^{5/2} (c d-b e)^{5/2}}",1,"(-(Sqrt[d]*Sqrt[c*d - b*e]*(4*c^3*d^2*x*(d + e*x) + b^3*e^2*(2*d + 3*e*x) + 2*b*c^2*d*(d^2 - d*e*x - 2*e^2*x^2) + b^2*c*e*(-4*d^2 - 2*d*e*x + 3*e^2*x^2))) - 3*b^2*e^2*(-2*c*d + b*e)*Sqrt[x]*Sqrt[b + c*x]*(d + e*x)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b^2*d^(5/2)*(c*d - b*e)^(5/2)*Sqrt[x*(b + c*x)]*(d + e*x))","A",1
328,1,289,296,0.7608903,"\int \frac{1}{(d+e x)^3 \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]","-\frac{2 b^2 d^{5/2} e (c d-b e)^{5/2}+(d+e x) \left(5 b^2 d^{3/2} e (c d-b e)^{3/2} (2 c d-b e)+(d+e x) \left(-3 b^2 e^2 \sqrt{x} \sqrt{b+c x} \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)+b \sqrt{d} (c d-b e)^{3/2} \left(15 b^2 e^2-28 b c d e+8 c^2 d^2\right)+c \sqrt{d} x \sqrt{c d-b e} \left(-15 b^3 e^3+38 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right)\right)\right)}{4 b^2 d^{7/2} \sqrt{x (b+c x)} (d+e x)^2 (c d-b e)^{7/2}}","-\frac{e \sqrt{b x+c x^2} \left(5 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{3 e^2 \left(5 b^2 e^2-16 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{e \sqrt{b x+c x^2} (2 c d-b e) \left(15 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)}",1,"-1/4*(2*b^2*d^(5/2)*e*(c*d - b*e)^(5/2) + (d + e*x)*(5*b^2*d^(3/2)*e*(c*d - b*e)^(3/2)*(2*c*d - b*e) + (d + e*x)*(b*Sqrt[d]*(c*d - b*e)^(3/2)*(8*c^2*d^2 - 28*b*c*d*e + 15*b^2*e^2) + c*Sqrt[d]*Sqrt[c*d - b*e]*(16*c^3*d^3 - 24*b*c^2*d^2*e + 38*b^2*c*d*e^2 - 15*b^3*e^3)*x - 3*b^2*e^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(b^2*d^(7/2)*(c*d - b*e)^(7/2)*Sqrt[x*(b + c*x)]*(d + e*x)^2)","A",1
329,1,196,208,5.1710163,"\int \frac{(d+e x)^4}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(b*x + c*x^2)^(5/2),x]","\frac{x^{5/2} (b+c x)^3 \left(-\frac{8 \sqrt{x} (b e-c d)^3 (b e+2 c d)}{3 b^4 c^2 (b+c x)}-\frac{8 d^3 (3 b e-2 c d)}{3 b^4 \sqrt{x}}+\frac{2 \sqrt{x} (b e-c d)^4}{3 b^3 c^2 (b+c x)^2}-\frac{2 d^4}{3 b^3 x^{3/2}}\right)}{(x (b+c x))^{5/2}}+\frac{2 e^4 x^{5/2} (b+c x)^{5/2} \log \left(\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right)}{c^{5/2} (x (b+c x))^{5/2}}","-\frac{2 (d+e x)^3 (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left(-3 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{3 b^4 c^2}+\frac{4 (d+e x) \left(x (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)+b c d^2 (4 c d-5 b e)\right)}{3 b^4 c \sqrt{b x+c x^2}}+\frac{2 e^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right)}{c^{5/2}}",1,"(x^(5/2)*(b + c*x)^3*((-2*d^4)/(3*b^3*x^(3/2)) - (8*d^3*(-2*c*d + 3*b*e))/(3*b^4*Sqrt[x]) + (2*(-(c*d) + b*e)^4*Sqrt[x])/(3*b^3*c^2*(b + c*x)^2) - (8*(-(c*d) + b*e)^3*(2*c*d + b*e)*Sqrt[x])/(3*b^4*c^2*(b + c*x))))/(x*(b + c*x))^(5/2) + (2*e^4*x^(5/2)*(b + c*x)^(5/2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(5/2)*(x*(b + c*x))^(5/2))","A",1
330,1,105,87,0.04279,"\int \frac{(d+e x)^3}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(b^3 \left(-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right)+6 b^2 c d x \left(d^2-6 d e x+e^2 x^2\right)+24 b c^2 d^2 x^2 (d-e x)+16 c^3 d^3 x^3\right)}{3 b^4 (x (b+c x))^{3/2}}","\frac{16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}",1,"(2*(16*c^3*d^3*x^3 + 24*b*c^2*d^2*x^2*(d - e*x) + 6*b^2*c*d*x*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3)))/(3*b^4*(x*(b + c*x))^(3/2))","A",1
331,1,95,78,0.0370198,"\int \frac{(d+e x)^2}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(b*x + c*x^2)^(5/2),x]","\frac{-2 b^3 \left(d^2+6 d e x-3 e^2 x^2\right)+4 b^2 c x \left(3 d^2-12 d e x+e^2 x^2\right)+16 b c^2 d x^2 (3 d-2 e x)+32 c^3 d^2 x^3}{3 b^4 (x (b+c x))^{3/2}}","\frac{8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 b^2 \left(b x+c x^2\right)^{3/2}}",1,"(32*c^3*d^2*x^3 + 16*b*c^2*d*x^2*(3*d - 2*e*x) - 2*b^3*(d^2 + 6*d*e*x - 3*e^2*x^2) + 4*b^2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2))/(3*b^4*(x*(b + c*x))^(3/2))","A",1
332,1,67,71,0.0221975,"\int \frac{d+e x}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(b*x + c*x^2)^(5/2),x]","-\frac{2 \left(b^3 (d+3 e x)-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (e x-3 d)-16 c^3 d x^3\right)}{3 b^4 (x (b+c x))^{3/2}}","\frac{8 (b+2 c x) (2 c d-b e)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}",1,"(-2*(-16*c^3*d*x^3 - 6*b^2*c*x*(d - 2*e*x) + 8*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 3*e*x)))/(3*b^4*(x*(b + c*x))^(3/2))","A",1
333,1,48,54,0.0116306,"\int \frac{1}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^(-5/2),x]","\frac{-2 b^3+12 b^2 c x+48 b c^2 x^2+32 c^3 x^3}{3 b^4 (x (b+c x))^{3/2}}","\frac{16 c (b+2 c x)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x)}{3 b^2 \left(b x+c x^2\right)^{3/2}}",1,"(-2*b^3 + 12*b^2*c*x + 48*b*c^2*x^2 + 32*c^3*x^3)/(3*b^4*(x*(b + c*x))^(3/2))","A",1
334,1,238,230,0.2801656,"\int \frac{1}{(d+e x) \left(b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]","\frac{6 b^4 e^4 x^{3/2} (b+c x)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)+2 \sqrt{d} \sqrt{c d-b e} \left(b^5 e^2 (3 e x-d)+2 b^4 c e \left(d^2+3 e^2 x^2\right)-b^3 c^2 \left(d^3+9 d^2 e x-3 d e^2 x^2-3 e^3 x^3\right)+2 b^2 c^3 d x \left(3 d^2-18 d e x+e^2 x^2\right)+24 b c^4 d^2 x^2 (d-e x)+16 c^5 d^3 x^3\right)}{3 b^4 d^{5/2} (x (b+c x))^{3/2} (c d-b e)^{5/2}}","-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left(b x+c x^2\right)^{3/2} (c d-b e)}+\frac{2 \left(c x (2 c d-b e) \left(-3 b^2 e^2-8 b c d e+8 c^2 d^2\right)+b (c d-b e) \left(-3 b^2 e^2-4 b c d e+8 c^2 d^2\right)\right)}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}+\frac{e^4 \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{d^{5/2} (c d-b e)^{5/2}}",1,"(2*Sqrt[d]*Sqrt[c*d - b*e]*(16*c^5*d^3*x^3 + 24*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-d + 3*e*x) + 2*b^2*c^3*d*x*(3*d^2 - 18*d*e*x + e^2*x^2) + 2*b^4*c*e*(d^2 + 3*e^2*x^2) - b^3*c^2*(d^3 + 9*d^2*e*x - 3*d*e^2*x^2 - 3*e^3*x^3)) + 6*b^4*e^4*x^(3/2)*(b + c*x)^(3/2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(3*b^4*d^(5/2)*(c*d - b*e)^(5/2)*(x*(b + c*x))^(3/2))","A",1
335,1,351,348,0.930165,"\int \frac{1}{(d+e x)^2 \left(b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]","-\frac{9 b^4 d^{5/2} e (c d-b e)^{5/2}+(d+e x) \left(3 b^3 d^{3/2} (2 c d-5 b e) (c d-b e)^{5/2}-9 b^2 \sqrt{d} x (c d-b e)^{5/2} \left(4 c^2 d^2-5 b^2 e^2\right)-3 b c \sqrt{d} x^2 (c d-b e)^{3/2} \left(15 b^3 e^3-10 b^2 c d e^2-16 b c^2 d^2 e+16 c^3 d^3\right)+x^{3/2} (b+c x) \left(45 b^4 e^4 \sqrt{b+c x} (b e-2 c d) \tanh ^{-1}\left(\frac{\sqrt{x} \sqrt{c d-b e}}{\sqrt{d} \sqrt{b+c x}}\right)-3 c \sqrt{d} \sqrt{x} \sqrt{c d-b e} \left(-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right)\right)\right)}{9 b^4 d^{7/2} (x (b+c x))^{3/2} (d+e x) (c d-b e)^{7/2}}","-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left(b x+c x^2\right)^{3/2} (d+e x) (c d-b e)}+\frac{2 \left(c x (2 c d-b e) \left(-5 b^2 e^2-8 b c d e+8 c^2 d^2\right)+b (c d-b e) \left(-5 b^2 e^2-2 b c d e+8 c^2 d^2\right)\right)}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}+\frac{e \sqrt{b x+c x^2} \left(-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right)}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left(\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right)}{2 d^{7/2} (c d-b e)^{7/2}}",1,"-1/9*(9*b^4*d^(5/2)*e*(c*d - b*e)^(5/2) + (d + e*x)*(3*b^3*d^(3/2)*(2*c*d - 5*b*e)*(c*d - b*e)^(5/2) - 9*b^2*Sqrt[d]*(c*d - b*e)^(5/2)*(4*c^2*d^2 - 5*b^2*e^2)*x - 3*b*c*Sqrt[d]*(c*d - b*e)^(3/2)*(16*c^3*d^3 - 16*b*c^2*d^2*e - 10*b^2*c*d*e^2 + 15*b^3*e^3)*x^2 + x^(3/2)*(b + c*x)*(-3*c*Sqrt[d]*Sqrt[c*d - b*e]*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e^3 - 15*b^4*e^4)*Sqrt[x] + 45*b^4*e^4*(-2*c*d + b*e)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(b^4*d^(7/2)*(c*d - b*e)^(7/2)*(x*(b + c*x))^(3/2)*(d + e*x))","A",1
336,1,11,17,0.0051192,"\int \frac{1}{(2+x) \sqrt{2 x+x^2}} \, dx","Integrate[1/((2 + x)*Sqrt[2*x + x^2]),x]","\frac{x}{\sqrt{x (x+2)}}","\frac{\sqrt{x^2+2 x}}{x+2}",1,"x/Sqrt[x*(2 + x)]","A",1
337,1,50,68,0.0439688,"\int (d+e x)^{7/2} \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^(7/2)*(b*x + c*x^2),x]","\frac{2 (d+e x)^{9/2} \left(13 b e (9 e x-2 d)+c \left(8 d^2-36 d e x+99 e^2 x^2\right)\right)}{1287 e^3}","-\frac{2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac{2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{13/2}}{13 e^3}",1,"(2*(d + e*x)^(9/2)*(13*b*e*(-2*d + 9*e*x) + c*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)","A",1
338,1,50,68,0.0379046,"\int (d+e x)^{5/2} \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^(5/2)*(b*x + c*x^2),x]","\frac{2 (d+e x)^{7/2} \left(11 b e (7 e x-2 d)+c \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)}{693 e^3}","-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 d (d+e x)^{7/2} (c d-b e)}{7 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}",1,"(2*(d + e*x)^(7/2)*(11*b*e*(-2*d + 7*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)","A",1
339,1,50,68,0.0338601,"\int (d+e x)^{3/2} \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^(3/2)*(b*x + c*x^2),x]","\frac{2 (d+e x)^{5/2} \left(9 b e (5 e x-2 d)+c \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)}{315 e^3}","-\frac{2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac{2 d (d+e x)^{5/2} (c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}",1,"(2*(d + e*x)^(5/2)*(9*b*e*(-2*d + 5*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)","A",1
340,1,50,68,0.0304909,"\int \sqrt{d+e x} \left(b x+c x^2\right) \, dx","Integrate[Sqrt[d + e*x]*(b*x + c*x^2),x]","\frac{2 (d+e x)^{3/2} \left(7 b e (3 e x-2 d)+c \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)}{105 e^3}","-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}",1,"(2*(d + e*x)^(3/2)*(7*b*e*(-2*d + 3*e*x) + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)","A",1
341,1,49,66,0.0319275,"\int \frac{b x+c x^2}{\sqrt{d+e x}} \, dx","Integrate[(b*x + c*x^2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(5 b e (e x-2 d)+c \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)}{15 e^3}","-\frac{2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac{2 d \sqrt{d+e x} (c d-b e)}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}",1,"(2*Sqrt[d + e*x]*(5*b*e*(-2*d + e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)","A",1
342,1,48,64,0.0311307,"\int \frac{b x+c x^2}{(d+e x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^(3/2),x]","\frac{2 \left(3 b e (2 d+e x)+c \left(-8 d^2-4 d e x+e^2 x^2\right)\right)}{3 e^3 \sqrt{d+e x}}","-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}",1,"(2*(3*b*e*(2*d + e*x) + c*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])","A",1
343,1,50,64,0.0336207,"\int \frac{b x+c x^2}{(d+e x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^(5/2),x]","\frac{2 \left(c \left(8 d^2+12 d e x+3 e^2 x^2\right)-b e (2 d+3 e x)\right)}{3 e^3 (d+e x)^{3/2}}","\frac{2 (2 c d-b e)}{e^3 \sqrt{d+e x}}-\frac{2 d (c d-b e)}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}",1,"(2*(-(b*e*(2*d + 3*e*x)) + c*(8*d^2 + 12*d*e*x + 3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2))","A",1
344,1,49,66,0.0309095,"\int \frac{b x+c x^2}{(d+e x)^{7/2}} \, dx","Integrate[(b*x + c*x^2)/(d + e*x)^(7/2),x]","-\frac{2 \left(b e (2 d+5 e x)+c \left(8 d^2+20 d e x+15 e^2 x^2\right)\right)}{15 e^3 (d+e x)^{5/2}}","\frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 d (c d-b e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}",1,"(-2*(b*e*(2*d + 5*e*x) + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))","A",1
345,1,124,147,0.0915964,"\int (d+e x)^{7/2} \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{9/2} \left(85 b^2 e^2 \left(8 d^2-36 d e x+99 e^2 x^2\right)+34 b c e \left(-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right)+c^2 \left(128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right)\right)}{109395 e^5}","\frac{2 (d+e x)^{13/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{13 e^5}+\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac{4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac{4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5}",1,"(2*(d + e*x)^(9/2)*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 34*b*c*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + c^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)))/(109395*e^5)","A",1
346,1,124,147,0.0780025,"\int (d+e x)^{5/2} \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{7/2} \left(65 b^2 e^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)+30 b c e \left(-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right)+c^2 \left(128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right)\right)}{45045 e^5}","\frac{2 (d+e x)^{11/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{11 e^5}+\frac{2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac{4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}",1,"(2*(d + e*x)^(7/2)*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5)","A",1
347,1,125,147,0.0723719,"\int (d+e x)^{3/2} \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{5/2} \left(143 b^2 e^2 \left(8 d^2-20 d e x+35 e^2 x^2\right)+78 b c e \left(-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right)+3 c^2 \left(128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right)\right)}{45045 e^5}","\frac{2 (d+e x)^{9/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}",1,"(2*(d + e*x)^(5/2)*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)","A",1
348,1,124,147,0.0668531,"\int \sqrt{d+e x} \left(b x+c x^2\right)^2 \, dx","Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{3/2} \left(33 b^2 e^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)+22 b c e \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)+c^2 \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)}{3465 e^5}","\frac{2 (d+e x)^{7/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{7 e^5}+\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac{4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}",1,"(2*(d + e*x)^(3/2)*(33*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*b*c*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5)","A",1
349,1,124,145,0.0745624,"\int \frac{\left(b x+c x^2\right)^2}{\sqrt{d+e x}} \, dx","Integrate[(b*x + c*x^2)^2/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(21 b^2 e^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)+18 b c e \left(-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right)+c^2 \left(128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right)\right)}{315 e^5}","\frac{2 (d+e x)^{5/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{5 e^5}+\frac{2 d^2 \sqrt{d+e x} (c d-b e)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac{4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}",1,"(2*Sqrt[d + e*x]*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)","A",1
350,1,123,143,0.070119,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^(3/2),x]","\frac{70 b^2 e^2 \left(-8 d^2-4 d e x+e^2 x^2\right)+84 b c e \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)-6 c^2 \left(128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right)}{105 e^5 \sqrt{d+e x}}","\frac{2 (d+e x)^{3/2} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{3 e^5}-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac{4 d \sqrt{d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}",1,"(70*b^2*e^2*(-8*d^2 - 4*d*e*x + e^2*x^2) + 84*b*c*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) - 6*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4))/(105*e^5*Sqrt[d + e*x])","A",1
351,1,123,143,0.0643696,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^(5/2),x]","\frac{2 \left(5 b^2 e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)+10 b c e \left(-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right)+c^2 \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{3/2}}","\frac{2 \sqrt{d+e x} \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{e^5}-\frac{2 d^2 (c d-b e)^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}",1,"(2*(5*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*b*c*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)))/(15*e^5*(d + e*x)^(3/2))","A",1
352,1,123,143,0.0639503,"\int \frac{\left(b x+c x^2\right)^2}{(d+e x)^{7/2}} \, dx","Integrate[(b*x + c*x^2)^2/(d + e*x)^(7/2),x]","-\frac{2 \left(b^2 e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)-6 b c e \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)+c^2 \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{5/2}}","-\frac{2 \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{e^5 \sqrt{d+e x}}-\frac{2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}",1,"(-2*(b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))","A",1
353,1,206,248,0.1809302,"\int (d+e x)^{7/2} \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]","\frac{2 (d+e x)^{9/2} \left(2567565 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-969969 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+3357585 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-2297295 c^2 (d+e x)^5 (2 c d-b e)+1616615 d^3 (c d-b e)^3-3968055 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+692835 c^3 (d+e x)^6\right)}{14549535 e^7}","\frac{6 c (d+e x)^{17/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{17 e^7}-\frac{2 (d+e x)^{15/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{15 e^7}+\frac{6 d (d+e x)^{13/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{13 e^7}-\frac{6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}+\frac{2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}-\frac{6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{21/2}}{21 e^7}",1,"(2*(d + e*x)^(9/2)*(1616615*d^3*(c*d - b*e)^3 - 3968055*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 3357585*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 969969*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 2567565*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 2297295*c^2*(2*c*d - b*e)*(d + e*x)^5 + 692835*c^3*(d + e*x)^6))/(14549535*e^7)","A",1
354,1,206,248,0.1490716,"\int (d+e x)^{5/2} \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(5/2)*(b*x + c*x^2)^3,x]","\frac{2 (d+e x)^{7/2} \left(969969 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-373065 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+1322685 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-855855 c^2 (d+e x)^5 (2 c d-b e)+692835 d^3 (c d-b e)^3-1616615 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+255255 c^3 (d+e x)^6\right)}{4849845 e^7}","\frac{2 c (d+e x)^{15/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{13 e^7}+\frac{6 d (d+e x)^{11/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{11 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}-\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}",1,"(2*(d + e*x)^(7/2)*(692835*d^3*(c*d - b*e)^3 - 1616615*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1322685*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 373065*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 969969*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 855855*c^2*(2*c*d - b*e)*(d + e*x)^5 + 255255*c^3*(d + e*x)^6))/(4849845*e^7)","A",1
355,1,206,248,0.1443334,"\int (d+e x)^{3/2} \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^3,x]","\frac{2 (d+e x)^{5/2} \left(58905 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-23205 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+85085 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-51051 c^2 (d+e x)^5 (2 c d-b e)+51051 d^3 (c d-b e)^3-109395 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+15015 c^3 (d+e x)^6\right)}{255255 e^7}","\frac{6 c (d+e x)^{13/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{11 e^7}+\frac{2 d (d+e x)^{9/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{3 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}-\frac{6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}",1,"(2*(d + e*x)^(5/2)*(51051*d^3*(c*d - b*e)^3 - 109395*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 85085*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 23205*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 58905*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 51051*c^2*(2*c*d - b*e)*(d + e*x)^5 + 15015*c^3*(d + e*x)^6))/(255255*e^7)","A",1
356,1,206,248,0.134784,"\int \sqrt{d+e x} \left(b x+c x^2\right)^3 \, dx","Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^3,x]","\frac{2 (d+e x)^{3/2} \left(12285 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-5005 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+19305 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-10395 c^2 (d+e x)^5 (2 c d-b e)+15015 d^3 (c d-b e)^3-27027 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+3003 c^3 (d+e x)^6\right)}{45045 e^7}","\frac{6 c (d+e x)^{11/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{9 e^7}+\frac{6 d (d+e x)^{7/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{7 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}-\frac{6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}",1,"(2*(d + e*x)^(3/2)*(15015*d^3*(c*d - b*e)^3 - 27027*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 19305*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 5005*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 12285*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 10395*c^2*(2*c*d - b*e)*(d + e*x)^5 + 3003*c^3*(d + e*x)^6))/(45045*e^7)","A",1
357,1,206,244,0.1314363,"\int \frac{\left(b x+c x^2\right)^3}{\sqrt{d+e x}} \, dx","Integrate[(b*x + c*x^2)^3/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(5005 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-2145 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+9009 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-4095 c^2 (d+e x)^5 (2 c d-b e)+15015 d^3 (c d-b e)^3-15015 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+1155 c^3 (d+e x)^6\right)}{15015 e^7}","\frac{2 c (d+e x)^{9/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{7 e^7}+\frac{6 d (d+e x)^{5/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{5 e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 d^3 \sqrt{d+e x} (c d-b e)^3}{e^7}-\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}",1,"(2*Sqrt[d + e*x]*(15015*d^3*(c*d - b*e)^3 - 15015*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 9009*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 2145*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 5005*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 4095*c^2*(2*c*d - b*e)*(d + e*x)^5 + 1155*c^3*(d + e*x)^6))/(15015*e^7)","A",1
358,1,206,242,0.1291841,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^(3/2),x]","\frac{2 \left(495 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-231 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+1155 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-385 c^2 (d+e x)^5 (2 c d-b e)-1155 d^3 (c d-b e)^3-3465 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+105 c^3 (d+e x)^6\right)}{1155 e^7 \sqrt{d+e x}}","\frac{6 c (d+e x)^{7/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{5 e^7}+\frac{2 d (d+e x)^{3/2} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac{2 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}-\frac{6 d^2 \sqrt{d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}",1,"(2*(-1155*d^3*(c*d - b*e)^3 - 3465*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1155*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 231*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 495*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 385*c^2*(2*c*d - b*e)*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(1155*e^7*Sqrt[d + e*x])","A",1
359,1,206,244,0.1260394,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^(5/2),x]","\frac{2 \left(189 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-105 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+945 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-135 c^2 (d+e x)^5 (2 c d-b e)-105 d^3 (c d-b e)^3+945 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+35 c^3 (d+e x)^6\right)}{315 e^7 (d+e x)^{3/2}}","\frac{6 c (d+e x)^{5/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{3 e^7}+\frac{6 d \sqrt{d+e x} (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}",1,"(2*(-105*d^3*(c*d - b*e)^3 + 945*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 945*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 105*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 189*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 135*c^2*(2*c*d - b*e)*(d + e*x)^5 + 35*c^3*(d + e*x)^6))/(315*e^7*(d + e*x)^(3/2))","A",1
360,1,206,240,0.1293622,"\int \frac{\left(b x+c x^2\right)^3}{(d+e x)^{7/2}} \, dx","Integrate[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]","\frac{2 \left(35 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-35 (d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)-105 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)-21 c^2 (d+e x)^5 (2 c d-b e)-7 d^3 (c d-b e)^3+35 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+5 c^3 (d+e x)^6\right)}{35 e^7 (d+e x)^{5/2}}","\frac{2 c (d+e x)^{3/2} \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{e^7}-\frac{6 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{e^7 \sqrt{d+e x}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}",1,"(2*(-7*d^3*(c*d - b*e)^3 + 35*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) - 105*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 35*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 35*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 21*c^2*(2*c*d - b*e)*(d + e*x)^5 + 5*c^3*(d + e*x)^6))/(35*e^7*(d + e*x)^(5/2))","A",1
361,1,138,157,0.2041994,"\int \frac{(d+e x)^{7/2}}{b x+c x^2} \, dx","Integrate[(d + e*x)^(7/2)/(b*x + c*x^2),x]","\frac{2 e \sqrt{d+e x} \left(15 b^2 e^2-5 b c e (10 d+e x)+c^2 \left(58 d^2+16 d e x+3 e^2 x^2\right)\right)}{15 c^3}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b c^{7/2}}-\frac{2 d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}","\frac{2 e \sqrt{d+e x} \left(b^2 e^2-3 b c d e+3 c^2 d^2\right)}{c^3}+\frac{2 (c d-b e)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b c^{7/2}}+\frac{2 e (d+e x)^{3/2} (2 c d-b e)}{3 c^2}-\frac{2 d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}+\frac{2 e (d+e x)^{5/2}}{5 c}",1,"(2*e*Sqrt[d + e*x]*(15*b^2*e^2 - 5*b*c*e*(10*d + e*x) + c^2*(58*d^2 + 16*d*e*x + 3*e^2*x^2)))/(15*c^3) - (2*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b + (2*(c*d - b*e)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(7/2))","A",1
362,1,107,118,0.1225315,"\int \frac{(d+e x)^{5/2}}{b x+c x^2} \, dx","Integrate[(d + e*x)^(5/2)/(b*x + c*x^2),x]","\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b c^{5/2}}+\frac{2 e \sqrt{d+e x} (-3 b e+7 c d+c e x)}{3 c^2}-\frac{2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}","\frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b c^{5/2}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}-\frac{2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}+\frac{2 e (d+e x)^{3/2}}{3 c}",1,"(2*e*Sqrt[d + e*x]*(7*c*d - 3*b*e + c*e*x))/(3*c^2) - (2*d^(5/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b + (2*(c*d - b*e)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(5/2))","A",1
363,1,97,92,0.0629946,"\int \frac{(d+e x)^{3/2}}{b x+c x^2} \, dx","Integrate[(d + e*x)^(3/2)/(b*x + c*x^2),x]","\frac{2 \left(b \sqrt{c} e \sqrt{d+e x}+(c d-b e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)-c^{3/2} d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)\right)}{b c^{3/2}}","\frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}+\frac{2 e \sqrt{d+e x}}{c}",1,"(2*(b*Sqrt[c]*e*Sqrt[d + e*x] - c^(3/2)*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (c*d - b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))/(b*c^(3/2))","A",1
364,1,75,77,0.0352212,"\int \frac{\sqrt{d+e x}}{b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]/(b*x + c*x^2),x]","\frac{2 \left(\frac{\sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{\sqrt{c}}-\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)\right)}{b}","\frac{2 \sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b \sqrt{c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b}",1,"(2*(-(Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) + (Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/Sqrt[c]))/b","A",1
365,1,75,77,0.0823416,"\int \frac{1}{\sqrt{d+e x} \left(b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)),x]","\frac{\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{\sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{b}","\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b \sqrt{d}}",1,"((-2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/Sqrt[c*d - b*e])/b","A",1
366,1,81,102,0.0266337,"\int \frac{1}{(d+e x)^{3/2} \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)),x]","-\frac{2 \left(c d \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)+(b e-c d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right)\right)}{b d \sqrt{d+e x} (c d-b e)}","\frac{2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b (c d-b e)^{3/2}}-\frac{2 e}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b d^{3/2}}",1,"(-2*(c*d*Hypergeometric2F1[-1/2, 1, 1/2, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (e*x)/d]))/(b*d*(c*d - b*e)*Sqrt[d + e*x])","C",1
367,1,83,138,0.024822,"\int \frac{1}{(d+e x)^{5/2} \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)),x]","-\frac{2 \left(c d \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)+(b e-c d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{e x}{d}+1\right)\right)}{3 b d (d+e x)^{3/2} (c d-b e)}","\frac{2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b (c d-b e)^{5/2}}-\frac{2 e (2 c d-b e)}{d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b d^{5/2}}",1,"(-2*(c*d*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))/(3*b*d*(c*d - b*e)*(d + e*x)^(3/2))","C",1
368,1,83,187,0.0274412,"\int \frac{1}{(d+e x)^{7/2} \left(b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)),x]","-\frac{2 \left(c d \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{c (d+e x)}{c d-b e}\right)+(b e-c d) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{e x}{d}+1\right)\right)}{5 b d (d+e x)^{5/2} (c d-b e)}","-\frac{2 e \left(b^2 e^2-3 b c d e+3 c^2 d^2\right)}{d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b (c d-b e)^{7/2}}-\frac{2 e (2 c d-b e)}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 e}{5 d (d+e x)^{5/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b d^{7/2}}",1,"(-2*(c*d*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b*d*(c*d - b*e)*(d + e*x)^(5/2))","C",1
369,1,202,251,0.3697328,"\int \frac{(d+e x)^{9/2}}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^2,x]","\frac{\frac{b \sqrt{d+e x} \left(-15 b^4 e^4 x+2 b^3 c e^3 x (19 d-5 e x)+2 b^2 c^2 e^2 x \left(-9 d^2+13 d e x+e^2 x^2\right)-3 b c^3 d^3 (d-4 e x)-6 c^4 d^4 x\right)}{c^3 x (b+c x)}-\frac{3 (c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{c^{7/2}}+3 d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{3 b^3}","-\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}+\frac{e (d+e x)^{3/2} \left(5 b^2 e^2-6 b c d e+6 c^2 d^2\right)}{3 b^2 c^2}+\frac{e \sqrt{d+e x} (2 c d-b e) \left(5 b^2 e^2-b c d e+c^2 d^2\right)}{b^2 c^3}-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left(b x+c x^2\right)}+\frac{e (d+e x)^{5/2} (2 c d-b e)}{b^2 c}",1,"((b*Sqrt[d + e*x]*(-6*c^4*d^4*x - 15*b^4*e^4*x + 2*b^3*c*e^3*x*(19*d - 5*e*x) - 3*b*c^3*d^3*(d - 4*e*x) + 2*b^2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2*x^2)))/(c^3*x*(b + c*x)) + 3*d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - (3*(c*d - b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(7/2))/(3*b^3)","A",1
370,1,167,200,0.3138773,"\int \frac{(d+e x)^{7/2}}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^2,x]","\frac{\frac{b \sqrt{d+e x} \left(3 b^3 e^3 x+b^2 c e^2 x (2 e x-3 d)-b c^2 d^2 (d-3 e x)-2 c^3 d^3 x\right)}{c^2 x (b+c x)}-\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{c^{5/2}}+d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}","-\frac{(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 c^{5/2}}+\frac{d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}+\frac{e \sqrt{d+e x} \left(3 b^2 e^2-2 b c d e+2 c^2 d^2\right)}{b^2 c^2}-\frac{(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left(b x+c x^2\right)}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{b^2 c}",1,"((b*Sqrt[d + e*x]*(-2*c^3*d^3*x + 3*b^3*e^3*x - b*c^2*d^2*(d - 3*e*x) + b^2*c*e^2*x*(-3*d + 2*e*x)))/(c^2*x*(b + c*x)) + d^(5/2)*(4*c*d - 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - ((c*d - b*e)^(5/2)*(4*c*d + 3*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(5/2))/b^3","A",1
371,1,159,159,0.2640841,"\int \frac{(d+e x)^{5/2}}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^2,x]","\frac{-\frac{b \sqrt{d+e x} \left(b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right)}{c x (b+c x)}-\frac{\sqrt{c d-b e} \left(-b^2 e^2-3 b c d e+4 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{c^{3/2}}+d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}","-\frac{(c d-b e)^{3/2} (b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 c^{3/2}}+\frac{d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}-\frac{(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left(b x+c x^2\right)}+\frac{e \sqrt{d+e x} (2 c d-b e)}{b^2 c}",1,"(-((b*Sqrt[d + e*x]*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x)))/(c*x*(b + c*x))) + d^(3/2)*(4*c*d - 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - (Sqrt[c*d - b*e]*(4*c^2*d^2 - 3*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(3/2))/b^3","A",1
372,1,127,134,0.1978637,"\int \frac{(d+e x)^{3/2}}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^2,x]","\frac{\frac{b \sqrt{d+e x} (-b d+b e x-2 c d x)}{x (b+c x)}+\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{\sqrt{c}}}{b^3}","\frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 \sqrt{c}}-\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{b^2 \left(b x+c x^2\right)}",1,"((b*Sqrt[d + e*x]*(-(b*d) - 2*c*d*x + b*e*x))/(x*(b + c*x)) + Sqrt[d]*(4*c*d - 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - (Sqrt[c*d - b*e]*(4*c*d - b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/Sqrt[c])/b^3","A",1
373,1,169,125,0.2081328,"\int \frac{\sqrt{d+e x}}{\left(b x+c x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(b*x + c*x^2)^2,x]","\frac{\sqrt{d} \left(b (b+2 c x) \sqrt{d+e x} (c d-b e)+\sqrt{c} x (b+c x) (4 c d-3 b e) \sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)\right)-x (b+c x) \left(b^2 e^2-5 b c d e+4 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 \sqrt{d} x (b+c x) (b e-c d)}","\frac{(4 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 \sqrt{d}}-\frac{\sqrt{c} (4 c d-3 b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 \sqrt{c d-b e}}-\frac{(b+2 c x) \sqrt{d+e x}}{b^2 \left(b x+c x^2\right)}",1,"(-((4*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*x*(b + c*x)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) + Sqrt[d]*(b*(c*d - b*e)*(b + 2*c*x)*Sqrt[d + e*x] + Sqrt[c]*(4*c*d - 3*b*e)*Sqrt[c*d - b*e]*x*(b + c*x)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))/(b^3*Sqrt[d]*(-(c*d) + b*e)*x*(b + c*x))","A",1
374,1,148,154,0.464324,"\int \frac{1}{\sqrt{d+e x} \left(b x+c x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^2),x]","\frac{\frac{b \sqrt{d+e x} \left(b^2 e+b c (e x-d)-2 c^2 d x\right)}{x (b+c x) (c d-b e)}+\frac{c^{3/2} d (5 b e-4 c d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{(c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{b^3 d}","-\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 d^{3/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left(b x+c x^2\right) (c d-b e)}",1,"((b*Sqrt[d + e*x]*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)))/((c*d - b*e)*x*(b + c*x)) + ((4*c*d + b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (c^(3/2)*d*(-4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(c*d - b*e)^(3/2))/(b^3*d)","A",1
375,1,166,206,0.1129648,"\int \frac{1}{(d+e x)^{3/2} \left(b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^2),x]","\frac{c^2 d^2 x (b+c x) (4 c d-7 b e) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)-(b e-c d) \left(x (b+c x) \left(3 b^2 e^2+b c d e-4 c^2 d^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right)+b d \left(b^2 e+b c (e x-d)-2 c^2 d x\right)\right)}{b^3 d^2 x (b+c x) \sqrt{d+e x} (c d-b e)^2}","-\frac{c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 (c d-b e)^{5/2}}+\frac{(3 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 d^{5/2}}-\frac{e \left(3 b^2 e^2-2 b c d e+2 c^2 d^2\right)}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{c x (2 c d-b e)+b (c d-b e)}{b^2 d \left(b x+c x^2\right) \sqrt{d+e x} (c d-b e)}",1,"(c^2*d^2*(4*c*d - 7*b*e)*x*(b + c*x)*Hypergeometric2F1[-1/2, 1, 1/2, (c*(d + e*x))/(c*d - b*e)] - (-(c*d) + b*e)*(b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (-4*c^2*d^2 + b*c*d*e + 3*b^2*e^2)*x*(b + c*x)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (e*x)/d]))/(b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*Sqrt[d + e*x])","C",1
376,1,170,267,0.1177282,"\int \frac{1}{(d+e x)^{5/2} \left(b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]","\frac{c^2 d^2 x (b+c x) (4 c d-9 b e) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)-(c d-b e) \left(x (b+c x) \left(-5 b^2 e^2+b c d e+4 c^2 d^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{e x}{d}+1\right)-3 b d \left(b^2 e+b c (e x-d)-2 c^2 d x\right)\right)}{3 b^3 d^2 x (b+c x) (d+e x)^{3/2} (c d-b e)^2}","-\frac{c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 (c d-b e)^{7/2}}+\frac{(5 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 d^{7/2}}-\frac{e \left(5 b^2 e^2-6 b c d e+6 c^2 d^2\right)}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left(5 b^2 e^2-b c d e+c^2 d^2\right)}{b^2 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{c x (2 c d-b e)+b (c d-b e)}{b^2 d \left(b x+c x^2\right) (d+e x)^{3/2} (c d-b e)}",1,"(c^2*d^2*(4*c*d - 9*b*e)*x*(b + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)*(-3*b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (4*c^2*d^2 + b*c*d*e - 5*b^2*e^2)*x*(b + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))/(3*b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*(d + e*x)^(3/2))","C",1
377,1,171,349,0.1197662,"\int \frac{1}{(d+e x)^{7/2} \left(b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]","\frac{c^2 d^2 x (b+c x) (4 c d-11 b e) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{c (d+e x)}{c d-b e}\right)-(c d-b e) \left(x (b+c x) \left(-7 b^2 e^2+3 b c d e+4 c^2 d^2\right) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{e x}{d}+1\right)-5 b d \left(b^2 e+b c (e x-d)-2 c^2 d x\right)\right)}{5 b^3 d^2 x (b+c x) (d+e x)^{5/2} (c d-b e)^2}","-\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{b^3 d^{9/2}}-\frac{e \left(7 b^2 e^2-10 b c d e+10 c^2 d^2\right)}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left(7 b^2 e^2-3 b c d e+3 c^2 d^2\right)}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{c x (2 c d-b e)+b (c d-b e)}{b^2 d \left(b x+c x^2\right) (d+e x)^{5/2} (c d-b e)}-\frac{e \left(7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right)}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}",1,"(c^2*d^2*(4*c*d - 11*b*e)*x*(b + c*x)*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)*(-5*b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (4*c^2*d^2 + 3*b*c*d*e - 7*b^2*e^2)*x*(b + c*x)*Hypergeometric2F1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*(d + e*x)^(5/2))","C",1
378,1,275,300,0.6284773,"\int \frac{(d+e x)^{9/2}}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]","\frac{-3 d^{5/2} \left(21 b^2 e^2-36 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)+\frac{3 (c d-b e)^{5/2} \left(b^2 e^2+4 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{c^{5/2}}+\frac{b \sqrt{d+e x} \left(-3 b^5 e^4 x^2-5 b^4 c e^3 x^2 (d+e x)+b^3 c^2 d \left(-2 d^3-17 d^2 e x+33 d e^2 x^2+3 e^3 x^3\right)+b^2 c^3 d^2 x \left(8 d^2-73 d e x+21 e^2 x^2\right)+12 b c^4 d^3 x^2 (3 d-4 e x)+24 c^5 d^4 x^3\right)}{c^2 x^2 (b+c x)^2}}{4 b^5}","-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left(b x+c x^2\right)^2}-\frac{3 d^{5/2} \left(21 b^2 e^2-36 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left(b^2 e^2+4 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 c^{5/2}}+\frac{(d+e x)^{3/2} \left(x (2 c d-b e) \left(-b^2 e^2-12 b c d e+12 c^2 d^2\right)+b c d^2 (12 c d-13 b e)\right)}{4 b^4 c \left(b x+c x^2\right)}-\frac{3 e \sqrt{d+e x} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)}{4 b^4 c^2}",1,"((b*Sqrt[d + e*x]*(-3*b^5*e^4*x^2 + 24*c^5*d^4*x^3 + 12*b*c^4*d^3*x^2*(3*d - 4*e*x) - 5*b^4*c*e^3*x^2*(d + e*x) + b^2*c^3*d^2*x*(8*d^2 - 73*d*e*x + 21*e^2*x^2) + b^3*c^2*d*(-2*d^3 - 17*d^2*e*x + 33*d*e^2*x^2 + 3*e^3*x^3)))/(c^2*x^2*(b + c*x)^2) - 3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(5/2))/(4*b^5)","A",1
379,1,263,248,0.4761931,"\int \frac{(d+e x)^{7/2}}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^3,x]","\frac{-\left(d^{3/2} \left(35 b^2 e^2-84 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)\right)+\frac{\sqrt{c d-b e} \left(b^3 e^3+11 b^2 c d e^2-60 b c^2 d^2 e+48 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{c^{3/2}}+\frac{b \sqrt{d+e x} \left(b^4 \left(-e^3\right) x^2+b^3 c \left(-2 d^3-13 d^2 e x+16 d e^2 x^2+e^3 x^3\right)+b^2 c^2 d x \left(8 d^2-55 d e x+10 e^2 x^2\right)+36 b c^3 d^2 x^2 (d-e x)+24 c^4 d^3 x^3\right)}{c x^2 (b+c x)^2}}{4 b^5}","-\frac{(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left(b x+c x^2\right)^2}-\frac{d^{3/2} \left(35 b^2 e^2-84 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5}+\frac{(c d-b e)^{3/2} \left(-b^2 e^2-12 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 c^{3/2}}+\frac{\sqrt{d+e x} \left(x (2 c d-b e) \left(b^2 e^2-12 b c d e+12 c^2 d^2\right)+b c d^2 (12 c d-11 b e)\right)}{4 b^4 c \left(b x+c x^2\right)}",1,"((b*Sqrt[d + e*x]*(-(b^4*e^3*x^2) + 24*c^4*d^3*x^3 + 36*b*c^3*d^2*x^2*(d - e*x) + b^2*c^2*d*x*(8*d^2 - 55*d*e*x + 10*e^2*x^2) + b^3*c*(-2*d^3 - 13*d^2*e*x + 16*d*e^2*x^2 + e^3*x^3)))/(c*x^2*(b + c*x)^2) - d^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[c*d - b*e]*(48*c^3*d^3 - 60*b*c^2*d^2*e + 11*b^2*c*d*e^2 + b^3*e^3)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(3/2))/(4*b^5)","A",1
380,1,222,232,0.4131635,"\int \frac{(d+e x)^{5/2}}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^3,x]","\frac{-3 \sqrt{d} \left(5 b^2 e^2-20 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)+\frac{3 \sqrt{c d-b e} \left(b^2 e^2-12 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{\sqrt{c}}+\frac{b \sqrt{d+e x} \left(b^3 \left(-2 d^2-9 d e x+5 e^2 x^2\right)+b^2 c x \left(8 d^2-37 d e x+3 e^2 x^2\right)+12 b c^2 d x^2 (3 d-2 e x)+24 c^3 d^2 x^3\right)}{x^2 (b+c x)^2}}{4 b^5}","-\frac{(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left(b x+c x^2\right)^2}-\frac{3 \sqrt{d} \left(5 b^2 e^2-20 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5}+\frac{3 \sqrt{c d-b e} \left(b^2 e^2-12 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 \sqrt{c}}+\frac{3 \sqrt{d+e x} \left(x \left(b^2 e^2-8 b c d e+8 c^2 d^2\right)+b d (4 c d-3 b e)\right)}{4 b^4 \left(b x+c x^2\right)}",1,"((b*Sqrt[d + e*x]*(24*c^3*d^2*x^3 + 12*b*c^2*d*x^2*(3*d - 2*e*x) + b^2*c*x*(8*d^2 - 37*d*e*x + 3*e^2*x^2) + b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2)))/(x^2*(b + c*x)^2) - 3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (3*Sqrt[c*d - b*e]*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/Sqrt[c])/(4*b^5)","A",1
381,1,289,246,0.525033,"\int \frac{(d+e x)^{3/2}}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^3,x]","\frac{3 x^2 (b+c x)^2 \left(-b^3 e^3+13 b^2 c d e^2-28 b c^2 d^2 e+16 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)-\sqrt{d} \left(3 \sqrt{c} x^2 (b+c x)^2 \sqrt{c d-b e} \left(5 b^2 e^2-20 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)+b \sqrt{d+e x} \left(b^4 e (2 d+5 e x)+b^3 c \left(-2 d^2-13 d e x+19 e^2 x^2\right)+b^2 c^2 x \left(8 d^2-55 d e x+12 e^2 x^2\right)+36 b c^3 d x^2 (d-e x)+24 c^4 d^2 x^3\right)\right)}{4 b^5 \sqrt{d} x^2 (b+c x)^2 (b e-c d)}","\frac{\sqrt{d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left(b x+c x^2\right) (c d-b e)}-\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left(b x+c x^2\right)^2}-\frac{3 \left(b^2 e^2-12 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5 \sqrt{d}}+\frac{3 \sqrt{c} \left(5 b^2 e^2-20 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 \sqrt{c d-b e}}",1,"(3*(16*c^3*d^3 - 28*b*c^2*d^2*e + 13*b^2*c*d*e^2 - b^3*e^3)*x^2*(b + c*x)^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - Sqrt[d]*(b*Sqrt[d + e*x]*(24*c^4*d^2*x^3 + 36*b*c^3*d*x^2*(d - e*x) + b^4*e*(2*d + 5*e*x) + b^2*c^2*x*(8*d^2 - 55*d*e*x + 12*e^2*x^2) + b^3*c*(-2*d^2 - 13*d*e*x + 19*e^2*x^2)) + 3*Sqrt[c]*Sqrt[c*d - b*e]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*x^2*(b + c*x)^2*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))/(4*b^5*Sqrt[d]*(-(c*d) + b*e)*x^2*(b + c*x)^2)","A",1
382,1,361,245,0.9640066,"\int \frac{\sqrt{d+e x}}{\left(b x+c x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(b*x + c*x^2)^3,x]","\frac{-\frac{2 c (d+e x)^{3/2} \left(-b^2 e^2-9 b c d e+12 c^2 d^2\right)}{b^2 d (b e-c d)}+\frac{(b+c x) \left((b+c x) \left(2 c^{3/2} (c d-b e)^2 \left(-b^2 e^2-12 b c d e+48 c^2 d^2\right) \left(\sqrt{d+e x}-\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)\right)-2 c^3 d^2 \left(35 b^2 e^2-84 b c d e+48 c^2 d^2\right) \left(\sqrt{c} \sqrt{d+e x}-\sqrt{c d-b e} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)\right)\right)+2 b c^{5/2} (d+e x)^{3/2} \left(b^3 e^3+10 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right)\right)}{b^4 c^{3/2} d (c d-b e)^2}+\frac{2 (d+e x)^{3/2} (b e+8 c d)}{b d x}-\frac{4 (d+e x)^{3/2}}{x^2}}{8 b d (b+c x)^2}","-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left(b x+c x^2\right)^2}-\frac{\left(-b^2 e^2-12 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5 d^{3/2}}+\frac{c^{3/2} \left(35 b^2 e^2-84 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 (c d-b e)^{3/2}}+\frac{\sqrt{d+e x} \left(c x \left(b^2 e^2-24 b c d e+24 c^2 d^2\right)+b (c d-b e) (12 c d-b e)\right)}{4 b^4 d \left(b x+c x^2\right) (c d-b e)}",1,"((-2*c*(12*c^2*d^2 - 9*b*c*d*e - b^2*e^2)*(d + e*x)^(3/2))/(b^2*d*(-(c*d) + b*e)) - (4*(d + e*x)^(3/2))/x^2 + (2*(8*c*d + b*e)*(d + e*x)^(3/2))/(b*d*x) + ((b + c*x)*(2*b*c^(5/2)*(24*c^3*d^3 - 36*b*c^2*d^2*e + 10*b^2*c*d*e^2 + b^3*e^3)*(d + e*x)^(3/2) + (b + c*x)*(2*c^(3/2)*(c*d - b*e)^2*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*(Sqrt[d + e*x] - Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 2*c^3*d^2*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*(Sqrt[c]*Sqrt[d + e*x] - Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))))/(b^4*c^(3/2)*d*(c*d - b*e)^2))/(8*b*d*(b + c*x)^2)","A",1
383,1,299,299,1.1454093,"\int \frac{1}{\sqrt{d+e x} \left(b x+c x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]","\frac{-3 \left(b^2 e^2+4 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)+\frac{3 c^{5/2} d^{5/2} \left(21 b^2 e^2-36 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{(c d-b e)^{5/2}}+\frac{b \sqrt{d} \sqrt{d+e x} \left(b^5 e^2 (3 e x-2 d)+2 b^4 c e \left(2 d^2+d e x+3 e^2 x^2\right)+b^3 c^2 \left(-2 d^3-13 d^2 e x+10 d e^2 x^2+3 e^3 x^3\right)+b^2 c^3 d x \left(8 d^2-55 d e x+6 e^2 x^2\right)+36 b c^4 d^2 x^2 (d-e x)+24 c^5 d^3 x^3\right)}{x^2 (b+c x)^2 (c d-b e)^2}}{4 b^5 d^{5/2}}","-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left(b x+c x^2\right)^2 (c d-b e)}-\frac{3 \left(b^2 e^2+4 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5 d^{5/2}}+\frac{3 c^{5/2} \left(21 b^2 e^2-36 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 (c d-b e)^{5/2}}+\frac{\sqrt{d+e x} \left(3 c x (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)+b (c d-b e) \left(-3 b^2 e^2-7 b c d e+12 c^2 d^2\right)\right)}{4 b^4 d^2 \left(b x+c x^2\right) (c d-b e)^2}",1,"((b*Sqrt[d]*Sqrt[d + e*x]*(24*c^5*d^3*x^3 + 36*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-2*d + 3*e*x) + 2*b^4*c*e*(2*d^2 + d*e*x + 3*e^2*x^2) + b^2*c^3*d*x*(8*d^2 - 55*d*e*x + 6*e^2*x^2) + b^3*c^2*(-2*d^3 - 13*d^2*e*x + 10*d*e^2*x^2 + 3*e^3*x^3)))/((c*d - b*e)^2*x^2*(b + c*x)^2) - 3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (3*c^(5/2)*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(c*d - b*e)^(5/2))/(4*b^5*d^(5/2))","A",1
384,1,299,370,0.2878629,"\int \frac{1}{(d+e x)^{3/2} \left(b x+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]","\frac{2 b^4 d^2 (b e-c d)^3+b^3 d x (c d-b e)^3 (5 b e+8 c d)-x^2 \left(b^2 c d \left(5 b^2 e^2+5 b c d e-12 c^2 d^2\right) (c d-b e)^2+(b+c x) \left(b c d (b e-c d) \left(5 b^3 e^3+2 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right)-(b+c x) \left(3 (c d-b e)^3 \left(5 b^2 e^2+12 b c d e+16 c^2 d^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right)-3 c^3 d^3 \left(33 b^2 e^2-44 b c d e+16 c^2 d^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)\right)\right)\right)}{4 b^5 d^3 x^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)^3}","-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left(b x+c x^2\right)^2 \sqrt{d+e x} (c d-b e)}-\frac{3 \left(5 b^2 e^2+12 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5 d^{7/2}}+\frac{3 c^{7/2} \left(33 b^2 e^2-44 b c d e+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 (c d-b e)^{7/2}}+\frac{3 e \left(-b^2 e^2-b c d e+c^2 d^2\right) \left(5 b^2 e^2-8 b c d e+8 c^2 d^2\right)}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{b \left(5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right)+c x (2 c d-b e) \left(-5 b^2 e^2-12 b c d e+12 c^2 d^2\right)}{4 b^4 d^2 \left(b x+c x^2\right) \sqrt{d+e x} (c d-b e)^2}",1,"(2*b^4*d^2*(-(c*d) + b*e)^3 + b^3*d*(c*d - b*e)^3*(8*c*d + 5*b*e)*x - x^2*(b^2*c*d*(c*d - b*e)^2*(-12*c^2*d^2 + 5*b*c*d*e + 5*b^2*e^2) + (b + c*x)*(b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e + 2*b^2*c*d*e^2 + 5*b^3*e^3) - (b + c*x)*(-3*c^3*d^3*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, (c*(d + e*x))/(c*d - b*e)] + 3*(c*d - b*e)^3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (e*x)/d]))))/(4*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sqrt[d + e*x])","C",1
385,1,301,470,0.3257937,"\int \frac{1}{(d+e x)^{5/2} \left(b x+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]","\frac{6 b^4 d^2 (b e-c d)^3+3 b^3 d x (c d-b e)^3 (7 b e+8 c d)-x^2 \left(3 b^2 c d \left(7 b^2 e^2+3 b c d e-12 c^2 d^2\right) (c d-b e)^2+(b+c x) \left(3 b c d (b e-c d) \left(7 b^3 e^3-2 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right)-(b+c x) \left((c d-b e)^3 \left(35 b^2 e^2+60 b c d e+48 c^2 d^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{e x}{d}+1\right)-c^3 d^3 \left(143 b^2 e^2-156 b c d e+48 c^2 d^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right)\right)\right)\right)}{12 b^5 d^3 x^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)^3}","-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left(b x+c x^2\right)^2 (d+e x)^{3/2} (c d-b e)}-\frac{\left(35 b^2 e^2+60 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 b^5 d^{9/2}}+\frac{c^{9/2} \left(143 b^2 e^2-156 b c d e+48 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right)}{4 b^5 (c d-b e)^{9/2}}+\frac{c x (2 c d-b e) \left(-7 b^2 e^2-12 b c d e+12 c^2 d^2\right)+b (c d-b e) \left(-7 b^2 e^2-3 b c d e+12 c^2 d^2\right)}{4 b^4 d^2 \left(b x+c x^2\right) (d+e x)^{3/2} (c d-b e)^2}+\frac{e (2 c d-b e) \left(-35 b^4 e^4+10 b^3 c d e^3+2 b^2 c^2 d^2 e^2-24 b c^3 d^3 e+12 c^4 d^4\right)}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left(-35 b^4 e^4+45 b^3 c d e^3+27 b^2 c^2 d^2 e^2-144 b c^3 d^3 e+72 c^4 d^4\right)}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}",1,"(6*b^4*d^2*(-(c*d) + b*e)^3 + 3*b^3*d*(c*d - b*e)^3*(8*c*d + 7*b*e)*x - x^2*(3*b^2*c*d*(c*d - b*e)^2*(-12*c^2*d^2 + 3*b*c*d*e + 7*b^2*e^2) + (b + c*x)*(3*b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e - 2*b^2*c*d*e^2 + 7*b^3*e^3) - (b + c*x)*(-(c^3*d^3*(48*c^2*d^2 - 156*b*c*d*e + 143*b^2*e^2)*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)]) + (c*d - b*e)^3*(48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))))/(12*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*(d + e*x)^(3/2))","C",1
386,1,372,362,1.8629034,"\int (d+e x)^{3/2} \sqrt{b x+c x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[b*x + c*x^2],x]","\frac{2 \left(b e x (b+c x) (d+e x) \left(-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left(d^2+8 d e x+5 e^2 x^2\right)\right)-\sqrt{\frac{b}{c}} \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right)\right)\right)}{105 b c^2 e^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right)}{105 c^2 e}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(2 b^2 e^2-3 b c d e+3 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(8 b^2 e^2-3 b c d e+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left(b x+c x^2\right)^{3/2} \sqrt{d+e x}}{7 c}",1,"(2*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^2*e^2 + 3*b*c*e*(3*d + e*x) + 3*c^2*(d^2 + 8*d*e*x + 5*e^2*x^2)) - Sqrt[b/c]*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(105*b*c^2*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
387,1,294,308,1.1430634,"\int \sqrt{d+e x} \sqrt{b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[b*x + c*x^2],x]","\frac{2 \left(b e x (b+c x) (d+e x) (b e+c (d+3 e x))+\sqrt{\frac{b}{c}} \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-3 b c d e+c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^2 e^2-b c d e+c^2 d^2\right)\right)\right)}{15 b c e^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 c^{3/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 c^{3/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c e}",1,"(2*(b*e*x*(b + c*x)*(d + e*x)*(b*e + c*(d + 3*e*x)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) - (2*I)*b*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*c*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
388,1,226,246,1.3980715,"\int \frac{\sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[b*x + c*x^2]/Sqrt[d + e*x],x]","\frac{2 \left(-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+(b+c x) (d+e x) (b e-2 c d+c e x)\right)}{3 c e^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x}}{3 e}",1,"(2*((b + c*x)*(d + e*x)*(-2*c*d + b*e + c*e*x) + I*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*c*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
389,1,195,231,0.5198562,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^(3/2),x]","\frac{2 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (2 d+e x)\right)}{e^2 \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{\sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2}}{e \sqrt{d+e x}}",1,"(2*(Sqrt[b/c]*(b + c*x)*(2*d + e*x) + (2*I)*b*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(Sqrt[b/c]*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
390,1,265,301,0.7325314,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^(5/2),x]","-\frac{2 \left(e x (b+c x) \left(b e^2 x-c d (d+2 e x)\right)+(d+e x) \left(i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+(b+c x) (d+e x) (2 c d-b e)\right)\right)}{3 d e^2 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)}","\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 d e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{3 d e \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{3 e (d+e x)^{3/2}}",1,"(-2*(e*x*(b + c*x)*(b*e^2*x - c*d*(d + 2*e*x)) + (d + e*x)*((2*c*d - b*e)*(b + c*x)*(d + e*x) - I*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*c*e*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*d*e^2*(c*d - b*e)*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))","C",1
391,1,362,398,0.9801788,"\int \frac{\sqrt{b x+c x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^(7/2),x]","-\frac{2 \left(b e x (b+c x) \left(-b^2 e^3 x (5 d+2 e x)+b c d e \left(-d^2+7 d e x+2 e^2 x^2\right)-c^2 d^2 \left(d^2+6 d e x+2 e^2 x^2\right)\right)+c \sqrt{\frac{b}{c}} (d+e x)^2 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-3 b c d e+c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^2 e^2-b c d e+c^2 d^2\right)\right)\right)}{15 b d^2 e^2 \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^2}","\frac{4 \sqrt{b x+c x^2} \left(b^2 e^2-b c d e+c^2 d^2\right)}{15 d^2 e \sqrt{d+e x} (c d-b e)^2}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 d^2 e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 d e^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{5 e (d+e x)^{5/2}}",1,"(-2*(b*e*x*(b + c*x)*(-(b^2*e^3*x*(5*d + 2*e*x)) - c^2*d^2*(d^2 + 6*d*e*x + 2*e^2*x^2) + b*c*d*e*(-d^2 + 7*d*e*x + 2*e^2*x^2)) + Sqrt[b/c]*c*(d + e*x)^2*(2*Sqrt[b/c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + (2*I)*b*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*d^2*e^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))","C",1
392,1,559,521,2.8961454,"\int (d+e x)^{3/2} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{3/2} \left(b e x (b+c x) (d+e x) \left(8 b^4 e^4-b^3 c e^3 (19 d+6 e x)+b^2 c^2 e^2 \left(6 d^2+14 d e x+5 e^2 x^2\right)+b c^3 e \left(-19 d^3+14 d^2 e x+205 d e^2 x^2+140 e^3 x^3\right)+c^4 \left(8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right)\right)+\sqrt{\frac{b}{c}} \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(16 b^5 e^5-48 b^4 c d e^4+35 b^3 c^2 d^2 e^3+10 b^2 c^3 d^3 e^2-21 b c^4 d^4 e+8 c^5 d^5\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-8 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^5 e^5-5 b^4 c d e^4+2 b^3 c^2 d^2 e^3+2 b^2 c^3 d^3 e^2-5 b c^4 d^4 e+2 c^5 d^5\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-8 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(2 b^5 e^5-5 b^4 c d e^4+2 b^3 c^2 d^2 e^3+2 b^2 c^3 d^3 e^2-5 b c^4 d^4 e+2 c^5 d^5\right)\right)\right)}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt{d+e x}}","\frac{2 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} \left(-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right)}{231 c^2 e}-\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left(8 b^2 e^2-b c d e+c^2 d^2\right)+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right)}{1155 c^3 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \left(b x+c x^2\right)^{5/2} \sqrt{d+e x}}{11 c}",1,"(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(8*b^4*e^4 - b^3*c*e^3*(19*d + 6*e*x) + b^2*c^2*e^2*(6*d^2 + 14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19*d^3 + 14*d^2*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*(2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*(b + c*x)*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(8*c^5*d^5 - 21*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])","C",1
393,1,463,457,2.3454557,"\int \sqrt{d+e x} \left(b x+c x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^(3/2),x]","\frac{2 (x (b+c x))^{3/2} \left(b e x (b+c x) (d+e x) \left(-4 b^3 e^3+3 b^2 c e^2 (d+e x)+b c^2 e \left(-15 d^2+11 d e x+50 e^2 x^2\right)+c^3 \left(8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right)\right)-\sqrt{\frac{b}{c}} \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right)\right)\right)}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt{d+e x}}","\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left(-b^2 e^2-2 b c d e+2 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(-4 b^3 e^3-6 c e x \left(2 b^2 e^2-b c d e+c^2 d^2\right)+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right)}{315 c^2 e^3}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left(b x+c x^2\right)^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e}",1,"(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^3*e^3 + 3*b^2*c*e^2*(d + e*x) + b*c^2*e*(-15*d^2 + 11*d*e*x + 50*e^2*x^2) + c^3*(8*d^3 - 6*d^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)) - Sqrt[b/c]*(Sqrt[b/c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(315*b*c^2*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])","C",1
394,1,380,360,1.9933093,"\int \frac{\left(b x+c x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/Sqrt[d + e*x],x]","\frac{2 (x (b+c x))^{3/2} \left(b e x (b+c x) (d+e x) \left(b^2 e^2+b c e (8 e x-11 d)+c^2 \left(8 d^2-6 d e x+5 e^2 x^2\right)\right)+\sqrt{\frac{b}{c}} \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right)\right)\right)}{35 b c e^4 x^2 (b+c x)^2 \sqrt{d+e x}}","\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(b^2 e^2-3 c e x (2 c d-b e)-11 b c d e+8 c^2 d^2\right)}{35 c e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(-b^2 e^2-16 b c d e+16 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x}}{7 e}",1,"(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(b^2*e^2 + b*c*e*(-11*d + 8*e*x) + c^2*(8*d^2 - 6*d*e*x + 5*e^2*x^2)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) - (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(35*b*c*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])","C",1
395,1,340,309,1.6331777,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(3/2),x]","\frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-9 b c d e+8 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \left(b^3 e^2 (d+e x)+b^2 c e \left(-16 d^2-8 d e x+3 e^2 x^2\right)+b c^2 \left(16 d^3-8 d^2 e x-11 d e^2 x^2+3 e^3 x^3\right)+c^3 x \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)\right)}{5 c e^4 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{16 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left(b x+c x^2\right)^{3/2}}{e \sqrt{d+e x}}",1,"(2*(b^3*e^2*(d + e*x) + b^2*c*e*(-16*d^2 - 8*d*e*x + 3*e^2*x^2) + c^3*x*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + b*c^2*(16*d^3 - 8*d^2*e*x - 11*d*e^2*x^2 + 3*e^3*x^3)) + (2*I)*Sqrt[b/c]*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(8*c^2*d^2 - 9*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(5*c*e^4*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
396,1,279,298,1.1803972,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]","\frac{2 (x (b+c x))^{3/2} \left(\frac{e x (b+c x) \left(c \left(8 d^2+10 d e x+e^2 x^2\right)-b e (3 d+4 e x)\right)}{d+e x}-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (5 b e-8 c d) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+8 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+8 (b+c x) (d+e x) (b e-2 c d)\right)}{3 e^4 x^2 (b+c x)^2 \sqrt{d+e x}}","\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left(b x+c x^2\right)^{3/2}}{3 e (d+e x)^{3/2}}",1,"(2*(x*(b + c*x))^(3/2)*(8*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (e*x*(b + c*x)*(-(b*e*(3*d + 4*e*x)) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(d + e*x) + (8*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(-8*c*d + 5*b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])","C",1
397,1,369,354,1.1640289,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]","-\frac{2 (x (b+c x))^{3/2} \left(b e x (b+c x) \left(b^2 e^4 x^2-b c d e \left(7 d^2+16 d e x+11 e^2 x^2\right)+c^2 d^2 \left(8 d^2+18 d e x+11 e^2 x^2\right)\right)-c \sqrt{\frac{b}{c}} (d+e x)^2 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-9 b c d e+8 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)\right)\right)}{5 b d e^4 x^2 (b+c x)^2 (d+e x)^{5/2} (c d-b e)}","\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{5 d e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left(e x \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)+c d^2 (8 c d-7 b e)\right)}{5 d e^3 (d+e x)^{3/2} (c d-b e)}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{5 e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left(b x+c x^2\right)^{3/2}}{5 e (d+e x)^{5/2}}",1,"(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(b^2*e^4*x^2 - b*c*d*e*(7*d^2 + 16*d*e*x + 11*e^2*x^2) + c^2*d^2*(8*d^2 + 18*d*e*x + 11*e^2*x^2)) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^2*d^2 - 9*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(5*b*d*e^4*(c*d - b*e)*x^2*(b + c*x)^2*(d + e*x)^(5/2))","C",1
398,1,479,476,1.5236608,"\int \frac{\left(b x+c x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]","-\frac{2 (x (b+c x))^{3/2} \left(b e x (b+c x) \left(d (d+e x)^2 \left(b^2 e^2-19 b c d e+19 c^2 d^2\right) (c d-b e)-2 (d+e x)^3 \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right)+5 d^3 (c d-b e)^3-8 d^2 (d+e x) (2 c d-b e) (c d-b e)^2\right)+c \sqrt{\frac{b}{c}} (d+e x)^3 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right)\right)\right)}{35 b d^2 e^4 x^2 (b+c x)^2 (d+e x)^{7/2} (c d-b e)^2}","\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left(-b^2 e^2-16 b c d e+16 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{35 d e^4 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{35 d^2 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left(e x \left(b^2 e^2-14 b c d e+14 c^2 d^2\right)+d \left(-2 b^2 e^2-5 b c d e+8 c^2 d^2\right)\right)}{35 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac{4 \sqrt{b x+c x^2} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)}{35 d^2 e^3 \sqrt{d+e x} (c d-b e)^2}-\frac{2 \left(b x+c x^2\right)^{3/2}}{7 e (d+e x)^{7/2}}",1,"(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(5*d^3*(c*d - b*e)^3 - 8*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + d*(c*d - b*e)*(19*c^2*d^2 - 19*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 2*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(d + e*x)^3) + Sqrt[b/c]*c*(d + e*x)^3*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(35*b*d^2*e^4*(c*d - b*e)^2*x^2*(b + c*x)^2*(d + e*x)^(7/2))","C",1
399,1,663,666,3.3554907,"\int \sqrt{d+e x} \left(b x+c x^2\right)^{5/2} \, dx","Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^(5/2),x]","\frac{2 (x (b+c x))^{5/2} \left(b e x (b+c x) (d+e x) \left(24 b^5 e^5-b^4 c e^4 (17 d+18 e x)+b^3 c^2 e^3 \left(-22 d^2+12 d e x+15 e^2 x^2\right)+b^2 c^3 e^2 \left(303 d^3-218 d^2 e x+178 d e^2 x^2+1113 e^3 x^3\right)+b c^4 e \left(-368 d^4+272 d^3 e x-225 d^2 e^2 x^2+196 d e^3 x^3+1701 e^4 x^4\right)+c^5 \left(128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right)\right)+\sqrt{\frac{b}{c}} \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(48 b^6 e^6-64 b^5 c d e^5-25 b^4 c^2 d^2 e^4-70 b^3 c^3 d^3 e^3+383 b^2 c^4 d^4 e^2-400 b c^5 d^5 e+128 c^6 d^6\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right)\right)\right)}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt{d+e x}}","\frac{10 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} \left(-18 b^3 e^3-14 c e x \left(3 b^2 e^2-b c d e+c^2 d^2\right)+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right)}{9009 c^2 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left(24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left(-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right)-368 b c^4 d^4 e+128 c^5 d^5\right)}{9009 c^3 e^5}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left(b x+c x^2\right)^{5/2} (d+e x)^{3/2}}{13 e}-\frac{10 \left(b x+c x^2\right)^{5/2} \sqrt{d+e x} (2 c d-b e)}{143 c e}",1,"(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(24*b^5*e^5 - b^4*c*e^4*(17*d + 18*e*x) + b^3*c^2*e^3*(-22*d^2 + 12*d*e*x + 15*e^2*x^2) + b^2*c^3*e^2*(303*d^3 - 218*d^2*e*x + 178*d*e^2*x^2 + 1113*e^3*x^3) + b*c^4*e*(-368*d^4 + 272*d^3*e*x - 225*d^2*e^2*x^2 + 196*d*e^3*x^3 + 1701*e^4*x^4) + c^5*(128*d^5 - 96*d^4*e*x + 80*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*(b + c*x)*(d + e*x) - (2*I)*b*e*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(128*c^6*d^6 - 400*b*c^5*d^5*e + 383*b^2*c^4*d^4*e^2 - 70*b^3*c^3*d^3*e^3 - 25*b^4*c^2*d^2*e^4 - 64*b^5*c*d*e^5 + 48*b^6*e^6)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(9009*b*c^3*e^6*x^3*(b + c*x)^3*Sqrt[d + e*x])","C",1
400,1,557,537,2.8560518,"\int \frac{\left(b x+c x^2\right)^{5/2}}{\sqrt{d+e x}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/Sqrt[d + e*x],x]","\frac{2 (x (b+c x))^{5/2} \left(b e x (b+c x) (d+e x) \left(-4 b^4 e^4+b^3 c e^3 (3 e x-7 d)+b^2 c^2 e^2 \left(195 d^2-139 d e x+113 e^2 x^2\right)+b c^3 e \left(-304 d^3+224 d^2 e x-185 d e^2 x^2+161 e^3 x^3\right)+c^4 \left(128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right)\right)+\sqrt{\frac{b}{c}} \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^5 e^5-9 b^4 c d e^4-34 b^3 c^2 d^2 e^3+259 b^2 c^3 d^3 e^2-336 b c^4 d^4 e+128 c^5 d^5\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^5 e^5-13 b^4 c d e^4-41 b^3 c^2 d^2 e^3+454 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(8 b^5 e^5+13 b^4 c d e^4+41 b^3 c^2 d^2 e^3-454 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right)\right)\right)}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt{d+e x}}","\frac{10 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} \left(3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right)}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(-4 b^4 e^4-7 b^3 c d e^3-12 c e x (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)+195 b^2 c^2 d^2 e^2-304 b c^3 d^3 e+128 c^4 d^4\right)}{693 c^2 e^5}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(8 b^4 e^4+29 b^3 c d e^3+99 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left(b x+c x^2\right)^{5/2} \sqrt{d+e x}}{11 e}",1,"(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^4*e^4 + b^3*c*e^3*(-7*d + 3*e*x) + b^2*c^2*e^2*(195*d^2 - 139*d*e*x + 113*e^2*x^2) + b*c^3*e*(-304*d^3 + 224*d^2*e*x - 185*d*e^2*x^2 + 161*e^3*x^3) + c^4*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 - 70*d*e^3*x^3 + 63*e^4*x^4)) + Sqrt[b/c]*(Sqrt[b/c]*(-256*c^5*d^5 + 640*b*c^4*d^4*e - 454*b^2*c^3*d^3*e^2 + 41*b^3*c^2*d^2*e^3 + 13*b^4*c*d*e^4 + 8*b^5*e^5)*(b + c*x)*(d + e*x) - I*b*e*(256*c^5*d^5 - 640*b*c^4*d^4*e + 454*b^2*c^3*d^3*e^2 - 41*b^3*c^2*d^2*e^3 - 13*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(128*c^5*d^5 - 336*b*c^4*d^4*e + 259*b^2*c^3*d^3*e^2 - 34*b^3*c^2*d^2*e^3 - 9*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(693*b*c^2*e^6*x^3*(b + c*x)^3*Sqrt[d + e*x])","C",1
401,1,498,457,2.2834243,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^{3/2}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]","\frac{2 (x (b+c x))^{5/2} \left(-e \sqrt{x} (b+c x) \left(-b^3 e^3 (d+e x)+3 b^2 c e^2 \left(37 d^2+11 d e x-5 e^2 x^2\right)-b c^2 e \left(240 d^3+64 d^2 e x-31 d e^2 x^2+19 e^3 x^3\right)+c^3 \left(128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right)\right)+\frac{2 (b+c x) (d+e x) \left(-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right)}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^4 e^4+13 b^3 c d e^3-159 b^2 c^2 d^2 e^2+272 b c^3 d^3 e-128 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)\right)}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}}","-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left(-b^2 e^2-128 b c d e+128 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(-b^3 e^3-3 c e x \left(b^2 e^2-32 b c d e+32 c^2 d^2\right)+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right)}{63 c e^5}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{10 \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left(b x+c x^2\right)^{5/2}}{e \sqrt{d+e x}}",1,"(2*(x*(b + c*x))^(5/2)*((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) - e*Sqrt[x]*(b + c*x)*(-(b^3*e^3*(d + e*x)) + 3*b^2*c*e^2*(37*d^2 + 11*d*e*x - 5*e^2*x^2) - b*c^2*e*(240*d^3 + 64*d^2*e*x - 31*d*e^2*x^2 + 19*e^3*x^3) + c^3*(128*d^4 + 32*d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)) - (2*I)*Sqrt[b/c]*e*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*e*(-128*c^4*d^4 + 272*b*c^3*d^3*e - 159*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 + 2*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(63*c*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])","C",1
402,1,442,401,2.1570236,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(5/2),x]","\frac{2 (x (b+c x))^{5/2} \left(\frac{e \sqrt{x} (b+c x) \left(b^2 e^2 \left(51 d^2+67 d e x+9 e^2 x^2\right)+b c e \left(-176 d^3-224 d^2 e x-25 d e^2 x^2+9 e^3 x^3\right)+c^2 \left(128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right)\right)}{d+e x}-\frac{(b+c x) (d+e x) \left(-3 b^3 e^3+134 b^2 c d e^2-384 b c^2 d^2 e+256 c^3 d^3\right)}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-3 b^3 e^3+83 b^2 c d e^2-208 b c^2 d^2 e+128 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(3 b^3 e^3-134 b^2 c d e^2+384 b c^2 d^2 e-256 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)\right)}{21 e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}}","\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(27 b^2 e^2-128 b c d e+128 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(3 b^2 e^2-128 b c d e+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left(51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right)}{21 e^5}+\frac{10 \left(b x+c x^2\right)^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left(b x+c x^2\right)^{5/2}}{3 e (d+e x)^{3/2}}",1,"(2*(x*(b + c*x))^(5/2)*(-(((256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x))/(c*Sqrt[x])) + (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(51*d^2 + 67*d*e*x + 9*e^2*x^2) + b*c*e*(-176*d^3 - 224*d^2*e*x - 25*d*e^2*x^2 + 9*e^3*x^3) + c^2*(128*d^4 + 160*d^3*e*x + 16*d^2*e^2*x^2 - 6*d*e^3*x^3 + 3*e^4*x^4)))/(d + e*x) + I*Sqrt[b/c]*e*(-256*c^3*d^3 + 384*b*c^2*d^2*e - 134*b^2*c*d*e^2 + 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(21*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])","C",1
403,1,401,392,1.6037827,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^{7/2}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]","\frac{2 (x (b+c x))^{5/2} \left(\frac{2 (b+c x) (d+e x) \left(23 b^2 e^2-128 b c d e+128 c^2 d^2\right)}{\sqrt{x}}-i c e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(31 b^2 e^2-144 b c d e+128 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i c e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(23 b^2 e^2-128 b c d e+128 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-\frac{e \sqrt{x} (b+c x) \left(b^2 e^2 \left(15 d^2+35 d e x+23 e^2 x^2\right)-b c e \left(112 d^3+256 d^2 e x+161 d e^2 x^2+11 e^3 x^3\right)+c^2 \left(128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right)\right)}{(d+e x)^2}\right)}{15 e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}}","-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left(15 b^2 e^2-128 b c d e+128 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(23 b^2 e^2-128 b c d e+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} \left(15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right)}{15 e^5 \sqrt{d+e x}}+\frac{2 \left(b x+c x^2\right)^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^3 (d+e x)^{3/2}}-\frac{2 \left(b x+c x^2\right)^{5/2}}{5 e (d+e x)^{5/2}}",1,"(2*(x*(b + c*x))^(5/2)*((2*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2)*(b + c*x)*(d + e*x))/Sqrt[x] - (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(15*d^2 + 35*d*e*x + 23*e^2*x^2) - b*c*e*(112*d^3 + 256*d^2*e*x + 161*d*e^2*x^2 + 11*e^3*x^3) + c^2*(128*d^4 + 288*d^3*e*x + 176*d^2*e^2*x^2 + 10*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)^2 + (2*I)*Sqrt[b/c]*c*e*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(128*c^2*d^2 - 144*b*c*d*e + 31*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(15*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])","C",1
404,1,500,474,1.6365944,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^{9/2}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]","-\frac{2 (x (b+c x))^{5/2} \left(b e x (b+c x) \left(3 b^3 e^6 x^3-b^2 c d e^2 \left(51 d^3+169 d^2 e x+194 d e^2 x^2+85 e^3 x^3\right)+b c^2 d e \left(176 d^4+576 d^3 e x+649 d^2 e^2 x^2+265 d e^3 x^3+7 e^4 x^4\right)-c^3 d^2 \left(128 d^4+416 d^3 e x+464 d^2 e^2 x^2+186 d e^3 x^3+7 e^4 x^4\right)\right)+c \sqrt{\frac{b}{c}} (d+e x)^3 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-3 b^3 e^3+83 b^2 c d e^2-208 b c^2 d^2 e+128 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-3 b^3 e^3+134 b^2 c d e^2-384 b c^2 d^2 e+256 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-3 b^3 e^3+134 b^2 c d e^2-384 b c^2 d^2 e+256 c^3 d^3\right)\right)\right)}{21 b d e^6 x^3 (b+c x)^3 (d+e x)^{7/2} (c d-b e)}","\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left(27 b^2 e^2-128 b c d e+128 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{21 e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(3 b^2 e^2-128 b c d e+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{21 d e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 c \sqrt{b x+c x^2} \left(e x \left(3 b^2 e^2-32 b c d e+32 c^2 d^2\right)+d \left(51 b^2 e^2-176 b c d e+128 c^2 d^2\right)\right)}{21 d e^5 \sqrt{d+e x} (c d-b e)}-\frac{2 \left(b x+c x^2\right)^{3/2} \left(e x \left(3 b^2 e^2-22 b c d e+22 c^2 d^2\right)+c d^2 (16 c d-13 b e)\right)}{21 d e^3 (d+e x)^{5/2} (c d-b e)}-\frac{2 \left(b x+c x^2\right)^{5/2}}{7 e (d+e x)^{7/2}}",1,"(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(3*b^3*e^6*x^3 - b^2*c*d*e^2*(51*d^3 + 169*d^2*e*x + 194*d*e^2*x^2 + 85*e^3*x^3) - c^3*d^2*(128*d^4 + 416*d^3*e*x + 464*d^2*e^2*x^2 + 186*d*e^3*x^3 + 7*e^4*x^4) + b*c^2*d*e*(176*d^4 + 576*d^3*e*x + 649*d^2*e^2*x^2 + 265*d*e^3*x^3 + 7*e^4*x^4)) + Sqrt[b/c]*c*(d + e*x)^3*(Sqrt[b/c]*(256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(21*b*d*e^6*(c*d - b*e)*x^3*(b + c*x)^3*(d + e*x)^(7/2))","C",1
405,1,610,570,3.5130812,"\int \frac{\left(b x+c x^2\right)^{5/2}}{(d+e x)^{11/2}} \, dx","Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]","-\frac{2 (x (b+c x))^{5/2} \left(b e x (b+c x) \left(d^2 (d+e x)^2 \left(15 b^2 e^2-88 b c d e+88 c^2 d^2\right) (c d-b e)^2-19 d^3 (d+e x) \left(b^2 e^2-3 b c d e+2 c^2 d^2\right) (c d-b e)^2-d (d+e x)^3 \left(-b^3 e^3+63 b^2 c d e^2-183 b c^2 d^2 e+122 c^3 d^3\right) (c d-b e)+(d+e x)^4 \left(-2 b^4 e^4-14 b^3 c d e^3+207 b^2 c^2 d^2 e^2-386 b c^3 d^3 e+193 c^4 d^4\right)+7 d^4 (c d-b e)^4\right)-c \sqrt{\frac{b}{c}} (d+e x)^4 \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-2 b^4 e^4-13 b^3 c d e^3+159 b^2 c^2 d^2 e^2-272 b c^3 d^3 e+128 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right)\right)\right)}{63 b d^2 e^6 x^3 (b+c x)^3 (d+e x)^{9/2} (c d-b e)^2}","-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left(-b^2 e^2-128 b c d e+128 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{63 d e^6 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \left(b x+c x^2\right)^{3/2} \left(e x \left(3 b^2 e^2-26 b c d e+26 c^2 d^2\right)+d \left(-2 b^2 e^2-11 b c d e+16 c^2 d^2\right)\right)}{63 d e^3 (d+e x)^{7/2} (c d-b e)}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{63 d^2 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left(c d^2 \left(-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right)+e x \left(-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right)\right)}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \left(b x+c x^2\right)^{5/2}}{9 e (d+e x)^{9/2}}",1,"(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(7*d^4*(c*d - b*e)^4 - 19*d^3*(c*d - b*e)^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*(d + e*x) + d^2*(c*d - b*e)^2*(88*c^2*d^2 - 88*b*c*d*e + 15*b^2*e^2)*(d + e*x)^2 - d*(c*d - b*e)*(122*c^3*d^3 - 183*b*c^2*d^2*e + 63*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^3 + (193*c^4*d^4 - 386*b*c^3*d^3*e + 207*b^2*c^2*d^2*e^2 - 14*b^3*c*d*e^3 - 2*b^4*e^4)*(d + e*x)^4) - Sqrt[b/c]*c*(d + e*x)^4*(-2*Sqrt[b/c]*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*(b + c*x)*(d + e*x) + (2*I)*b*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(128*c^4*d^4 - 272*b*c^3*d^3*e + 159*b^2*c^2*d^2*e^2 - 13*b^3*c*d*e^3 - 2*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(63*b*d^2*e^6*(c*d - b*e)^2*x^3*(b + c*x)^3*(d + e*x)^(9/2))","C",1
406,1,388,379,2.2608508,"\int \frac{(d+e x)^{7/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{x} \left(e \sqrt{x} (b+c x) (d+e x) \left(24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left(122 d^2+66 d e x+15 e^2 x^2\right)\right)+\frac{8 (b+c x) (d+e x) \left(-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right)}{c \sqrt{x}}+8 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(48 b^4 e^4-208 b^3 c d e^3+353 b^2 c^2 d^2 e^2-298 b c^3 d^3 e+105 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)}{b}\right)}{105 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(24 b^2 e^2-71 b c d e+71 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(6 b^2 e^2-11 b c d e+11 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left(24 b^2 e^2-71 b c d e+71 c^2 d^2\right)}{105 c^3}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c}",1,"(2*Sqrt[x]*((8*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) + e*Sqrt[x]*(b + c*x)*(d + e*x)*(24*b^2*e^2 - b*c*e*(89*d + 18*e*x) + c^2*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + (8*I)*Sqrt[b/c]*e*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (I*Sqrt[b/c]*(105*c^4*d^4 - 298*b*c^3*d^3*e + 353*b^2*c^2*d^2*e^2 - 208*b^3*c*d*e^3 + 48*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(105*c^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
407,1,314,303,1.2432229,"\int \frac{(d+e x)^{5/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[b*x + c*x^2],x]","\frac{2 \sqrt{x} \left(\frac{(b+c x) (d+e x) \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right)}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(8 b^3 e^3-27 b^2 c d e^2+34 b c^2 d^2 e-15 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)}{b}+e \sqrt{x} (b+c x) (d+e x) (-4 b e+11 c d+3 c e x)\right)}{15 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c}",1,"(2*Sqrt[x]*(((23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) + e*Sqrt[x]*(b + c*x)*(d + e*x)*(11*c*d - 4*b*e + 3*c*e*x) + I*Sqrt[b/c]*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (I*Sqrt[b/c]*(-15*c^3*d^3 + 34*b*c^2*d^2*e - 27*b^2*c*d*e^2 + 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(15*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
408,1,246,241,1.005273,"\int \frac{(d+e x)^{3/2}}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[b*x + c*x^2],x]","\frac{2 i c x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-5 b c d e+3 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-4 i b c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 b (b+c x) (d+e x) (2 b e-c (4 d+e x))}{3 b c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c}",1,"(-2*b*(b + c*x)*(d + e*x)*(2*b*e - c*(4*d + e*x)) - (4*I)*b*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (2*I)*Sqrt[b/c]*c*(3*c^2*d^2 - 5*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(3*b*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
409,1,121,94,0.5366662,"\int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[b*x + c*x^2],x]","-\frac{2 \sqrt{x} \left(\frac{b}{x}+c\right) \sqrt{d+e x} \left(\frac{d \sqrt{\frac{d}{e x}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right)|\frac{b e}{c d}\right)}{\sqrt{-\frac{d}{e}} \sqrt{\frac{b}{c x}+1} \left(\frac{d}{x}+e\right)}-\sqrt{x}\right)}{c \sqrt{x (b+c x)}}","\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}",1,"(-2*(c + b/x)*Sqrt[x]*Sqrt[d + e*x]*(-Sqrt[x] + (d*Sqrt[1 + d/(e*x)]*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (b*e)/(c*d)])/(Sqrt[-(d/e)]*Sqrt[1 + b/(c*x)]*(e + d/x))))/(c*Sqrt[x*(b + c*x)])","A",1
410,1,94,94,0.116918,"\int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[b*x + c*x^2]),x]","-\frac{2 x^{3/2} \sqrt{\frac{\frac{b}{x}+c}{c}} \sqrt{\frac{\frac{d}{x}+e}{e}} F\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)}{\sqrt{-\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}",1,"(-2*Sqrt[(c + b/x)/c]*Sqrt[(e + d/x)/e]*x^(3/2)*EllipticF[ArcSin[Sqrt[-(b/c)]/Sqrt[x]], (c*d)/(b*e)])/(Sqrt[-(b/c)]*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","A",1
411,1,127,146,0.161801,"\int \frac{1}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[b*x + c*x^2]),x]","\frac{2 \sqrt{x (b+c x)} \left(e \sqrt{x} \sqrt{-\frac{d}{e}} \sqrt{\frac{d}{e x}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right)|\frac{b e}{c d}\right)+d \sqrt{\frac{b}{c x}+1}\right)}{d x \sqrt{\frac{b}{c x}+1} \sqrt{d+e x} (c d-b e)}","\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)}",1,"(2*Sqrt[x*(b + c*x)]*(d*Sqrt[1 + b/(c*x)] + Sqrt[-(d/e)]*e*Sqrt[1 + d/(e*x)]*Sqrt[x]*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (b*e)/(c*d)]))/(d*(c*d - b*e)*Sqrt[1 + b/(c*x)]*x*Sqrt[d + e*x])","A",1
412,1,290,317,1.1510696,"\int \frac{1}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[b*x + c*x^2]),x]","-\frac{2 \left(-b e x (b+c x) (b e (3 d+2 e x)-c d (5 d+4 e x))-c \sqrt{\frac{b}{c}} (d+e x) \left(i x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-5 b c d e+3 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 c d-b e) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) (b e-2 c d)\right)\right)}{3 b d^2 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^2}","-\frac{4 e \sqrt{b x+c x^2} (2 c d-b e)}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}",1,"(-2*(-(b*e*x*(b + c*x)*(b*e*(3*d + 2*e*x) - c*d*(5*d + 4*e*x))) - Sqrt[b/c]*c*(d + e*x)*(-2*Sqrt[b/c]*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (2*I)*b*e*(2*c*d - b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*(3*c^2*d^2 - 5*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b*d^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))","C",1
413,1,381,403,1.2468876,"\int \frac{1}{(d+e x)^{7/2} \sqrt{b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(7/2)*Sqrt[b*x + c*x^2]),x]","-\frac{2 \left(b e x (b+c x) \left((d+e x)^2 \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right)+3 d^2 (c d-b e)^2+4 d (d+e x) (2 c d-b e) (c d-b e)\right)-c \sqrt{\frac{b}{c}} (d+e x)^2 \left(i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right)+i x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+27 b^2 c d e^2-34 b c^2 d^2 e+15 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)\right)\right)}{15 b d^3 \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^3}","-\frac{2 e \sqrt{b x+c x^2} \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right)}{15 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(8 b^2 e^2-23 b c d e+23 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{8 e \sqrt{b x+c x^2} (2 c d-b e)}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{8 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{15 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{5 d (d+e x)^{5/2} (c d-b e)}",1,"(-2*(b*e*x*(b + c*x)*(3*d^2*(c*d - b*e)^2 + 4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x) + (23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*(15*c^3*d^3 - 34*b*c^2*d^2*e + 27*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))","C",1
414,1,360,395,0.8898174,"\int \frac{(d+e x)^{7/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x]","-\frac{2 \left(b (d+e x) \left(-b^2 e^3 x (b+c x)+3 c^2 d^3 (b+c x)+3 x (c d-b e)^3\right)-\sqrt{\frac{b}{c}} \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right)\right)\right)}{3 b^3 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{4 e \sqrt{b x+c x^2} \sqrt{d+e x} \left(2 b^2 e^2-3 b c d e+3 c^2 d^2\right)}{3 b^2 c^2}-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(2 b^2 e^2-3 b c d e+3 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(8 b^2 e^2-3 b c d e+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c}",1,"(-2*(b*(d + e*x)*(3*(c*d - b*e)^3*x + 3*c^2*d^3*(b + c*x) - b^2*e^3*x*(b + c*x)) - Sqrt[b/c]*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
415,1,262,310,1.1117075,"\int \frac{(d+e x)^{5/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^(3/2),x]","\frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-3 b c d e+c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+4 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 b (d+e x) \left(2 b^2 e^2+b c e (e x-2 d)+c^2 d^2\right)}{b^2 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{4 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{b^2 c}-\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}",1,"(2*b*(d + e*x)*(c^2*d^2 + 2*b^2*e^2 + b*c*e*(-2*d + e*x)) + (4*I)*Sqrt[b/c]*c*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b^2*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
416,1,210,249,0.4731175,"\int \frac{(d+e x)^{3/2}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^(3/2),x]","\frac{2 (c d-b e) \left(b (d+e x)-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)\right)-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)}{b^2 c \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}",1,"((-2*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + 2*(c*d - b*e)*(b*(d + e*x) - I*Sqrt[b/c]*c*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(b^2*c*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
417,1,186,231,0.3906058,"\int \frac{\sqrt{d+e x}}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(b*x + c*x^2)^(3/2),x]","\frac{-2 i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+4 i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \sqrt{\frac{b}{c}} (d+e x)}{b \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}}","-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}",1,"(2*Sqrt[b/c]*(d + e*x) + (4*I)*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b*Sqrt[b/c]*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
418,1,220,274,0.5281121,"\int \frac{1}{\sqrt{d+e x} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)),x]","\frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-2 b c d (d+e x)}{b^2 d \sqrt{x (b+c x)} \sqrt{d+e x} (b e-c d)}","-\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}",1,"(-2*b*c*d*(d + e*x) + (2*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b^2*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
419,1,266,370,0.7533603,"\int \frac{1}{(d+e x)^{3/2} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(3/2)),x]","\frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^2 e^2-3 b c d e+c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+4 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 b d \left(b^2 e^2+b c e^2 x+c^2 d (d+e x)\right)}{b^2 d^2 \sqrt{x (b+c x)} \sqrt{d+e x} (c d-b e)^2}","-\frac{4 e \sqrt{b x+c x^2} \left(b^2 e^2-b c d e+c^2 d^2\right)}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-b c d e+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}",1,"(2*b*d*(b^2*e^2 + b*c*e^2*x + c^2*d*(d + e*x)) + (4*I)*Sqrt[b/c]*c*e*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(b^2*d^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
420,1,420,478,1.1480838,"\int \frac{1}{(d+e x)^{5/2} \left(b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]","-\frac{2 \left(b \left(b^2 d e^3 x (b+c x) (c d-b e)-5 b^2 e^3 x (b+c x) (d+e x) (b e-2 c d)+3 (b+c x) (d+e x)^2 (c d-b e)^3+3 c^4 d^3 x (d+e x)^2\right)-c \sqrt{\frac{b}{c}} (d+e x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right)\right)\right)}{3 b^3 d^3 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^3}","-\frac{4 e \sqrt{b x+c x^2} \left(2 b^2 e^2-3 b c d e+3 c^2 d^2\right)}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left(2 b^2 e^2-3 b c d e+3 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left(8 b^2 e^2-3 b c d e+3 c^2 d^2\right)}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(8 b^2 e^2-3 b c d e+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)}",1,"(-2*(b*(b^2*d*e^3*(c*d - b*e)*x*(b + c*x) - 5*b^2*e^3*(-2*c*d + b*e)*x*(b + c*x)*(d + e*x) + 3*c^4*d^3*x*(d + e*x)^2 + 3*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))","C",1
421,1,451,470,2.6074989,"\int \frac{(d+e x)^{9/2}}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(b (d+e x) \left(-b c^2 d^4 (b+c x)^2+c^2 d^3 x (b+c x)^2 (8 c d-13 b e)+b x^2 (c d-b e)^4+x^2 (b+c x) (c d-b e)^3 (5 b e+8 c d)\right)-x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right)\right)\right)}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt{d+e x}}","\frac{8 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left(-b^2 e^2-2 b c d e+2 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 (d+e x)^{7/2} (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}-\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} \left(b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right)}{3 b^4 c^2}+\frac{2 (d+e x)^{3/2} \left(x (2 c d-b e) \left(-3 b^2 e^2-8 b c d e+8 c^2 d^2\right)+b c d^2 (8 c d-11 b e)\right)}{3 b^4 c \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}",1,"(2*(b*(d + e*x)*(b*(c*d - b*e)^4*x^2 + (c*d - b*e)^3*(8*c*d + 5*b*e)*x^2*(b + c*x) - b*c^2*d^4*(b + c*x)^2 + c^2*d^3*(8*c*d - 13*b*e)*x*(b + c*x)^2) - Sqrt[b/c]*x*(b + c*x)*(Sqrt[b/c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*c^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
422,1,405,383,1.1428512,"\int \frac{(d+e x)^{7/2}}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(b (d+e x) \left(-b c d^3 (b+c x)^2+2 c d^2 x (b+c x)^2 (4 c d-5 b e)+b x^2 (c d-b e)^3+2 x^2 (b+c x) (c d-b e)^2 (b e+4 c d)\right)-x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right)\right)\right)}{3 b^5 c (x (b+c x))^{3/2} \sqrt{d+e x}}","\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left(-b^2 e^2-16 b c d e+16 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}+\frac{2 \sqrt{d+e x} \left(x (2 c d-b e) \left(-b^2 e^2-8 b c d e+8 c^2 d^2\right)+b c d^2 (8 c d-9 b e)\right)}{3 b^4 c \sqrt{b x+c x^2}}",1,"(2*(b*(d + e*x)*(b*(c*d - b*e)^3*x^2 + 2*(c*d - b*e)^2*(4*c*d + b*e)*x^2*(b + c*x) - b*c*d^3*(b + c*x)^2 + 2*c*d^2*(4*c*d - 5*b*e)*x*(b + c*x)^2) - Sqrt[b/c]*x*(b + c*x)*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*c*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
423,1,353,343,1.0394292,"\int \frac{(d+e x)^{5/2}}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(b (d+e x) \left(-\left(b^3 \left(d^2+7 d e x-2 e^2 x^2\right)\right)+b^2 c x \left(6 d^2-25 d e x+e^2 x^2\right)+8 b c^2 d x^2 (3 d-2 e x)+16 c^3 d^2 x^3\right)-x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-9 b c d e+8 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)\right)\right)}{3 b^5 (x (b+c x))^{3/2} \sqrt{d+e x}}","-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}+\frac{2 \sqrt{d+e x} \left(x \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)+b d (8 c d-7 b e)\right)}{3 b^4 \sqrt{b x+c x^2}}+\frac{16 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}",1,"(2*(b*(d + e*x)*(16*c^3*d^2*x^3 + 8*b*c^2*d*x^2*(3*d - 2*e*x) - b^3*(d^2 + 7*d*e*x - 2*e^2*x^2) + b^2*c*x*(6*d^2 - 25*d*e*x + e^2*x^2)) - Sqrt[b/c]*x*(b + c*x)*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^2*d^2 - 9*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
424,1,290,344,1.1042504,"\int \frac{(d+e x)^{3/2}}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(\frac{(d+e x) \left(-\left(b^3 (d+4 e x)\right)+b^2 c x (6 d-13 e x)-8 b c^2 x^2 (e x-3 d)+16 c^3 d x^3\right)}{b+c x}+i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (8 c d-5 b e) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+8 i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)-8 x (b+c x) (d+e x) (2 c d-b e)\right)}{3 b^4 x \sqrt{x (b+c x)} \sqrt{d+e x}}","\frac{2 \sqrt{d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt{b x+c x^2} (c d-b e)}-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left(b x+c x^2\right)^{3/2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}",1,"(2*(-8*(2*c*d - b*e)*x*(b + c*x)*(d + e*x) + ((d + e*x)*(16*c^3*d*x^3 + b^2*c*x*(6*d - 13*e*x) - 8*b*c^2*x^2*(-3*d + e*x) - b^3*(d + 4*e*x)))/(b + c*x) + (8*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*c*e*(8*c*d - 5*b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*b^4*x*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])","C",1
425,1,375,359,0.9110579,"\int \frac{\sqrt{d+e x}}{\left(b x+c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[d + e*x]/(b*x + c*x^2)^(5/2),x]","\frac{2 \left(b (d+e x) \left(b c^2 d x^2 (c d-b e)+c^2 d x^2 (b+c x) (8 c d-7 b e)+x (b+c x)^2 (c d-b e) (8 c d-b e)+b d (b+c x)^2 (b e-c d)\right)-c x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-9 b c d e+8 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)\right)\right)}{3 b^5 d (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)}","-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(b^2 e^2-16 b c d e+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 (b+2 c x) \sqrt{d+e x}}{3 b^2 \left(b x+c x^2\right)^{3/2}}+\frac{2 \sqrt{d+e x} \left(c x \left(b^2 e^2-16 b c d e+16 c^2 d^2\right)+b (c d-b e) (8 c d-b e)\right)}{3 b^4 d \sqrt{b x+c x^2} (c d-b e)}+\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}",1,"(2*(b*(d + e*x)*(b*c^2*d*(c*d - b*e)*x^2 + c^2*d*(8*c*d - 7*b*e)*x^2*(b + c*x) + b*d*(-(c*d) + b*e)*(b + c*x)^2 + (c*d - b*e)*(8*c*d - b*e)*x*(b + c*x)^2) - Sqrt[b/c]*c*x*(b + c*x)*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^2*d^2 - 9*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d*(c*d - b*e)*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
426,1,429,451,1.1802789,"\int \frac{1}{\sqrt{d+e x} \left(b x+c x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]","\frac{2 \left(b (d+e x) \left(b c^3 d^2 x^2 (c d-b e)+2 c^3 d^2 x^2 (b+c x) (4 c d-5 b e)-b d (b+c x)^2 (c d-b e)^2+2 x (b+c x)^2 (c d-b e)^2 (b e+4 c d)\right)-c x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right)\right)\right)}{3 b^5 d^2 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^2}","\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left(-b^2 e^2-16 b c d e+16 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left(b x+c x^2\right)^{3/2} (c d-b e)}+\frac{2 \sqrt{d+e x} \left(2 c x (2 c d-b e) \left(-b^2 e^2-4 b c d e+4 c^2 d^2\right)+b (c d-b e) \left(-2 b^2 e^2-5 b c d e+8 c^2 d^2\right)\right)}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}",1,"(2*(b*(d + e*x)*(b*c^3*d^2*(c*d - b*e)*x^2 + 2*c^3*d^2*(4*c*d - 5*b*e)*x^2*(b + c*x) - b*d*(c*d - b*e)^2*(b + c*x)^2 + 2*(c*d - b*e)^2*(4*c*d + b*e)*x*(b + c*x)^2) - Sqrt[b/c]*c*x*(b + c*x)*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d^2*(c*d - b*e)^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
427,1,504,567,1.6624961,"\int \frac{1}{(d+e x)^{3/2} \left(b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]","-\frac{2 \left(b \left(3 b^4 e^5 x^2 (b+c x)^2+b c^4 d^3 x^2 (d+e x) (b e-c d)-c^4 d^3 x^2 (b+c x) (d+e x) (8 c d-13 b e)+b d (b+c x)^2 (d+e x) (c d-b e)^3-x (b+c x)^2 (d+e x) (c d-b e)^3 (5 b e+8 c d)\right)+c x \sqrt{\frac{b}{c}} (b+c x) \left(-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right)|\frac{c d}{b e}\right)+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right)\right)\right)}{3 b^5 d^3 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^3}","\frac{8 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left(-b^2 e^2-2 b c d e+2 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left(b x+c x^2\right)^{3/2} \sqrt{d+e x} (c d-b e)}+\frac{2 \left(4 c x \left(b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right)+b (c d-b e) \left(-4 b^2 e^2-3 b c d e+8 c^2 d^2\right)\right)}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 e \sqrt{b x+c x^2} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right)}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left(-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right)|\frac{b e}{c d}\right)}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}",1,"(-2*(b*(3*b^4*e^5*x^2*(b + c*x)^2 + b*c^4*d^3*(-(c*d) + b*e)*x^2*(d + e*x) - c^4*d^3*(8*c*d - 13*b*e)*x^2*(b + c*x)*(d + e*x) + b*d*(c*d - b*e)^3*(b + c*x)^2*(d + e*x) - (c*d - b*e)^3*(8*c*d + 5*b*e)*x*(b + c*x)^2*(d + e*x)) + Sqrt[b/c]*c*x*(b + c*x)*(Sqrt[b/c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d^3*(c*d - b*e)^3*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])","C",1
428,1,117,51,0.3100959,"\int \frac{\sqrt{d+e x}}{\sqrt{2 x-3 x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[2*x - 3*x^2],x]","\frac{2 (3 x-2) \sqrt{-\frac{d}{e}} (d+e x)-2 d \sqrt{9-\frac{6}{x}} x^{3/2} \sqrt{\frac{d}{e x}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right)|-\frac{2 e}{3 d}\right)}{3 \sqrt{-x (3 x-2)} \sqrt{-\frac{d}{e}} \sqrt{d+e x}}","\frac{2 \sqrt{d+e x} E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} \sqrt{x}\right)|-\frac{2 e}{3 d}\right)}{\sqrt{3} \sqrt{\frac{e x}{d}+1}}",1,"(2*Sqrt[-(d/e)]*(-2 + 3*x)*(d + e*x) - 2*d*Sqrt[9 - 6/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (-2*e)/(3*d)])/(3*Sqrt[-(d/e)]*Sqrt[-(x*(-2 + 3*x))]*Sqrt[d + e*x])","B",1
429,1,76,51,0.1120658,"\int \frac{1}{\sqrt{d+e x} \sqrt{2 x-3 x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[2*x - 3*x^2]),x]","-\frac{\sqrt{6-\frac{4}{x}} x^{3/2} \sqrt{\frac{d}{e x}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right)|-\frac{3 d}{2 e}\right)}{\sqrt{-x (3 x-2)} \sqrt{d+e x}}","\frac{2 \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} \sqrt{x}\right)|-\frac{2 e}{3 d}\right)}{\sqrt{3} \sqrt{d+e x}}",1,"-((Sqrt[6 - 4/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[ArcSin[Sqrt[2/3]/Sqrt[x]], (-3*d)/(2*e)])/(Sqrt[-(x*(-2 + 3*x))]*Sqrt[d + e*x]))","A",1
430,1,117,53,0.2139765,"\int \frac{\sqrt{d+e x}}{\sqrt{-2 x-3 x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],x]","\frac{2 (3 x+2) \sqrt{-\frac{d}{e}} (d+e x)-2 d \sqrt{\frac{6}{x}+9} x^{3/2} \sqrt{\frac{d}{e x}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right)|\frac{2 e}{3 d}\right)}{3 \sqrt{-x (3 x+2)} \sqrt{-\frac{d}{e}} \sqrt{d+e x}}","-\frac{2 \sqrt{d+e x} E\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} \sqrt{-x}\right)|\frac{2 e}{3 d}\right)}{\sqrt{3} \sqrt{\frac{e x}{d}+1}}",1,"(2*Sqrt[-(d/e)]*(2 + 3*x)*(d + e*x) - 2*d*Sqrt[9 + 6/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (2*e)/(3*d)])/(3*Sqrt[-(d/e)]*Sqrt[-(x*(2 + 3*x))]*Sqrt[d + e*x])","B",1
431,1,82,53,0.1040538,"\int \frac{1}{\sqrt{d+e x} \sqrt{-2 x-3 x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[-2*x - 3*x^2]),x]","\frac{i \sqrt{\frac{4}{x}+6} x^{3/2} \sqrt{\frac{d}{e x}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right)|\frac{3 d}{2 e}\right)}{\sqrt{-x (3 x+2)} \sqrt{d+e x}}","-\frac{2 \sqrt{\frac{e x}{d}+1} F\left(\sin ^{-1}\left(\sqrt{\frac{3}{2}} \sqrt{-x}\right)|\frac{2 e}{3 d}\right)}{\sqrt{3} \sqrt{d+e x}}",1,"(I*Sqrt[6 + 4/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], (3*d)/(2*e)])/(Sqrt[-(x*(2 + 3*x))]*Sqrt[d + e*x])","C",1
432,1,66,12,0.0279171,"\int \frac{\sqrt{1-x}}{\sqrt{-x} \sqrt{1+x}} \, dx","Integrate[Sqrt[1 - x]/(Sqrt[-x]*Sqrt[1 + x]),x]","-\frac{2 x \sqrt{1-x^2} \left(x \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right)-3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right)\right)}{3 \sqrt{1-x} \sqrt{-x (x+1)}}","-2 E\left(\left.\sin ^{-1}\left(\sqrt{-x}\right)\right|-1\right)",1,"(-2*x*Sqrt[1 - x^2]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x*Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/(3*Sqrt[1 - x]*Sqrt[-(x*(1 + x))])","C",1
433,1,66,12,0.0114003,"\int \frac{\sqrt{1-x}}{\sqrt{-x-x^2}} \, dx","Integrate[Sqrt[1 - x]/Sqrt[-x - x^2],x]","-\frac{2 x \sqrt{1-x^2} \left(x \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right)-3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right)\right)}{3 \sqrt{1-x} \sqrt{-x (x+1)}}","-2 E\left(\left.\sin ^{-1}\left(\sqrt{-x}\right)\right|-1\right)",1,"(-2*x*Sqrt[1 - x^2]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x*Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/(3*Sqrt[1 - x]*Sqrt[-(x*(1 + x))])","C",1
434,1,70,95,0.0575454,"\int (d+e x)^m \left(c d x+c e x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(c*d*x + c*e*x^2)^3,x]","\frac{c^3 (d+e x)^{m+4} \left(-\frac{d^3}{m+4}+\frac{3 d^2 (d+e x)}{m+5}-\frac{3 d (d+e x)^2}{m+6}+\frac{(d+e x)^3}{m+7}\right)}{e^4}","-\frac{c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac{3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^4 (m+7)}",1,"(c^3*(d + e*x)^(4 + m)*(-(d^3/(4 + m)) + (3*d^2*(d + e*x))/(5 + m) - (3*d*(d + e*x)^2)/(6 + m) + (d + e*x)^3/(7 + m)))/e^4","A",1
435,1,60,69,0.0324232,"\int (d+e x)^m \left(c d x+c e x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(c*d*x + c*e*x^2)^2,x]","\frac{c^2 (d+e x)^{m+3} \left(2 d^2-2 d e (m+3) x+e^2 \left(m^2+7 m+12\right) x^2\right)}{e^3 (m+3) (m+4) (m+5)}","\frac{c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^3 (m+5)}",1,"(c^2*(d + e*x)^(3 + m)*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))/(e^3*(3 + m)*(4 + m)*(5 + m))","A",1
436,1,34,41,0.019918,"\int (d+e x)^m \left(c d x+c e x^2\right) \, dx","Integrate[(d + e*x)^m*(c*d*x + c*e*x^2),x]","\frac{c (d+e x)^{m+2} (e (m+2) x-d)}{e^2 (m+2) (m+3)}","\frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)}",1,"(c*(d + e*x)^(2 + m)*(-d + e*(2 + m)*x))/(e^2*(2 + m)*(3 + m))","A",1
437,1,17,18,0.0096805,"\int (d+e x)^m \, dx","Integrate[(d + e*x)^m,x]","\frac{(d+e x)^{m+1}}{e m+e}","\frac{(d+e x)^{m+1}}{e (m+1)}",1,"(d + e*x)^(1 + m)/(e + e*m)","A",1
438,1,32,32,0.0078676,"\int \frac{(d+e x)^m}{c d x+c e x^2} \, dx","Integrate[(d + e*x)^m/(c*d*x + c*e*x^2),x]","-\frac{(d+e x)^m \, _2F_1\left(1,m;m+1;\frac{e x}{d}+1\right)}{c d m}","-\frac{(d+e x)^m \, _2F_1\left(1,m;m+1;\frac{e x}{d}+1\right)}{c d m}",1,"-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, 1 + (e*x)/d])/(c*d*m))","A",1
439,1,36,39,0.0124531,"\int \frac{(d+e x)^m}{\left(c d x+c e x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(c*d*x + c*e*x^2)^2,x]","\frac{e (d+e x)^{m-1} \, _2F_1\left(2,m-1;m;\frac{e x}{d}+1\right)}{c^2 d^2 (m-1)}","-\frac{e (d+e x)^{m-1} \, _2F_1\left(2,m-1;m;\frac{e x}{d}+1\right)}{c^2 d^2 (1-m)}",1,"(e*(d + e*x)^(-1 + m)*Hypergeometric2F1[2, -1 + m, m, 1 + (e*x)/d])/(c^2*d^2*(-1 + m))","A",1
440,1,236,267,0.1910693,"\int (d+e x)^m \left(b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(b*x + c*x^2)^3,x]","\frac{(d+e x)^{m+1} \left(\frac{3 c (d+e x)^4 \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{m+5}-\frac{(d+e x)^3 (2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)}{m+4}+\frac{3 d (d+e x)^2 (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right)}{m+3}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{m+6}+\frac{d^3 (c d-b e)^3}{m+1}-\frac{3 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)}{m+2}+\frac{c^3 (d+e x)^6}{m+7}\right)}{e^7}","\frac{3 d (c d-b e) \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{(2 c d-b e) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c \left(b^2 e^2-5 b c d e+5 c^2 d^2\right) (d+e x)^{m+5}}{e^7 (m+5)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2*c*d - b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)))/e^7","A",1
441,1,138,159,0.0915695,"\int (d+e x)^m \left(b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(b*x + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{(d+e x)^2 \left(b^2 e^2-6 b c d e+6 c^2 d^2\right)}{m+3}+\frac{d^2 (c d-b e)^2}{m+1}-\frac{2 c (d+e x)^3 (2 c d-b e)}{m+4}-\frac{2 d (d+e x) (c d-b e) (2 c d-b e)}{m+2}+\frac{c^2 (d+e x)^4}{m+5}\right)}{e^5}","\frac{\left(b^2 e^2-6 b c d e+6 c^2 d^2\right) (d+e x)^{m+3}}{e^5 (m+3)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)}",1,"((d + e*x)^(1 + m)*((d^2*(c*d - b*e)^2)/(1 + m) - (2*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x))/(2 + m) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - (2*c*(2*c*d - b*e)*(d + e*x)^3)/(4 + m) + (c^2*(d + e*x)^4)/(5 + m)))/e^5","A",1
442,1,75,75,0.0524495,"\int (d+e x)^m \left(b x+c x^2\right) \, dx","Integrate[(d + e*x)^m*(b*x + c*x^2),x]","\frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)}","\frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)}",1,"(d*(c*d - b*e)*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d + e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))","A",1
443,1,86,93,0.0262855,"\int \frac{(d+e x)^m}{b x+c x^2} \, dx","Integrate[(d + e*x)^m/(b*x + c*x^2),x]","-\frac{(d+e x)^{m+1} \left(c d \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)+(b e-c d) \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)\right)}{b d (m+1) (b e-c d)}","\frac{c (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)}{b (m+1) (c d-b e)}-\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)}{b d (m+1)}",1,"-(((d + e*x)^(1 + m)*(c*d*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d]))/(b*d*(-(c*d) + b*e)*(1 + m)))","A",1
444,1,174,180,0.132174,"\int \frac{(d+e x)^m}{\left(b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(b*x + c*x^2)^2,x]","-\frac{(d+e x)^{m+1} \left(b^2 d (m+1) (c d-b e)^2+x (b+c x) \left(c^2 d^2 (b e (m-2)+2 c d) \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)-(c d-b e)^2 (2 c d-b e m) \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)\right)+b c d (m+1) x (b e-2 c d) (b e-c d)\right)}{b^3 d^2 (m+1) x (b+c x) (c d-b e)^2}","-\frac{c^2 (d+e x)^{m+1} (2 c d-b e (2-m)) \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)}{b^3 (m+1) (c d-b e)^2}+\frac{(d+e x)^{m+1} (2 c d-b e m) \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)}{b^3 d^2 (m+1)}-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left(b x+c x^2\right) (c d-b e)}",1,"-(((d + e*x)^(1 + m)*(b^2*d*(c*d - b*e)^2*(1 + m) + b*c*d*(-2*c*d + b*e)*(-(c*d) + b*e)*(1 + m)*x + x*(b + c*x)*(c^2*d^2*(2*c*d + b*e*(-2 + m))*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)^2*(2*c*d - b*e*m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d])))/(b^3*d^2*(c*d - b*e)^2*(1 + m)*x*(b + c*x)))","A",1
445,1,337,350,0.4829165,"\int \frac{(d+e x)^m}{\left(b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^m/(b*x + c*x^2)^3,x]","-\frac{(d+e x)^{m+1} \left(2 b^4 d^2 (m+1) (c d-b e)^3-2 b^3 d (m+1) x (c d-b e)^3 (4 c d-b e (m-1))+x^2 \left(-2 b^2 c d (m+1) (c d-b e)^2 \left(b^2 e^2 (m-1)-b c d e (m+4)+6 c^2 d^2\right)-(b+c x) \left((b+c x) \left(2 c^3 d^3 \left(b^2 e^2 \left(m^2-7 m+12\right)+6 b c d e (m-4)+12 c^2 d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)-2 (c d-b e)^3 \left(b^2 e^2 (m-1) m-6 b c d e m+12 c^2 d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)\right)-2 b c d (m+1) (2 c d-b e) (b e-c d) \left(b^2 e^2 (m-1)-6 b c d e+6 c^2 d^2\right)\right)\right)\right)}{4 b^5 d^3 (m+1) x^2 (b+c x)^2 (c d-b e)^3}","-\frac{(d+e x)^{m+1} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left(b x+c x^2\right)^2 (c d-b e)}-\frac{(d+e x)^{m+1} \left(-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{e x}{d}+1\right)}{2 b^5 d^3 (m+1)}+\frac{c^3 (d+e x)^{m+1} \left(b^2 e^2 \left(m^2-7 m+12\right)-6 b c d e (4-m)+12 c^2 d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right)}{2 b^5 (m+1) (c d-b e)^3}+\frac{(d+e x)^{m+1} \left(c x (2 c d-b e) \left(-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right)+b (c d-b e) \left(-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right)\right)}{2 b^4 d^2 \left(b x+c x^2\right) (c d-b e)^2}",1,"-1/4*((d + e*x)^(1 + m)*(2*b^4*d^2*(c*d - b*e)^3*(1 + m) - 2*b^3*d*(c*d - b*e)^3*(4*c*d - b*e*(-1 + m))*(1 + m)*x + x^2*(-2*b^2*c*d*(c*d - b*e)^2*(1 + m)*(6*c^2*d^2 + b^2*e^2*(-1 + m) - b*c*d*e*(4 + m)) - (b + c*x)*(-2*b*c*d*(2*c*d - b*e)*(-(c*d) + b*e)*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2*(-1 + m))*(1 + m) + (b + c*x)*(2*c^3*d^3*(12*c^2*d^2 + 6*b*c*d*e*(-4 + m) + b^2*e^2*(12 - 7*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e)] - 2*(c*d - b*e)^3*(12*c^2*d^2 - 6*b*c*d*e*m + b^2*e^2*(-1 + m)*m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d])))))/(b^5*d^3*(c*d - b*e)^3*(1 + m)*x^2*(b + c*x)^2)","A",1
446,1,111,105,0.0799011,"\int (d+e x)^m \left(b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(b*x + c*x^2)^(3/2),x]","\frac{2 x^2 \sqrt{x (b+c x)} (d+e x)^m \left(\frac{e x}{d}+1\right)^{-m} \left(7 b F_1\left(\frac{5}{2};-\frac{1}{2},-m;\frac{7}{2};-\frac{c x}{b},-\frac{e x}{d}\right)+5 c x F_1\left(\frac{7}{2};-\frac{1}{2},-m;\frac{9}{2};-\frac{c x}{b},-\frac{e x}{d}\right)\right)}{35 \sqrt{\frac{c x}{b}+1}}","\frac{\left(b x+c x^2\right)^{3/2} (d+e x)^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right)}{e (m+1) \left(-\frac{e x}{d}\right)^{3/2} \left(1-\frac{c (d+e x)}{c d-b e}\right)^{3/2}}",1,"(2*x^2*Sqrt[x*(b + c*x)]*(d + e*x)^m*(7*b*AppellF1[5/2, -1/2, -m, 7/2, -((c*x)/b), -((e*x)/d)] + 5*c*x*AppellF1[7/2, -1/2, -m, 9/2, -((c*x)/b), -((e*x)/d)]))/(35*Sqrt[1 + (c*x)/b]*(1 + (e*x)/d)^m)","A",0
447,1,76,105,0.0351978,"\int (d+e x)^m \sqrt{b x+c x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[b*x + c*x^2],x]","\frac{2 x \sqrt{x (b+c x)} (d+e x)^m \left(\frac{d+e x}{d}\right)^{-m} F_1\left(\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right)}{3 \sqrt{\frac{b+c x}{b}}}","\frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right)}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}}",1,"(2*x*Sqrt[x*(b + c*x)]*(d + e*x)^m*AppellF1[3/2, -1/2, -m, 5/2, -((c*x)/b), -((e*x)/d)])/(3*Sqrt[(b + c*x)/b]*((d + e*x)/d)^m)","A",0
448,1,74,105,0.0396643,"\int \frac{(d+e x)^m}{\sqrt{b x+c x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[b*x + c*x^2],x]","\frac{2 x \sqrt{\frac{b+c x}{b}} (d+e x)^m \left(\frac{d+e x}{d}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right)}{\sqrt{x (b+c x)}}","\frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right)}{e (m+1) \sqrt{b x+c x^2}}",1,"(2*x*Sqrt[(b + c*x)/b]*(d + e*x)^m*AppellF1[1/2, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)])/(Sqrt[x*(b + c*x)]*((d + e*x)/d)^m)","A",0
449,1,138,105,0.1260772,"\int \frac{(d+e x)^m}{\left(b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(b*x + c*x^2)^(3/2),x]","-\frac{2 \sqrt{x (b+c x)} (d+e x)^m \left(\frac{e x}{d}+1\right)^{-m} \left(b F_1\left(-\frac{1}{2};-\frac{1}{2},-m;\frac{1}{2};-\frac{c x}{b},-\frac{e x}{d}\right)+c x \left(F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right)+F_1\left(\frac{1}{2};\frac{3}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right)\right)\right)}{b^3 x \sqrt{\frac{c x}{b}+1}}","\frac{\left(-\frac{e x}{d}\right)^{3/2} (d+e x)^{m+1} \left(1-\frac{c (d+e x)}{c d-b e}\right)^{3/2} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right)}{e (m+1) \left(b x+c x^2\right)^{3/2}}",1,"(-2*Sqrt[x*(b + c*x)]*(d + e*x)^m*(b*AppellF1[-1/2, -1/2, -m, 1/2, -((c*x)/b), -((e*x)/d)] + c*x*(AppellF1[1/2, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)] + AppellF1[1/2, 3/2, -m, 3/2, -((c*x)/b), -((e*x)/d)])))/(b^3*x*Sqrt[1 + (c*x)/b]*(1 + (e*x)/d)^m)","A",0
450,1,76,103,0.0950227,"\int (d+e x)^m \left(b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(b*x + c*x^2)^p,x]","\frac{x \left(\frac{b+c x}{b}\right)^{-p} (x (b+c x))^p (d+e x)^m \left(\frac{d+e x}{d}\right)^{-m} F_1\left(p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right)}{p+1}","\frac{\left(b x+c x^2\right)^p (d+e x)^{m+1} \left(-\frac{e x}{d}\right)^{-p} \left(1-\frac{c (d+e x)}{c d-b e}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right)}{e (m+1)}",1,"(x*(x*(b + c*x))^p*(d + e*x)^m*AppellF1[1 + p, -p, -m, 2 + p, -((c*x)/b), -((e*x)/d)])/((1 + p)*((b + c*x)/b)^p*((d + e*x)/d)^m)","A",0
451,1,101,57,0.0153414,"\int (d+e x)^4 \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^4*(a + c*x^2),x]","\frac{1}{5} e^2 x^5 \left(a e^2+6 c d^2\right)+d e x^4 \left(a e^2+c d^2\right)+\frac{1}{3} d^2 x^3 \left(6 a e^2+c d^2\right)+a d^4 x+2 a d^3 e x^2+\frac{2}{3} c d e^3 x^6+\frac{1}{7} c e^4 x^7","\frac{(d+e x)^5 \left(a e^2+c d^2\right)}{5 e^3}+\frac{c (d+e x)^7}{7 e^3}-\frac{c d (d+e x)^6}{3 e^3}",1,"a*d^4*x + 2*a*d^3*e*x^2 + (d^2*(c*d^2 + 6*a*e^2)*x^3)/3 + d*e*(c*d^2 + a*e^2)*x^4 + (e^2*(6*c*d^2 + a*e^2)*x^5)/5 + (2*c*d*e^3*x^6)/3 + (c*e^4*x^7)/7","A",1
452,1,74,57,0.0134959,"\int (d+e x)^3 \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^3*(a + c*x^2),x]","\frac{1}{4} a x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+\frac{1}{60} c x^3 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)","\frac{(d+e x)^4 \left(a e^2+c d^2\right)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}-\frac{2 c d (d+e x)^5}{5 e^3}",1,"(a*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))/4 + (c*x^3*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3))/60","A",1
453,1,53,57,0.0087139,"\int (d+e x)^2 \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^2*(a + c*x^2),x]","\frac{1}{3} x^3 \left(a e^2+c d^2\right)+a d^2 x+a d e x^2+\frac{1}{2} c d e x^4+\frac{1}{5} c e^2 x^5","\frac{(d+e x)^3 \left(a e^2+c d^2\right)}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3}",1,"a*d^2*x + a*d*e*x^2 + ((c*d^2 + a*e^2)*x^3)/3 + (c*d*e*x^4)/2 + (c*e^2*x^5)/5","A",1
454,1,32,31,0.0015738,"\int (d+e x) \left(a+c x^2\right) \, dx","Integrate[(d + e*x)*(a + c*x^2),x]","a d x+\frac{1}{2} a e x^2+\frac{1}{3} c d x^3+\frac{1}{4} c e x^4","\frac{e \left(a+c x^2\right)^2}{4 c}+a d x+\frac{1}{3} c d x^3",1,"a*d*x + (a*e*x^2)/2 + (c*d*x^3)/3 + (c*e*x^4)/4","A",1
455,1,38,41,0.0113501,"\int \frac{a+c x^2}{d+e x} \, dx","Integrate[(a + c*x^2)/(d + e*x),x]","\frac{2 \left(a e^2+c d^2\right) \log (d+e x)+c e x (e x-2 d)}{2 e^3}","\frac{\left(a e^2+c d^2\right) \log (d+e x)}{e^3}-\frac{c d x}{e^2}+\frac{c x^2}{2 e}",1,"(c*e*x*(-2*d + e*x) + 2*(c*d^2 + a*e^2)*Log[d + e*x])/(2*e^3)","A",1
456,1,39,43,0.0206497,"\int \frac{a+c x^2}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)/(d + e*x)^2,x]","\frac{-\frac{a e^2+c d^2}{d+e x}-2 c d \log (d+e x)+c e x}{e^3}","-\frac{a e^2+c d^2}{e^3 (d+e x)}-\frac{2 c d \log (d+e x)}{e^3}+\frac{c x}{e^2}",1,"(c*e*x - (c*d^2 + a*e^2)/(d + e*x) - 2*c*d*Log[d + e*x])/e^3","A",1
457,1,48,53,0.0151938,"\int \frac{a+c x^2}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)/(d + e*x)^3,x]","\frac{-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2}","\frac{-a e^2-c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3}",1,"(-(a*e^2) + c*d*(3*d + 4*e*x) + 2*c*(d + e*x)^2*Log[d + e*x])/(2*e^3*(d + e*x)^2)","A",1
458,1,39,54,0.013426,"\int \frac{a+c x^2}{(d+e x)^4} \, dx","Integrate[(a + c*x^2)/(d + e*x)^4,x]","-\frac{a e^2+c \left(d^2+3 d e x+3 e^2 x^2\right)}{3 e^3 (d+e x)^3}","\frac{-a e^2-c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2}",1,"-1/3*(a*e^2 + c*(d^2 + 3*d*e*x + 3*e^2*x^2))/(e^3*(d + e*x)^3)","A",1
459,1,40,59,0.0145895,"\int \frac{a+c x^2}{(d+e x)^5} \, dx","Integrate[(a + c*x^2)/(d + e*x)^5,x]","-\frac{3 a e^2+c \left(d^2+4 d e x+6 e^2 x^2\right)}{12 e^3 (d+e x)^4}","\frac{-a e^2-c d^2}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}+\frac{2 c d}{3 e^3 (d+e x)^3}",1,"-1/12*(3*a*e^2 + c*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)","A",1
460,1,167,117,0.0199844,"\int (d+e x)^4 \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^4*(a + c*x^2)^2,x]","\frac{1}{5} x^5 \left(a^2 e^4+12 a c d^2 e^2+c^2 d^4\right)+a^2 d^4 x+2 a^2 d^3 e x^2+\frac{2}{7} c e^2 x^7 \left(a e^2+3 c d^2\right)+\frac{2}{3} c d e x^6 \left(2 a e^2+c d^2\right)+a d e x^4 \left(a e^2+2 c d^2\right)+\frac{2}{3} a d^2 x^3 \left(3 a e^2+c d^2\right)+\frac{1}{2} c^2 d e^3 x^8+\frac{1}{9} c^2 e^4 x^9","\frac{2 c (d+e x)^7 \left(a e^2+3 c d^2\right)}{7 e^5}-\frac{2 c d (d+e x)^6 \left(a e^2+c d^2\right)}{3 e^5}+\frac{(d+e x)^5 \left(a e^2+c d^2\right)^2}{5 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5}",1,"a^2*d^4*x + 2*a^2*d^3*e*x^2 + (2*a*d^2*(c*d^2 + 3*a*e^2)*x^3)/3 + a*d*e*(2*c*d^2 + a*e^2)*x^4 + ((c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (2*c*d*e*(c*d^2 + 2*a*e^2)*x^6)/3 + (2*c*e^2*(3*c*d^2 + a*e^2)*x^7)/7 + (c^2*d*e^3*x^8)/2 + (c^2*e^4*x^9)/9","A",1
461,1,117,117,0.0173593,"\int (d+e x)^3 \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^2,x]","\frac{1}{4} a^2 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+\frac{1}{30} a c x^3 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+\frac{1}{280} c^2 x^5 \left(56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right)","\frac{c (d+e x)^6 \left(a e^2+3 c d^2\right)}{3 e^5}-\frac{4 c d (d+e x)^5 \left(a e^2+c d^2\right)}{5 e^5}+\frac{(d+e x)^4 \left(a e^2+c d^2\right)^2}{4 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}-\frac{4 c^2 d (d+e x)^7}{7 e^5}",1,"(a^2*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))/4 + (a*c*x^3*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3))/30 + (c^2*x^5*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3))/280","A",1
462,1,91,80,0.0115019,"\int (d+e x)^2 \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^2,x]","a^2 d^2 x+a^2 d e x^2+\frac{1}{5} c x^5 \left(2 a e^2+c d^2\right)+\frac{1}{3} a x^3 \left(a e^2+2 c d^2\right)+a c d e x^4+\frac{1}{3} c^2 d e x^6+\frac{1}{7} c^2 e^2 x^7","a^2 d^2 x+\frac{1}{5} c x^5 \left(2 a e^2+c d^2\right)+\frac{1}{3} a x^3 \left(a e^2+2 c d^2\right)+\frac{d e \left(a+c x^2\right)^3}{3 c}+\frac{1}{7} c^2 e^2 x^7",1,"a^2*d^2*x + a^2*d*e*x^2 + (a*(2*c*d^2 + a*e^2)*x^3)/3 + a*c*d*e*x^4 + (c*(c*d^2 + 2*a*e^2)*x^5)/5 + (c^2*d*e*x^6)/3 + (c^2*e^2*x^7)/7","A",1
463,1,60,45,0.0025151,"\int (d+e x) \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)*(a + c*x^2)^2,x]","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a c d x^3+\frac{1}{2} a c e x^4+\frac{1}{5} c^2 d x^5+\frac{1}{6} c^2 e x^6","a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left(a+c x^2\right)^3}{6 c}+\frac{1}{5} c^2 d x^5",1,"a^2*d*x + (a^2*e*x^2)/2 + (2*a*c*d*x^3)/3 + (a*c*e*x^4)/2 + (c^2*d*x^5)/5 + (c^2*e*x^6)/6","A",1
464,1,79,94,0.0302709,"\int \frac{\left(a+c x^2\right)^2}{d+e x} \, dx","Integrate[(a + c*x^2)^2/(d + e*x),x]","\frac{12 \left(a e^2+c d^2\right)^2 \log (d+e x)+c e x \left(12 a e^2 (e x-2 d)+c \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)}{12 e^5}","\frac{\left(a e^2+c d^2\right)^2 \log (d+e x)}{e^5}-\frac{c d x \left(2 a e^2+c d^2\right)}{e^4}+\frac{c x^2 \left(2 a e^2+c d^2\right)}{2 e^3}-\frac{c^2 d x^3}{3 e^2}+\frac{c^2 x^4}{4 e}",1,"(c*e*x*(12*a*e^2*(-2*d + e*x) + c*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 12*(c*d^2 + a*e^2)^2*Log[d + e*x])/(12*e^5)","A",1
465,1,91,94,0.0534767,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^2,x]","\frac{3 c e x \left(2 a e^2+3 c d^2\right)-\frac{3 \left(a e^2+c d^2\right)^2}{d+e x}-12 c d \left(a e^2+c d^2\right) \log (d+e x)-3 c^2 d e^2 x^2+c^2 e^3 x^3}{3 e^5}","-\frac{\left(a e^2+c d^2\right)^2}{e^5 (d+e x)}-\frac{4 c d \left(a e^2+c d^2\right) \log (d+e x)}{e^5}+\frac{c x \left(2 a e^2+3 c d^2\right)}{e^4}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2}",1,"(3*c*e*(3*c*d^2 + 2*a*e^2)*x - 3*c^2*d*e^2*x^2 + c^2*e^3*x^3 - (3*(c*d^2 + a*e^2)^2)/(d + e*x) - 12*c*d*(c*d^2 + a*e^2)*Log[d + e*x])/(3*e^5)","A",1
466,1,111,100,0.0368504,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^3,x]","\frac{-a^2 e^4+4 c (d+e x)^2 \left(a e^2+3 c d^2\right) \log (d+e x)+2 a c d e^2 (3 d+4 e x)+c^2 \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)}{2 e^5 (d+e x)^2}","\frac{4 c d \left(a e^2+c d^2\right)}{e^5 (d+e x)}-\frac{\left(a e^2+c d^2\right)^2}{2 e^5 (d+e x)^2}+\frac{2 c \left(a e^2+3 c d^2\right) \log (d+e x)}{e^5}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^2}{2 e^3}",1,"(-(a^2*e^4) + 2*a*c*d*e^2*(3*d + 4*e*x) + c^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*c*(3*c*d^2 + a*e^2)*(d + e*x)^2*Log[d + e*x])/(2*e^5*(d + e*x)^2)","A",1
467,1,110,101,0.053616,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^4,x]","-\frac{a^2 e^4+2 a c e^2 \left(d^2+3 d e x+3 e^2 x^2\right)+c^2 \left(13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right)+12 c^2 d (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3}","-\frac{2 c \left(a e^2+3 c d^2\right)}{e^5 (d+e x)}+\frac{2 c d \left(a e^2+c d^2\right)}{e^5 (d+e x)^2}-\frac{\left(a e^2+c d^2\right)^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4}",1,"-1/3*(a^2*e^4 + 2*a*c*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) + c^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4) + 12*c^2*d*(d + e*x)^3*Log[d + e*x])/(e^5*(d + e*x)^3)","A",1
468,1,100,109,0.0346268,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^5,x]","\frac{-3 a^2 e^4-2 a c e^2 \left(d^2+4 d e x+6 e^2 x^2\right)+c^2 d \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}","-\frac{c \left(a e^2+3 c d^2\right)}{e^5 (d+e x)^2}+\frac{4 c d \left(a e^2+c d^2\right)}{3 e^5 (d+e x)^3}-\frac{\left(a e^2+c d^2\right)^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}",1,"(-3*a^2*e^4 - 2*a*c*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)","A",1
469,1,90,110,0.0362305,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^6,x]","-\frac{3 a^2 e^4+a c e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+3 c^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)}{15 e^5 (d+e x)^5}","-\frac{2 c \left(a e^2+3 c d^2\right)}{3 e^5 (d+e x)^3}+\frac{c d \left(a e^2+c d^2\right)}{e^5 (d+e x)^4}-\frac{\left(a e^2+c d^2\right)^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2}",1,"-1/15*(3*a^2*e^4 + a*c*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(e^5*(d + e*x)^5)","A",1
470,1,89,117,0.0303757,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^7,x]","-\frac{5 a^2 e^4+a c e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+c^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)}{30 e^5 (d+e x)^6}","-\frac{c \left(a e^2+3 c d^2\right)}{2 e^5 (d+e x)^4}+\frac{4 c d \left(a e^2+c d^2\right)}{5 e^5 (d+e x)^5}-\frac{\left(a e^2+c d^2\right)^2}{6 e^5 (d+e x)^6}-\frac{c^2}{2 e^5 (d+e x)^2}+\frac{4 c^2 d}{3 e^5 (d+e x)^3}",1,"-1/30*(5*a^2*e^4 + a*c*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))/(e^5*(d + e*x)^6)","A",1
471,1,90,114,0.035367,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^8,x]","-\frac{15 a^2 e^4+2 a c e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+c^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)}{105 e^5 (d+e x)^7}","-\frac{2 c \left(a e^2+3 c d^2\right)}{5 e^5 (d+e x)^5}+\frac{2 c d \left(a e^2+c d^2\right)}{3 e^5 (d+e x)^6}-\frac{\left(a e^2+c d^2\right)^2}{7 e^5 (d+e x)^7}-\frac{c^2}{3 e^5 (d+e x)^3}+\frac{c^2 d}{e^5 (d+e x)^4}",1,"-1/105*(15*a^2*e^4 + 2*a*c*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(e^5*(d + e*x)^7)","A",1
472,1,338,190,0.0488844,"\int (d+e x)^6 \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^6*(a + c*x^2)^3,x]","a^3 d^6 x+3 a^3 d^5 e x^2+\frac{1}{3} c e^2 x^9 \left(a^2 e^4+15 a c d^2 e^2+5 c^2 d^4\right)+\frac{3}{4} c d e x^8 \left(3 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right)+a d e x^6 \left(a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right)+\frac{3}{5} a d^2 x^5 \left(5 a^2 e^4+15 a c d^2 e^2+c^2 d^4\right)+a^2 d^4 x^3 \left(5 a e^2+c d^2\right)+\frac{1}{2} a^2 d^3 e x^4 \left(10 a e^2+9 c d^2\right)+\frac{1}{7} x^7 \left(a^3 e^6+45 a^2 c d^2 e^4+45 a c^2 d^4 e^2+c^3 d^6\right)+\frac{3}{11} c^2 e^4 x^{11} \left(a e^2+5 c d^2\right)+\frac{1}{5} c^2 d e^3 x^{10} \left(9 a e^2+10 c d^2\right)+\frac{1}{2} c^3 d e^5 x^{12}+\frac{1}{13} c^3 e^6 x^{13}","\frac{3 c^2 (d+e x)^{11} \left(a e^2+5 c d^2\right)}{11 e^7}-\frac{2 c^2 d (d+e x)^{10} \left(3 a e^2+5 c d^2\right)}{5 e^7}+\frac{c (d+e x)^9 \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{3 e^7}-\frac{3 c d (d+e x)^8 \left(a e^2+c d^2\right)^2}{4 e^7}+\frac{(d+e x)^7 \left(a e^2+c d^2\right)^3}{7 e^7}+\frac{c^3 (d+e x)^{13}}{13 e^7}-\frac{c^3 d (d+e x)^{12}}{2 e^7}",1,"a^3*d^6*x + 3*a^3*d^5*e*x^2 + a^2*d^4*(c*d^2 + 5*a*e^2)*x^3 + (a^2*d^3*e*(9*c*d^2 + 10*a*e^2)*x^4)/2 + (3*a*d^2*(c^2*d^4 + 15*a*c*d^2*e^2 + 5*a^2*e^4)*x^5)/5 + a*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^6 + ((c^3*d^6 + 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)*x^7)/7 + (3*c*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 + 3*a^2*e^4)*x^8)/4 + (c*e^2*(5*c^2*d^4 + 15*a*c*d^2*e^2 + a^2*e^4)*x^9)/3 + (c^2*d*e^3*(10*c*d^2 + 9*a*e^2)*x^10)/5 + (3*c^2*e^4*(5*c*d^2 + a*e^2)*x^11)/11 + (c^3*d*e^5*x^12)/2 + (c^3*e^6*x^13)/13","A",1
473,1,252,190,0.0549557,"\int (d+e x)^5 \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^5*(a + c*x^2)^3,x]","\frac{1}{6} a^3 x \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+a^2 c \left(d^5 x^3+\frac{15}{4} d^4 e x^4+6 d^3 e^2 x^5+5 d^2 e^3 x^6+\frac{15}{7} d e^4 x^7+\frac{3 e^5 x^8}{8}\right)+\frac{1}{420} a c^2 x^5 \left(252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right)+\frac{c^3 x^7 \left(792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right)}{5544}","\frac{3 c^2 (d+e x)^{10} \left(a e^2+5 c d^2\right)}{10 e^7}-\frac{4 c^2 d (d+e x)^9 \left(3 a e^2+5 c d^2\right)}{9 e^7}+\frac{3 c (d+e x)^8 \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{8 e^7}-\frac{6 c d (d+e x)^7 \left(a e^2+c d^2\right)^2}{7 e^7}+\frac{(d+e x)^6 \left(a e^2+c d^2\right)^3}{6 e^7}+\frac{c^3 (d+e x)^{12}}{12 e^7}-\frac{6 c^3 d (d+e x)^{11}}{11 e^7}",1,"(a^3*x*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5))/6 + (a*c^2*x^5*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5))/420 + (c^3*x^7*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5))/5544 + a^2*c*(d^5*x^3 + (15*d^4*e*x^4)/4 + 6*d^3*e^2*x^5 + 5*d^2*e^3*x^6 + (15*d*e^4*x^7)/7 + (3*e^5*x^8)/8)","A",1
474,1,206,188,0.0464113,"\int (d+e x)^4 \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^4*(a + c*x^2)^3,x]","a^3 \left(d^4 x+2 d^3 e x^2+2 d^2 e^2 x^3+d e^3 x^4+\frac{e^4 x^5}{5}\right)+a^2 c \left(d^4 x^3+3 d^3 e x^4+\frac{18}{5} d^2 e^2 x^5+2 d e^3 x^6+\frac{3 e^4 x^7}{7}\right)+\frac{1}{210} a c^2 x^5 \left(126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right)+\frac{c^3 x^7 \left(330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right)}{2310}","\frac{c^2 (d+e x)^9 \left(a e^2+5 c d^2\right)}{3 e^7}-\frac{c^2 d (d+e x)^8 \left(3 a e^2+5 c d^2\right)}{2 e^7}+\frac{3 c (d+e x)^7 \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{7 e^7}-\frac{c d (d+e x)^6 \left(a e^2+c d^2\right)^2}{e^7}+\frac{(d+e x)^5 \left(a e^2+c d^2\right)^3}{5 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7}-\frac{3 c^3 d (d+e x)^{10}}{5 e^7}",1,"(a*c^2*x^5*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4))/210 + (c^3*x^7*(330*d^4 + 1155*d^3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4))/2310 + a^3*(d^4*x + 2*d^3*e*x^2 + 2*d^2*e^2*x^3 + d*e^3*x^4 + (e^4*x^5)/5) + a^2*c*(d^4*x^3 + 3*d^3*e*x^4 + (18*d^2*e^2*x^5)/5 + 2*d*e^3*x^6 + (3*e^4*x^7)/7)","A",1
475,1,155,161,0.046804,"\int (d+e x)^3 \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^3,x]","\frac{1}{840} x \left(210 a^3 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+42 a^2 c x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+9 a c^2 x^4 \left(56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right)+c^3 x^6 \left(120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right)\right)","a^3 d^3 x+\frac{1}{4} a^3 e^3 x^4+a^2 d x^3 \left(a e^2+c d^2\right)+\frac{1}{2} a^2 c e^3 x^6+\frac{1}{7} c^2 d x^7 \left(9 a e^2+c d^2\right)+\frac{3}{8} a c^2 e^3 x^8+\frac{3}{5} a c d x^5 \left(3 a e^2+c d^2\right)+\frac{3 d^2 e \left(a+c x^2\right)^4}{8 c}+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10}",1,"(x*(210*a^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 42*a^2*c*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 9*a*c^2*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + c^3*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3)))/840","A",1
476,1,116,104,0.0220923,"\int (d+e x)^2 \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^3,x]","a^3 \left(d^2 x+d e x^2+\frac{e^2 x^3}{3}\right)+\frac{1}{10} a^2 c x^3 \left(10 d^2+15 d e x+6 e^2 x^2\right)+\frac{1}{35} a c^2 x^5 \left(21 d^2+35 d e x+15 e^2 x^2\right)+\frac{1}{252} c^3 x^7 \left(36 d^2+63 d e x+28 e^2 x^2\right)","a^3 d^2 x+\frac{1}{3} a^2 x^3 \left(a e^2+3 c d^2\right)+\frac{1}{7} c^2 x^7 \left(3 a e^2+c d^2\right)+\frac{3}{5} a c x^5 \left(a e^2+c d^2\right)+\frac{d e \left(a+c x^2\right)^4}{4 c}+\frac{1}{9} c^3 e^2 x^9",1,"(a^2*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/10 + (a*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (c^3*x^7*(36*d^2 + 63*d*e*x + 28*e^2*x^2))/252 + a^3*(d^2*x + d*e*x^2 + (e^2*x^3)/3)","A",1
477,1,85,56,0.0025298,"\int (d+e x) \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)*(a + c*x^2)^3,x]","a^3 d x+\frac{1}{2} a^3 e x^2+a^2 c d x^3+\frac{3}{4} a^2 c e x^4+\frac{3}{5} a c^2 d x^5+\frac{1}{2} a c^2 e x^6+\frac{1}{7} c^3 d x^7+\frac{1}{8} c^3 e x^8","a^3 d x+a^2 c d x^3+\frac{3}{5} a c^2 d x^5+\frac{e \left(a+c x^2\right)^4}{8 c}+\frac{1}{7} c^3 d x^7",1,"a^3*d*x + (a^3*e*x^2)/2 + a^2*c*d*x^3 + (3*a^2*c*e*x^4)/4 + (3*a*c^2*d*x^5)/5 + (a*c^2*e*x^6)/2 + (c^3*d*x^7)/7 + (c^3*e*x^8)/8","A",1
478,1,142,173,0.0527277,"\int \frac{\left(a+c x^2\right)^3}{d+e x} \, dx","Integrate[(a + c*x^2)^3/(d + e*x),x]","\frac{c e x \left(90 a^2 e^4 (e x-2 d)+15 a c e^2 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+c^2 \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)+60 \left(a e^2+c d^2\right)^3 \log (d+e x)}{60 e^7}","-\frac{c d x \left(3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right)}{e^6}+\frac{c x^2 \left(3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right)}{2 e^5}-\frac{c^2 d x^3 \left(3 a e^2+c d^2\right)}{3 e^4}+\frac{c^2 x^4 \left(3 a e^2+c d^2\right)}{4 e^3}+\frac{\left(a e^2+c d^2\right)^3 \log (d+e x)}{e^7}-\frac{c^3 d x^5}{5 e^2}+\frac{c^3 x^6}{6 e}",1,"(c*e*x*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + c^2*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 60*(c*d^2 + a*e^2)^3*Log[d + e*x])/(60*e^7)","A",1
479,1,193,158,0.0544111,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^2,x]","\frac{-10 a^3 e^6+30 a^2 c e^4 \left(-d^2+d e x+e^2 x^2\right)+10 a c^2 e^2 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)-60 c d (d+e x) \left(a e^2+c d^2\right)^2 \log (d+e x)+c^3 \left(-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right)}{10 e^7 (d+e x)}","\frac{c x \left(3 a^2 e^4+9 a c d^2 e^2+5 c^2 d^4\right)}{e^6}-\frac{c^2 d x^2 \left(3 a e^2+2 c d^2\right)}{e^5}+\frac{c^2 x^3 \left(a e^2+c d^2\right)}{e^4}-\frac{\left(a e^2+c d^2\right)^3}{e^7 (d+e x)}-\frac{6 c d \left(a e^2+c d^2\right)^2 \log (d+e x)}{e^7}-\frac{c^3 d x^4}{2 e^3}+\frac{c^3 x^5}{5 e^2}",1,"(-10*a^3*e^6 + 30*a^2*c*e^4*(-d^2 + d*e*x + e^2*x^2) + 10*a*c^2*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + c^3*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*c*d*(c*d^2 + a*e^2)^2*(d + e*x)*Log[d + e*x])/(10*e^7*(d + e*x))","A",1
480,1,198,163,0.0643997,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^3,x]","\frac{-2 a^3 e^6+12 c (d+e x)^2 \left(a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right) \log (d+e x)+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)+c^3 \left(22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right)}{4 e^7 (d+e x)^2}","-\frac{c^2 d x \left(9 a e^2+10 c d^2\right)}{e^6}+\frac{3 c^2 x^2 \left(a e^2+2 c d^2\right)}{2 e^5}+\frac{6 c d \left(a e^2+c d^2\right)^2}{e^7 (d+e x)}-\frac{\left(a e^2+c d^2\right)^3}{2 e^7 (d+e x)^2}+\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right) \log (d+e x)}{e^7}-\frac{c^3 d x^3}{e^4}+\frac{c^3 x^4}{4 e^3}",1,"(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 4*e*x) + 6*a*c^2*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + c^3*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + 12*c*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*(d + e*x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)","A",1
481,1,197,165,0.0634347,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^4} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^4,x]","\frac{-a^3 e^6-3 a^2 c e^4 \left(d^2+3 d e x+3 e^2 x^2\right)-12 c^2 d (d+e x)^3 \left(3 a e^2+5 c d^2\right) \log (d+e x)+3 a c^2 e^2 \left(-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right)+c^3 \left(-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right)}{3 e^7 (d+e x)^3}","-\frac{4 c^2 d \left(3 a e^2+5 c d^2\right) \log (d+e x)}{e^7}+\frac{c^2 x \left(3 a e^2+10 c d^2\right)}{e^6}-\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{e^7 (d+e x)}+\frac{3 c d \left(a e^2+c d^2\right)^2}{e^7 (d+e x)^2}-\frac{\left(a e^2+c d^2\right)^3}{3 e^7 (d+e x)^3}-\frac{2 c^3 d x^2}{e^5}+\frac{c^3 x^3}{3 e^4}",1,"(-(a^3*e^6) - 3*a^2*c*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 3*a*c^2*e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + c^3*(-37*d^6 - 51*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 12*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)","A",1
482,1,185,171,0.0603223,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^5} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^5,x]","\frac{-a^3 e^6-a^2 c e^4 \left(d^2+4 d e x+6 e^2 x^2\right)+12 c^2 (d+e x)^4 \left(a e^2+5 c d^2\right) \log (d+e x)+a c^2 d e^2 \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+c^3 \left(57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right)}{4 e^7 (d+e x)^4}","\frac{4 c^2 d \left(3 a e^2+5 c d^2\right)}{e^7 (d+e x)}+\frac{3 c^2 \left(a e^2+5 c d^2\right) \log (d+e x)}{e^7}-\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{2 e^7 (d+e x)^2}+\frac{2 c d \left(a e^2+c d^2\right)^2}{e^7 (d+e x)^3}-\frac{\left(a e^2+c d^2\right)^3}{4 e^7 (d+e x)^4}-\frac{5 c^3 d x}{e^6}+\frac{c^3 x^2}{2 e^5}",1,"(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) + 12*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)","A",1
483,1,182,172,0.0780983,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^6} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^6,x]","-\frac{2 a^3 e^6+a^2 c e^4 \left(d^2+5 d e x+10 e^2 x^2\right)+6 a c^2 e^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)+c^3 \left(87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right)+60 c^3 d (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5}","-\frac{3 c^2 \left(a e^2+5 c d^2\right)}{e^7 (d+e x)}+\frac{2 c^2 d \left(3 a e^2+5 c d^2\right)}{e^7 (d+e x)^2}-\frac{c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{e^7 (d+e x)^3}+\frac{3 c d \left(a e^2+c d^2\right)^2}{2 e^7 (d+e x)^4}-\frac{\left(a e^2+c d^2\right)^3}{5 e^7 (d+e x)^5}-\frac{6 c^3 d \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}",1,"-1/10*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*c^3*d*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)","A",1
484,1,172,184,0.057692,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^7} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^7,x]","\frac{-10 a^3 e^6-3 a^2 c e^4 \left(d^2+6 d e x+15 e^2 x^2\right)-6 a c^2 e^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)+c^3 d \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}","-\frac{3 c^2 \left(a e^2+5 c d^2\right)}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left(3 a e^2+5 c d^2\right)}{3 e^7 (d+e x)^3}-\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{4 e^7 (d+e x)^4}+\frac{6 c d \left(a e^2+c d^2\right)^2}{5 e^7 (d+e x)^5}-\frac{\left(a e^2+c d^2\right)^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}",1,"(-10*a^3*e^6 - 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) - 6*a*c^2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)","A",1
485,1,161,178,0.0508509,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^8} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^8,x]","-\frac{5 a^3 e^6+a^2 c e^4 \left(d^2+7 d e x+21 e^2 x^2\right)+a c^2 e^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+5 c^3 \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)}{35 e^7 (d+e x)^7}","-\frac{c^2 \left(a e^2+5 c d^2\right)}{e^7 (d+e x)^3}+\frac{c^2 d \left(3 a e^2+5 c d^2\right)}{e^7 (d+e x)^4}-\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{5 e^7 (d+e x)^5}+\frac{c d \left(a e^2+c d^2\right)^2}{e^7 (d+e x)^6}-\frac{\left(a e^2+c d^2\right)^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)}+\frac{3 c^3 d}{e^7 (d+e x)^2}",1,"-1/35*(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)","A",1
486,1,163,188,0.05637,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^9} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^9,x]","-\frac{35 a^3 e^6+5 a^2 c e^4 \left(d^2+8 d e x+28 e^2 x^2\right)+3 a c^2 e^2 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+5 c^3 \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)}{280 e^7 (d+e x)^8}","-\frac{3 c^2 \left(a e^2+5 c d^2\right)}{4 e^7 (d+e x)^4}+\frac{4 c^2 d \left(3 a e^2+5 c d^2\right)}{5 e^7 (d+e x)^5}-\frac{c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{2 e^7 (d+e x)^6}+\frac{6 c d \left(a e^2+c d^2\right)^2}{7 e^7 (d+e x)^7}-\frac{\left(a e^2+c d^2\right)^3}{8 e^7 (d+e x)^8}-\frac{c^3}{2 e^7 (d+e x)^2}+\frac{2 c^3 d}{e^7 (d+e x)^3}",1,"-1/280*(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c^2*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)","A",1
487,1,163,190,0.0533941,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^{10}} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^10,x]","-\frac{140 a^3 e^6+15 a^2 c e^4 \left(d^2+9 d e x+36 e^2 x^2\right)+6 a c^2 e^2 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+5 c^3 \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)}{1260 e^7 (d+e x)^9}","-\frac{3 c^2 \left(a e^2+5 c d^2\right)}{5 e^7 (d+e x)^5}+\frac{2 c^2 d \left(3 a e^2+5 c d^2\right)}{3 e^7 (d+e x)^6}-\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{7 e^7 (d+e x)^7}+\frac{3 c d \left(a e^2+c d^2\right)^2}{4 e^7 (d+e x)^8}-\frac{\left(a e^2+c d^2\right)^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3}+\frac{3 c^3 d}{2 e^7 (d+e x)^4}",1,"-1/1260*(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)","A",1
488,1,423,278,0.083816,"\int (d+e x)^7 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^7*(a + c*x^2)^4,x]","\frac{1}{8} a^4 x \left(8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right)+\frac{1}{90} a^3 c x^3 \left(120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right)+\frac{1}{660} a^2 c^2 x^5 \left(792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right)+\frac{a c^3 x^7 \left(3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right)}{6006}+\frac{c^4 x^9 \left(11440 d^7+72072 d^6 e x+196560 d^5 e^2 x^2+300300 d^4 e^3 x^3+277200 d^3 e^4 x^4+154440 d^2 e^5 x^5+48048 d e^6 x^6+6435 e^7 x^7\right)}{102960}","\frac{c^2 (d+e x)^{12} \left(3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right)}{6 e^9}+\frac{2 c^3 (d+e x)^{14} \left(a e^2+7 c d^2\right)}{7 e^9}-\frac{8 c^3 d (d+e x)^{13} \left(3 a e^2+7 c d^2\right)}{13 e^9}-\frac{8 c^2 d (d+e x)^{11} \left(a e^2+c d^2\right) \left(3 a e^2+7 c d^2\right)}{11 e^9}+\frac{2 c (d+e x)^{10} \left(a e^2+c d^2\right)^2 \left(a e^2+7 c d^2\right)}{5 e^9}-\frac{8 c d (d+e x)^9 \left(a e^2+c d^2\right)^3}{9 e^9}+\frac{(d+e x)^8 \left(a e^2+c d^2\right)^4}{8 e^9}+\frac{c^4 (d+e x)^{16}}{16 e^9}-\frac{8 c^4 d (d+e x)^{15}}{15 e^9}",1,"(a^4*x*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7))/8 + (a^3*c*x^3*(120*d^7 + 630*d^6*e*x + 1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7))/90 + (a^2*c^2*x^5*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7))/660 + (a*c^3*x^7*(3432*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4*x^4 + 42042*d^2*e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7))/6006 + (c^4*x^9*(11440*d^7 + 72072*d^6*e*x + 196560*d^5*e^2*x^2 + 300300*d^4*e^3*x^3 + 277200*d^3*e^4*x^4 + 154440*d^2*e^5*x^5 + 48048*d*e^6*x^6 + 6435*e^7*x^7))/102960","A",1
489,1,361,276,0.1167683,"\int (d+e x)^6 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^6*(a + c*x^2)^4,x]","\frac{6435 a^4 x \left(7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right)+715 a^3 c x^3 \left(84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right)+117 a^2 c^2 x^5 \left(462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right)+15 a c^3 x^7 \left(1716 d^6+9009 d^5 e x+20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+6006 d e^5 x^5+924 e^6 x^6\right)+c^4 x^9 \left(5005 d^6+27027 d^5 e x+61425 d^4 e^2 x^2+75075 d^3 e^3 x^3+51975 d^2 e^4 x^4+19305 d e^5 x^5+3003 e^6 x^6\right)}{45045}","\frac{2 c^2 (d+e x)^{11} \left(3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right)}{11 e^9}+\frac{4 c^3 (d+e x)^{13} \left(a e^2+7 c d^2\right)}{13 e^9}-\frac{2 c^3 d (d+e x)^{12} \left(3 a e^2+7 c d^2\right)}{3 e^9}-\frac{4 c^2 d (d+e x)^{10} \left(a e^2+c d^2\right) \left(3 a e^2+7 c d^2\right)}{5 e^9}+\frac{4 c (d+e x)^9 \left(a e^2+c d^2\right)^2 \left(a e^2+7 c d^2\right)}{9 e^9}-\frac{c d (d+e x)^8 \left(a e^2+c d^2\right)^3}{e^9}+\frac{(d+e x)^7 \left(a e^2+c d^2\right)^4}{7 e^9}+\frac{c^4 (d+e x)^{15}}{15 e^9}-\frac{4 c^4 d (d+e x)^{14}}{7 e^9}",1,"(6435*a^4*x*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5*x^5 + e^6*x^6) + 715*a^3*c*x^3*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 117*a^2*c^2*x^5*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6) + 15*a*c^3*x^7*(1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^2*x^2 + 24024*d^3*e^3*x^3 + 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6*x^6) + c^4*x^9*(5005*d^6 + 27027*d^5*e*x + 61425*d^4*e^2*x^2 + 75075*d^3*e^3*x^3 + 51975*d^2*e^4*x^4 + 19305*d*e^5*x^5 + 3003*e^6*x^6))/45045","A",1
490,1,307,278,0.0887676,"\int (d+e x)^5 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^5*(a + c*x^2)^4,x]","\frac{x \left(15015 a^4 \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+2145 a^3 c x^2 \left(56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right)+429 a^2 c^2 x^4 \left(252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right)+65 a c^3 x^6 \left(792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right)+5 c^4 x^8 \left(2002 d^5+9009 d^4 e x+16380 d^3 e^2 x^2+15015 d^2 e^3 x^3+6930 d e^4 x^4+1287 e^5 x^5\right)\right)}{90090}","\frac{c^2 (d+e x)^{10} \left(3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right)}{5 e^9}+\frac{c^3 (d+e x)^{12} \left(a e^2+7 c d^2\right)}{3 e^9}-\frac{8 c^3 d (d+e x)^{11} \left(3 a e^2+7 c d^2\right)}{11 e^9}-\frac{8 c^2 d (d+e x)^9 \left(a e^2+c d^2\right) \left(3 a e^2+7 c d^2\right)}{9 e^9}+\frac{c (d+e x)^8 \left(a e^2+c d^2\right)^2 \left(a e^2+7 c d^2\right)}{2 e^9}-\frac{8 c d (d+e x)^7 \left(a e^2+c d^2\right)^3}{7 e^9}+\frac{(d+e x)^6 \left(a e^2+c d^2\right)^4}{6 e^9}+\frac{c^4 (d+e x)^{14}}{14 e^9}-\frac{8 c^4 d (d+e x)^{13}}{13 e^9}",1,"(x*(15015*a^4*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 2145*a^3*c*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 429*a^2*c^2*x^4*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 65*a*c^3*x^6*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + 5*c^4*x^8*(2002*d^5 + 9009*d^4*e*x + 16380*d^3*e^2*x^2 + 15015*d^2*e^3*x^3 + 6930*d*e^4*x^4 + 1287*e^5*x^5)))/90090","A",1
491,1,300,270,0.0407628,"\int (d+e x)^4 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^4*(a + c*x^2)^4,x]","a^4 d^4 x+2 a^4 d^3 e x^2+a^3 d e x^4 \left(a e^2+4 c d^2\right)+\frac{2}{3} a^3 d^2 x^3 \left(3 a e^2+2 c d^2\right)+\frac{1}{9} c^2 x^9 \left(6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right)+\frac{4}{7} a c x^7 \left(a^2 e^4+9 a c d^2 e^2+c^2 d^4\right)+\frac{1}{5} a^2 x^5 \left(a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right)+\frac{4}{3} a^2 c d e x^6 \left(2 a e^2+3 c d^2\right)+\frac{2}{11} c^3 e^2 x^{11} \left(2 a e^2+3 c d^2\right)+\frac{2}{5} c^3 d e x^{10} \left(4 a e^2+c d^2\right)+a c^2 d e x^8 \left(3 a e^2+2 c d^2\right)+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}","a^4 d^4 x+a^4 d e^3 x^4+\frac{2}{3} a^3 d^2 x^3 \left(3 a e^2+2 c d^2\right)+\frac{8}{3} a^3 c d e^3 x^6+\frac{1}{9} c^2 x^9 \left(6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right)+\frac{4}{7} a c x^7 \left(a^2 e^4+9 a c d^2 e^2+c^2 d^4\right)+\frac{1}{5} a^2 x^5 \left(a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right)+3 a^2 c^2 d e^3 x^8+\frac{2}{11} c^3 e^2 x^{11} \left(2 a e^2+3 c d^2\right)+\frac{8}{5} a c^3 d e^3 x^{10}+\frac{2 d^3 e \left(a+c x^2\right)^5}{5 c}+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}",1,"a^4*d^4*x + 2*a^4*d^3*e*x^2 + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^3*d*e*(4*c*d^2 + a*e^2)*x^4 + (a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (4*a^2*c*d*e*(3*c*d^2 + 2*a*e^2)*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4)*x^7)/7 + a*c^2*d*e*(2*c*d^2 + 3*a*e^2)*x^8 + (c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x^9)/9 + (2*c^3*d*e*(c*d^2 + 4*a*e^2)*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (c^4*d*e^3*x^12)/3 + (c^4*e^4*x^13)/13","A",1
492,1,197,209,0.0619545,"\int (d+e x)^3 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^4,x]","\frac{x \left(3465 a^4 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+924 a^3 c x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+297 a^2 c^2 x^4 \left(56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right)+66 a c^3 x^6 \left(120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right)+7 c^4 x^8 \left(220 d^3+594 d^2 e x+540 d e^2 x^2+165 e^3 x^3\right)\right)}{13860}","a^4 d^3 x+\frac{1}{4} a^4 e^3 x^4+\frac{1}{3} a^3 d x^3 \left(3 a e^2+4 c d^2\right)+\frac{2}{3} a^3 c e^3 x^6+\frac{3}{4} a^2 c^2 e^3 x^8+\frac{6}{5} a^2 c d x^5 \left(2 a e^2+c d^2\right)+\frac{1}{9} c^3 d x^9 \left(12 a e^2+c d^2\right)+\frac{2}{5} a c^3 e^3 x^{10}+\frac{2}{7} a c^2 d x^7 \left(9 a e^2+2 c d^2\right)+\frac{3 d^2 e \left(a+c x^2\right)^5}{10 c}+\frac{3}{11} c^4 d e^2 x^{11}+\frac{1}{12} c^4 e^3 x^{12}",1,"(x*(3465*a^4*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 924*a^3*c*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 297*a^2*c^2*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 66*a*c^3*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3) + 7*c^4*x^8*(220*d^3 + 594*d^2*e*x + 540*d*e^2*x^2 + 165*e^3*x^3)))/13860","A",1
493,1,148,132,0.0271264,"\int (d+e x)^2 \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^4,x]","a^4 \left(d^2 x+d e x^2+\frac{e^2 x^3}{3}\right)+\frac{2}{15} a^3 c x^3 \left(10 d^2+15 d e x+6 e^2 x^2\right)+\frac{2}{35} a^2 c^2 x^5 \left(21 d^2+35 d e x+15 e^2 x^2\right)+\frac{1}{63} a c^3 x^7 \left(36 d^2+63 d e x+28 e^2 x^2\right)+\frac{1}{495} c^4 x^9 \left(55 d^2+99 d e x+45 e^2 x^2\right)","a^4 d^2 x+\frac{1}{3} a^3 x^3 \left(a e^2+4 c d^2\right)+\frac{2}{5} a^2 c x^5 \left(2 a e^2+3 c d^2\right)+\frac{1}{9} c^3 x^9 \left(4 a e^2+c d^2\right)+\frac{2}{7} a c^2 x^7 \left(3 a e^2+2 c d^2\right)+\frac{d e \left(a+c x^2\right)^5}{5 c}+\frac{1}{11} c^4 e^2 x^{11}",1,"(2*a^3*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/15 + (2*a^2*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (a*c^3*x^7*(36*d^2 + 63*d*e*x + 28*e^2*x^2))/63 + (c^4*x^9*(55*d^2 + 99*d*e*x + 45*e^2*x^2))/495 + a^4*(d^2*x + d*e*x^2 + (e^2*x^3)/3)","A",1
494,1,110,73,0.002945,"\int (d+e x) \left(a+c x^2\right)^4 \, dx","Integrate[(d + e*x)*(a + c*x^2)^4,x]","a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{3} a^3 c d x^3+a^3 c e x^4+\frac{6}{5} a^2 c^2 d x^5+a^2 c^2 e x^6+\frac{4}{7} a c^3 d x^7+\frac{1}{2} a c^3 e x^8+\frac{1}{9} c^4 d x^9+\frac{1}{10} c^4 e x^{10}","a^4 d x+\frac{4}{3} a^3 c d x^3+\frac{6}{5} a^2 c^2 d x^5+\frac{4}{7} a c^3 d x^7+\frac{e \left(a+c x^2\right)^5}{10 c}+\frac{1}{9} c^4 d x^9",1,"a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*c*d*x^3)/3 + a^3*c*e*x^4 + (6*a^2*c^2*d*x^5)/5 + a^2*c^2*e*x^6 + (4*a*c^3*d*x^7)/7 + (a*c^3*e*x^8)/2 + (c^4*d*x^9)/9 + (c^4*e*x^10)/10","A",1
495,1,227,264,0.0983526,"\int \frac{\left(a+c x^2\right)^4}{d+e x} \, dx","Integrate[(a + c*x^2)^4/(d + e*x),x]","\frac{c x \left(1680 a^3 e^6 (e x-2 d)+420 a^2 c e^4 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+56 a c^2 e^2 \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)+c^3 \left(-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right)\right)}{840 e^8}+\frac{\left(a e^2+c d^2\right)^4 \log (d+e x)}{e^9}","\frac{c^2 (d+e x)^4 \left(3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right)}{2 e^9}+\frac{2 c^3 (d+e x)^6 \left(a e^2+7 c d^2\right)}{3 e^9}-\frac{8 c^3 d (d+e x)^5 \left(3 a e^2+7 c d^2\right)}{5 e^9}-\frac{8 c^2 d (d+e x)^3 \left(a e^2+c d^2\right) \left(3 a e^2+7 c d^2\right)}{3 e^9}+\frac{2 c (d+e x)^2 \left(a e^2+c d^2\right)^2 \left(a e^2+7 c d^2\right)}{e^9}+\frac{\left(a e^2+c d^2\right)^4 \log (d+e x)}{e^9}-\frac{8 c d x \left(a e^2+c d^2\right)^3}{e^8}+\frac{c^4 (d+e x)^8}{8 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9}",1,"(c*x*(1680*a^3*e^6*(-2*d + e*x) + 420*a^2*c*e^4*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 56*a*c^2*e^2*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + c^3*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7)))/(840*e^8) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9","A",1
496,1,289,255,0.0889554,"\int \frac{\left(a+c x^2\right)^4}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^4/(d + e*x)^2,x]","\frac{-105 a^4 e^8+420 a^3 c e^6 \left(-d^2+d e x+e^2 x^2\right)+210 a^2 c^2 e^4 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+42 a c^3 e^2 \left(-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right)-840 c d (d+e x) \left(a e^2+c d^2\right)^3 \log (d+e x)+c^4 \left(-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right)}{105 e^9 (d+e x)}","-\frac{c^2 d x^2 \left(6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right)}{e^7}+\frac{c^2 x^3 \left(6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right)}{3 e^6}+\frac{c x \left(4 a^3 e^6+18 a^2 c d^2 e^4+20 a c^2 d^4 e^2+7 c^3 d^6\right)}{e^8}-\frac{c^3 d x^4 \left(2 a e^2+c d^2\right)}{e^5}+\frac{c^3 x^5 \left(4 a e^2+3 c d^2\right)}{5 e^4}-\frac{\left(a e^2+c d^2\right)^4}{e^9 (d+e x)}-\frac{8 c d \left(a e^2+c d^2\right)^3 \log (d+e x)}{e^9}-\frac{c^4 d x^6}{3 e^3}+\frac{c^4 x^7}{7 e^2}",1,"(-105*a^4*e^8 + 420*a^3*c*e^6*(-d^2 + d*e*x + e^2*x^2) + 210*a^2*c^2*e^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 42*a*c^3*e^2*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) - 840*c*d*(c*d^2 + a*e^2)^3*(d + e*x)*Log[d + e*x])/(105*e^9*(d + e*x))","A",1
497,1,110,73,0.0040546,"\int (c+d x) \left(a+b x^2\right)^4 \, dx","Integrate[(c + d*x)*(a + b*x^2)^4,x]","a^4 c x+\frac{1}{2} a^4 d x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{1}{9} b^4 c x^9+\frac{1}{10} b^4 d x^{10}","a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{d \left(a+b x^2\right)^5}{10 b}+\frac{1}{9} b^4 c x^9",1,"a^4*c*x + (a^4*d*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (6*a^2*b^2*c*x^5)/5 + a^2*b^2*d*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (b^4*c*x^9)/9 + (b^4*d*x^10)/10","A",1
498,1,111,123,0.0833804,"\int \frac{(d+e x)^4}{a+c x^2} \, dx","Integrate[(d + e*x)^4/(a + c*x^2),x]","\frac{\left(a^2 e^4-6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{5/2}}+\frac{e \left(6 \left(c d^3-a d e^2\right) \log \left(a+c x^2\right)-3 a e^3 x+c e x \left(18 d^2+6 d e x+e^2 x^2\right)\right)}{3 c^2}","\frac{\left(a^2 e^4-6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{5/2}}+\frac{2 d e \left(c d^2-a e^2\right) \log \left(a+c x^2\right)}{c^2}+\frac{e^2 x \left(6 c d^2-a e^2\right)}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c}",1,"((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/2)) + (e*(-3*a*e^3*x + c*e*x*(18*d^2 + 6*d*e*x + e^2*x^2) + 6*(c*d^3 - a*d*e^2)*Log[a + c*x^2]))/(3*c^2)","A",1
499,1,80,90,0.0541643,"\int \frac{(d+e x)^3}{a+c x^2} \, dx","Integrate[(d + e*x)^3/(a + c*x^2),x]","\frac{d \left(c d^2-3 a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{e \left(\left(3 c d^2-a e^2\right) \log \left(a+c x^2\right)+c e x (6 d+e x)\right)}{2 c^2}","\frac{d \left(c d^2-3 a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{e \left(3 c d^2-a e^2\right) \log \left(a+c x^2\right)}{2 c^2}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}",1,"(d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(c*e*x*(6*d + e*x) + (3*c*d^2 - a*e^2)*Log[a + c*x^2]))/(2*c^2)","A",1
500,1,56,59,0.0402136,"\int \frac{(d+e x)^2}{a+c x^2} \, dx","Integrate[(d + e*x)^2/(a + c*x^2),x]","\frac{\left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{e \left(d \log \left(a+c x^2\right)+e x\right)}{c}","\frac{\left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{d e \log \left(a+c x^2\right)}{c}+\frac{e^2 x}{c}",1,"((c*d^2 - a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(e*x + d*Log[a + c*x^2]))/c","A",1
501,1,42,42,0.0136805,"\int \frac{d+e x}{a+c x^2} \, dx","Integrate[(d + e*x)/(a + c*x^2),x]","\frac{d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{c}}+\frac{e \log \left(a+c x^2\right)}{2 c}","\frac{d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{c}}+\frac{e \log \left(a+c x^2\right)}{2 c}",1,"(d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (e*Log[a + c*x^2])/(2*c)","A",1
502,1,63,86,0.0356591,"\int \frac{1}{(d+e x) \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)),x]","\frac{\frac{2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a}}-e \log \left(a+c x^2\right)+2 e \log (d+e x)}{2 a e^2+2 c d^2}","-\frac{e \log \left(a+c x^2\right)}{2 \left(a e^2+c d^2\right)}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} \left(a e^2+c d^2\right)}",1,"((2*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + 2*e*Log[d + e*x] - e*Log[a + c*x^2])/(2*c*d^2 + 2*a*e^2)","A",1
503,1,113,123,0.0843023,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)),x]","\frac{\sqrt{c} (d+e x) \left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)-\sqrt{a} e \left(c d (d+e x) \log \left(a+c x^2\right)+a e^2+c d^2-2 c d (d+e x) \log (d+e x)\right)}{\sqrt{a} (d+e x) \left(a e^2+c d^2\right)^2}","-\frac{c d e \log \left(a+c x^2\right)}{\left(a e^2+c d^2\right)^2}-\frac{e}{(d+e x) \left(a e^2+c d^2\right)}+\frac{2 c d e \log (d+e x)}{\left(a e^2+c d^2\right)^2}+\frac{\sqrt{c} \left(c d^2-a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} \left(a e^2+c d^2\right)^2}",1,"(Sqrt[c]*(c*d^2 - a*e^2)*(d + e*x)*ArcTan[(Sqrt[c]*x)/Sqrt[a]] - Sqrt[a]*e*(c*d^2 + a*e^2 - 2*c*d*(d + e*x)*Log[d + e*x] + c*d*(d + e*x)*Log[a + c*x^2]))/(Sqrt[a]*(c*d^2 + a*e^2)^2*(d + e*x))","A",1
504,1,140,176,0.2860804,"\int \frac{1}{(d+e x)^3 \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(a + c*x^2)),x]","\frac{\frac{2 c^{3/2} d \left(c d^2-3 a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a}}+e \left(c \left(a e^2-3 c d^2\right) \log \left(a+c x^2\right)-\frac{\left(a e^2+c d^2\right) \left(a e^2+c d (5 d+4 e x)\right)}{(d+e x)^2}+2 c \left(3 c d^2-a e^2\right) \log (d+e x)\right)}{2 \left(a e^2+c d^2\right)^3}","\frac{c^{3/2} d \left(c d^2-3 a e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} \left(a e^2+c d^2\right)^3}-\frac{c e \left(3 c d^2-a e^2\right) \log \left(a+c x^2\right)}{2 \left(a e^2+c d^2\right)^3}-\frac{2 c d e}{(d+e x) \left(a e^2+c d^2\right)^2}-\frac{e}{2 (d+e x)^2 \left(a e^2+c d^2\right)}+\frac{c e \left(3 c d^2-a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^3}",1,"((2*c^(3/2)*d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + e*(-(((c*d^2 + a*e^2)*(a*e^2 + c*d*(5*d + 4*e*x)))/(d + e*x)^2) + 2*c*(3*c*d^2 - a*e^2)*Log[d + e*x] + c*(-3*c*d^2 + a*e^2)*Log[a + c*x^2]))/(2*(c*d^2 + a*e^2)^3)","A",1
505,1,164,190,0.1197075,"\int \frac{(d+e x)^5}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(a + c*x^2)^2,x]","\frac{\frac{\sqrt{c} d \left(-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{-a^3 e^5+5 a^2 c d e^3 (2 d+e x)-5 a c^2 d^3 e (d+2 e x)+c^3 d^5 x}{a \left(a+c x^2\right)}+2 \left(5 c d^2 e^3-a e^5\right) \log \left(a+c x^2\right)+10 c d e^4 x+c e^5 x^2}{2 c^3}","\frac{d \left(-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{5/2}}+\frac{e^3 \left(5 c d^2-a e^2\right) \log \left(a+c x^2\right)}{c^3}-\frac{3 d e^2 x \left(2 c d^2-5 a e^2\right)}{2 a c^2}-\frac{e^3 x^2 \left(2 c d^2-a e^2\right)}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(d+e x)^4 (a e-c d x)}{2 a c \left(a+c x^2\right)}",1,"(10*c*d*e^4*x + c*e^5*x^2 + (-(a^3*e^5) + c^3*d^5*x + 5*a^2*c*d*e^3*(2*d + e*x) - 5*a*c^2*d^3*e*(d + 2*e*x))/(a*(a + c*x^2)) + (Sqrt[c]*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*(5*c*d^2*e^3 - a*e^5)*Log[a + c*x^2])/(2*c^3)","A",1
506,1,137,149,0.0801986,"\int \frac{(d+e x)^4}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(a + c*x^2)^2,x]","\frac{a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x}{2 a c^2 \left(a+c x^2\right)}+\frac{\left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{5/2}}+\frac{2 d e^3 \log \left(a+c x^2\right)}{c^2}+\frac{e^4 x}{c^2}","\frac{\left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{5/2}}-\frac{3 e^2 x \left(c d^2-a e^2\right)}{2 a c^2}+\frac{2 d e^3 \log \left(a+c x^2\right)}{c^2}-\frac{d e^3 x^2}{2 a c}-\frac{(d+e x)^3 (a e-c d x)}{2 a c \left(a+c x^2\right)}",1,"(e^4*x)/c^2 + (c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))/(2*a*c^2*(a + c*x^2)) + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2","A",1
507,1,107,109,0.0751108,"\int \frac{(d+e x)^3}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(a + c*x^2)^2,x]","\frac{\frac{\sqrt{a} \left(a^2 e^3-3 a c d e (d+e x)+a e^3 \left(a+c x^2\right) \log \left(a+c x^2\right)+c^2 d^3 x\right)}{a+c x^2}+\sqrt{c} d \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^2}","\frac{d \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{3/2}}+\frac{e^3 \log \left(a+c x^2\right)}{2 c^2}-\frac{d e^2 x}{2 a c}-\frac{(d+e x)^2 (a e-c d x)}{2 a c \left(a+c x^2\right)}",1,"(Sqrt[c]*d*(c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]] + (Sqrt[a]*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x) + a*e^3*(a + c*x^2)*Log[a + c*x^2]))/(a + c*x^2))/(2*a^(3/2)*c^2)","A",1
508,1,77,72,0.0582667,"\int \frac{(d+e x)^2}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(a + c*x^2)^2,x]","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{2 a c \left(a+c x^2\right)}","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a e-c d x)}{2 a c \left(a+c x^2\right)}",1,"(-2*a*d*e + c*d^2*x - a*e^2*x)/(2*a*c*(a + c*x^2)) + ((c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))","A",1
509,1,57,57,0.0263505,"\int \frac{d+e x}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)/(a + c*x^2)^2,x]","\frac{d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{c}}+\frac{c d x-a e}{2 a c \left(a+c x^2\right)}","\frac{d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{c}}+\frac{c d x-a e}{2 a c \left(a+c x^2\right)}",1,"(-(a*e) + c*d*x)/(2*a*c*(a + c*x^2)) + (d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[c])","A",1
510,1,138,142,0.0968949,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^2),x]","\frac{\sqrt{c} d \left(a+c x^2\right) \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{a} \left(\left(a e^2+c d^2\right) (a e+c d x)+2 a e^3 \left(a+c x^2\right) \log (d+e x)-a e^3 \left(a+c x^2\right) \log \left(a+c x^2\right)\right)}{2 a^{3/2} \left(a+c x^2\right) \left(a e^2+c d^2\right)^2}","\frac{\sqrt{c} d \left(3 a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)}-\frac{e^3 \log \left(a+c x^2\right)}{2 \left(a e^2+c d^2\right)^2}+\frac{e^3 \log (d+e x)}{\left(a e^2+c d^2\right)^2}",1,"(Sqrt[c]*d*(c*d^2 + 3*a*e^2)*(a + c*x^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[a]*((c*d^2 + a*e^2)*(a*e + c*d*x) + 2*a*e^3*(a + c*x^2)*Log[d + e*x] - a*e^3*(a + c*x^2)*Log[a + c*x^2]))/(2*a^(3/2)*(c*d^2 + a*e^2)^2*(a + c*x^2))","A",1
511,1,162,205,0.2002658,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)^2),x]","\frac{\frac{\sqrt{c} \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{3/2}}+\frac{c \left(a e^2+c d^2\right) \left(a e (2 d-e x)+c d^2 x\right)}{a \left(a+c x^2\right)}-\frac{2 e^3 \left(a e^2+c d^2\right)}{d+e x}-4 c d e^3 \log \left(a+c x^2\right)+8 c d e^3 \log (d+e x)}{2 \left(a e^2+c d^2\right)^3}","\frac{\sqrt{c} \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} \left(a e^2+c d^2\right)^3}+\frac{a e+c d x}{2 a \left(a+c x^2\right) (d+e x) \left(a e^2+c d^2\right)}+\frac{e \left(c d^2-3 a e^2\right)}{2 a (d+e x) \left(a e^2+c d^2\right)^2}-\frac{2 c d e^3 \log \left(a+c x^2\right)}{\left(a e^2+c d^2\right)^3}+\frac{4 c d e^3 \log (d+e x)}{\left(a e^2+c d^2\right)^3}",1,"((-2*e^3*(c*d^2 + a*e^2))/(d + e*x) + (c*(c*d^2 + a*e^2)*(c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 8*c*d*e^3*Log[d + e*x] - 4*c*d*e^3*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)","A",1
512,1,199,198,0.1446669,"\int \frac{(d+e x)^5}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(a + c*x^2)^3,x]","\frac{\frac{\sqrt{c} d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{5/2}}-\frac{2 \left(a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right)}{a \left(a+c x^2\right)^2}+\frac{8 a^3 e^5-5 a^2 c d e^3 (8 d+5 e x)+10 a c^2 d^3 e^2 x+3 c^3 d^5 x}{a^2 \left(a+c x^2\right)}+4 e^5 \log \left(a+c x^2\right)}{8 c^3}","-\frac{(d+e x)^2 \left(2 a e \left(2 a e^2+c d^2\right)-c d x \left(5 a e^2+3 c d^2\right)\right)}{8 a^2 c^2 \left(a+c x^2\right)}-\frac{d e^2 x \left(7 a e^2+3 c d^2\right)}{8 a^2 c^2}+\frac{d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left(a+c x^2\right)}{2 c^3}-\frac{(d+e x)^4 (a e-c d x)}{4 a c \left(a+c x^2\right)^2}",1,"((-2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)))/(a*(a + c*x^2)^2) + (8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(8*d + 5*e*x))/(a^2*(a + c*x^2)) + (Sqrt[c]*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*e^5*Log[a + c*x^2])/(8*c^3)","A",1
513,1,148,120,0.1009189,"\int \frac{(d+e x)^4}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(a + c*x^2)^3,x]","\frac{3 \left(a e^2+c d^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{5/2}}+\frac{-a^3 e^3 (8 d+3 e x)-a^2 c e \left(8 d^3+6 d^2 e x+16 d e^2 x^2+5 e^3 x^3\right)+a c^2 d^2 x \left(5 d^2+6 e^2 x^2\right)+3 c^3 d^4 x^3}{8 a^2 c^2 \left(a+c x^2\right)^2}","\frac{3 \left(a e^2+c d^2\right)^2 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{5/2}}-\frac{3 (d+e x) \left(a e^2+c d^2\right) (a e-c d x)}{8 a^2 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^3 (a e-c d x)}{4 a c \left(a+c x^2\right)^2}",1,"(3*c^3*d^4*x^3 - a^3*e^3*(8*d + 3*e*x) + a*c^2*d^2*x*(5*d^2 + 6*e^2*x^2) - a^2*c*e*(8*d^3 + 6*d^2*e*x + 16*d*e^2*x^2 + 5*e^3*x^3))/(8*a^2*c^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^2)^2*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))","A",1
514,1,127,98,0.1032428,"\int \frac{(d+e x)^3}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(a + c*x^2)^3,x]","\frac{\frac{\sqrt{a} \left(-2 a^3 e^3-a^2 c e \left(6 d^2+3 d e x+4 e^2 x^2\right)+a c^2 d x \left(5 d^2+3 e^2 x^2\right)+3 c^3 d^3 x^3\right)}{\left(a+c x^2\right)^2}+3 \sqrt{c} d \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^2}","\frac{3 d \left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left(a+c x^2\right)}+\frac{x (d+e x)^3}{4 a \left(a+c x^2\right)^2}",1,"((Sqrt[a]*(-2*a^3*e^3 + 3*c^3*d^3*x^3 + a*c^2*d*x*(5*d^2 + 3*e^2*x^2) - a^2*c*e*(6*d^2 + 3*d*e*x + 4*e^2*x^2)))/(a + c*x^2)^2 + 3*Sqrt[c]*d*(c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^2)","A",1
515,1,101,113,0.0683921,"\int \frac{(d+e x)^2}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(a + c*x^2)^3,x]","\frac{\left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{3/2}}+\frac{-a^2 e (4 d+e x)+a c x \left(5 d^2+e^2 x^2\right)+3 c^2 d^2 x^3}{8 a^2 c \left(a+c x^2\right)^2}","\frac{\left(a e^2+3 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{3/2}}-\frac{2 a d e-x \left(a e^2+3 c d^2\right)}{8 a^2 c \left(a+c x^2\right)}-\frac{(d+e x) (a e-c d x)}{4 a c \left(a+c x^2\right)^2}",1,"(3*c^2*d^2*x^3 - a^2*e*(4*d + e*x) + a*c*x*(5*d^2 + e^2*x^2))/(8*a^2*c*(a + c*x^2)^2) + ((3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))","A",1
516,1,71,75,0.0485237,"\int \frac{d+e x}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)/(a + c*x^2)^3,x]","\frac{\frac{\sqrt{a} \left(-2 a^2 e+5 a c d x+3 c^2 d x^3\right)}{\left(a+c x^2\right)^2}+3 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c}","\frac{3 d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{c}}+\frac{3 d x}{8 a^2 \left(a+c x^2\right)}+\frac{c d x-a e}{4 a c \left(a+c x^2\right)^2}",1,"((Sqrt[a]*(-2*a^2*e + 5*a*c*d*x + 3*c^2*d*x^3))/(a + c*x^2)^2 + 3*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c)","A",1
517,1,180,212,0.1357595,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^3),x]","\frac{\frac{\left(a e^2+c d^2\right) \left(4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right)}{a^2 \left(a+c x^2\right)}+\frac{\sqrt{c} d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 \left(a e^2+c d^2\right)^2 (a e+c d x)}{a \left(a+c x^2\right)^2}-4 e^5 \log \left(a+c x^2\right)+8 e^5 \log (d+e x)}{8 \left(a e^2+c d^2\right)^3}","\frac{4 a^2 e^3+c d x \left(7 a e^2+3 c d^2\right)}{8 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)^2}+\frac{\sqrt{c} d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} \left(a e^2+c d^2\right)^3}+\frac{a e+c d x}{4 a \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)}-\frac{e^5 \log \left(a+c x^2\right)}{2 \left(a e^2+c d^2\right)^3}+\frac{e^5 \log (d+e x)}{\left(a e^2+c d^2\right)^3}",1,"((2*(c*d^2 + a*e^2)^2*(a*e + c*d*x))/(a*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(4*a^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x))/(a^2*(a + c*x^2)) + (Sqrt[c]*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*e^5*Log[d + e*x] - 4*e^5*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^3)","A",1
518,1,241,300,0.3556242,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)^3),x]","\frac{\frac{c \left(a e^2+c d^2\right) \left(a^2 e^3 (16 d-7 e x)+12 a c d^2 e^2 x+3 c^2 d^4 x\right)}{a^2 \left(a+c x^2\right)}+\frac{3 \sqrt{c} \left(-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 c \left(a e^2+c d^2\right)^2 \left(a e (2 d-e x)+c d^2 x\right)}{a \left(a+c x^2\right)^2}-\frac{8 e^5 \left(a e^2+c d^2\right)}{d+e x}-24 c d e^5 \log \left(a+c x^2\right)+48 c d e^5 \log (d+e x)}{8 \left(a e^2+c d^2\right)^4}","-\frac{a e \left(c d^2-5 a e^2\right)-3 c d x \left(3 a e^2+c d^2\right)}{8 a^2 \left(a+c x^2\right) (d+e x) \left(a e^2+c d^2\right)^2}+\frac{3 e \left(c d^2-a e^2\right) \left(5 a e^2+c d^2\right)}{8 a^2 (d+e x) \left(a e^2+c d^2\right)^3}+\frac{3 \sqrt{c} \left(-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} \left(a e^2+c d^2\right)^4}+\frac{a e+c d x}{4 a \left(a+c x^2\right)^2 (d+e x) \left(a e^2+c d^2\right)}-\frac{3 c d e^5 \log \left(a+c x^2\right)}{\left(a e^2+c d^2\right)^4}+\frac{6 c d e^5 \log (d+e x)}{\left(a e^2+c d^2\right)^4}",1,"((-8*e^5*(c*d^2 + a*e^2))/(d + e*x) + (c*(c*d^2 + a*e^2)*(3*c^2*d^4*x + 12*a*c*d^2*e^2*x + a^2*e^3*(16*d - 7*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*x^2)^2) + (3*Sqrt[c]*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 48*c*d*e^5*Log[d + e*x] - 24*c*d*e^5*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)","A",1
519,1,197,155,0.1472449,"\int \frac{(d+e x)^4}{\left(a+c x^2\right)^4} \, dx","Integrate[(d + e*x)^4/(a + c*x^2)^4,x]","\frac{\left(a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{5/2}}+\frac{-a^4 e^3 (16 d+3 e x)-2 a^3 c e \left(16 d^3+9 d^2 e x+24 d e^2 x^2+4 e^3 x^3\right)+3 a^2 c^2 x \left(11 d^4+16 d^2 e^2 x^2+e^4 x^4\right)+2 a c^3 d^2 x^3 \left(20 d^2+9 e^2 x^2\right)+15 c^4 d^4 x^5}{48 a^3 c^2 \left(a+c x^2\right)^3}","\frac{\left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{5/2}}-\frac{(d+e x) \left(a e^2+5 c d^2\right) (a e-c d x)}{16 a^3 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left(a+c x^2\right)^2}+\frac{x (d+e x)^4}{6 a \left(a+c x^2\right)^3}",1,"(15*c^4*d^4*x^5 - a^4*e^3*(16*d + 3*e*x) + 2*a*c^3*d^2*x^3*(20*d^2 + 9*e^2*x^2) - 2*a^3*c*e*(16*d^3 + 9*d^2*e*x + 24*d*e^2*x^2 + 4*e^3*x^3) + 3*a^2*c^2*x*(11*d^4 + 16*d^2*e^2*x^2 + e^4*x^4))/(48*a^3*c^2*(a + c*x^2)^3) + ((5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(5/2))","A",1
520,1,155,156,0.1496633,"\int \frac{(d+e x)^3}{\left(a+c x^2\right)^4} \, dx","Integrate[(d + e*x)^3/(a + c*x^2)^4,x]","\frac{\frac{\sqrt{a} \left(-4 a^4 e^3-3 a^3 c e \left(8 d^2+3 d e x+4 e^2 x^2\right)+3 a^2 c^2 d x \left(11 d^2+8 e^2 x^2\right)+a c^3 d x^3 \left(40 d^2+9 e^2 x^2\right)+15 c^4 d^3 x^5\right)}{\left(a+c x^2\right)^3}+3 \sqrt{c} d \left(3 a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{48 a^{7/2} c^2}","\frac{d \left(3 a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{3/2}}-\frac{4 a e \left(a e^2+5 c d^2\right)-c d x \left(15 c d^2-a e^2\right)}{48 a^3 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left(a+c x^2\right)^2}+\frac{x (d+e x)^3}{6 a \left(a+c x^2\right)^3}",1,"((Sqrt[a]*(-4*a^4*e^3 + 15*c^4*d^3*x^5 - 3*a^3*c*e*(8*d^2 + 3*d*e*x + 4*e^2*x^2) + 3*a^2*c^2*d*x*(11*d^2 + 8*e^2*x^2) + a*c^3*d*x^3*(40*d^2 + 9*e^2*x^2)))/(a + c*x^2)^3 + 3*Sqrt[c]*d*(5*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7/2)*c^2)","A",1
521,1,127,145,0.0920275,"\int \frac{(d+e x)^2}{\left(a+c x^2\right)^4} \, dx","Integrate[(d + e*x)^2/(a + c*x^2)^4,x]","\frac{\left(a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{3/2}}+\frac{-a^3 e (16 d+3 e x)+a^2 c x \left(33 d^2+8 e^2 x^2\right)+a c^2 x^3 \left(40 d^2+3 e^2 x^2\right)+15 c^3 d^2 x^5}{48 a^3 c \left(a+c x^2\right)^3}","\frac{\left(a e^2+5 c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{3/2}}+\frac{x \left(a e^2+5 c d^2\right)}{16 a^3 c \left(a+c x^2\right)}-\frac{4 a d e-x \left(a e^2+5 c d^2\right)}{24 a^2 c \left(a+c x^2\right)^2}-\frac{(d+e x) (a e-c d x)}{6 a c \left(a+c x^2\right)^3}",1,"(15*c^3*d^2*x^5 - a^3*e*(16*d + 3*e*x) + a*c^2*x^3*(40*d^2 + 3*e^2*x^2) + a^2*c*x*(33*d^2 + 8*e^2*x^2))/(48*a^3*c*(a + c*x^2)^3) + ((5*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(3/2))","A",1
522,1,83,93,0.0476528,"\int \frac{d+e x}{\left(a+c x^2\right)^4} \, dx","Integrate[(d + e*x)/(a + c*x^2)^4,x]","\frac{\frac{\sqrt{a} \left(-8 a^3 e+33 a^2 c d x+40 a c^2 d x^3+15 c^3 d x^5\right)}{\left(a+c x^2\right)^3}+15 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{48 a^{7/2} c}","\frac{5 d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} \sqrt{c}}+\frac{5 d x}{16 a^3 \left(a+c x^2\right)}+\frac{5 d x}{24 a^2 \left(a+c x^2\right)^2}+\frac{c d x-a e}{6 a c \left(a+c x^2\right)^3}",1,"((Sqrt[a]*(-8*a^3*e + 33*a^2*c*d*x + 40*a*c^2*d*x^3 + 15*c^3*d*x^5))/(a + c*x^2)^3 + 15*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7/2)*c)","A",1
523,1,265,295,0.2185618,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^4} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^4),x]","\frac{\frac{2 \left(a e^2+c d^2\right)^2 \left(6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right)}{a^2 \left(a+c x^2\right)^2}+\frac{3 \left(a e^2+c d^2\right) \left(8 a^3 e^5+19 a^2 c d e^4 x+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right)}{a^3 \left(a+c x^2\right)}+\frac{3 \sqrt{c} d \left(35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{7/2}}+\frac{8 \left(a e^2+c d^2\right)^3 (a e+c d x)}{a \left(a+c x^2\right)^3}-24 e^7 \log \left(a+c x^2\right)+48 e^7 \log (d+e x)}{48 \left(a e^2+c d^2\right)^4}","\frac{6 a^2 e^3+c d x \left(11 a e^2+5 c d^2\right)}{24 a^2 \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)^2}+\frac{8 a^3 e^5+c d x \left(19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right)}{16 a^3 \left(a+c x^2\right) \left(a e^2+c d^2\right)^3}+\frac{\sqrt{c} d \left(35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} \left(a e^2+c d^2\right)^4}+\frac{a e+c d x}{6 a \left(a+c x^2\right)^3 \left(a e^2+c d^2\right)}-\frac{e^7 \log \left(a+c x^2\right)}{2 \left(a e^2+c d^2\right)^4}+\frac{e^7 \log (d+e x)}{\left(a e^2+c d^2\right)^4}",1,"((8*(c*d^2 + a*e^2)^3*(a*e + c*d*x))/(a*(a + c*x^2)^3) + (2*(c*d^2 + a*e^2)^2*(6*a^2*e^3 + 5*c^2*d^3*x + 11*a*c*d*e^2*x))/(a^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^2)*(8*a^3*e^5 + 5*c^3*d^5*x + 16*a*c^2*d^3*e^2*x + 19*a^2*c*d*e^4*x))/(a^3*(a + c*x^2)) + (3*Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3*e^6)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 48*e^7*Log[d + e*x] - 24*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^4)","A",1
524,1,336,430,0.4348296,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)^4} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)^4),x]","\frac{\frac{2 c \left(a e^2+c d^2\right)^2 \left(a^2 e^3 (24 d-11 e x)+18 a c d^2 e^2 x+5 c^2 d^4 x\right)}{a^2 \left(a+c x^2\right)^2}+\frac{3 c \left(a e^2+c d^2\right) \left(a^3 e^5 (48 d-19 e x)+47 a^2 c d^2 e^4 x+23 a c^2 d^4 e^2 x+5 c^3 d^6 x\right)}{a^3 \left(a+c x^2\right)}+\frac{3 \sqrt{c} \left(-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{a^{7/2}}+\frac{8 c \left(a e^2+c d^2\right)^3 \left(a e (2 d-e x)+c d^2 x\right)}{a \left(a+c x^2\right)^3}-\frac{48 e^7 \left(a e^2+c d^2\right)}{d+e x}-192 c d e^7 \log \left(a+c x^2\right)+384 c d e^7 \log (d+e x)}{48 \left(a e^2+c d^2\right)^5}","-\frac{a e \left(c d^2-7 a e^2\right)-c d x \left(13 a e^2+5 c d^2\right)}{24 a^2 \left(a+c x^2\right)^2 (d+e x) \left(a e^2+c d^2\right)^2}-\frac{a e \left(5 c d^2-7 a e^2\right) \left(5 a e^2+c d^2\right)-3 c d x \left(29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right)}{48 a^3 \left(a+c x^2\right) (d+e x) \left(a e^2+c d^2\right)^3}+\frac{e \left(-35 a^3 e^6+47 a^2 c d^2 e^4+23 a c^2 d^4 e^2+5 c^3 d^6\right)}{16 a^3 (d+e x) \left(a e^2+c d^2\right)^4}+\frac{\sqrt{c} \left(-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} \left(a e^2+c d^2\right)^5}+\frac{a e+c d x}{6 a \left(a+c x^2\right)^3 (d+e x) \left(a e^2+c d^2\right)}-\frac{4 c d e^7 \log \left(a+c x^2\right)}{\left(a e^2+c d^2\right)^5}+\frac{8 c d e^7 \log (d+e x)}{\left(a e^2+c d^2\right)^5}",1,"((-48*e^7*(c*d^2 + a*e^2))/(d + e*x) + (3*c*(c*d^2 + a*e^2)*(5*c^3*d^6*x + 23*a*c^2*d^4*e^2*x + 47*a^2*c*d^2*e^4*x + a^3*e^5*(48*d - 19*e*x)))/(a^3*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(5*c^2*d^4*x + 18*a*c*d^2*e^2*x + a^2*e^3*(24*d - 11*e*x)))/(a^2*(a + c*x^2)^2) + (8*c*(c*d^2 + a*e^2)^3*(c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*x^2)^3) + (3*Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 384*c*d*e^7*Log[d + e*x] - 192*c*d*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^5)","A",1
525,1,177,207,0.1372709,"\int (d+e x)^4 \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^4*Sqrt[a + c*x^2],x]","\frac{15 a \left(a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\sqrt{c} \sqrt{a+c x^2} \left(-a^2 e^3 (128 d+15 e x)+2 a c e \left(160 d^3+90 d^2 e x+32 d e^2 x^2+5 e^3 x^3\right)+8 c^2 x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)\right)}{240 c^{5/2}}","\frac{x \sqrt{a+c x^2} \left(a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right)}{16 c^2}+\frac{a \left(a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{16 c^{5/2}}+\frac{e \left(a+c x^2\right)^{3/2} \left(3 e x \left(16 c d^2-5 a e^2\right)+8 d \left(13 c d^2-8 a e^2\right)\right)}{120 c^2}+\frac{e \left(a+c x^2\right)^{3/2} (d+e x)^3}{6 c}+\frac{3 d e \left(a+c x^2\right)^{3/2} (d+e x)^2}{10 c}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(-(a^2*e^3*(128*d + 15*e*x)) + 2*a*c*e*(160*d^3 + 90*d^2*e*x + 32*d*e^2*x^2 + 5*e^3*x^3) + 8*c^2*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + 15*a*(8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(240*c^(5/2))","A",1
526,1,132,144,0.0801808,"\int (d+e x)^3 \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[a + c*x^2],x]","\frac{\sqrt{a+c x^2} \left(-16 a^2 e^3+a c e \left(120 d^2+45 d e x+8 e^2 x^2\right)+6 c^2 x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)\right)+15 a \sqrt{c} d \left(4 c d^2-3 a e^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{120 c^2}","\frac{a d \left(4 c d^2-3 a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{3/2}}+\frac{e \left(a+c x^2\right)^{3/2} \left(8 \left(6 c d^2-a e^2\right)+21 c d e x\right)}{60 c^2}+\frac{d x \sqrt{a+c x^2} \left(4 c d^2-3 a e^2\right)}{8 c}+\frac{e \left(a+c x^2\right)^{3/2} (d+e x)^2}{5 c}",1,"(Sqrt[a + c*x^2]*(-16*a^2*e^3 + a*c*e*(120*d^2 + 45*d*e*x + 8*e^2*x^2) + 6*c^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) + 15*a*Sqrt[c]*d*(4*c*d^2 - 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(120*c^2)","A",1
527,1,99,119,0.0602913,"\int (d+e x)^2 \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[a + c*x^2],x]","\frac{\sqrt{c} \sqrt{a+c x^2} \left(a e (16 d+3 e x)+2 c x \left(6 d^2+8 d e x+3 e^2 x^2\right)\right)-3 a \left(a e^2-4 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{24 c^{3/2}}","\frac{a \left(4 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} \left(4 c d^2-a e^2\right)}{8 c}+\frac{5 d e \left(a+c x^2\right)^{3/2}}{12 c}+\frac{e \left(a+c x^2\right)^{3/2} (d+e x)}{4 c}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(a*e*(16*d + 3*e*x) + 2*c*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) - 3*a*(-4*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(24*c^(3/2))","A",1
528,1,67,67,0.0344723,"\int (d+e x) \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)*Sqrt[a + c*x^2],x]","\frac{\sqrt{a+c x^2} (2 a e+c x (3 d+2 e x))+3 a \sqrt{c} d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{6 c}","\frac{1}{2} d x \sqrt{a+c x^2}+\frac{a d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 \sqrt{c}}+\frac{e \left(a+c x^2\right)^{3/2}}{3 c}",1,"(Sqrt[a + c*x^2]*(2*a*e + c*x*(3*d + 2*e*x)) + 3*a*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c)","A",1
529,1,99,103,0.0464943,"\int \frac{\sqrt{a+c x^2}}{d+e x} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x),x]","\frac{-\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)-\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+e \sqrt{a+c x^2}}{e^2}","-\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2}-\frac{\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^2}+\frac{\sqrt{a+c x^2}}{e}",1,"(e*Sqrt[a + c*x^2] - Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^2","A",1
530,1,134,110,0.1316813,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^2,x]","\frac{\frac{c d \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\sqrt{a e^2+c d^2}}-\frac{c d \log (d+e x)}{\sqrt{a e^2+c d^2}}-\frac{e \sqrt{a+c x^2}}{d+e x}+\sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{e^2}","\frac{c d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^2}",1,"(-((e*Sqrt[a + c*x^2])/(d + e*x)) - (c*d*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] + Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (c*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/e^2","A",1
531,1,127,103,0.1068084,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^3,x]","\frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} (c d x-a e)-a c (d+e x)^2 \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)+a c (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left(a e^2+c d^2\right)^{3/2}}","-\frac{\sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left(a e^2+c d^2\right)}-\frac{a c \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{3/2}}",1,"(Sqrt[c*d^2 + a*e^2]*(-(a*e) + c*d*x)*Sqrt[a + c*x^2] + a*c*(d + e*x)^2*Log[d + e*x] - a*c*(d + e*x)^2*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(2*(c*d^2 + a*e^2)^(3/2)*(d + e*x)^2)","A",1
532,1,173,144,0.1737986,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^4,x]","\frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left(-2 a^2 e^3-a c e \left(5 d^2+3 d e x+2 e^2 x^2\right)+c^2 d^2 x (3 d+e x)\right)-3 a c^2 d (d+e x)^3 \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)+3 a c^2 d (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left(a e^2+c d^2\right)^{5/2}}","-\frac{a c^2 d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{5/2}}-\frac{c d \sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left(a e^2+c d^2\right)^2}-\frac{e \left(a+c x^2\right)^{3/2}}{3 (d+e x)^3 \left(a e^2+c d^2\right)}",1,"(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-2*a^2*e^3 + c^2*d^2*x*(3*d + e*x) - a*c*e*(5*d^2 + 3*d*e*x + 2*e^2*x^2)) + 3*a*c^2*d*(d + e*x)^3*Log[d + e*x] - 3*a*c^2*d*(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(5/2)*(d + e*x)^3)","A",1
533,1,248,206,0.2460589,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^5,x]","\frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left(c^2 d (d+e x)^3 \left(2 c d^2-13 a e^2\right)+2 c d (d+e x) \left(a e^2+c d^2\right)^2+c (d+e x)^2 \left(2 c d^2-3 a e^2\right) \left(a e^2+c d^2\right)-6 \left(a e^2+c d^2\right)^3\right)+3 a c^2 e (d+e x)^4 \left(a e^2-4 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)-3 a c^2 e (d+e x)^4 \left(a e^2-4 c d^2\right) \log (d+e x)}{24 e (d+e x)^4 \left(a e^2+c d^2\right)^{7/2}}","-\frac{a c^2 \left(4 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{8 \left(a e^2+c d^2\right)^{7/2}}-\frac{5 c d e \left(a+c x^2\right)^{3/2}}{12 (d+e x)^3 \left(a e^2+c d^2\right)^2}-\frac{c \sqrt{a+c x^2} \left(4 c d^2-a e^2\right) (a e-c d x)}{8 (d+e x)^2 \left(a e^2+c d^2\right)^3}-\frac{e \left(a+c x^2\right)^{3/2}}{4 (d+e x)^4 \left(a e^2+c d^2\right)}",1,"(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-6*(c*d^2 + a*e^2)^3 + 2*c*d*(c*d^2 + a*e^2)^2*(d + e*x) + c*(2*c*d^2 - 3*a*e^2)*(c*d^2 + a*e^2)*(d + e*x)^2 + c^2*d*(2*c*d^2 - 13*a*e^2)*(d + e*x)^3) - 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[d + e*x] + 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(24*e*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^4)","A",1
534,1,231,255,0.1606144,"\int (d+e x)^4 \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^4*(a + c*x^2)^(3/2),x]","\frac{105 a^2 \left(a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\sqrt{c} \sqrt{a+c x^2} \left(-a^3 e^3 (1024 d+105 e x)+2 a^2 c e \left(1792 d^3+840 d^2 e x+256 d e^2 x^2+35 e^3 x^3\right)+8 a c^2 x \left(350 d^4+896 d^3 e x+980 d^2 e^2 x^2+512 d e^3 x^3+105 e^4 x^4\right)+16 c^3 x^3 \left(70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right)\right)}{4480 c^{5/2}}","\frac{x \left(a+c x^2\right)^{3/2} \left(a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right)}{64 c^2}+\frac{3 a x \sqrt{a+c x^2} \left(a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right)}{128 c^2}+\frac{3 a^2 \left(a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{128 c^{5/2}}+\frac{e \left(a+c x^2\right)^{5/2} \left(5 e x \left(26 c d^2-7 a e^2\right)+4 d \left(67 c d^2-32 a e^2\right)\right)}{560 c^2}+\frac{e \left(a+c x^2\right)^{5/2} (d+e x)^3}{8 c}+\frac{11 d e \left(a+c x^2\right)^{5/2} (d+e x)^2}{56 c}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(-(a^3*e^3*(1024*d + 105*e*x)) + 2*a^2*c*e*(1792*d^3 + 840*d^2*e*x + 256*d*e^2*x^2 + 35*e^3*x^3) + 16*c^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 8*a*c^2*x*(350*d^4 + 896*d^3*e*x + 980*d^2*e^2*x^2 + 512*d*e^3*x^3 + 105*e^4*x^4)) + 105*a^2*(16*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(4480*c^(5/2))","A",1
535,1,174,180,0.1171197,"\int (d+e x)^3 \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^(3/2),x]","\frac{\sqrt{a+c x^2} \left(-32 a^3 e^3+a^2 c e \left(336 d^2+105 d e x+16 e^2 x^2\right)+2 a c^2 x \left(175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right)+4 c^3 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)\right)-105 a^2 \sqrt{c} d \left(a e^2-2 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{560 c^2}","\frac{3 a^2 d \left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{16 c^{3/2}}+\frac{e \left(a+c x^2\right)^{5/2} \left(4 \left(8 c d^2-a e^2\right)+15 c d e x\right)}{70 c^2}+\frac{d x \left(a+c x^2\right)^{3/2} \left(2 c d^2-a e^2\right)}{8 c}+\frac{3 a d x \sqrt{a+c x^2} \left(2 c d^2-a e^2\right)}{16 c}+\frac{e \left(a+c x^2\right)^{5/2} (d+e x)^2}{7 c}",1,"(Sqrt[a + c*x^2]*(-32*a^3*e^3 + a^2*c*e*(336*d^2 + 105*d*e*x + 16*e^2*x^2) + 4*c^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 2*a*c^2*x*(175*d^3 + 336*d^2*e*x + 245*d*e^2*x^2 + 64*e^3*x^3)) - 105*a^2*Sqrt[c]*d*(-2*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(560*c^2)","A",1
536,1,132,154,0.0884662,"\int (d+e x)^2 \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^(3/2),x]","\frac{\sqrt{c} \sqrt{a+c x^2} \left(3 a^2 e (32 d+5 e x)+2 a c x \left(75 d^2+96 d e x+35 e^2 x^2\right)+4 c^2 x^3 \left(15 d^2+24 d e x+10 e^2 x^2\right)\right)-15 a^2 \left(a e^2-6 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{240 c^{3/2}}","\frac{a^2 \left(6 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{16 c^{3/2}}+\frac{x \left(a+c x^2\right)^{3/2} \left(6 c d^2-a e^2\right)}{24 c}+\frac{a x \sqrt{a+c x^2} \left(6 c d^2-a e^2\right)}{16 c}+\frac{7 d e \left(a+c x^2\right)^{5/2}}{30 c}+\frac{e \left(a+c x^2\right)^{5/2} (d+e x)}{6 c}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^2*e*(32*d + 5*e*x) + 4*c^2*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 2*a*c*x*(75*d^2 + 96*d*e*x + 35*e^2*x^2)) - 15*a^2*(-6*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(240*c^(3/2))","A",1
537,1,88,87,0.0509989,"\int (d+e x) \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(a + c*x^2)^(3/2),x]","\frac{\sqrt{a+c x^2} \left(8 a^2 e+a c x (25 d+16 e x)+2 c^2 x^3 (5 d+4 e x)\right)+15 a^2 \sqrt{c} d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{40 c}","\frac{3 a^2 d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 \sqrt{c}}+\frac{1}{4} d x \left(a+c x^2\right)^{3/2}+\frac{3}{8} a d x \sqrt{a+c x^2}+\frac{e \left(a+c x^2\right)^{5/2}}{5 c}",1,"(Sqrt[a + c*x^2]*(8*a^2*e + 2*c^2*x^3*(5*d + 4*e*x) + a*c*x*(25*d + 16*e*x)) + 15*a^2*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(40*c)","A",1
538,1,195,159,0.447648,"\int \frac{\left(a+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x),x]","\frac{e \sqrt{a+c x^2} \left(8 a e^2+c \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-6 \sqrt{c} d \left(a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-6 \left(a e^2+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)-\frac{3 \sqrt{a} \sqrt{c} d e^2 \sqrt{a+c x^2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}}{6 e^4}","-\frac{\left(a e^2+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^4}-\frac{\sqrt{c} d \left(3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 e^4}+\frac{\sqrt{a+c x^2} \left(2 \left(a e^2+c d^2\right)-c d e x\right)}{2 e^3}+\frac{\left(a+c x^2\right)^{3/2}}{3 e}",1,"(e*Sqrt[a + c*x^2]*(8*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - (3*Sqrt[a]*Sqrt[c]*d*e^2*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a] - 6*Sqrt[c]*d*(c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - 6*(c*d^2 + a*e^2)^(3/2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(6*e^4)","A",1
539,1,179,153,0.1717583,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^2,x]","\frac{-\frac{e \sqrt{a+c x^2} \left(2 a e^2+c \left(6 d^2+3 d e x-e^2 x^2\right)\right)}{d+e x}+3 \sqrt{c} \left(a e^2+2 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+6 c d \sqrt{a e^2+c d^2} \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)-6 c d \sqrt{a e^2+c d^2} \log (d+e x)}{2 e^4}","\frac{3 \sqrt{c} \left(a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 e^4}+\frac{3 c d \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^4}-\frac{3 c \sqrt{a+c x^2} (2 d-e x)}{2 e^3}-\frac{\left(a+c x^2\right)^{3/2}}{e (d+e x)}",1,"(-((e*Sqrt[a + c*x^2]*(2*a*e^2 + c*(6*d^2 + 3*d*e*x - e^2*x^2)))/(d + e*x)) - 6*c*d*Sqrt[c*d^2 + a*e^2]*Log[d + e*x] + 3*Sqrt[c]*(2*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + 6*c*d*Sqrt[c*d^2 + a*e^2]*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(2*e^4)","A",1
540,1,189,161,0.2060454,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^3,x]","\frac{-6 c^{3/2} d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\frac{e \sqrt{a+c x^2} \left(c \left(6 d^2+9 d e x+2 e^2 x^2\right)-a e^2\right)}{(d+e x)^2}-\frac{3 c \left(a e^2+2 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\sqrt{a e^2+c d^2}}+\frac{3 c \left(a e^2+2 c d^2\right) \log (d+e x)}{\sqrt{a e^2+c d^2}}}{2 e^4}","-\frac{3 c^{3/2} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^4}-\frac{3 c \left(a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 e^4 \sqrt{a e^2+c d^2}}+\frac{3 c \sqrt{a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac{\left(a+c x^2\right)^{3/2}}{2 e (d+e x)^2}",1,"((e*Sqrt[a + c*x^2]*(-(a*e^2) + c*(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (3*c*(2*c*d^2 + a*e^2)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] - 6*c^(3/2)*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*c*(2*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(2*e^4)","A",1
541,1,242,200,0.2578131,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^4} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^4,x]","\frac{-\frac{e \sqrt{a+c x^2} \left(2 a^2 e^4+a c e^2 \left(5 d^2+9 d e x+8 e^2 x^2\right)+c^2 d^2 \left(6 d^2+15 d e x+11 e^2 x^2\right)\right)}{(d+e x)^3 \left(a e^2+c d^2\right)}+6 c^{3/2} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\frac{3 c^2 d \left(3 a e^2+2 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{3 c^2 d \left(3 a e^2+2 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}}{6 e^4}","\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^4}+\frac{c^2 d \left(3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 e^4 \left(a e^2+c d^2\right)^{3/2}}-\frac{c \sqrt{a+c x^2} \left(e x \left(2 a e^2+3 c d^2\right)+d \left(a e^2+2 c d^2\right)\right)}{2 e^3 (d+e x)^2 \left(a e^2+c d^2\right)}-\frac{\left(a+c x^2\right)^{3/2}}{3 e (d+e x)^3}",1,"(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 9*d*e*x + 8*e^2*x^2) + c^2*d^2*(6*d^2 + 15*d*e*x + 11*e^2*x^2)))/((c*d^2 + a*e^2)*(d + e*x)^3)) - (3*c^2*d*(2*c*d^2 + 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + 6*c^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c^2*d*(2*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(6*e^4)","A",1
542,1,198,153,0.2845039,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^5} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^5,x]","\frac{1}{8} \left(-\frac{3 a^2 c^2 \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{5/2}}+\frac{3 a^2 c^2 \log (d+e x)}{\left(a e^2+c d^2\right)^{5/2}}+\frac{\sqrt{a+c x^2} \left(-2 a^3 e^3-a^2 c e \left(5 d^2+4 d e x+5 e^2 x^2\right)+a c^2 d x \left(5 d^2+4 d e x+5 e^2 x^2\right)+2 c^3 d^3 x^3\right)}{(d+e x)^4 \left(a e^2+c d^2\right)^2}\right)","-\frac{3 a^2 c^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{8 \left(a e^2+c d^2\right)^{5/2}}-\frac{3 a c \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left(a e^2+c d^2\right)^2}-\frac{\left(a+c x^2\right)^{3/2} (a e-c d x)}{4 (d+e x)^4 \left(a e^2+c d^2\right)}",1,"((Sqrt[a + c*x^2]*(-2*a^3*e^3 + 2*c^3*d^3*x^3 - a^2*c*e*(5*d^2 + 4*d*e*x + 5*e^2*x^2) + a*c^2*d*x*(5*d^2 + 4*d*e*x + 5*e^2*x^2)))/((c*d^2 + a*e^2)^2*(d + e*x)^4) + (3*a^2*c^2*Log[d + e*x])/(c*d^2 + a*e^2)^(5/2) - (3*a^2*c^2*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(5/2))/8","A",1
543,1,272,195,0.4241971,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^6} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^6,x]","\frac{1}{40} \left(-\frac{15 a^2 c^3 d \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{7/2}}+\frac{15 a^2 c^3 d \log (d+e x)}{\left(a e^2+c d^2\right)^{7/2}}+\frac{\sqrt{a+c x^2} \left(-8 a^4 e^5-2 a^3 c e^3 \left(13 d^2+5 d e x+8 e^2 x^2\right)-a^2 c^2 e \left(33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right)+a c^3 d^2 x \left(25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right)+2 c^4 d^4 x^3 (5 d+e x)\right)}{(d+e x)^5 \left(a e^2+c d^2\right)^3}\right)","-\frac{3 a^2 c^3 d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{8 \left(a e^2+c d^2\right)^{7/2}}-\frac{3 a c^2 d \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left(a e^2+c d^2\right)^3}-\frac{c d \left(a+c x^2\right)^{3/2} (a e-c d x)}{4 (d+e x)^4 \left(a e^2+c d^2\right)^2}-\frac{e \left(a+c x^2\right)^{5/2}}{5 (d+e x)^5 \left(a e^2+c d^2\right)}",1,"((Sqrt[a + c*x^2]*(-8*a^4*e^5 + 2*c^4*d^4*x^3*(5*d + e*x) - 2*a^3*c*e^3*(13*d^2 + 5*d*e*x + 8*e^2*x^2) + a*c^3*d^2*x*(25*d^3 + 29*d^2*e*x + 45*d*e^2*x^2 + 9*e^3*x^3) - a^2*c^2*e*(33*d^4 + 45*d^3*e*x + 77*d^2*e^2*x^2 + 25*d*e^3*x^3 + 8*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d + e*x)^5) + (15*a^2*c^3*d*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^2*c^3*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/40","A",1
544,1,358,269,0.7363279,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^7} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^7,x]","\frac{1}{240} \left(\frac{15 a^2 c^3 \left(a e^2-6 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{9/2}}+\frac{15 a^2 c^3 \left(6 c d^2-a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{9/2}}-\frac{\sqrt{a+c x^2} \left(-c^2 (d+e x)^4 \left(-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right) \left(a e^2+c d^2\right)-c^3 d (d+e x)^5 \left(-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right)-2 c^2 d (d+e x)^3 \left(9 a e^2+2 c d^2\right) \left(a e^2+c d^2\right)^2-104 c d (d+e x) \left(a e^2+c d^2\right)^4+2 c (d+e x)^2 \left(35 a e^2+38 c d^2\right) \left(a e^2+c d^2\right)^3+40 \left(a e^2+c d^2\right)^5\right)}{e^3 (d+e x)^6 \left(a e^2+c d^2\right)^4}\right)","-\frac{a^2 c^3 \left(6 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{16 \left(a e^2+c d^2\right)^{9/2}}-\frac{a c^2 \sqrt{a+c x^2} \left(6 c d^2-a e^2\right) (a e-c d x)}{16 (d+e x)^2 \left(a e^2+c d^2\right)^4}-\frac{7 c d e \left(a+c x^2\right)^{5/2}}{30 (d+e x)^5 \left(a e^2+c d^2\right)^2}-\frac{c \left(a+c x^2\right)^{3/2} \left(6 c d^2-a e^2\right) (a e-c d x)}{24 (d+e x)^4 \left(a e^2+c d^2\right)^3}-\frac{e \left(a+c x^2\right)^{5/2}}{6 (d+e x)^6 \left(a e^2+c d^2\right)}",1,"(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x) + 2*c*(c*d^2 + a*e^2)^3*(38*c*d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*(d + e*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x)^6)) + (15*a^2*c^3*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) + (15*a^2*c^3*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/240","A",1
545,1,283,307,0.2439364,"\int (d+e x)^4 \left(a+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^4*(a + c*x^2)^(5/2),x]","\frac{a^3 \left(3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{256 c^{5/2}}+\frac{\sqrt{a+c x^2} \left(-5 a^4 e^3 (2048 d+189 e x)+10 a^3 c e \left(4608 d^3+1890 d^2 e x+512 d e^2 x^2+63 e^3 x^3\right)+24 a^2 c^2 x \left(2310 d^4+5760 d^3 e x+6195 d^2 e^2 x^2+3200 d e^3 x^3+651 e^4 x^4\right)+16 a c^3 x^3 \left(2730 d^4+8640 d^3 e x+10710 d^2 e^2 x^2+6080 d e^3 x^3+1323 e^4 x^4\right)+64 c^4 x^5 \left(210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right)\right)}{80640 c^2}","\frac{x \left(a+c x^2\right)^{5/2} \left(3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right)}{480 c^2}+\frac{a x \left(a+c x^2\right)^{3/2} \left(3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right)}{384 c^2}+\frac{a^2 x \sqrt{a+c x^2} \left(3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right)}{256 c^2}+\frac{a^3 \left(3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{256 c^{5/2}}+\frac{e \left(a+c x^2\right)^{7/2} \left(7 e x \left(116 c d^2-27 a e^2\right)+16 d \left(103 c d^2-40 a e^2\right)\right)}{5040 c^2}+\frac{e \left(a+c x^2\right)^{7/2} (d+e x)^3}{10 c}+\frac{13 d e \left(a+c x^2\right)^{7/2} (d+e x)^2}{90 c}",1,"(Sqrt[a + c*x^2]*(-5*a^4*e^3*(2048*d + 189*e*x) + 10*a^3*c*e*(4608*d^3 + 1890*d^2*e*x + 512*d*e^2*x^2 + 63*e^3*x^3) + 64*c^4*x^5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 24*a^2*c^2*x*(2310*d^4 + 5760*d^3*e*x + 6195*d^2*e^2*x^2 + 3200*d*e^3*x^3 + 651*e^4*x^4) + 16*a*c^3*x^3*(2730*d^4 + 8640*d^3*e*x + 10710*d^2*e^2*x^2 + 6080*d*e^3*x^3 + 1323*e^4*x^4)))/(80640*c^2) + (a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(256*c^(5/2))","A",1
546,1,216,216,0.1600956,"\int (d+e x)^3 \left(a+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^(5/2),x]","\frac{\sqrt{a+c x^2} \left(-256 a^4 e^3+a^3 c e \left(3456 d^2+945 d e x+128 e^2 x^2\right)+6 a^2 c^2 x \left(924 d^3+1728 d^2 e x+1239 d e^2 x^2+320 e^3 x^3\right)+8 a c^3 x^3 \left(546 d^3+1296 d^2 e x+1071 d e^2 x^2+304 e^3 x^3\right)+16 c^4 x^5 \left(84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right)\right)-315 a^3 \sqrt{c} d \left(3 a e^2-8 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{8064 c^2}","\frac{5 a^3 d \left(8 c d^2-3 a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{128 c^{3/2}}+\frac{5 a^2 d x \sqrt{a+c x^2} \left(8 c d^2-3 a e^2\right)}{128 c}+\frac{e \left(a+c x^2\right)^{7/2} \left(16 \left(10 c d^2-a e^2\right)+77 c d e x\right)}{504 c^2}+\frac{d x \left(a+c x^2\right)^{5/2} \left(8 c d^2-3 a e^2\right)}{48 c}+\frac{5 a d x \left(a+c x^2\right)^{3/2} \left(8 c d^2-3 a e^2\right)}{192 c}+\frac{e \left(a+c x^2\right)^{7/2} (d+e x)^2}{9 c}",1,"(Sqrt[a + c*x^2]*(-256*a^4*e^3 + a^3*c*e*(3456*d^2 + 945*d*e*x + 128*e^2*x^2) + 16*c^4*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 8*a*c^3*x^3*(546*d^3 + 1296*d^2*e*x + 1071*d*e^2*x^2 + 304*e^3*x^3) + 6*a^2*c^2*x*(924*d^3 + 1728*d^2*e*x + 1239*d*e^2*x^2 + 320*e^3*x^3)) - 315*a^3*Sqrt[c]*d*(-8*c*d^2 + 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(8064*c^2)","A",1
547,1,162,189,0.1207792,"\int (d+e x)^2 \left(a+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^(5/2),x]","\frac{\sqrt{c} \sqrt{a+c x^2} \left(3 a^3 e (256 d+35 e x)+2 a^2 c x \left(924 d^2+1152 d e x+413 e^2 x^2\right)+8 a c^2 x^3 \left(182 d^2+288 d e x+119 e^2 x^2\right)+16 c^3 x^5 \left(28 d^2+48 d e x+21 e^2 x^2\right)\right)-105 a^3 \left(a e^2-8 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{2688 c^{3/2}}","\frac{5 a^3 \left(8 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} \left(8 c d^2-a e^2\right)}{128 c}+\frac{x \left(a+c x^2\right)^{5/2} \left(8 c d^2-a e^2\right)}{48 c}+\frac{5 a x \left(a+c x^2\right)^{3/2} \left(8 c d^2-a e^2\right)}{192 c}+\frac{9 d e \left(a+c x^2\right)^{7/2}}{56 c}+\frac{e \left(a+c x^2\right)^{7/2} (d+e x)}{8 c}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2)) - 105*a^3*(-8*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2688*c^(3/2))","A",1
548,1,108,107,0.0720559,"\int (d+e x) \left(a+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(a + c*x^2)^(5/2),x]","\frac{105 a^3 \sqrt{c} d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\sqrt{a+c x^2} \left(48 a^3 e+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)+8 c^3 x^5 (7 d+6 e x)\right)}{336 c}","\frac{5 a^3 d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{16 \sqrt{c}}+\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{1}{6} d x \left(a+c x^2\right)^{5/2}+\frac{5}{24} a d x \left(a+c x^2\right)^{3/2}+\frac{e \left(a+c x^2\right)^{7/2}}{7 c}",1,"(Sqrt[a + c*x^2]*(48*a^3*e + 8*c^3*x^5*(7*d + 6*e*x) + 3*a^2*c*x*(77*d + 48*e*x) + 2*a*c^2*x^3*(91*d + 72*e*x)) + 105*a^3*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(336*c)","A",1
549,1,332,226,0.9264757,"\int \frac{\left(a+c x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x),x]","\frac{-\frac{5 \sqrt{c} d \sqrt{a+c x^2} \left(3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{c} x \left(5 a+2 c x^2\right) \sqrt{\frac{c x^2}{a}+1}\right)}{8 e \sqrt{\frac{c x^2}{a}+1}}-\frac{5 \left(a e^2+c d^2\right) \left(\sqrt{\frac{c x^2}{a}+1} \left(-e \sqrt{a+c x^2} \left(8 a e^2+6 c d^2-3 c d e x+2 c e^2 x^2\right)+6 \left(a e^2+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+6 \sqrt{c} d \left(a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)+3 \sqrt{a} \sqrt{c} d e^2 \sqrt{a+c x^2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{6 e^5 \sqrt{\frac{c x^2}{a}+1}}+\left(a+c x^2\right)^{5/2}}{5 e}","-\frac{\sqrt{c} d \left(15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 e^6}-\frac{\left(a e^2+c d^2\right)^{5/2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^6}+\frac{\sqrt{a+c x^2} \left(8 \left(a e^2+c d^2\right)^2-c d e x \left(7 a e^2+4 c d^2\right)\right)}{8 e^5}+\frac{\left(a+c x^2\right)^{3/2} \left(4 \left(a e^2+c d^2\right)-3 c d e x\right)}{12 e^3}+\frac{\left(a+c x^2\right)^{5/2}}{5 e}",1,"((a + c*x^2)^(5/2) - (5*Sqrt[c]*d*Sqrt[a + c*x^2]*(Sqrt[c]*x*(5*a + 2*c*x^2)*Sqrt[1 + (c*x^2)/a] + 3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(8*e*Sqrt[1 + (c*x^2)/a]) - (5*(c*d^2 + a*e^2)*(3*Sqrt[a]*Sqrt[c]*d*e^2*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[1 + (c*x^2)/a]*(-(e*Sqrt[a + c*x^2]*(6*c*d^2 + 8*a*e^2 - 3*c*d*e*x + 2*c*e^2*x^2)) + 6*Sqrt[c]*d*(c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + 6*(c*d^2 + a*e^2)^(3/2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])))/(6*e^5*Sqrt[1 + (c*x^2)/a]))/(5*e)","A",1
550,1,239,219,0.2329527,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^2,x]","\frac{15 \sqrt{c} \left(3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+e \sqrt{a+c x^2} \left(9 c e x \left(3 a e^2+4 c d^2\right)-\frac{24 \left(a e^2+c d^2\right)^2}{d+e x}-16 c d \left(7 a e^2+6 c d^2\right)-16 c^2 d e^2 x^2+6 c^2 e^3 x^3\right)+120 c d \left(a e^2+c d^2\right)^{3/2} \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)-120 c d \left(a e^2+c d^2\right)^{3/2} \log (d+e x)}{24 e^6}","\frac{5 \sqrt{c} \left(3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 e^6}+\frac{5 c d \left(a e^2+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^6}-\frac{5 c \sqrt{a+c x^2} \left(8 d \left(a e^2+c d^2\right)-e x \left(3 a e^2+4 c d^2\right)\right)}{8 e^5}-\frac{5 c \left(a+c x^2\right)^{3/2} (4 d-3 e x)}{12 e^3}-\frac{\left(a+c x^2\right)^{5/2}}{e (d+e x)}",1,"(e*Sqrt[a + c*x^2]*(-16*c*d*(6*c*d^2 + 7*a*e^2) + 9*c*e*(4*c*d^2 + 3*a*e^2)*x - 16*c^2*d*e^2*x^2 + 6*c^2*e^3*x^3 - (24*(c*d^2 + a*e^2)^2)/(d + e*x)) - 120*c*d*(c*d^2 + a*e^2)^(3/2)*Log[d + e*x] + 15*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + 120*c*d*(c*d^2 + a*e^2)^(3/2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(24*e^6)","A",1
551,1,281,213,0.3207473,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^3,x]","\frac{-\frac{15 c \left(a^2 e^4+5 a c d^2 e^2+4 c^2 d^4\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\sqrt{a e^2+c d^2}}+\frac{15 c \left(a^2 e^4+5 a c d^2 e^2+4 c^2 d^4\right) \log (d+e x)}{\sqrt{a e^2+c d^2}}+\frac{e \sqrt{a+c x^2} \left(-3 a^2 e^4+a c e^2 \left(35 d^2+55 d e x+14 e^2 x^2\right)+c^2 \left(60 d^4+90 d^3 e x+20 d^2 e^2 x^2-5 d e^3 x^3+2 e^4 x^4\right)\right)}{(d+e x)^2}-15 c^{3/2} d \left(3 a e^2+4 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{6 e^6}","-\frac{5 c^{3/2} d \left(3 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 e^6}-\frac{5 c \sqrt{a e^2+c d^2} \left(a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 e^6}+\frac{5 c \sqrt{a+c x^2} \left(a e^2+4 c d^2-2 c d e x\right)}{2 e^5}+\frac{5 c \left(a+c x^2\right)^{3/2} (4 d+e x)}{6 e^3 (d+e x)}-\frac{\left(a+c x^2\right)^{5/2}}{2 e (d+e x)^2}",1,"((e*Sqrt[a + c*x^2]*(-3*a^2*e^4 + a*c*e^2*(35*d^2 + 55*d*e*x + 14*e^2*x^2) + c^2*(60*d^4 + 90*d^3*e*x + 20*d^2*e^2*x^2 - 5*d*e^3*x^3 + 2*e^4*x^4)))/(d + e*x)^2 + (15*c*(4*c^2*d^4 + 5*a*c*d^2*e^2 + a^2*e^4)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] - 15*c^(3/2)*d*(4*c*d^2 + 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (15*c*(4*c^2*d^4 + 5*a*c*d^2*e^2 + a^2*e^4)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(6*e^6)","A",1
552,1,260,222,0.2862475,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^4,x]","\frac{-\frac{e \sqrt{a+c x^2} \left(2 a^2 e^4+a c e^2 \left(5 d^2+15 d e x+14 e^2 x^2\right)+c^2 \left(60 d^4+150 d^3 e x+110 d^2 e^2 x^2+15 d e^3 x^3-3 e^4 x^4\right)\right)}{(d+e x)^3}+15 c^{3/2} \left(a e^2+4 c d^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\frac{15 c^2 d \left(3 a e^2+4 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\sqrt{a e^2+c d^2}}-\frac{15 c^2 d \left(3 a e^2+4 c d^2\right) \log (d+e x)}{\sqrt{a e^2+c d^2}}}{6 e^6}","\frac{5 c^{3/2} \left(a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 e^6}+\frac{5 c^2 d \left(3 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 e^6 \sqrt{a e^2+c d^2}}-\frac{5 c \sqrt{a+c x^2} \left(a e^2+4 c d^2+2 c d e x\right)}{2 e^5 (d+e x)}+\frac{5 c \left(a+c x^2\right)^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac{\left(a+c x^2\right)^{5/2}}{3 e (d+e x)^3}",1,"(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 15*d*e*x + 14*e^2*x^2) + c^2*(60*d^4 + 150*d^3*e*x + 110*d^2*e^2*x^2 + 15*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)^3) - (15*c^2*d*(4*c*d^2 + 3*a*e^2)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] + 15*c^(3/2)*(4*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*d*(4*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(6*e^6)","A",1
553,1,299,287,0.4965581,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^5} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^5,x]","\frac{-\frac{15 c^2 \left(3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{15 c^2 \left(3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}-120 c^{5/2} d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+e \sqrt{a+c x^2} \left(\frac{c^2 d \left(139 a e^2+154 c d^2\right)}{(d+e x) \left(a e^2+c d^2\right)}-\frac{c \left(27 a e^2+86 c d^2\right)}{(d+e x)^2}+\frac{34 c d \left(a e^2+c d^2\right)}{(d+e x)^3}-\frac{6 \left(a e^2+c d^2\right)^2}{(d+e x)^4}+24 c^2\right)}{24 e^6}","-\frac{5 c^2 \left(3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{8 e^6 \left(a e^2+c d^2\right)^{3/2}}-\frac{5 c^{5/2} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^6}+\frac{5 c^2 \sqrt{a+c x^2} \left(e x \left(3 a e^2+4 c d^2\right)+8 d \left(a e^2+c d^2\right)\right)}{8 e^5 (d+e x) \left(a e^2+c d^2\right)}-\frac{5 c \left(a+c x^2\right)^{3/2} \left(3 e x \left(a e^2+2 c d^2\right)+d \left(a e^2+4 c d^2\right)\right)}{24 e^3 (d+e x)^3 \left(a e^2+c d^2\right)}-\frac{\left(a+c x^2\right)^{5/2}}{4 e (d+e x)^4}",1,"(e*Sqrt[a + c*x^2]*(24*c^2 - (6*(c*d^2 + a*e^2)^2)/(d + e*x)^4 + (34*c*d*(c*d^2 + a*e^2))/(d + e*x)^3 - (c*(86*c*d^2 + 27*a*e^2))/(d + e*x)^2 + (c^2*d*(154*c*d^2 + 139*a*e^2))/((c*d^2 + a*e^2)*(d + e*x))) + (15*c^2*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - 120*c^(5/2)*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (15*c^2*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(24*e^6)","A",1
554,1,359,314,0.5141413,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^6} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^6,x]","\frac{-\frac{e \sqrt{a+c x^2} \left(c^2 (d+e x)^4 \left(184 a^2 e^4+503 a c d^2 e^2+274 c^2 d^4\right)-c^2 d (d+e x)^3 \left(311 a e^2+326 c d^2\right) \left(a e^2+c d^2\right)-126 c d (d+e x) \left(a e^2+c d^2\right)^3+2 c (d+e x)^2 \left(44 a e^2+137 c d^2\right) \left(a e^2+c d^2\right)^2+24 \left(a e^2+c d^2\right)^4\right)}{(d+e x)^5 \left(a e^2+c d^2\right)^2}+\frac{15 c^3 d \left(15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{5/2}}-\frac{15 c^3 d \left(15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{5/2}}+120 c^{5/2} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{120 e^6}","-\frac{c^2 \sqrt{a+c x^2} \left(e x \left(8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right)+d \left(a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right)\right)}{8 e^5 (d+e x)^2 \left(a e^2+c d^2\right)^2}+\frac{c^3 d \left(15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{8 e^6 \left(a e^2+c d^2\right)^{5/2}}+\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^6}-\frac{c \left(a+c x^2\right)^{3/2} \left(e x \left(4 a e^2+7 c d^2\right)+d \left(a e^2+4 c d^2\right)\right)}{12 e^3 (d+e x)^4 \left(a e^2+c d^2\right)}-\frac{\left(a+c x^2\right)^{5/2}}{5 e (d+e x)^5}",1,"(-((e*Sqrt[a + c*x^2]*(24*(c*d^2 + a*e^2)^4 - 126*c*d*(c*d^2 + a*e^2)^3*(d + e*x) + 2*c*(c*d^2 + a*e^2)^2*(137*c*d^2 + 44*a*e^2)*(d + e*x)^2 - c^2*d*(c*d^2 + a*e^2)*(326*c*d^2 + 311*a*e^2)*(d + e*x)^3 + c^2*(274*c^2*d^4 + 503*a*c*d^2*e^2 + 184*a^2*e^4)*(d + e*x)^4))/((c*d^2 + a*e^2)^2*(d + e*x)^5)) - (15*c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[d + e*x])/(c*d^2 + a*e^2)^(5/2) + 120*c^(5/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(5/2))/(120*e^6)","A",1
555,1,305,203,0.5711359,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^7} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]","\frac{1}{48} \left(-\frac{15 a^3 c^3 \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{7/2}}+\frac{15 a^3 c^3 \log (d+e x)}{\left(a e^2+c d^2\right)^{7/2}}+\frac{\sqrt{a+c x^2} \left(-8 a^5 e^5-2 a^4 c e^3 \left(13 d^2+6 d e x+13 e^2 x^2\right)-a^3 c^2 e \left(33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right)+a^2 c^3 d x \left(33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right)+2 a c^4 d^3 x^3 \left(13 d^2+6 d e x+13 e^2 x^2\right)+8 c^5 d^5 x^5\right)}{(d+e x)^6 \left(a e^2+c d^2\right)^3}\right)","-\frac{5 a^3 c^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{16 \left(a e^2+c d^2\right)^{7/2}}-\frac{5 a^2 c^2 \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left(a e^2+c d^2\right)^3}-\frac{5 a c \left(a+c x^2\right)^{3/2} (a e-c d x)}{24 (d+e x)^4 \left(a e^2+c d^2\right)^2}-\frac{\left(a+c x^2\right)^{5/2} (a e-c d x)}{6 (d+e x)^6 \left(a e^2+c d^2\right)}",1,"((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) + 2*a*c^4*d^3*x^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4) + a^2*c^3*d*x*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d + e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/48","A",1
556,1,403,246,0.6141071,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^8} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^8,x]","-\frac{5 a^3 c^4 d \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{16 \left(a e^2+c d^2\right)^{9/2}}+\frac{5 a^3 c^4 d \log (d+e x)}{16 \left(a e^2+c d^2\right)^{9/2}}-\frac{\sqrt{a+c x^2} \left(2 c^2 (d+e x)^4 \left(72 a^2 e^4+159 a c d^2 e^2+80 c^2 d^4\right) \left(a e^2+c d^2\right)^2-c^3 d (d+e x)^5 \left(57 a^2 e^4+30 a c d^2 e^2+8 c^2 d^4\right) \left(a e^2+c d^2\right)-c^3 (d+e x)^6 \left(-48 a^3 e^6+87 a^2 c d^2 e^4+38 a c^2 d^4 e^2+8 c^3 d^6\right)-2 c^2 d (d+e x)^3 \left(197 a e^2+200 c d^2\right) \left(a e^2+c d^2\right)^3-232 c d (d+e x) \left(a e^2+c d^2\right)^5+8 c (d+e x)^2 \left(18 a e^2+55 c d^2\right) \left(a e^2+c d^2\right)^4+48 \left(a e^2+c d^2\right)^6\right)}{336 e^5 (d+e x)^7 \left(a e^2+c d^2\right)^4}","-\frac{5 a^3 c^4 d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{16 \left(a e^2+c d^2\right)^{9/2}}-\frac{5 a^2 c^3 d \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left(a e^2+c d^2\right)^4}-\frac{5 a c^2 d \left(a+c x^2\right)^{3/2} (a e-c d x)}{24 (d+e x)^4 \left(a e^2+c d^2\right)^3}-\frac{c d \left(a+c x^2\right)^{5/2} (a e-c d x)}{6 (d+e x)^6 \left(a e^2+c d^2\right)^2}-\frac{e \left(a+c x^2\right)^{7/2}}{7 (d+e x)^7 \left(a e^2+c d^2\right)}",1,"-1/336*(Sqrt[a + c*x^2]*(48*(c*d^2 + a*e^2)^6 - 232*c*d*(c*d^2 + a*e^2)^5*(d + e*x) + 8*c*(c*d^2 + a*e^2)^4*(55*c*d^2 + 18*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^3*(200*c*d^2 + 197*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2 + a*e^2)^2*(80*c^2*d^4 + 159*a*c*d^2*e^2 + 72*a^2*e^4)*(d + e*x)^4 - c^3*d*(c*d^2 + a*e^2)*(8*c^2*d^4 + 30*a*c*d^2*e^2 + 57*a^2*e^4)*(d + e*x)^5 - c^3*(8*c^3*d^6 + 38*a*c^2*d^4*e^2 + 87*a^2*c*d^2*e^4 - 48*a^3*e^6)*(d + e*x)^6))/(e^5*(c*d^2 + a*e^2)^4*(d + e*x)^7) + (5*a^3*c^4*d*Log[d + e*x])/(16*(c*d^2 + a*e^2)^(9/2)) - (5*a^3*c^4*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(16*(c*d^2 + a*e^2)^(9/2))","A",1
557,1,489,332,1.1708529,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^9} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^9,x]","\frac{\frac{105 a^3 c^4 \left(a e^2-8 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{11/2}}+\frac{105 a^3 c^4 \left(8 c d^2-a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{11/2}}-\frac{\sqrt{a+c x^2} \left(2 c^2 (d+e x)^4 \left(413 a^2 e^4+880 a c d^2 e^2+440 c^2 d^4\right) \left(a e^2+c d^2\right)^3-2 c^3 d (d+e x)^5 \left(87 a^2 e^4+32 a c d^2 e^2+8 c^2 d^4\right) \left(a e^2+c d^2\right)^2-c^3 (d+e x)^6 \left(-105 a^3 e^6+282 a^2 c d^2 e^4+88 a c^2 d^4 e^2+16 c^3 d^6\right) \left(a e^2+c d^2\right)-c^4 d (d+e x)^7 \left(-663 a^3 e^6+370 a^2 c d^2 e^4+104 a c^2 d^4 e^2+16 c^3 d^6\right)-8 c^2 d (d+e x)^3 \left(307 a e^2+310 c d^2\right) \left(a e^2+c d^2\right)^4-1584 c d (d+e x) \left(a e^2+c d^2\right)^6+8 c (d+e x)^2 \left(119 a e^2+362 c d^2\right) \left(a e^2+c d^2\right)^5+336 \left(a e^2+c d^2\right)^7\right)}{(d+e x)^8 \left(a e^3+c d^2 e\right)^5}}{2688}","-\frac{5 a^3 c^4 \left(8 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{128 \left(a e^2+c d^2\right)^{11/2}}-\frac{5 a^2 c^3 \sqrt{a+c x^2} \left(8 c d^2-a e^2\right) (a e-c d x)}{128 (d+e x)^2 \left(a e^2+c d^2\right)^5}-\frac{5 a c^2 \left(a+c x^2\right)^{3/2} \left(8 c d^2-a e^2\right) (a e-c d x)}{192 (d+e x)^4 \left(a e^2+c d^2\right)^4}-\frac{9 c d e \left(a+c x^2\right)^{7/2}}{56 (d+e x)^7 \left(a e^2+c d^2\right)^2}-\frac{c \left(a+c x^2\right)^{5/2} \left(8 c d^2-a e^2\right) (a e-c d x)}{48 (d+e x)^6 \left(a e^2+c d^2\right)^3}-\frac{e \left(a+c x^2\right)^{7/2}}{8 (d+e x)^8 \left(a e^2+c d^2\right)}",1,"(-((Sqrt[a + c*x^2]*(336*(c*d^2 + a*e^2)^7 - 1584*c*d*(c*d^2 + a*e^2)^6*(d + e*x) + 8*c*(c*d^2 + a*e^2)^5*(362*c*d^2 + 119*a*e^2)*(d + e*x)^2 - 8*c^2*d*(c*d^2 + a*e^2)^4*(310*c*d^2 + 307*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2 + a*e^2)^3*(440*c^2*d^4 + 880*a*c*d^2*e^2 + 413*a^2*e^4)*(d + e*x)^4 - 2*c^3*d*(c*d^2 + a*e^2)^2*(8*c^2*d^4 + 32*a*c*d^2*e^2 + 87*a^2*e^4)*(d + e*x)^5 - c^3*(c*d^2 + a*e^2)*(16*c^3*d^6 + 88*a*c^2*d^4*e^2 + 282*a^2*c*d^2*e^4 - 105*a^3*e^6)*(d + e*x)^6 - c^4*d*(16*c^3*d^6 + 104*a*c^2*d^4*e^2 + 370*a^2*c*d^2*e^4 - 663*a^3*e^6)*(d + e*x)^7))/((c*d^2*e + a*e^3)^5*(d + e*x)^8)) + (105*a^3*c^4*(8*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(11/2) + (105*a^3*c^4*(-8*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(11/2))/2688","A",1
558,1,56,56,0.0307536,"\int \frac{\sqrt{2+x^2}}{1+4 x} \, dx","Integrate[Sqrt[2 + x^2]/(1 + 4*x),x]","\frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left(\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right)-\frac{1}{16} \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)","\frac{\sqrt{x^2+2}}{4}-\frac{1}{16} \sqrt{33} \tanh ^{-1}\left(\frac{8-x}{\sqrt{33} \sqrt{x^2+2}}\right)-\frac{1}{16} \sinh ^{-1}\left(\frac{x}{\sqrt{2}}\right)",1,"Sqrt[2 + x^2]/4 - ArcSinh[x/Sqrt[2]]/16 - (Sqrt[33]*ArcTanh[(8 - x)/(Sqrt[33]*Sqrt[2 + x^2])])/16","A",1
559,1,57,67,0.0448869,"\int \frac{\sqrt{2+4 x^2}}{5+4 x} \, dx","Integrate[Sqrt[2 + 4*x^2]/(5 + 4*x),x]","\frac{1}{4} \sqrt{4 x^2+2}-\frac{1}{8} \sqrt{33} \tanh ^{-1}\left(\frac{2-5 x}{\sqrt{33 x^2+\frac{33}{2}}}\right)-\frac{5}{8} \sinh ^{-1}\left(\sqrt{2} x\right)","\frac{\sqrt{2 x^2+1}}{2 \sqrt{2}}-\frac{1}{8} \sqrt{33} \tanh ^{-1}\left(\frac{\sqrt{\frac{2}{33}} (2-5 x)}{\sqrt{2 x^2+1}}\right)-\frac{5}{8} \sinh ^{-1}\left(\sqrt{2} x\right)",1,"Sqrt[2 + 4*x^2]/4 - (5*ArcSinh[Sqrt[2]*x])/8 - (Sqrt[33]*ArcTanh[(2 - 5*x)/Sqrt[33/2 + 33*x^2]])/8","A",1
560,1,50,55,0.0416696,"\int (2+3 x) \sqrt{-5+7 x^2} \, dx","Integrate[(2 + 3*x)*Sqrt[-5 + 7*x^2],x]","\left(x^2+x-\frac{5}{7}\right) \sqrt{7 x^2-5}-\frac{5 \log \left(\sqrt{7} \sqrt{7 x^2-5}+7 x\right)}{\sqrt{7}}","\frac{1}{7} \left(7 x^2-5\right)^{3/2}+x \sqrt{7 x^2-5}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{7} x}{\sqrt{7 x^2-5}}\right)}{\sqrt{7}}",1,"(-5/7 + x + x^2)*Sqrt[-5 + 7*x^2] - (5*Log[7*x + Sqrt[7]*Sqrt[-5 + 7*x^2]])/Sqrt[7]","A",1
561,1,126,161,0.0937864,"\int \frac{(d+e x)^4}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^4/Sqrt[a + c*x^2],x]","\frac{3 \left(3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\sqrt{c} e \sqrt{a+c x^2} \left(c \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)-a e^2 (64 d+9 e x)\right)}{24 c^{5/2}}","\frac{\left(3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{5/2}}+\frac{e \sqrt{a+c x^2} \left(e x \left(26 c d^2-9 a e^2\right)+4 d \left(19 c d^2-16 a e^2\right)\right)}{24 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^3}{4 c}+\frac{7 d e \sqrt{a+c x^2} (d+e x)^2}{12 c}",1,"(Sqrt[c]*e*Sqrt[a + c*x^2]*(-(a*e^2*(64*d + 9*e*x)) + c*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 3*(8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(24*c^(5/2))","A",1
562,1,92,110,0.0604587,"\int \frac{(d+e x)^3}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[a + c*x^2],x]","\frac{e \sqrt{a+c x^2} \left(c \left(18 d^2+9 d e x+2 e^2 x^2\right)-4 a e^2\right)+3 \sqrt{c} d \left(2 c d^2-3 a e^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{6 c^2}","\frac{d \left(2 c d^2-3 a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} \left(4 \left(4 c d^2-a e^2\right)+5 c d e x\right)}{6 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c}",1,"(e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*Sqrt[c]*d*(2*c*d^2 - 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c^2)","A",1
563,1,71,86,0.0411353,"\int \frac{(d+e x)^2}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[a + c*x^2],x]","\frac{\left(2 c d^2-a e^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)+\sqrt{c} e \sqrt{a+c x^2} (4 d+e x)}{2 c^{3/2}}","\frac{\left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c}",1,"(Sqrt[c]*e*(4*d + e*x)*Sqrt[a + c*x^2] + (2*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2*c^(3/2))","A",1
564,1,46,43,0.021598,"\int \frac{d+e x}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)/Sqrt[a + c*x^2],x]","\frac{d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}",1,"(e*Sqrt[a + c*x^2])/c + (d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c]","A",1
565,1,54,54,0.0086139,"\int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + c*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}","-\frac{\tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}",1,"-(ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]/Sqrt[c*d^2 + a*e^2])","A",1
566,1,115,91,0.0847298,"\int \frac{1}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + c*x^2]),x]","-\frac{e \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{c d \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{c d \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}","-\frac{e \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{c d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"-((e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x))) + (c*d*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - (c*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2)","A",1
567,1,161,145,0.1388514,"\int \frac{1}{(d+e x)^3 \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[a + c*x^2]),x]","\frac{-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left(a e^2+c d (4 d+3 e x)\right)-c (d+e x)^2 \left(2 c d^2-a e^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)+c (d+e x)^2 \left(2 c d^2-a e^2\right) \log (d+e x)}{2 (d+e x)^2 \left(a e^2+c d^2\right)^{5/2}}","-\frac{3 c d e \sqrt{a+c x^2}}{2 (d+e x) \left(a e^2+c d^2\right)^2}-\frac{e \sqrt{a+c x^2}}{2 (d+e x)^2 \left(a e^2+c d^2\right)}-\frac{c \left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{5/2}}",1,"(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(a*e^2 + c*d*(4*d + 3*e*x))) + c*(2*c*d^2 - a*e^2)*(d + e*x)^2*Log[d + e*x] - c*(2*c*d^2 - a*e^2)*(d + e*x)^2*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(2*(c*d^2 + a*e^2)^(5/2)*(d + e*x)^2)","A",1
568,1,209,198,0.183185,"\int \frac{1}{(d+e x)^4 \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]","\frac{-3 c^2 d (d+e x)^3 \left(2 c d^2-3 a e^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)+3 c^2 d (d+e x)^3 \left(2 c d^2-3 a e^2\right) \log (d+e x)-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left(5 c d (d+e x) \left(a e^2+c d^2\right)+c (d+e x)^2 \left(11 c d^2-4 a e^2\right)+2 \left(a e^2+c d^2\right)^2\right)}{6 (d+e x)^3 \left(a e^2+c d^2\right)^{7/2}}","-\frac{c^2 d \left(2 c d^2-3 a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{7/2}}-\frac{c e \sqrt{a+c x^2} \left(11 c d^2-4 a e^2\right)}{6 (d+e x) \left(a e^2+c d^2\right)^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left(a e^2+c d^2\right)^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left(a e^2+c d^2\right)}",1,"(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2 - 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^3)","A",1
569,1,127,162,0.1388272,"\int \frac{(d+e x)^4}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^4/(a + c*x^2)^(3/2),x]","\frac{a^2 e^3 (16 d+3 e x)+a c e \left(-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right)+2 c^2 d^4 x}{2 a c^2 \sqrt{a+c x^2}}+\frac{3 \left(4 c d^2 e^2-a e^4\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{2 c^{5/2}}","\frac{3 e^2 \left(4 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{5/2}}-\frac{e \sqrt{a+c x^2} \left(e x \left(2 c d^2-3 a e^2\right)+4 d \left(c d^2-4 a e^2\right)\right)}{2 a c^2}-\frac{(d+e x)^3 (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^2}{a c}",1,"(2*c^2*d^4*x + a^2*e^3*(16*d + 3*e*x) + a*c*e*(-8*d^3 - 12*d^2*e*x + 8*d*e^2*x^2 + e^3*x^3))/(2*a*c^2*Sqrt[a + c*x^2]) + (3*(4*c*d^2*e^2 - a*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(2*c^(5/2))","A",1
570,1,91,106,0.1130328,"\int \frac{(d+e x)^3}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(a + c*x^2)^(3/2),x]","\frac{2 a^2 e^3+a c e \left(-3 d^2-3 d e x+e^2 x^2\right)+c^2 d^3 x}{a c^2 \sqrt{a+c x^2}}+\frac{3 d e^2 \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{3/2}}","\frac{3 d e^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2}}-\frac{e \sqrt{a+c x^2} \left(2 \left(c d^2-a e^2\right)+c d e x\right)}{a c^2}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}}",1,"(2*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + e^2*x^2))/(a*c^2*Sqrt[a + c*x^2]) + (3*d*e^2*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(3/2)","A",1
571,1,69,83,0.0709169,"\int \frac{(d+e x)^2}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(a + c*x^2)^(3/2),x]","\frac{e^2 \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{a c \sqrt{a+c x^2}}","\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}}",1,"(-2*a*d*e + c*d^2*x - a*e^2*x)/(a*c*Sqrt[a + c*x^2]) + (e^2*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(3/2)","A",1
572,1,27,28,0.0142511,"\int \frac{d+e x}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(a + c*x^2)^(3/2),x]","\frac{c d x-a e}{a c \sqrt{a+c x^2}}","-\frac{a e-c d x}{a c \sqrt{a+c x^2}}",1,"(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2])","A",1
573,1,94,94,0.0557582,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{a e+c d x}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}","\frac{a e+c d x}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (e^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2)","A",1
574,1,139,151,0.1205348,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)^(3/2)),x]","\frac{-a^2 e^3+a c e \left(2 d^2+d e x-2 e^2 x^2\right)+c^2 d^2 x (d+e x)}{a \sqrt{a+c x^2} (d+e x) \left(a e^2+c d^2\right)^2}-\frac{3 c d e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{5/2}}","\frac{e \sqrt{a+c x^2} \left(c d^2-2 a e^2\right)}{a (d+e x) \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x) \left(a e^2+c d^2\right)}-\frac{3 c d e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{5/2}}",1,"(-(a^2*e^3) + c^2*d^2*x*(d + e*x) + a*c*e*(2*d^2 + d*e*x - 2*e^2*x^2))/(a*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a + c*x^2]) - (3*c*d*e^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(5/2)","A",1
575,1,240,223,0.4227698,"\int \frac{1}{(d+e x)^3 \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(a + c*x^2)^(3/2)),x]","\frac{1}{2} \left(\frac{-a^3 e^5-a^2 c e^3 \left(10 d^2+11 d e x+3 e^2 x^2\right)+a c^2 d e \left(6 d^3+6 d^2 e x-14 d e^2 x^2-13 e^3 x^3\right)+2 c^3 d^3 x (d+e x)^2}{a \sqrt{a+c x^2} (d+e x)^2 \left(a e^2+c d^2\right)^3}+\frac{3 c e^2 \left(a e^2-4 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{7/2}}+\frac{3 c e^2 \left(4 c d^2-a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{7/2}}\right)","\frac{c d e \sqrt{a+c x^2} \left(2 c d^2-13 a e^2\right)}{2 a (d+e x) \left(a e^2+c d^2\right)^3}+\frac{e \sqrt{a+c x^2} \left(2 c d^2-3 a e^2\right)}{2 a (d+e x)^2 \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^2 \left(a e^2+c d^2\right)}-\frac{3 c e^2 \left(4 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{7/2}}",1,"((-(a^3*e^5) + 2*c^3*d^3*x*(d + e*x)^2 - a^2*c*e^3*(10*d^2 + 11*d*e*x + 3*e^2*x^2) + a*c^2*d*e*(6*d^3 + 6*d^2*e*x - 14*d*e^2*x^2 - 13*e^3*x^3))/(a*(c*d^2 + a*e^2)^3*(d + e*x)^2*Sqrt[a + c*x^2]) + (3*c*e^2*(4*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) + (3*c*e^2*(-4*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/2","A",1
576,1,279,293,0.6837365,"\int \frac{1}{(d+e x)^4 \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^4*(a + c*x^2)^(3/2)),x]","\frac{1}{6} \left(\frac{\sqrt{a+c x^2} \left(\frac{6 c^2 \left(a^2 e^3 (e x-4 d)+2 a c d^2 e (2 d-3 e x)+c^2 d^4 x\right)}{a \left(a+c x^2\right)}+\frac{c e^3 \left(10 a e^2-47 c d^2\right)}{d+e x}-\frac{11 c d e^3 \left(a e^2+c d^2\right)}{(d+e x)^2}-\frac{2 e^3 \left(a e^2+c d^2\right)^2}{(d+e x)^3}\right)}{\left(a e^2+c d^2\right)^4}-\frac{15 c^2 d e^2 \left(4 c d^2-3 a e^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{9/2}}+\frac{15 c^2 d e^2 \left(4 c d^2-3 a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{9/2}}\right)","\frac{c e \sqrt{a+c x^2} \left(16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right)}{6 a (d+e x) \left(a e^2+c d^2\right)^4}-\frac{5 c^2 d e^2 \left(4 c d^2-3 a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{9/2}}+\frac{c d e \sqrt{a+c x^2} \left(6 c d^2-29 a e^2\right)}{6 a (d+e x)^2 \left(a e^2+c d^2\right)^3}+\frac{e \sqrt{a+c x^2} \left(3 c d^2-4 a e^2\right)}{3 a (d+e x)^3 \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^3 \left(a e^2+c d^2\right)}",1,"((Sqrt[a + c*x^2]*((-2*e^3*(c*d^2 + a*e^2)^2)/(d + e*x)^3 - (11*c*d*e^3*(c*d^2 + a*e^2))/(d + e*x)^2 + (c*e^3*(-47*c*d^2 + 10*a*e^2))/(d + e*x) + (6*c^2*(c^2*d^4*x + 2*a*c*d^2*e*(2*d - 3*e*x) + a^2*e^3*(-4*d + e*x)))/(a*(a + c*x^2))))/(c*d^2 + a*e^2)^4 + (15*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) - (15*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/6","A",1
577,1,167,191,0.2416007,"\int \frac{(d+e x)^5}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(a + c*x^2)^(5/2),x]","\frac{8 a^4 e^5+a^3 c e^3 \left(-20 d^2-15 d e x+12 e^2 x^2\right)+a^2 c^2 e \left(-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right)+a c^3 d^3 x \left(3 d^2+10 e^2 x^2\right)+2 c^4 d^5 x^3}{3 a^2 c^3 \left(a+c x^2\right)^{3/2}}+\frac{5 d e^4 \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{5/2}}","-\frac{2 (d+e x)^2 \left(2 a^2 e^3-c d x \left(3 a e^2+c d^2\right)\right)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{e \sqrt{a+c x^2} \left(4 \left(-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right)+c d e x \left(7 a e^2+2 c d^2\right)\right)}{3 a^2 c^3}+\frac{5 d e^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{5/2}}-\frac{(d+e x)^4 (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*x^3 + 3*e^4*x^4))/(3*a^2*c^3*(a + c*x^2)^(3/2)) + (5*d*e^4*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(5/2)","A",1
578,1,130,161,0.2046645,"\int \frac{(d+e x)^4}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(a + c*x^2)^(5/2),x]","\frac{-a^3 e^3 (8 d+3 e x)-4 a^2 c e \left(d^3+3 d e^2 x^2+e^3 x^3\right)+3 a c^2 d^2 x \left(d^2+2 e^2 x^2\right)+2 c^3 d^4 x^3}{3 a^2 c^2 \left(a+c x^2\right)^{3/2}}+\frac{e^4 \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{5/2}}","-\frac{d e \sqrt{a+c x^2} \left(5 a e^2+2 c d^2\right)}{3 a^2 c^2}-\frac{(d+e x) \left(a e \left(3 a e^2+c d^2\right)-2 c d x \left(2 a e^2+c d^2\right)\right)}{3 a^2 c^2 \sqrt{a+c x^2}}+\frac{e^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(2*c^3*d^4*x^3 - a^3*e^3*(8*d + 3*e*x) + 3*a*c^2*d^2*x*(d^2 + 2*e^2*x^2) - 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3))/(3*a^2*c^2*(a + c*x^2)^(3/2)) + (e^4*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(5/2)","A",1
579,1,78,79,0.0964236,"\int \frac{(d+e x)^3}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(a + c*x^2)^(5/2),x]","\frac{-2 a^3 e^3-3 a^2 c e \left(d^2+e^2 x^2\right)+3 a c^2 d x \left(d^2+e^2 x^2\right)+2 c^3 d^3 x^3}{3 a^2 c^2 \left(a+c x^2\right)^{3/2}}","-\frac{2 \left(a e^2+c d^2\right) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(-2*a^3*e^3 + 2*c^3*d^3*x^3 - 3*a^2*c*e*(d^2 + e^2*x^2) + 3*a*c^2*d*x*(d^2 + e^2*x^2))/(3*a^2*c^2*(a + c*x^2)^(3/2))","A",1
580,1,57,58,0.0768557,"\int \frac{(d+e x)^2}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(a + c*x^2)^(5/2),x]","\frac{-2 a^2 d e+a c x \left(3 d^2+e^2 x^2\right)+2 c^2 d^2 x^3}{3 a^2 c \left(a+c x^2\right)^{3/2}}","\frac{x (d+e x)^2}{3 a \left(a+c x^2\right)^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}}",1,"(-2*a^2*d*e + 2*c^2*d^2*x^3 + a*c*x*(3*d^2 + e^2*x^2))/(3*a^2*c*(a + c*x^2)^(3/2))","A",1
581,1,43,51,0.017535,"\int \frac{d+e x}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(a + c*x^2)^(5/2),x]","\frac{-a^2 e+3 a c d x+2 c^2 d x^3}{3 a^2 c \left(a+c x^2\right)^{3/2}}","\frac{2 d x}{3 a^2 \sqrt{a+c x^2}}+\frac{c d x-a e}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(-(a^2*e) + 3*a*c*d*x + 2*c^2*d*x^3)/(3*a^2*c*(a + c*x^2)^(3/2))","A",1
582,1,153,154,0.1359821,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^(5/2)),x]","\frac{4 a^3 e^3+a^2 c e \left(d^2+6 d e x+3 e^2 x^2\right)+a c^2 d x \left(3 d^2+5 e^2 x^2\right)+2 c^3 d^3 x^3}{3 a^2 \left(a+c x^2\right)^{3/2} \left(a e^2+c d^2\right)^2}-\frac{e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{5/2}}","\frac{3 a^2 e^3+c d x \left(5 a e^2+2 c d^2\right)}{3 a^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{3 a \left(a+c x^2\right)^{3/2} \left(a e^2+c d^2\right)}-\frac{e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{5/2}}",1,"(4*a^3*e^3 + 2*c^3*d^3*x^3 + a^2*c*e*(d^2 + 6*d*e*x + 3*e^2*x^2) + a*c^2*d*x*(3*d^2 + 5*e^2*x^2))/(3*a^2*(c*d^2 + a*e^2)^2*(a + c*x^2)^(3/2)) - (e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(5/2)","A",1
583,1,237,244,0.2799129,"\int \frac{1}{(d+e x)^2 \left(a+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]","\frac{-3 a^4 e^5+2 a^3 c e^3 \left(7 d^2+4 d e x-6 e^2 x^2\right)+a^2 c^2 e \left(2 d^4+11 d^3 e x+21 d^2 e^2 x^2+7 d e^3 x^3-8 e^4 x^4\right)+3 a c^3 d^2 x \left(d^3+d^2 e x+3 d e^2 x^2+3 e^3 x^3\right)+2 c^4 d^4 x^3 (d+e x)}{3 a^2 \left(a+c x^2\right)^{3/2} (d+e x) \left(a e^2+c d^2\right)^3}-\frac{5 c d e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{7/2}}","\frac{e \sqrt{a+c x^2} \left(-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right)}{3 a^2 (d+e x) \left(a e^2+c d^2\right)^3}-\frac{a e \left(c d^2-4 a e^2\right)-c d x \left(7 a e^2+2 c d^2\right)}{3 a^2 \sqrt{a+c x^2} (d+e x) \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{3 a \left(a+c x^2\right)^{3/2} (d+e x) \left(a e^2+c d^2\right)}-\frac{5 c d e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{7/2}}",1,"(-3*a^4*e^5 + 2*c^4*d^4*x^3*(d + e*x) + 2*a^3*c*e^3*(7*d^2 + 4*d*e*x - 6*e^2*x^2) + 3*a*c^3*d^2*x*(d^3 + d^2*e*x + 3*d*e^2*x^2 + 3*e^3*x^3) + a^2*c^2*e*(2*d^4 + 11*d^3*e*x + 21*d^2*e^2*x^2 + 7*d*e^3*x^3 - 8*e^4*x^4))/(3*a^2*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)^(3/2)) - (5*c*d*e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(7/2)","A",1
584,1,296,327,0.898702,"\int \frac{1}{(d+e x)^3 \left(a+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]","\frac{1}{6} \left(\frac{\sqrt{a+c x^2} \left(\frac{2 c \left(a e^2+c d^2\right) \left(-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right)}{a \left(a+c x^2\right)^2}+\frac{4 c \left(-3 a^3 e^5+3 a^2 c d e^3 (5 d-4 e x)+7 a c^2 d^3 e^2 x+c^3 d^5 x\right)}{a^2 \left(a+c x^2\right)}-\frac{3 e^5 \left(a e^2+c d^2\right)}{(d+e x)^2}-\frac{33 c d e^5}{d+e x}\right)}{\left(a e^2+c d^2\right)^4}+\frac{15 c e^4 \left(a e^2-6 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{9/2}}+\frac{15 c e^4 \left(6 c d^2-a e^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{9/2}}\right)","\frac{c d e \sqrt{a+c x^2} \left(-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right)}{6 a^2 (d+e x) \left(a e^2+c d^2\right)^4}+\frac{e \sqrt{a+c x^2} \left(-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right)}{6 a^2 (d+e x)^2 \left(a e^2+c d^2\right)^3}-\frac{a e \left(2 c d^2-5 a e^2\right)-c d x \left(9 a e^2+2 c d^2\right)}{3 a^2 \sqrt{a+c x^2} (d+e x)^2 \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{3 a \left(a+c x^2\right)^{3/2} (d+e x)^2 \left(a e^2+c d^2\right)}-\frac{5 c e^4 \left(6 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{2 \left(a e^2+c d^2\right)^{9/2}}",1,"((Sqrt[a + c*x^2]*((-3*e^5*(c*d^2 + a*e^2))/(d + e*x)^2 - (33*c*d*e^5)/(d + e*x) + (4*c*(-3*a^3*e^5 + c^3*d^5*x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)*(-(a^2*e^3) + c^2*d^3*x + 3*a*c*d*e*(d - e*x)))/(a*(a + c*x^2)^2)))/(c*d^2 + a*e^2)^4 + (15*c*e^4*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) + (15*c*e^4*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/6","A",1
585,1,18,18,0.012897,"\int \frac{3+x}{\sqrt{1-x^2}} \, dx","Integrate[(3 + x)/Sqrt[1 - x^2],x]","3 \sin ^{-1}(x)-\sqrt{1-x^2}","3 \sin ^{-1}(x)-\sqrt{1-x^2}",1,"-Sqrt[1 - x^2] + 3*ArcSin[x]","A",1
586,1,20,20,0.0120816,"\int \frac{1+x}{\sqrt{4-x^2}} \, dx","Integrate[(1 + x)/Sqrt[4 - x^2],x]","\sin ^{-1}\left(\frac{x}{2}\right)-\sqrt{4-x^2}","\sin ^{-1}\left(\frac{x}{2}\right)-\sqrt{4-x^2}",1,"-Sqrt[4 - x^2] + ArcSin[x/2]","A",1
587,1,18,18,0.0193585,"\int \frac{2+x}{\sqrt{9+x^2}} \, dx","Integrate[(2 + x)/Sqrt[9 + x^2],x]","\sqrt{x^2+9}+2 \sinh ^{-1}\left(\frac{x}{3}\right)","\sqrt{x^2+9}+2 \sinh ^{-1}\left(\frac{x}{3}\right)",1,"Sqrt[9 + x^2] + 2*ArcSinh[x/3]","A",1
588,1,38,54,0.0379383,"\int \frac{(a+b x)^2}{\sqrt{1-x^2}} \, dx","Integrate[(a + b*x)^2/Sqrt[1 - x^2],x]","\frac{1}{2} \left(\left(2 a^2+b^2\right) \sin ^{-1}(x)-b \sqrt{1-x^2} (4 a+b x)\right)","\frac{1}{2} \left(2 a^2+b^2\right) \sin ^{-1}(x)-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b \sqrt{1-x^2} (a+b x)",1,"(-(b*(4*a + b*x)*Sqrt[1 - x^2]) + (2*a^2 + b^2)*ArcSin[x])/2","A",1
589,1,36,52,0.0267483,"\int \frac{(a+b x)^2}{\sqrt{1+x^2}} \, dx","Integrate[(a + b*x)^2/Sqrt[1 + x^2],x]","\left(a^2-\frac{b^2}{2}\right) \sinh ^{-1}(x)+\frac{1}{2} b \sqrt{x^2+1} (4 a+b x)","\frac{1}{2} \left(2 a^2-b^2\right) \sinh ^{-1}(x)+\frac{3}{2} a b \sqrt{x^2+1}+\frac{1}{2} b \sqrt{x^2+1} (a+b x)",1,"(b*(4*a + b*x)*Sqrt[1 + x^2])/2 + (a^2 - b^2/2)*ArcSinh[x]","A",1
590,1,16,18,0.0088522,"\int \frac{2+3 x}{\left(4+x^2\right)^{3/2}} \, dx","Integrate[(2 + 3*x)/(4 + x^2)^(3/2),x]","\frac{x-6}{2 \sqrt{x^2+4}}","-\frac{6-x}{2 \sqrt{x^2+4}}",1,"(-6 + x)/(2*Sqrt[4 + x^2])","A",1
591,1,44,63,0.0461349,"\int (d+e x)^{5/2} \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^(5/2)*(a + c*x^2),x]","\frac{2 (d+e x)^{7/2} \left(99 a e^2+c \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)}{693 e^3}","\frac{2 (d+e x)^{7/2} \left(a e^2+c d^2\right)}{7 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}-\frac{4 c d (d+e x)^{9/2}}{9 e^3}",1,"(2*(d + e*x)^(7/2)*(99*a*e^2 + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)","A",1
592,1,44,63,0.0372635,"\int (d+e x)^{3/2} \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + c*x^2),x]","\frac{2 (d+e x)^{5/2} \left(63 a e^2+c \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)}{315 e^3}","\frac{2 (d+e x)^{5/2} \left(a e^2+c d^2\right)}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3}",1,"(2*(d + e*x)^(5/2)*(63*a*e^2 + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)","A",1
593,1,44,63,0.0303375,"\int \sqrt{d+e x} \left(a+c x^2\right) \, dx","Integrate[Sqrt[d + e*x]*(a + c*x^2),x]","\frac{2 (d+e x)^{3/2} \left(35 a e^2+c \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)}{105 e^3}","\frac{2 (d+e x)^{3/2} \left(a e^2+c d^2\right)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}-\frac{4 c d (d+e x)^{5/2}}{5 e^3}",1,"(2*(d + e*x)^(3/2)*(35*a*e^2 + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)","A",1
594,1,44,61,0.034097,"\int \frac{a+c x^2}{\sqrt{d+e x}} \, dx","Integrate[(a + c*x^2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(15 a e^2+c \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)}{15 e^3}","\frac{2 \sqrt{d+e x} \left(a e^2+c d^2\right)}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3}",1,"(2*Sqrt[d + e*x]*(15*a*e^2 + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)","A",1
595,1,43,59,0.0339525,"\int \frac{a+c x^2}{(d+e x)^{3/2}} \, dx","Integrate[(a + c*x^2)/(d + e*x)^(3/2),x]","\frac{2 \left(c \left(-8 d^2-4 d e x+e^2 x^2\right)-3 a e^2\right)}{3 e^3 \sqrt{d+e x}}","-\frac{2 \left(a e^2+c d^2\right)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3}",1,"(2*(-3*a*e^2 + c*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])","A",1
596,1,44,59,0.0374606,"\int \frac{a+c x^2}{(d+e x)^{5/2}} \, dx","Integrate[(a + c*x^2)/(d + e*x)^(5/2),x]","\frac{2 \left(c \left(8 d^2+12 d e x+3 e^2 x^2\right)-a e^2\right)}{3 e^3 (d+e x)^{3/2}}","-\frac{2 \left(a e^2+c d^2\right)}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}}",1,"(2*(-(a*e^2) + c*(8*d^2 + 12*d*e*x + 3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2))","A",1
597,1,44,61,0.0371702,"\int \frac{a+c x^2}{(d+e x)^{7/2}} \, dx","Integrate[(a + c*x^2)/(d + e*x)^(7/2),x]","-\frac{2 \left(3 a e^2+c \left(8 d^2+20 d e x+15 e^2 x^2\right)\right)}{15 e^3 (d+e x)^{5/2}}","-\frac{2 \left(a e^2+c d^2\right)}{5 e^3 (d+e x)^{5/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}+\frac{4 c d}{3 e^3 (d+e x)^{3/2}}",1,"(-2*(3*a*e^2 + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))","A",1
598,1,96,127,0.1060486,"\int (d+e x)^{5/2} \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(5/2)*(a + c*x^2)^2,x]","\frac{2 (d+e x)^{7/2} \left(6435 a^2 e^4+130 a c e^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)+c^2 \left(128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right)\right)}{45045 e^5}","\frac{4 c (d+e x)^{11/2} \left(a e^2+3 c d^2\right)}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left(a e^2+c d^2\right)}{9 e^5}+\frac{2 (d+e x)^{7/2} \left(a e^2+c d^2\right)^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5}",1,"(2*(d + e*x)^(7/2)*(6435*a^2*e^4 + 130*a*c*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5)","A",1
599,1,97,127,0.0808005,"\int (d+e x)^{3/2} \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(3/2)*(a + c*x^2)^2,x]","\frac{2 (d+e x)^{5/2} \left(9009 a^2 e^4+286 a c e^2 \left(8 d^2-20 d e x+35 e^2 x^2\right)+3 c^2 \left(128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right)\right)}{45045 e^5}","\frac{4 c (d+e x)^{9/2} \left(a e^2+3 c d^2\right)}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left(a e^2+c d^2\right)}{7 e^5}+\frac{2 (d+e x)^{5/2} \left(a e^2+c d^2\right)^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5}",1,"(2*(d + e*x)^(5/2)*(9009*a^2*e^4 + 286*a*c*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)","A",1
600,1,96,127,0.0557837,"\int \sqrt{d+e x} \left(a+c x^2\right)^2 \, dx","Integrate[Sqrt[d + e*x]*(a + c*x^2)^2,x]","\frac{2 (d+e x)^{3/2} \left(1155 a^2 e^4+66 a c e^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)+c^2 \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)}{3465 e^5}","\frac{4 c (d+e x)^{7/2} \left(a e^2+3 c d^2\right)}{7 e^5}-\frac{8 c d (d+e x)^{5/2} \left(a e^2+c d^2\right)}{5 e^5}+\frac{2 (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}{3 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{9/2}}{9 e^5}",1,"(2*(d + e*x)^(3/2)*(1155*a^2*e^4 + 66*a*c*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5)","A",1
601,1,96,125,0.0585661,"\int \frac{\left(a+c x^2\right)^2}{\sqrt{d+e x}} \, dx","Integrate[(a + c*x^2)^2/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(315 a^2 e^4+42 a c e^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)+c^2 \left(128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right)\right)}{315 e^5}","\frac{4 c (d+e x)^{5/2} \left(a e^2+3 c d^2\right)}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left(a e^2+c d^2\right)}{3 e^5}+\frac{2 \sqrt{d+e x} \left(a e^2+c d^2\right)^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5}",1,"(2*Sqrt[d + e*x]*(315*a^2*e^4 + 42*a*c*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)","A",1
602,1,97,123,0.0542002,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^{3/2}} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^(3/2),x]","-\frac{2 \left(105 a^2 e^4+70 a c e^2 \left(8 d^2+4 d e x-e^2 x^2\right)+3 c^2 \left(128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right)\right)}{105 e^5 \sqrt{d+e x}}","\frac{4 c (d+e x)^{3/2} \left(a e^2+3 c d^2\right)}{3 e^5}-\frac{8 c d \sqrt{d+e x} \left(a e^2+c d^2\right)}{e^5}-\frac{2 \left(a e^2+c d^2\right)^2}{e^5 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c^2 d (d+e x)^{5/2}}{5 e^5}",1,"(-2*(105*a^2*e^4 + 70*a*c*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + 3*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)))/(105*e^5*Sqrt[d + e*x])","A",1
603,1,96,123,0.0543961,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^{5/2}} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^(5/2),x]","\frac{2 \left(-5 a^2 e^4+10 a c e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)+c^2 \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{3/2}}","\frac{4 c \sqrt{d+e x} \left(a e^2+3 c d^2\right)}{e^5}+\frac{8 c d \left(a e^2+c d^2\right)}{e^5 \sqrt{d+e x}}-\frac{2 \left(a e^2+c d^2\right)^2}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}-\frac{8 c^2 d (d+e x)^{3/2}}{3 e^5}",1,"(2*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)))/(15*e^5*(d + e*x)^(3/2))","A",1
604,1,96,123,0.0576725,"\int \frac{\left(a+c x^2\right)^2}{(d+e x)^{7/2}} \, dx","Integrate[(a + c*x^2)^2/(d + e*x)^(7/2),x]","-\frac{2 \left(3 a^2 e^4+2 a c e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)+c^2 \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{5/2}}","-\frac{4 c \left(a e^2+3 c d^2\right)}{e^5 \sqrt{d+e x}}+\frac{8 c d \left(a e^2+c d^2\right)}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left(a e^2+c d^2\right)^2}{5 e^5 (d+e x)^{5/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac{8 c^2 d \sqrt{d+e x}}{e^5}",1,"(-2*(3*a^2*e^4 + 2*a*c*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))","A",1
605,1,188,204,0.2347803,"\int (d+e x)^{5/2} \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(5/2)*(a + c*x^2)^3,x]","\frac{2 \left(\frac{1}{5} c^2 (d+e x)^{15/2} \left(a e^2+5 c d^2\right)-\frac{4}{13} c^2 d (d+e x)^{13/2} \left(3 a e^2+5 c d^2\right)+\frac{3}{11} c (d+e x)^{11/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)-\frac{2}{3} c d (d+e x)^{9/2} \left(a e^2+c d^2\right)^2+\frac{1}{7} (d+e x)^{7/2} \left(a e^2+c d^2\right)^3+\frac{1}{19} c^3 (d+e x)^{19/2}-\frac{6}{17} c^3 d (d+e x)^{17/2}\right)}{e^7}","\frac{2 c^2 (d+e x)^{15/2} \left(a e^2+5 c d^2\right)}{5 e^7}-\frac{8 c^2 d (d+e x)^{13/2} \left(3 a e^2+5 c d^2\right)}{13 e^7}+\frac{6 c (d+e x)^{11/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{11 e^7}-\frac{4 c d (d+e x)^{9/2} \left(a e^2+c d^2\right)^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left(a e^2+c d^2\right)^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7}",1,"(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/7 - (2*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/3 + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/11 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/13 + (c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(15/2))/5 - (6*c^3*d*(d + e*x)^(17/2))/17 + (c^3*(d + e*x)^(19/2))/19))/e^7","A",1
606,1,188,204,0.1758303,"\int (d+e x)^{3/2} \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(3/2)*(a + c*x^2)^3,x]","\frac{2 \left(\frac{3}{13} c^2 (d+e x)^{13/2} \left(a e^2+5 c d^2\right)-\frac{4}{11} c^2 d (d+e x)^{11/2} \left(3 a e^2+5 c d^2\right)+\frac{1}{3} c (d+e x)^{9/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)-\frac{6}{7} c d (d+e x)^{7/2} \left(a e^2+c d^2\right)^2+\frac{1}{5} (d+e x)^{5/2} \left(a e^2+c d^2\right)^3+\frac{1}{17} c^3 (d+e x)^{17/2}-\frac{2}{5} c^3 d (d+e x)^{15/2}\right)}{e^7}","\frac{6 c^2 (d+e x)^{13/2} \left(a e^2+5 c d^2\right)}{13 e^7}-\frac{8 c^2 d (d+e x)^{11/2} \left(3 a e^2+5 c d^2\right)}{11 e^7}+\frac{2 c (d+e x)^{9/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{3 e^7}-\frac{12 c d (d+e x)^{7/2} \left(a e^2+c d^2\right)^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left(a e^2+c d^2\right)^3}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac{4 c^3 d (d+e x)^{15/2}}{5 e^7}",1,"(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/5 - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/7 + (c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/3 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/11 + (3*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(13/2))/13 - (2*c^3*d*(d + e*x)^(15/2))/5 + (c^3*(d + e*x)^(17/2))/17))/e^7","A",1
607,1,170,204,0.1237642,"\int \sqrt{d+e x} \left(a+c x^2\right)^3 \, dx","Integrate[Sqrt[d + e*x]*(a + c*x^2)^3,x]","\frac{2 (d+e x)^{3/2} \left(15015 a^3 e^6+1287 a^2 c e^4 \left(8 d^2-12 d e x+15 e^2 x^2\right)+39 a c^2 e^2 \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)+c^3 \left(1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right)\right)}{45045 e^7}","\frac{6 c^2 (d+e x)^{11/2} \left(a e^2+5 c d^2\right)}{11 e^7}-\frac{8 c^2 d (d+e x)^{9/2} \left(3 a e^2+5 c d^2\right)}{9 e^7}+\frac{6 c (d+e x)^{7/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{7 e^7}-\frac{12 c d (d+e x)^{5/2} \left(a e^2+c d^2\right)^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left(a e^2+c d^2\right)^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7}",1,"(2*(d + e*x)^(3/2)*(15015*a^3*e^6 + 1287*a^2*c*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 39*a*c^2*e^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)","A",1
608,1,171,200,0.106274,"\int \frac{\left(a+c x^2\right)^3}{\sqrt{d+e x}} \, dx","Integrate[(a + c*x^2)^3/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(15015 a^3 e^6+3003 a^2 c e^4 \left(8 d^2-4 d e x+3 e^2 x^2\right)+143 a c^2 e^2 \left(128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right)+5 c^3 \left(1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right)\right)}{15015 e^7}","\frac{2 c^2 (d+e x)^{9/2} \left(a e^2+5 c d^2\right)}{3 e^7}-\frac{8 c^2 d (d+e x)^{7/2} \left(3 a e^2+5 c d^2\right)}{7 e^7}+\frac{6 c (d+e x)^{5/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{5 e^7}-\frac{4 c d (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}{e^7}+\frac{2 \sqrt{d+e x} \left(a e^2+c d^2\right)^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}-\frac{12 c^3 d (d+e x)^{11/2}}{11 e^7}",1,"(2*Sqrt[d + e*x]*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)))/(15015*e^7)","A",1
609,1,171,198,0.108022,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^{3/2}} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^(3/2),x]","-\frac{2 \left(1155 a^3 e^6+1155 a^2 c e^4 \left(8 d^2+4 d e x-e^2 x^2\right)+99 a c^2 e^2 \left(128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right)+5 c^3 \left(1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right)\right)}{1155 e^7 \sqrt{d+e x}}","\frac{6 c^2 (d+e x)^{7/2} \left(a e^2+5 c d^2\right)}{7 e^7}-\frac{8 c^2 d (d+e x)^{5/2} \left(3 a e^2+5 c d^2\right)}{5 e^7}+\frac{2 c (d+e x)^{3/2} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{e^7}-\frac{12 c d \sqrt{d+e x} \left(a e^2+c d^2\right)^2}{e^7}-\frac{2 \left(a e^2+c d^2\right)^3}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}-\frac{4 c^3 d (d+e x)^{9/2}}{3 e^7}",1,"(-2*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a*c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)))/(1155*e^7*Sqrt[d + e*x])","A",1
610,1,171,200,0.1063432,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^{5/2}} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^(5/2),x]","\frac{2 \left(-105 a^3 e^6+315 a^2 c e^4 \left(8 d^2+12 d e x+3 e^2 x^2\right)+63 a c^2 e^2 \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)+5 c^3 \left(1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right)\right)}{315 e^7 (d+e x)^{3/2}}","\frac{6 c^2 (d+e x)^{5/2} \left(a e^2+5 c d^2\right)}{5 e^7}-\frac{8 c^2 d (d+e x)^{3/2} \left(3 a e^2+5 c d^2\right)}{3 e^7}+\frac{6 c \sqrt{d+e x} \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{e^7}+\frac{12 c d \left(a e^2+c d^2\right)^2}{e^7 \sqrt{d+e x}}-\frac{2 \left(a e^2+c d^2\right)^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7}",1,"(2*(-105*a^3*e^6 + 315*a^2*c*e^4*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 63*a*c^2*e^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)))/(315*e^7*(d + e*x)^(3/2))","A",1
611,1,170,196,0.1115842,"\int \frac{\left(a+c x^2\right)^3}{(d+e x)^{7/2}} \, dx","Integrate[(a + c*x^2)^3/(d + e*x)^(7/2),x]","-\frac{2 \left(7 a^3 e^6+7 a^2 c e^4 \left(8 d^2+20 d e x+15 e^2 x^2\right)+7 a c^2 e^2 \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)+c^3 \left(1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right)\right)}{35 e^7 (d+e x)^{5/2}}","\frac{2 c^2 (d+e x)^{3/2} \left(a e^2+5 c d^2\right)}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left(3 a e^2+5 c d^2\right)}{e^7}-\frac{6 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right)}{e^7 \sqrt{d+e x}}+\frac{4 c d \left(a e^2+c d^2\right)^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left(a e^2+c d^2\right)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7}",1,"(-2*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))","A",1
612,1,158,167,0.2602393,"\int \frac{(d+e x)^{5/2}}{a-c x^2} \, dx","Integrate[(d + e*x)^(5/2)/(a - c*x^2),x]","\frac{-2 \sqrt{a} c^{3/4} e \sqrt{d+e x} (7 d+e x)-3 \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)+3 \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{3 \sqrt{a} c^{7/4}}","-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} c^{7/4}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} c^{7/4}}-\frac{2 e (d+e x)^{3/2}}{3 c}-\frac{4 d e \sqrt{d+e x}}{c}",1,"(-2*Sqrt[a]*c^(3/4)*e*Sqrt[d + e*x]*(7*d + e*x) - 3*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(3*Sqrt[a]*c^(7/4))","A",1
613,1,147,149,0.1146754,"\int \frac{(d+e x)^{3/2}}{a-c x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a - c*x^2),x]","\frac{-2 \sqrt{a} \sqrt[4]{c} e \sqrt{d+e x}+\left(\sqrt{c} d-\sqrt{a} e\right)^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)\right)+\left(\sqrt{a} e+\sqrt{c} d\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} c^{5/4}}","-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} c^{5/4}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} c^{5/4}}-\frac{2 e \sqrt{d+e x}}{c}",1,"(-2*Sqrt[a]*c^(1/4)*e*Sqrt[d + e*x] - (Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + (Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(5/4))","A",1
614,1,125,134,0.0581751,"\int \frac{\sqrt{d+e x}}{a-c x^2} \, dx","Integrate[Sqrt[d + e*x]/(a - c*x^2),x]","\frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)-\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} c^{3/4}}","\frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} c^{3/4}}-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} c^{3/4}}",1,"(-(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]) + Sqrt[Sqrt[c]*d + Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(3/4))","A",1
615,1,125,134,0.0827164,"\int \frac{1}{\sqrt{d+e x} \left(a-c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)),x]","\frac{\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{\sqrt{c} d-\sqrt{a} e}}}{\sqrt{a} \sqrt[4]{c}}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} \sqrt[4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} \sqrt[4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}",1,"(-(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e])/(Sqrt[a]*c^(1/4))","A",1
616,1,136,160,0.0860484,"\int \frac{1}{(d+e x)^{3/2} \left(a-c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(a - c*x^2)),x]","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right)+\left(\sqrt{a} e-\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right)}{\sqrt{a} \sqrt{d+e x} \left(c d^2-a e^2\right)}","\frac{2 e}{\sqrt{d+e x} \left(c d^2-a e^2\right)}-\frac{\sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} \left(\sqrt{c} d-\sqrt{a} e\right)^{3/2}}+\frac{\sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} \left(\sqrt{a} e+\sqrt{c} d\right)^{3/2}}",1,"((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(c*d^2 - a*e^2)*Sqrt[d + e*x])","C",1
617,1,130,190,0.0634388,"\int \frac{1}{(d+e x)^{5/2} \left(a-c x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)),x]","\frac{\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right)}{\sqrt{c} d-\sqrt{a} e}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right)}{\sqrt{a} e+\sqrt{c} d}}{3 \sqrt{a} (d+e x)^{3/2}}","-\frac{c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2}}+\frac{c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2}}+\frac{4 c d e}{\sqrt{d+e x} \left(c d^2-a e^2\right)^2}+\frac{2 e}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)}",1,"(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)]/(Sqrt[c]*d - Sqrt[a]*e) - Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]/(Sqrt[c]*d + Sqrt[a]*e))/(3*Sqrt[a]*(d + e*x)^(3/2))","C",1
618,1,226,781,0.3503658,"\int \frac{(d+e x)^{5/2}}{a+c x^2} \, dx","Integrate[(d + e*x)^(5/2)/(a + c*x^2),x]","\frac{2 \sqrt{-a} c^{3/4} e \sqrt{d+e x} (7 d+e x)+3 \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(-2 \sqrt{-a} \sqrt{c} d e-a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-3 \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(2 \sqrt{-a} \sqrt{c} d e-a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{3 \sqrt{-a} c^{7/4}}","\frac{e \left(\left(3 c d^2-a e^2\right) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left(\left(3 c d^2-a e^2\right) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left(-\left(3 c d^2-a e^2\right) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{e \left(-\left(3 c d^2-a e^2\right) \sqrt{a e^2+c d^2}+2 a \sqrt{c} d e^2+2 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{7/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{4 d e \sqrt{d+e x}}{c}",1,"(2*Sqrt[-a]*c^(3/4)*e*Sqrt[d + e*x]*(7*d + e*x) + 3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d^2 - 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - 3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(3*Sqrt[-a]*c^(7/4))","A",1
619,1,159,689,0.1443998,"\int \frac{(d+e x)^{3/2}}{a+c x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a + c*x^2),x]","\frac{2 \sqrt{-a} \sqrt[4]{c} e \sqrt{d+e x}+\left(\sqrt{c} d-\sqrt{-a} e\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-\left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} c^{5/4}}","\frac{e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{2 e \sqrt{d+e x}}{c}",1,"(2*Sqrt[-a]*c^(1/4)*e*Sqrt[d + e*x] + (Sqrt[c]*d - Sqrt[-a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - (Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(5/4))","A",1
620,1,135,478,0.079281,"\int \frac{\sqrt{d+e x}}{a+c x^2} \, dx","Integrate[Sqrt[d + e*x]/(a + c*x^2),x]","\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} c^{3/4}}","\frac{e \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}",1,"(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(3/4))","A",1
621,1,135,538,0.108635,"\int \frac{1}{\sqrt{d+e x} \left(a+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)),x]","\frac{\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}}{\sqrt{-a} \sqrt[4]{c}}","-\frac{e \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}",1,"(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[-a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[-a]*e])/(Sqrt[-a]*c^(1/4))","A",1
622,1,132,663,0.1286252,"\int \frac{1}{(d+e x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)),x]","\frac{\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d+a e}}{\sqrt{d+e x}}","-\frac{2 e}{\sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{\sqrt[4]{c} e \left(\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt[4]{c} e \left(\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt[4]{c} e \left(2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt[4]{c} e \left(2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}",1,"(-(Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e))/Sqrt[d + e*x]","C",1
623,1,135,736,0.0618493,"\int \frac{1}{(d+e x)^{5/2} \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)),x]","\frac{\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d+a e}}{3 (d+e x)^{3/2}}","-\frac{c^{3/4} e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \left(a e^2+c d^2\right)^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{2 \sqrt{2} \left(a e^2+c d^2\right)^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \left(a e^2+c d^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{c^{3/4} e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{\sqrt{2} \left(a e^2+c d^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{4 c d e}{\sqrt{d+e x} \left(a e^2+c d^2\right)^2}-\frac{2 e}{3 (d+e x)^{3/2} \left(a e^2+c d^2\right)}",1,"(-(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e))/(3*(d + e*x)^(3/2))","C",1
624,1,251,263,0.4920291,"\int \frac{(d+e x)^{7/2}}{\left(a-c x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(a - c*x^2)^2,x]","\frac{2 \sqrt{a} \sqrt[4]{c} \sqrt{d+e x} \left(5 a^2 e^3+a c e \left(3 d^2+3 d e x-4 e^2 x^2\right)+c^2 d^3 x\right)+\left(c x^2-a\right) \left(5 \sqrt{a} e+2 \sqrt{c} d\right) \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)-\left(c x^2-a\right) \left(2 \sqrt{c} d-5 \sqrt{a} e\right) \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} c^{9/4} \left(a-c x^2\right)}","-\frac{\left(5 \sqrt{a} e+2 \sqrt{c} d\right) \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} c^{9/4}}+\frac{\left(2 \sqrt{c} d-5 \sqrt{a} e\right) \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} c^{9/4}}+\frac{e \sqrt{d+e x} \left(5 a e^2+c d^2\right)}{2 a c^2}+\frac{(d+e x)^{5/2} (a e+c d x)}{2 a c \left(a-c x^2\right)}+\frac{d e (d+e x)^{3/2}}{2 a c}",1,"(2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(3*d^2 + 3*d*e*x - 4*e^2*x^2)) + (Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - (2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)*(a - c*x^2))","A",1
625,1,247,231,0.347509,"\int \frac{(d+e x)^{5/2}}{\left(a-c x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(a - c*x^2)^2,x]","\frac{2 \sqrt{a} c^{3/4} \sqrt{d+e x} \left(a e (2 d+e x)+c d^2 x\right)+\left(c x^2-a\right) \sqrt{\sqrt{c} d-\sqrt{a} e} \left(\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)-\left(c x^2-a\right) \sqrt{\sqrt{a} e+\sqrt{c} d} \left(-\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} c^{7/4} \left(a-c x^2\right)}","-\frac{\left(3 \sqrt{a} e+2 \sqrt{c} d\right) \left(\sqrt{c} d-\sqrt{a} e\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} c^{7/4}}+\frac{\left(2 \sqrt{c} d-3 \sqrt{a} e\right) \left(\sqrt{a} e+\sqrt{c} d\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} c^{7/4}}+\frac{(d+e x)^{3/2} (a e+c d x)}{2 a c \left(a-c x^2\right)}+\frac{d e \sqrt{d+e x}}{2 a c}",1,"(2*Sqrt[a]*c^(3/4)*Sqrt[d + e*x]*(c*d^2*x + a*e*(2*d + e*x)) + Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(2*c*d^2 - Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4)*(a - c*x^2))","A",1
626,1,218,209,0.2453157,"\int \frac{(d+e x)^{3/2}}{\left(a-c x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(a - c*x^2)^2,x]","\frac{\left(c x^2-a\right) \sqrt{\sqrt{c} d-\sqrt{a} e} \left(\sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)-\left(c x^2-a\right) \left(2 \sqrt{c} d-\sqrt{a} e\right) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)+2 \sqrt{a} \sqrt[4]{c} \sqrt{d+e x} (a e+c d x)}{4 a^{3/2} c^{5/4} \left(a-c x^2\right)}","-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left(\sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} c^{5/4}}+\frac{\left(2 \sqrt{c} d-\sqrt{a} e\right) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} c^{5/4}}+\frac{\sqrt{d+e x} (a e+c d x)}{2 a c \left(a-c x^2\right)}",1,"(2*Sqrt[a]*c^(1/4)*(a*e + c*d*x)*Sqrt[d + e*x] + Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*Sqrt[c]*d + Sqrt[a]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - (2*Sqrt[c]*d - Sqrt[a]*e)*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(5/4)*(a - c*x^2))","A",1
627,1,267,194,0.2857836,"\int \frac{\sqrt{d+e x}}{\left(a-c x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(a - c*x^2)^2,x]","\frac{\left(\sqrt{c} d-\sqrt{a} e\right) \left(\left(c x^2-a\right) \sqrt{\sqrt{a} e+\sqrt{c} d} \left(\sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)-2 \sqrt{a} c^{3/4} x \sqrt{d+e x} \left(\sqrt{a} e+\sqrt{c} d\right)\right)-\left(c x^2-a\right) \sqrt{\sqrt{c} d-\sqrt{a} e} \left(\sqrt{a} \sqrt{c} d e-a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} c^{3/4} \left(a-c x^2\right) \left(a e^2-c d^2\right)}","-\frac{\left(\frac{2 \sqrt{c} d}{\sqrt{a}}-e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\left(\frac{2 \sqrt{c} d}{\sqrt{a}}+e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a c^{3/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left(a-c x^2\right)}",1,"(-(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - a*e^2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]) + (Sqrt[c]*d - Sqrt[a]*e)*(-2*Sqrt[a]*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*x*Sqrt[d + e*x] + Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(2*Sqrt[c]*d + Sqrt[a]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*a^(3/2)*c^(3/4)*(-(c*d^2) + a*e^2)*(a - c*x^2))","A",1
628,1,248,222,0.4643481,"\int \frac{1}{\sqrt{d+e x} \left(a-c x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)^2),x]","\frac{\frac{\frac{\left(c x^2-a\right) \left(\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{a} \sqrt[4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}+2 \sqrt{d+e x} (a e-c d x)}{a-c x^2}+\frac{\left(-\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} \sqrt[4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}}{4 a \left(a e^2-c d^2\right)}","-\frac{\left(2 \sqrt{c} d-3 \sqrt{a} e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} \sqrt[4]{c} \left(\sqrt{c} d-\sqrt{a} e\right)^{3/2}}+\frac{\left(3 \sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} \sqrt[4]{c} \left(\sqrt{a} e+\sqrt{c} d\right)^{3/2}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a \left(a-c x^2\right) \left(c d^2-a e^2\right)}",1,"(((2*c*d^2 - Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (2*(a*e - c*d*x)*Sqrt[d + e*x] + ((2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(a - c*x^2))/(4*a*(-(c*d^2) + a*e^2))","A",1
629,1,331,265,0.4251173,"\int \frac{1}{(d+e x)^{3/2} \left(a-c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(a - c*x^2)^2),x]","\frac{\frac{3 c^{3/4} d \left(\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{\sqrt{c} d-\sqrt{a} e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{2 \sqrt{a}}+\frac{\left(5 a e^2+c d^2\right) \left(\left(\sqrt{a} e+\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right)+\left(\sqrt{a} e-\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right)\right)}{2 \sqrt{a} \sqrt{d+e x} \left(c d^2-a e^2\right)}+\frac{a e-c d x}{\left(a-c x^2\right) \sqrt{d+e x}}}{2 a \left(a e^2-c d^2\right)}","-\frac{\sqrt[4]{c} \left(2 \sqrt{c} d-5 \sqrt{a} e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2}}+\frac{\sqrt[4]{c} \left(5 \sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2}}-\frac{a e-c d x}{2 a \left(a-c x^2\right) \sqrt{d+e x} \left(c d^2-a e^2\right)}-\frac{e \left(5 a e^2+c d^2\right)}{2 a \sqrt{d+e x} \left(c d^2-a e^2\right)^2}",1,"((a*e - c*d*x)/(Sqrt[d + e*x]*(a - c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(2*Sqrt[a]) + ((c*d^2 + 5*a*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(2*Sqrt[a]*(c*d^2 - a*e^2)*Sqrt[d + e*x]))/(2*a*(-(c*d^2) + a*e^2))","C",1
630,1,327,311,0.3069281,"\int \frac{1}{(d+e x)^{5/2} \left(a-c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)^2),x]","\frac{\frac{\left(7 a e^2+3 c d^2\right) \left(\left(\sqrt{a} e+\sqrt{c} d\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right)+\left(\sqrt{a} e-\sqrt{c} d\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right)\right)}{\sqrt{a} \left(c d^2-a e^2\right)}+\frac{15 c d (d+e x) \left(\left(\sqrt{a} e+\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right)+\left(\sqrt{a} e-\sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right)\right)}{\sqrt{a} \left(a e^2-c d^2\right)}+\frac{6 a e-6 c d x}{a-c x^2}}{12 a (d+e x)^{3/2} \left(a e^2-c d^2\right)}","-\frac{c^{3/4} \left(2 \sqrt{c} d-7 \sqrt{a} e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{4 a^{3/2} \left(\sqrt{c} d-\sqrt{a} e\right)^{7/2}}+\frac{c^{3/4} \left(7 \sqrt{a} e+2 \sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{4 a^{3/2} \left(\sqrt{a} e+\sqrt{c} d\right)^{7/2}}-\frac{a e-c d x}{2 a \left(a-c x^2\right) (d+e x)^{3/2} \left(c d^2-a e^2\right)}-\frac{e \left(7 a e^2+3 c d^2\right)}{6 a (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}-\frac{c d e \left(19 a e^2+c d^2\right)}{2 a \sqrt{d+e x} \left(c d^2-a e^2\right)^3}",1,"((6*a*e - 6*c*d*x)/(a - c*x^2) + ((3*c*d^2 + 7*a*e^2)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(Sqrt[a]*(c*d^2 - a*e^2)) + (15*c*d*(d + e*x)*((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/(Sqrt[a]*(-(c*d^2) + a*e^2)))/(12*a*(-(c*d^2) + a*e^2)*(d + e*x)^(3/2))","C",1
631,1,311,887,0.8242475,"\int \frac{(d+e x)^{7/2}}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(a + c*x^2)^2,x]","\frac{\frac{2 \sqrt[4]{c} \sqrt{d+e x} \left(5 a^2 e^3+a c e \left(-3 d^2-3 d e x+4 e^2 x^2\right)+c^2 d^3 x\right)}{a \left(a+c x^2\right)}+\frac{a \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(\sqrt{-a} c d^2 e+8 a \sqrt{c} d e^2+5 (-a)^{3/2} e^3+2 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{(-a)^{5/2}}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(-\sqrt{-a} c d^2 e+8 a \sqrt{c} d e^2+5 \sqrt{-a} a e^3+2 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{3/2}}}{4 c^{9/4}}","-\frac{(a e-c d x) (d+e x)^{5/2}}{2 a c \left(c x^2+a\right)}-\frac{d e (d+e x)^{3/2}}{2 a c}-\frac{e \left(c d^2-5 a e^2\right) \sqrt{d+e x}}{2 a c^2}+\frac{e \left(c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+13 a e^2\right) d-5 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+13 a e^2\right) d-5 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+13 a e^2\right) d-5 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left(c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+13 a e^2\right) d-5 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}",1,"((2*c^(1/4)*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2)))/(a*(a + c*x^2)) + (a*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*c^(3/2)*d^3 + Sqrt[-a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + 5*(-a)^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*c^(3/2)*d^3 - Sqrt[-a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + 5*Sqrt[-a]*a*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(3/2))/(4*c^(9/4))","A",1
632,1,248,811,0.56889,"\int \frac{(d+e x)^{5/2}}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(a + c*x^2)^2,x]","\frac{\frac{2 c^{3/4} \sqrt{d+e x} \left(c d^2 x-a e (2 d+e x)\right)}{a \left(a+c x^2\right)}+\frac{a \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{(-a)^{5/2}}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(-\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{3/2}}}{4 c^{7/4}}","-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left(c x^2+a\right)}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left(c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(c d^2+3 a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(c d^2+3 a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(c d^2+3 a e^2\right)\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left(c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(c d^2+3 a e^2\right)\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}",1,"((2*c^(3/4)*Sqrt[d + e*x]*(c*d^2*x - a*e*(2*d + e*x)))/(a*(a + c*x^2)) + (a*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*c*d^2 + Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*c*d^2 - Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(3/2))/(4*c^(7/4))","A",1
633,1,208,726,0.5480237,"\int \frac{(d+e x)^{3/2}}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(a + c*x^2)^2,x]","\frac{\frac{2 a \sqrt[4]{c} \sqrt{d+e x} (c d x-a e)}{a+c x^2}-\sqrt{\sqrt{c} d-\sqrt{-a} e} \left(2 \sqrt{-a} \sqrt{c} d-a e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)+\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(2 \sqrt{-a} \sqrt{c} d+a e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{4 a^2 c^{5/4}}","-\frac{e \left(-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left(-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left(\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left(\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a c \left(a+c x^2\right)}",1,"((2*a*c^(1/4)*(-(a*e) + c*d*x)*Sqrt[d + e*x])/(a + c*x^2) - Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*Sqrt[-a]*Sqrt[c]*d - a*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] + Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*Sqrt[-a]*Sqrt[c]*d + a*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(4*a^2*c^(5/4))","A",1
634,1,265,675,0.5207515,"\int \frac{\sqrt{d+e x}}{\left(a+c x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(a + c*x^2)^2,x]","\frac{-\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left(2 \sqrt{-a} c d^2-a \sqrt{c} d e+\sqrt{-a} a e^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{2 a c^{3/4}}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(2 \sqrt{-a} c d^2+a \sqrt{c} d e+\sqrt{-a} a e^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{2 a c^{3/4}}+\frac{x \sqrt{d+e x} \left(a e^2+c d^2\right)}{a+c x^2}}{2 a \left(a e^2+c d^2\right)}","\frac{e \left(\sqrt{a e^2+c d^2}+\sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left(\sqrt{a e^2+c d^2}+\sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left(d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left(d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a \sqrt[4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left(a+c x^2\right)}",1,"(((c*d^2 + a*e^2)*x*Sqrt[d + e*x])/(a + c*x^2) - (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 - a*Sqrt[c]*d*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(2*a*c^(3/4)) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 + a*Sqrt[c]*d*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(2*a*c^(3/4)))/(2*a*(c*d^2 + a*e^2))","A",1
635,1,255,739,0.5622302,"\int \frac{1}{\sqrt{d+e x} \left(a+c x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)^2),x]","\frac{\frac{\left(-\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt[4]{c} \sqrt{\sqrt{c} d-\sqrt{-a} e}}+\frac{\left(2 \sqrt{-a} c d^2-a \sqrt{c} d e+3 \sqrt{-a} a e^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{a \sqrt[4]{c} \sqrt{\sqrt{-a} e+\sqrt{c} d}}+\frac{2 \sqrt{d+e x} (a e+c d x)}{a+c x^2}}{4 a \left(a e^2+c d^2\right)}","\frac{\sqrt{d+e x} (a e+c d x)}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)}-\frac{e \left(-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a \sqrt[4]{c} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left(-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{8 \sqrt{2} a \sqrt[4]{c} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left(\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a \sqrt[4]{c} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left(\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{4 \sqrt{2} a \sqrt[4]{c} \left(a e^2+c d^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}",1,"((2*(a*e + c*d*x)*Sqrt[d + e*x])/(a + c*x^2) + ((2*c*d^2 - Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(Sqrt[-a]*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) + ((2*Sqrt[-a]*c*d^2 - a*Sqrt[c]*d*e + 3*Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(a*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/(4*a*(c*d^2 + a*e^2))","A",1
636,1,346,845,0.4971118,"\int \frac{1}{(d+e x)^{3/2} \left(a+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]","\frac{\frac{3 c^{3/4} d \left(\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{2 \sqrt{-a}}-\frac{\left(5 a e^2-c d^2\right) \left(\left(a e-\sqrt{-a} \sqrt{c} d\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)+\left(\sqrt{-a} \sqrt{c} d+a e\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)\right)}{2 a \sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{a e+c d x}{\left(a+c x^2\right) \sqrt{d+e x}}}{2 a \left(a e^2+c d^2\right)}","\frac{e \left(c d^2-5 a e^2\right)}{2 a \left(c d^2+a e^2\right)^2 \sqrt{d+e x}}+\frac{\sqrt[4]{c} e \left(c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left(c d^2-5 a e^2\right) \sqrt{c d^2+a e^2}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt[4]{c} e \left(c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left(c d^2-5 a e^2\right) \sqrt{c d^2+a e^2}\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt[4]{c} e \left(c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left(c d^2-5 a e^2\right) \sqrt{c d^2+a e^2}\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt[4]{c} e \left(c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left(c d^2-5 a e^2\right) \sqrt{c d^2+a e^2}\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left(c d^2+a e^2\right) \sqrt{d+e x} \left(c x^2+a\right)}",1,"((a*e + c*d*x)/(Sqrt[d + e*x]*(a + c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[-a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/(2*Sqrt[-a]) - ((-(c*d^2) + 5*a*e^2)*((-(Sqrt[-a]*Sqrt[c]*d) + a*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] + (Sqrt[-a]*Sqrt[c]*d + a*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]))/(2*a*(c*d^2 + a*e^2))","C",1
637,1,321,930,0.4494319,"\int \frac{1}{(d+e x)^{5/2} \left(a+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]","\frac{-\frac{\left(3 c d^2 e-7 a e^3\right) \left(\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d+a e}\right)}{e}+15 c d (d+e x) \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d+a e}\right)+\frac{6 (a e+c d x)}{a+c x^2}}{12 a (d+e x)^{3/2} \left(a e^2+c d^2\right)}","\frac{c d e \left(c d^2-19 a e^2\right)}{2 a \left(c d^2+a e^2\right)^3 \sqrt{d+e x}}+\frac{c^{3/4} e \left(c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left(c d^2-19 a e^2\right) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a \left(c d^2+a e^2\right)^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left(c^2 d^4+34 a c e^2 d^2+\sqrt{c} \left(c d^2-19 a e^2\right) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{4 \sqrt{2} a \left(c d^2+a e^2\right)^{7/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{c^{3/4} e \left(c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left(c d^2-19 a e^2\right) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a \left(c d^2+a e^2\right)^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{c^{3/4} e \left(c^2 d^4+34 a c e^2 d^2-\sqrt{c} \left(c d^2-19 a e^2\right) \sqrt{c d^2+a e^2} d-7 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{8 \sqrt{2} a \left(c d^2+a e^2\right)^{7/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left(c d^2+a e^2\right) (d+e x)^{3/2} \left(c x^2+a\right)}+\frac{e \left(3 c d^2-7 a e^2\right)}{6 a \left(c d^2+a e^2\right)^2 (d+e x)^{3/2}}",1,"((6*(a*e + c*d*x))/(a + c*x^2) - ((3*c*d^2*e - 7*a*e^3)*(-(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e)))/e + 15*c*d*(d + e*x)*(-(Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e)))/(12*a*(c*d^2 + a*e^2)*(d + e*x)^(3/2))","C",1
638,1,319,294,0.6708388,"\int \frac{(d+e x)^{7/2}}{\left(a-c x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(a - c*x^2)^3,x]","\frac{-\sqrt{\sqrt{c} d-\sqrt{a} e} \left(-5 a^{3/2} e^3+6 \sqrt{a} c d^2 e-13 a \sqrt{c} d e^2+12 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)+\sqrt{\sqrt{a} e+\sqrt{c} d} \left(5 a^{3/2} e^3-6 \sqrt{a} c d^2 e-13 a \sqrt{c} d e^2+12 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)+\frac{2 \sqrt{a} \sqrt[4]{c} \sqrt{d+e x} \left(-5 a^3 e^3+a^2 c e \left(11 d^2+4 d e x+9 e^2 x^2\right)+a c^2 d x \left(10 d^2+d e x+8 e^2 x^2\right)-6 c^3 d^3 x^3\right)}{\left(a-c x^2\right)^2}}{32 a^{5/2} c^{9/4}}","-\frac{\left(\sqrt{c} d-\sqrt{a} e\right)^{3/2} \left(18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{32 a^{5/2} c^{9/4}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right)^{3/2} \left(-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} c^{9/4}}+\frac{\sqrt{d+e x} \left(2 c d x \left(3 c d^2-2 a e^2\right)+a e \left(7 c d^2-5 a e^2\right)\right)}{16 a^2 c^2 \left(a-c x^2\right)}+\frac{(d+e x)^{5/2} (a e+c d x)}{4 a c \left(a-c x^2\right)^2}",1,"((2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(-5*a^3*e^3 - 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) + a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a - c*x^2)^2 - Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(12*c^(3/2)*d^3 + 6*Sqrt[a]*c*d^2*e - 13*a*Sqrt[c]*d*e^2 - 5*a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(12*c^(3/2)*d^3 - 6*Sqrt[a]*c*d^2*e - 13*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))","A",1
639,1,260,279,0.4971075,"\int \frac{(d+e x)^{5/2}}{\left(a-c x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(a - c*x^2)^3,x]","\frac{\frac{2 \sqrt{a} c^{3/4} \sqrt{d+e x} \left(a^2 e (7 d+e x)+a c x \left(10 d^2+d e x+3 e^2 x^2\right)-6 c^2 d^2 x^3\right)}{\left(a-c x^2\right)^2}-3 \sqrt{\sqrt{c} d-\sqrt{a} e} \left(2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)+3 \sqrt{\sqrt{a} e+\sqrt{c} d} \left(-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} c^{7/4}}","-\frac{3 \sqrt{\sqrt{c} d-\sqrt{a} e} \left(2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{\sqrt{a} e+\sqrt{c} d} \left(-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{d+e x} \left(x \left(2 c d^2-a e^2\right)+a d e\right)}{16 a^2 c \left(a-c x^2\right)}+\frac{(d+e x)^{3/2} (a e+c d x)}{4 a c \left(a-c x^2\right)^2}",1,"((2*Sqrt[a]*c^(3/4)*Sqrt[d + e*x]*(-6*c^2*d^2*x^3 + a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)))/(a - c*x^2)^2 - 3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4))","A",1
640,1,428,268,0.9455046,"\int \frac{(d+e x)^{3/2}}{\left(a-c x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(a - c*x^2)^3,x]","\frac{\frac{2 (d+e x)^{5/2} \left(3 a^2 e^3-a c d e (5 d+4 e x)+6 c^2 d^3 x\right)}{a-c x^2}+\frac{3 \sqrt{\sqrt{a} e+\sqrt{c} d} \left(-a^{3/2} e^3+6 \sqrt{a} c d^2 e+a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \left(\sqrt{c} d-\sqrt{a} e\right)^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)-3 \left(\sqrt{a} e+\sqrt{c} d\right)^2 \left(a^{3/2} e^3-6 \sqrt{a} c d^2 e+a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)+2 \sqrt{a} \sqrt[4]{c} e \sqrt{d+e x} \left(3 a^2 e^4-a c d e^2 (13 d+4 e x)+6 c^2 d^3 (2 d+e x)\right)}{\sqrt{a} c^{5/4}}+\frac{8 a (d+e x)^{5/2} \left(c d^2-a e^2\right) (c d x-a e)}{\left(a-c x^2\right)^2}}{32 a^2 \left(c d^2-a e^2\right)^2}","-\frac{3 \left(-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{3 \left(2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\sqrt{d+e x} (a e-6 c d x)}{16 a^2 c \left(a-c x^2\right)}+\frac{\sqrt{d+e x} (a e+c d x)}{4 a c \left(a-c x^2\right)^2}",1,"((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*(d + e*x)^(5/2))/(a - c*x^2)^2 + (2*(d + e*x)^(5/2)*(3*a^2*e^3 + 6*c^2*d^3*x - a*c*d*e*(5*d + 4*e*x)))/(a - c*x^2) + (2*Sqrt[a]*c^(1/4)*e*Sqrt[d + e*x]*(3*a^2*e^4 + 6*c^2*d^3*(2*d + e*x) - a*c*d*e^2*(13*d + 4*e*x)) - 3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(Sqrt[c]*d + Sqrt[a]*e)^2*(4*c^(3/2)*d^3 - 6*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4*c^(3/2)*d^3 + 6*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 - a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(5/4)))/(32*a^2*(c*d^2 - a*e^2)^2)","A",1
641,1,368,281,0.7065035,"\int \frac{\sqrt{d+e x}}{\left(a-c x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(a - c*x^2)^3,x]","\frac{\frac{2 (d+e x)^{3/2} \left(5 a^2 e^3-a c d e (3 d+8 e x)+6 c^2 d^3 x\right)}{a-c x^2}+\frac{4 \sqrt{a} c^{3/4} d e \sqrt{d+e x} \left(3 c d^2-4 a e^2\right)+\sqrt{\sqrt{a} e+\sqrt{c} d} \left(18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \left(\sqrt{c} d-\sqrt{a} e\right)^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)-\left(\sqrt{a} e+\sqrt{c} d\right)^2 \left(-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} c^{3/4}}+\frac{8 a (d+e x)^{3/2} \left(c d^2-a e^2\right) (c d x-a e)}{\left(a-c x^2\right)^2}}{32 a^2 \left(c d^2-a e^2\right)^2}","-\frac{\left(-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{32 a^{5/2} c^{3/4} \left(\sqrt{c} d-\sqrt{a} e\right)^{3/2}}+\frac{\left(18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} c^{3/4} \left(\sqrt{a} e+\sqrt{c} d\right)^{3/2}}-\frac{\sqrt{d+e x} \left(a d e-x \left(6 c d^2-5 a e^2\right)\right)}{16 a^2 \left(a-c x^2\right) \left(c d^2-a e^2\right)}+\frac{x \sqrt{d+e x}}{4 a \left(a-c x^2\right)^2}",1,"((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*(d + e*x)^(3/2))/(a - c*x^2)^2 + (2*(d + e*x)^(3/2)*(5*a^2*e^3 + 6*c^2*d^3*x - a*c*d*e*(3*d + 8*e*x)))/(a - c*x^2) + (4*Sqrt[a]*c^(3/4)*d*e*(3*c*d^2 - 4*a*e^2)*Sqrt[d + e*x] - Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(Sqrt[c]*d + Sqrt[a]*e)^2*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + (Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(3/4)))/(32*a^2*(c*d^2 - a*e^2)^2)","A",1
642,1,335,315,0.6719787,"\int \frac{1}{\sqrt{d+e x} \left(a-c x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)^3),x]","\frac{\frac{2 \sqrt{d+e x} \left(7 a^2 e^3-a c d e (d+12 e x)+6 c^2 d^3 x\right)}{a-c x^2}+\frac{8 a \sqrt{d+e x} \left(c d^2-a e^2\right) (c d x-a e)}{\left(a-c x^2\right)^2}+\frac{3 \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2} \left(10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)-3 \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2} \left(-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{\sqrt{a} \sqrt[4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e} \sqrt{\sqrt{a} e+\sqrt{c} d}}}{32 a^2 \left(c d^2-a e^2\right)^2}","-\frac{3 \left(-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right)}{32 a^{5/2} \sqrt[4]{c} \left(\sqrt{c} d-\sqrt{a} e\right)^{5/2}}+\frac{3 \left(10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right)}{32 a^{5/2} \sqrt[4]{c} \left(\sqrt{a} e+\sqrt{c} d\right)^{5/2}}-\frac{\sqrt{d+e x} \left(a e \left(c d^2-7 a e^2\right)-6 c d x \left(c d^2-2 a e^2\right)\right)}{16 a^2 \left(a-c x^2\right) \left(c d^2-a e^2\right)^2}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a \left(a-c x^2\right)^2 \left(c d^2-a e^2\right)}",1,"((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*Sqrt[d + e*x])/(a - c*x^2)^2 + (2*Sqrt[d + e*x]*(7*a^2*e^3 + 6*c^2*d^3*x - a*c*d*e*(d + 12*e*x)))/(a - c*x^2) + (-3*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(32*a^2*(c*d^2 - a*e^2)^2)","A",1
643,1,343,905,1.028241,"\int \frac{(d+e x)^{7/2}}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(a + c*x^2)^3,x]","\frac{\frac{2 \sqrt[4]{c} \sqrt{d+e x} \left(-5 a^3 e^3-a^2 c e \left(11 d^2+4 d e x+9 e^2 x^2\right)+a c^2 d x \left(10 d^2+d e x+8 e^2 x^2\right)+6 c^3 d^3 x^3\right)}{a^2 \left(a+c x^2\right)^2}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left(6 \sqrt{-a} c d^2 e+13 a \sqrt{c} d e^2+5 \sqrt{-a} a e^3+12 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{(-a)^{5/2}}+\frac{a \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(-6 \sqrt{-a} c d^2 e+13 a \sqrt{c} d e^2+5 (-a)^{3/2} e^3+12 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{7/2}}}{32 c^{9/4}}","-\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left(c x^2+a\right)^2}-\frac{\left(a e \left(7 c d^2+5 a e^2\right)-2 c d \left(3 c d^2+2 a e^2\right) x\right) \sqrt{d+e x}}{16 a^2 c^2 \left(c x^2+a\right)}+\frac{e \left(6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left(6 c d^2+8 a e^2\right) d+5 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left(6 c d^2+8 a e^2\right) d+5 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left(3 c d^2+4 a e^2\right) d+5 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left(6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left(3 c d^2+4 a e^2\right) d+5 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}",1,"((2*c^(1/4)*Sqrt[d + e*x]*(-5*a^3*e^3 + 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) - a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a^2*(a + c*x^2)^2) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(12*c^(3/2)*d^3 + 6*Sqrt[-a]*c*d^2*e + 13*a*Sqrt[c]*d*e^2 + 5*Sqrt[-a]*a*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) + (a*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(12*c^(3/2)*d^3 - 6*Sqrt[-a]*c*d^2*e + 13*a*Sqrt[c]*d*e^2 + 5*(-a)^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(7/2))/(32*c^(9/4))","A",1
644,1,277,846,0.6893874,"\int \frac{(d+e x)^{5/2}}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(a + c*x^2)^3,x]","\frac{\frac{2 c^{3/4} \sqrt{d+e x} \left(-a^2 e (7 d+e x)+a c x \left(10 d^2+d e x+3 e^2 x^2\right)+6 c^2 d^2 x^3\right)}{a^2 \left(a+c x^2\right)^2}+\frac{3 \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(2 \sqrt{-a} \sqrt{c} d e+a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{(-a)^{5/2}}-\frac{3 \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(-2 \sqrt{-a} \sqrt{c} d e+a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{5/2}}}{32 c^{7/4}}","-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left(c x^2+a\right)^2}-\frac{3 \left(a d e-\left(2 c d^2+a e^2\right) x\right) \sqrt{d+e x}}{16 a^2 c \left(c x^2+a\right)}+\frac{3 e \left(2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(2 c d^2+a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left(2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(2 c d^2+a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left(2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(2 c d^2+a e^2\right)\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left(2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(2 c d^2+a e^2\right)\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}",1,"((2*c^(3/4)*Sqrt[d + e*x]*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)))/(a^2*(a + c*x^2)^2) + (3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(4*c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e + a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) - (3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(4*c*d^2 - 2*Sqrt[-a]*Sqrt[c]*d*e + a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(5/2))/(32*c^(7/4))","A",1
645,1,430,769,1.164476,"\int \frac{(d+e x)^{3/2}}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(a + c*x^2)^3,x]","\frac{\frac{2 (d+e x)^{5/2} \left(3 a^2 e^3+a c d e (5 d+4 e x)+6 c^2 d^3 x\right)}{a+c x^2}+\frac{-2 \sqrt{-a} \sqrt[4]{c} e \sqrt{d+e x} \left(3 a^2 e^4+a c d e^2 (13 d+4 e x)+6 c^2 d^3 (2 d+e x)\right)+3 \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(a e^2+c d^2\right) \left(2 \sqrt{-a} c d^2 e+3 a \sqrt{c} d e^2+\sqrt{-a} a e^3+4 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-3 \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(a e^2+c d^2\right) \left(-2 \sqrt{-a} c d^2 e+3 a \sqrt{c} d e^2+(-a)^{3/2} e^3+4 c^{3/2} d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} c^{5/4}}+\frac{8 a (d+e x)^{5/2} \left(a e^2+c d^2\right) (a e+c d x)}{\left(a+c x^2\right)^2}}{32 a^2 \left(a e^2+c d^2\right)^2}","-\frac{3 e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left(-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right)}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{3 e \left(2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right)}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{\sqrt{d+e x} (a e+6 c d x)}{16 a^2 c \left(a+c x^2\right)}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a c \left(a+c x^2\right)^2}",1,"((8*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*(d + e*x)^(5/2))/(a + c*x^2)^2 + (2*(d + e*x)^(5/2)*(3*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(5*d + 4*e*x)))/(a + c*x^2) + (-2*Sqrt[-a]*c^(1/4)*e*Sqrt[d + e*x]*(3*a^2*e^4 + 6*c^2*d^3*(2*d + e*x) + a*c*d*e^2*(13*d + 4*e*x)) + 3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d^2 + a*e^2)*(4*c^(3/2)*d^3 + 2*Sqrt[-a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + Sqrt[-a]*a*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - 3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d^2 + a*e^2)*(4*c^(3/2)*d^3 - 2*Sqrt[-a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + (-a)^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(5/4)))/(32*a^2*(c*d^2 + a*e^2)^2)","A",1
646,1,412,849,0.8022473,"\int \frac{\sqrt{d+e x}}{\left(a+c x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(a + c*x^2)^3,x]","\frac{\frac{2 (d+e x)^{3/2} \left(5 a^2 e^3+a c d e (3 d+8 e x)+6 c^2 d^3 x\right)}{a+c x^2}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left(5 a^2 e^4+6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 \sqrt{-a} a \sqrt{c} d e^3+12 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(5 a^2 e^4-6 \sqrt{-a} c^{3/2} d^3 e+19 a c d^2 e^2+8 (-a)^{3/2} \sqrt{c} d e^3+12 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)-4 \sqrt{-a} c^{3/4} d e \sqrt{d+e x} \left(4 a e^2+3 c d^2\right)}{\sqrt{-a} c^{3/4}}+\frac{8 a (d+e x)^{3/2} \left(a e^2+c d^2\right) (a e+c d x)}{\left(a+c x^2\right)^2}}{32 a^2 \left(a e^2+c d^2\right)^2}","\frac{\sqrt{d+e x} x}{4 a \left(c x^2+a\right)^2}+\frac{e \left(6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(6 c d^2+5 a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{3/4} \left(c d^2+a e^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left(6 c d^2+5 a e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 c^{3/4} \left(c d^2+a e^2\right)^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left(6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(6 c d^2+5 a e^2\right)\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{3/4} \left(c d^2+a e^2\right)^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left(6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left(6 c d^2+5 a e^2\right)\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 c^{3/4} \left(c d^2+a e^2\right)^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left(a d e+\left(6 c d^2+5 a e^2\right) x\right)}{16 a^2 \left(c d^2+a e^2\right) \left(c x^2+a\right)}",1,"((8*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*(d + e*x)^(3/2))/(a + c*x^2)^2 + (2*(d + e*x)^(3/2)*(5*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(3*d + 8*e*x)))/(a + c*x^2) + (-4*Sqrt[-a]*c^(3/4)*d*e*(3*c*d^2 + 4*a*e^2)*Sqrt[d + e*x] + Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(12*c^2*d^4 + 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*Sqrt[-a]*a*Sqrt[c]*d*e^3 + 5*a^2*e^4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(12*c^2*d^4 - 6*Sqrt[-a]*c^(3/2)*d^3*e + 19*a*c*d^2*e^2 + 8*(-a)^(3/2)*Sqrt[c]*d*e^3 + 5*a^2*e^4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(3/4)))/(32*a^2*(c*d^2 + a*e^2)^2)","A",1
647,1,464,920,0.7098343,"\int \frac{1}{\sqrt{d+e x} \left(a+c x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)^3),x]","\frac{\frac{\sqrt{d+e x} \left(7 a^2 e^3+a c d e (d+12 e x)+6 c^2 d^3 x\right)}{2 \left(a+c x^2\right)}+\frac{3 \left(\frac{\left(7 a^2 e^4+5 a c d^2 e^2+2 c^2 d^4\right) \left(\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)\right)}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{-a} e+\sqrt{c} d}}+2 \sqrt{c} d \left(2 a e^2+c d^2\right) \left(\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)-\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)\right)\right)}{4 \sqrt{-a} \sqrt[4]{c}}+\frac{2 a \sqrt{d+e x} \left(a e^2+c d^2\right) (a e+c d x)}{\left(a+c x^2\right)^2}}{8 a^2 \left(a e^2+c d^2\right)^2}","\frac{\sqrt{d+e x} (a e+c d x)}{4 a \left(c d^2+a e^2\right) \left(c x^2+a\right)^2}+\frac{3 e \left(2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+2 a e^2\right) d+7 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 \sqrt[4]{c} \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left(2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+2 a e^2\right) d+7 a^2 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt[4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right)}{32 \sqrt{2} a^2 \sqrt[4]{c} \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left(2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+2 a e^2\right) d+7 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)-\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 \sqrt[4]{c} \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left(2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left(c d^2+2 a e^2\right) d+7 a^2 e^4\right) \log \left(\sqrt{c} (d+e x)+\sqrt{2} \sqrt[4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right)}{64 \sqrt{2} a^2 \sqrt[4]{c} \left(c d^2+a e^2\right)^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left(a e \left(c d^2+7 a e^2\right)+6 c d \left(c d^2+2 a e^2\right) x\right)}{16 a^2 \left(c d^2+a e^2\right)^2 \left(c x^2+a\right)}",1,"((2*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*Sqrt[d + e*x])/(a + c*x^2)^2 + (Sqrt[d + e*x]*(7*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(d + 12*e*x)))/(2*(a + c*x^2)) + (3*(((2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4)*(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]))/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]) + 2*Sqrt[c]*d*(c*d^2 + 2*a*e^2)*(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])))/(4*Sqrt[-a]*c^(1/4)))/(8*a^2*(c*d^2 + a*e^2)^2)","A",1
648,1,59,214,0.0297938,"\int \frac{\sqrt{2+3 x}}{1+x^2} \, dx","Integrate[Sqrt[2 + 3*x]/(1 + x^2),x]","i \sqrt{2+3 i} \tanh ^{-1}\left(\frac{\sqrt{3 x+2}}{\sqrt{2+3 i}}\right)-i \sqrt{2-3 i} \tanh ^{-1}\left(\frac{\sqrt{3 x+2}}{\sqrt{2-3 i}}\right)","\frac{3 \log \left(3 x-\sqrt{2 \left(2+\sqrt{13}\right)} \sqrt{3 x+2}+\sqrt{13}+2\right)}{2 \sqrt{2 \left(2+\sqrt{13}\right)}}-\frac{3 \log \left(3 x+\sqrt{2 \left(2+\sqrt{13}\right)} \sqrt{3 x+2}+\sqrt{13}+2\right)}{2 \sqrt{2 \left(2+\sqrt{13}\right)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{2 \left(2+\sqrt{13}\right)}-2 \sqrt{3 x+2}}{\sqrt{2 \left(\sqrt{13}-2\right)}}\right)}{\sqrt{2 \left(\sqrt{13}-2\right)}}+\frac{3 \tan ^{-1}\left(\frac{2 \sqrt{3 x+2}+\sqrt{2 \left(2+\sqrt{13}\right)}}{\sqrt{2 \left(\sqrt{13}-2\right)}}\right)}{\sqrt{2 \left(\sqrt{13}-2\right)}}",1,"(-I)*Sqrt[2 - 3*I]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[2 - 3*I]] + I*Sqrt[2 + 3*I]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[2 + 3*I]]","C",1
649,1,75,316,0.0405775,"\int \frac{\sqrt{c+d x}}{1+x^2} \, dx","Integrate[Sqrt[c + d*x]/(1 + x^2),x]","i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c+i d}}\right)-i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c-i d}}\right)","\frac{d \log \left(-\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right)}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}-\frac{d \log \left(\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right)}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c^2+d^2}+c}-\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right)}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{\sqrt{c^2+d^2}+c}+\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right)}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}}",1,"(-I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - I*d]] + I*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c + I*d]]","C",1
650,1,35,35,0.0176161,"\int \frac{\sqrt{2+3 x}}{1-x^2} \, dx","Integrate[Sqrt[2 + 3*x]/(1 - x^2),x]","\sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{3 x+2}}{\sqrt{5}}\right)-\tan ^{-1}\left(\sqrt{3 x+2}\right)","\sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{3 x+2}}{\sqrt{5}}\right)-\tan ^{-1}\left(\sqrt{3 x+2}\right)",1,"-ArcTan[Sqrt[2 + 3*x]] + Sqrt[5]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[5]]","A",1
651,1,58,58,0.0357427,"\int \frac{\sqrt{c+d x}}{1-x^2} \, dx","Integrate[Sqrt[c + d*x]/(1 - x^2),x]","\sqrt{c+d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right)-\sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right)","\sqrt{c+d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right)-\sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right)",1,"-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c + d]]","A",1
652,1,133,427,0.1373496,"\int \frac{\sqrt{2+3 x}}{a+b x^2} \, dx","Integrate[Sqrt[2 + 3*x]/(a + b*x^2),x]","\frac{\sqrt{3 \sqrt{-a}-2 \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{-a}-2 \sqrt{b}}}\right)-\sqrt{3 \sqrt{-a}+2 \sqrt{b}} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{-a}+2 \sqrt{b}}}\right)}{\sqrt{-a} b^{3/4}}","\frac{3 \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right)}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}-\frac{3 \log \left(\sqrt{2} \sqrt[4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right)}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}-\sqrt{2} \sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right)}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{2} \sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right)}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}",1,"(Sqrt[3*Sqrt[-a] - 2*Sqrt[b]]*ArcTan[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[-a] - 2*Sqrt[b]]] - Sqrt[3*Sqrt[-a] + 2*Sqrt[b]]*ArcTanh[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[-a] + 2*Sqrt[b]]])/(Sqrt[-a]*b^(3/4))","A",1
653,1,123,132,0.0931797,"\int \frac{\sqrt{2+3 x}}{a-b x^2} \, dx","Integrate[Sqrt[2 + 3*x]/(a - b*x^2),x]","\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right)-\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right)}{\sqrt{a} b^{3/4}}","\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right)}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right)}{\sqrt{a} b^{3/4}}",1,"(-(Sqrt[3*Sqrt[a] - 2*Sqrt[b]]*ArcTan[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a] - 2*Sqrt[b]]]) + Sqrt[3*Sqrt[a] + 2*Sqrt[b]]*ArcTanh[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a] + 2*Sqrt[b]]])/(Sqrt[a]*b^(3/4))","A",1
654,1,55,205,0.027116,"\int \frac{\sqrt{1+x}}{1+x^2} \, dx","Integrate[Sqrt[1 + x]/(1 + x^2),x]","i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1+i}}\right)-i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1-i}}\right)","\frac{\log \left(x-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)}}-\frac{\log \left(x+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)}}-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{x+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{x+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"(-I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + x]/Sqrt[1 - I]] + I*Sqrt[1 + I]*ArcTanh[Sqrt[1 + x]/Sqrt[1 + I]]","C",1
655,1,55,198,0.0224871,"\int \frac{1}{\sqrt{1+x} \left(1+x^2\right)} \, dx","Integrate[1/(Sqrt[1 + x]*(1 + x^2)),x]","\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1-i}}\right)+\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1+i}}\right)","-\frac{\log \left(x-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\log \left(x+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{x+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)+\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{x+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)",1,"((1 - I)^(3/2)*ArcTanh[Sqrt[1 + x]/Sqrt[1 - I]])/2 + ((1 + I)^(3/2)*ArcTanh[Sqrt[1 + x]/Sqrt[1 + I]])/2","C",1
656,1,90,272,0.0881284,"\int \frac{\sqrt{-1+x}}{\left(1+x^2\right)^3} \, dx","Integrate[Sqrt[-1 + x]/(1 + x^2)^3,x]","\frac{1}{64} \left(\frac{2 \sqrt{x-1} \left(11 x^3-x^2+19 x-1\right)}{\left(x^2+1\right)^2}-(7-18 i) \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{1-i}}\right)-(7+18 i) \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{1+i}}\right)\right)","-\frac{\sqrt{x-1} (1-11 x)}{32 \left(x^2+1\right)}+\frac{\sqrt{x-1} x}{4 \left(x^2+1\right)^2}-\frac{1}{128} \sqrt{\frac{1}{2} \left(527+373 \sqrt{2}\right)} \log \left(-x-\sqrt{2 \left(\sqrt{2}-1\right)} \sqrt{x-1}-\sqrt{2}+1\right)+\frac{1}{128} \sqrt{\frac{1}{2} \left(527+373 \sqrt{2}\right)} \log \left(-x+\sqrt{2 \left(\sqrt{2}-1\right)} \sqrt{x-1}-\sqrt{2}+1\right)-\frac{1}{64} \sqrt{\frac{1}{2} \left(373 \sqrt{2}-527\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{2}-1\right)}-2 \sqrt{x-1}}{\sqrt{2 \left(1+\sqrt{2}\right)}}\right)+\frac{1}{64} \sqrt{\frac{1}{2} \left(373 \sqrt{2}-527\right)} \tan ^{-1}\left(\frac{2 \sqrt{x-1}+\sqrt{2 \left(\sqrt{2}-1\right)}}{\sqrt{2 \left(1+\sqrt{2}\right)}}\right)",1,"((2*Sqrt[-1 + x]*(-1 + 19*x - x^2 + 11*x^3))/(1 + x^2)^2 - (7 - 18*I)*Sqrt[1 - I]*ArcTan[Sqrt[-1 + x]/Sqrt[1 - I]] - (7 + 18*I)*Sqrt[1 + I]*ArcTan[Sqrt[-1 + x]/Sqrt[1 + I]])/64","C",1
657,1,582,398,3.4406564,"\int (d+e x)^{3/2} \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[a + c*x^2],x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) \left(10 a e^2+3 c \left(d^2+8 d e x+5 e^2 x^2\right)\right)}{c e}+\frac{4 \left(\sqrt{a} e (d+e x)^{3/2} \left(-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-29 a^2 e^2+a c \left(3 d^2-29 e^2 x^2\right)+3 c^2 d^2 x^2\right)\right)}{c e^3 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{105 \sqrt{a+c x^2}}","-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(3 c d^2-5 a e^2\right) \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{105 c^{3/2} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left(-5 a e^2+3 c d^2+24 c d e x\right)}{105 c e}+\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 c d^2-29 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{105 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left(a+c x^2\right)^{3/2} \sqrt{d+e x}}{7 c}",1,"(Sqrt[d + e*x]*((2*(a + c*x^2)*(10*a*e^2 + 3*c*(d^2 + 8*d*e*x + 5*e^2*x^2)))/(c*e) + (4*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^2*x^2 + a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29*a*Sqrt[c]*d*e^2 - (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^3*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(105*Sqrt[a + c*x^2])","C",1
658,1,536,362,2.898612,"\int \sqrt{d+e x} \sqrt{a+c x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[a + c*x^2],x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) (d+3 e x)}{e}-\frac{4 \left(\sqrt{c} (d+e x)^{3/2} \left(-3 a^{3/2} e^3+\sqrt{a} c d^2 e+3 i a \sqrt{c} d e^2-i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-3 a^2 e^2+a c \left(d^2-3 e^2 x^2\right)+c^2 d^2 x^2\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{c e^3 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{15 \sqrt{a+c x^2}}","-\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(c d^2-3 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} (d+e x)^{3/2}}{5 e}-\frac{4 d \sqrt{a+c x^2} \sqrt{d+e x}}{15 e}",1,"(Sqrt[d + e*x]*((2*(d + 3*e*x)*(a + c*x^2))/e - (4*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3*e^2*x^2)) + Sqrt[c]*((-I)*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + (3*I)*a*Sqrt[c]*d*e^2 - 3*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^3*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])","C",1
659,1,456,322,1.957199,"\int \frac{\sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[a + c*x^2]/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(e^2 \left(a+c x^2\right)-\frac{2 \left(d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}-\sqrt{a} e (d+e x)^{3/2} \left(\sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{3 e^3 \sqrt{a+c x^2}}","-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x}}{3 e}",1,"(2*Sqrt[d + e*x]*(e^2*(a + c*x^2) - (2*(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(3*e^3*Sqrt[a + c*x^2])","C",1
660,1,419,305,1.2758671,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^(3/2),x]","\frac{2 \left(-\frac{2 \sqrt{a} \sqrt{c} e (d+e x)^{3/2} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{2 \sqrt{c} (d+e x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+e^2 \left(a+c x^2\right)\right)}{e^3 \sqrt{a+c x^2} \sqrt{d+e x}}","\frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}}",1,"(2*(e^2*(a + c*x^2) + (2*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]] - (2*Sqrt[a]*Sqrt[c]*e*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(e^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
661,1,504,366,1.0879747,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^(5/2),x]","\frac{2 \sqrt{a+c x^2} \left(c d (d+2 e x)-a e^2\right)}{3 (d+e x)^{3/2} \left(a e^3+c d^2 e\right)}-\frac{4 c \left(d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}-\sqrt{a} e (d+e x)^{3/2} \left(\sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{3 e^3 \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(a e^2+c d^2\right)}","\frac{4 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 e^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c d \sqrt{a+c x^2}}{3 e \sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{a+c x^2}}{3 e (d+e x)^{3/2}}",1,"(2*Sqrt[a + c*x^2]*(-(a*e^2) + c*d*(d + 2*e*x)))/(3*(c*d^2*e + a*e^3)*(d + e*x)^(3/2)) - (4*c*(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(3*e^3*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
662,1,602,444,2.1057614,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x)^(7/2),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(3 a^2 e^4+2 a c e^2 \left(5 d^2+5 d e x+3 e^2 x^2\right)-c^2 d^2 \left(d^2+6 d e x+2 e^2 x^2\right)\right)+\frac{2 c (d+e x)^2 \left(\sqrt{c} (d+e x)^{3/2} \left(3 a^{3/2} e^3-\sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2+i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-3 a^2 e^2+a c \left(d^2-3 e^2 x^2\right)+c^2 d^2 x^2\right)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{15 e^3 \sqrt{a+c x^2} (d+e x)^{5/2} \left(a e^2+c d^2\right)^2}","-\frac{4 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 e^2 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{4 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(c d^2-3 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 e^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 c \sqrt{a+c x^2} \left(c d^2-3 a e^2\right)}{15 e \sqrt{d+e x} \left(a e^2+c d^2\right)^2}+\frac{4 c d \sqrt{a+c x^2}}{15 e (d+e x)^{3/2} \left(a e^2+c d^2\right)}-\frac{2 \sqrt{a+c x^2}}{5 e (d+e x)^{5/2}}",1,"(2*(-(e^2*(a + c*x^2)*(3*a^2*e^4 - c^2*d^2*(d^2 + 6*d*e*x + 2*e^2*x^2) + 2*a*c*e^2*(5*d^2 + 5*d*e*x + 3*e^2*x^2))) + (2*c*(d + e*x)^2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3*e^2*x^2))) + Sqrt[c]*(I*c^(3/2)*d^3 - Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(15*e^3*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2)*Sqrt[a + c*x^2])","C",1
663,1,695,497,3.9824606,"\int (d+e x)^{3/2} \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(a + c*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x} \left(e^2 \left(a+c x^2\right) \left(60 a^2 e^4+a c e^2 \left(47 d^2+326 d e x+195 e^2 x^2\right)+c^2 \left(8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right)\right)+\frac{4 \left(-4 d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-27 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right)+\sqrt{a} e (d+e x)^{3/2} \left(114 i a^{3/2} c d^2 e^3-15 i a^{5/2} e^5-108 a^2 \sqrt{c} d e^4+24 a c^{3/2} d^3 e^2+i \sqrt{a} c^2 d^4 e+4 c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 \sqrt{c} d (d+e x)^{3/2} \left(-6 a^{3/2} c d^2 e^3+27 a^{5/2} e^5-27 i a^2 \sqrt{c} d e^4+6 i a c^{3/2} d^3 e^2-\sqrt{a} c^2 d^4 e+i c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{1155 c e^5 \sqrt{a+c x^2}}","\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left(-15 a^2 e^4-3 c d e x \left(c d^2-31 a e^2\right)+21 a c d^2 e^2+4 c^2 d^4\right)}{1155 c e^3}-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left(a+c x^2\right)^{3/2} \sqrt{d+e x} \left(-3 a e^2+c d^2+28 c d e x\right)}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(c d^2-3 a e^2\right) \left(9 a e^2+c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left(a+c x^2\right)^{5/2} \sqrt{d+e x}}{11 c}",1,"(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(60*a^2*e^4 + a*c*e^2*(47*d^2 + 326*d*e*x + 195*e^2*x^2) + c^2*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + (4*(-4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c^2*d^4 + 6*a*c*d^2*e^2 - 27*a^2*e^4)*(a + c*x^2) + 4*Sqrt[c]*d*(I*c^(5/2)*d^5 - Sqrt[a]*c^2*d^4*e + (6*I)*a*c^(3/2)*d^3*e^2 - 6*a^(3/2)*c*d^2*e^3 - (27*I)*a^2*Sqrt[c]*d*e^4 + 27*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(4*c^(5/2)*d^5 + I*Sqrt[a]*c^2*d^4*e + 24*a*c^(3/2)*d^3*e^2 + (114*I)*a^(3/2)*c*d^2*e^3 - 108*a^2*Sqrt[c]*d*e^4 - (15*I)*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(1155*c*e^5*Sqrt[a + c*x^2])","C",1
664,1,646,448,4.1493696,"\int \sqrt{d+e x} \left(a+c x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(a + c*x^2)^(3/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) \left(a e^2 (29 d+77 e x)+c \left(8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right)\right)}{e^3}+\frac{8 \left(e^2 \left(a+c x^2\right) \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(-15 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt{c} d e^4+15 i a c^{3/2} d^3 e^2-4 \sqrt{a} c^2 d^4 e+4 i c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{c e^5 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{315 \sqrt{a+c x^2}}","\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(3 a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left(4 d \left(3 a e^2+c d^2\right)-3 e x \left(c d^2-7 a e^2\right)\right)}{315 e^3}+\frac{2 \left(a+c x^2\right)^{3/2} (d+e x)^{3/2}}{9 e}-\frac{4 d \left(a+c x^2\right)^{3/2} \sqrt{d+e x}}{21 e}",1,"(Sqrt[d + e*x]*((2*(a + c*x^2)*(a*e^2*(29*d + 77*e*x) + c*(8*d^3 - 6*d^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)))/e^3 + (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(a + c*x^2)) + Sqrt[c]*((4*I)*c^(5/2)*d^5 - 4*Sqrt[a]*c^2*d^4*e + (15*I)*a*c^(3/2)*d^3*e^2 - 15*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(315*Sqrt[a + c*x^2])","C",1
665,1,575,393,3.1815537,"\int \frac{\left(a+c x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(a + c*x^2)^(3/2)/Sqrt[d + e*x],x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) \left(15 a e^2+c \left(8 d^2-6 d e x+5 e^2 x^2\right)\right)}{e^3}-\frac{8 \left(-\sqrt{a} e (d+e x)^{3/2} \left(5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 \sqrt{c} d (d+e x)^{3/2} \left(2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(2 a^2 e^2+a c \left(d^2+2 e^2 x^2\right)+c^2 d^2 x^2\right)\right)}{e^5 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{35 \sqrt{a+c x^2}}","-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(5 a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{35 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{32 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(2 a e^2+c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{35 e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left(5 a e^2+4 c d^2-3 c d e x\right)}{35 e^3}+\frac{2 \left(a+c x^2\right)^{3/2} \sqrt{d+e x}}{7 e}",1,"(Sqrt[d + e*x]*((2*(a + c*x^2)*(15*a*e^2 + c*(8*d^2 - 6*d*e*x + 5*e^2*x^2)))/e^3 - (8*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(35*Sqrt[a + c*x^2])","C",1
666,1,565,369,2.9551323,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(3/2),x]","\frac{2 \sqrt{a+c x^2} \left(c \left(-8 d^2-2 d e x+e^2 x^2\right)-5 a e^2\right)}{5 e^3 \sqrt{d+e x}}+\frac{8 \left(\sqrt{c} (d+e x)^{3/2} \left(3 a^{3/2} e^3+4 \sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2-4 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(3 a^2 e^2+a c \left(4 d^2+3 e^2 x^2\right)+4 c^2 d^2 x^2\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{5 e^5 \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}","\frac{32 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{5 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 a e^2+4 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{5 e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \sqrt{a+c x^2} (4 d-3 e x) \sqrt{d+e x}}{5 e^3}-\frac{2 \left(a+c x^2\right)^{3/2}}{e \sqrt{d+e x}}",1,"(2*Sqrt[a + c*x^2]*(-5*a*e^2 + c*(-8*d^2 - 2*d*e*x + e^2*x^2)))/(5*e^3*Sqrt[d + e*x]) + (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)) + Sqrt[c]*((-4*I)*c^(3/2)*d^3 + 4*Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(5*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
667,1,494,358,2.2877831,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(5/2),x]","\frac{2 \sqrt{a+c x^2} \left(c \left(8 d^2+10 d e x+e^2 x^2\right)-a e^2\right)}{3 e^3 (d+e x)^{3/2}}-\frac{8 c \left(4 d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}-\sqrt{a} e (d+e x)^{3/2} \left(4 \sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 \sqrt{c} d (d+e x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{3 e^5 \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}","\frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 c \sqrt{a+c x^2} (4 d+e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left(a+c x^2\right)^{3/2}}{3 e (d+e x)^{3/2}}",1,"(2*Sqrt[a + c*x^2]*(-(a*e^2) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(3*e^3*(d + e*x)^(3/2)) - (8*c*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(3*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
668,1,602,410,2.0684882,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(7/2),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(-4 c d (d+e x) \left(a e^2+c d^2\right)+c (d+e x)^2 \left(7 a e^2+11 c d^2\right)+\left(a e^2+c d^2\right)^2\right)+\frac{4 c (d+e x)^2 \left(\sqrt{c} (d+e x)^{3/2} \left(3 a^{3/2} e^3+4 \sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2-4 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(3 a^2 e^2+a c \left(4 d^2+3 e^2 x^2\right)+4 c^2 d^2 x^2\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{5 e^5 \sqrt{a+c x^2} (d+e x)^{5/2} \left(a e^2+c d^2\right)}","-\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 a e^2+4 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{5 e^4 \sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{5 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 c \sqrt{a+c x^2} \left(e x \left(3 a e^2+5 c d^2\right)+2 d \left(a e^2+2 c d^2\right)\right)}{5 e^3 (d+e x)^{3/2} \left(a e^2+c d^2\right)}-\frac{2 \left(a+c x^2\right)^{3/2}}{5 e (d+e x)^{5/2}}",1,"(2*(-(e^2*(a + c*x^2)*((c*d^2 + a*e^2)^2 - 4*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2 + 7*a*e^2)*(d + e*x)^2)) + (4*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)) + Sqrt[c]*((-4*I)*c^(3/2)*d^3 + 4*Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(5*e^5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)*Sqrt[a + c*x^2])","C",1
669,1,659,491,3.2207108,"\int \frac{\left(a+c x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(9/2),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(-16 c^2 d (d+e x)^3 \left(2 a e^2+c d^2\right)-16 c d (d+e x) \left(a e^2+c d^2\right)^2+c (d+e x)^2 \left(15 a e^2+19 c d^2\right) \left(a e^2+c d^2\right)+5 \left(a e^2+c d^2\right)^3\right)-\frac{4 c^2 (d+e x)^3 \left(-\sqrt{a} e (d+e x)^{3/2} \left(5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 \sqrt{c} d (d+e x)^{3/2} \left(2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(2 a^2 e^2+a c \left(d^2+2 e^2 x^2\right)+c^2 d^2 x^2\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{35 e^5 \sqrt{a+c x^2} (d+e x)^{7/2} \left(a e^2+c d^2\right)^2}","-\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left(5 a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{35 e^4 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{32 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(2 a e^2+c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{35 e^4 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 c^2 d \sqrt{a+c x^2} \left(2 a e^2+c d^2\right)}{35 e^3 \sqrt{d+e x} \left(a e^2+c d^2\right)^2}-\frac{4 c \sqrt{a+c x^2} \left(e x \left(5 a e^2+7 c d^2\right)+2 d \left(a e^2+2 c d^2\right)\right)}{35 e^3 (d+e x)^{5/2} \left(a e^2+c d^2\right)}-\frac{2 \left(a+c x^2\right)^{3/2}}{7 e (d+e x)^{7/2}}",1,"(2*(-(e^2*(a + c*x^2)*(5*(c*d^2 + a*e^2)^3 - 16*c*d*(c*d^2 + a*e^2)^2*(d + e*x) + c*(c*d^2 + a*e^2)*(19*c*d^2 + 15*a*e^2)*(d + e*x)^2 - 16*c^2*d*(c*d^2 + 2*a*e^2)*(d + e*x)^3)) - (4*c^2*(d + e*x)^3*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(35*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2)*Sqrt[a + c*x^2])","C",1
670,1,790,566,4.8995734,"\int \sqrt{d+e x} \left(a+c x^2\right)^{5/2} \, dx","Integrate[Sqrt[d + e*x]*(a + c*x^2)^(5/2),x]","\frac{2 \sqrt{d+e x} \left(e^2 \left(a+c x^2\right) \left(a^2 e^4 (971 d+2387 e x)+2 a c e^2 \left(266 d^3-197 d^2 e x+163 d e^2 x^2+1078 e^3 x^3\right)+c^2 \left(128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right)\right)+\frac{8 \left(e^2 \left(a+c x^2\right) \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(32 i a^{3/2} c^{3/2} d^3 e^3+408 i a^{5/2} \sqrt{c} d e^5-231 a^3 e^6+258 a^2 c d^2 e^4+8 i \sqrt{a} c^{5/2} d^5 e+137 a c^2 d^4 e^2+32 c^3 d^6\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(-137 a^{3/2} c^2 d^4 e^3-258 a^{5/2} c d^2 e^5+231 a^{7/2} e^7-231 i a^3 \sqrt{c} d e^6+258 i a^2 c^{3/2} d^3 e^4+137 i a c^{5/2} d^5 e^2-32 \sqrt{a} c^3 d^6 e+32 i c^{7/2} d^7\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{c (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{9009 e^7 \sqrt{a+c x^2}}","-\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left(d \left(177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right)-3 e x \left(-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right)\right)}{9009 e^5}+\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left(a+c x^2\right)^{3/2} \sqrt{d+e x} \left(4 d \left(5 a e^2+2 c d^2\right)-7 e x \left(c d^2-11 a e^2\right)\right)}{9009 e^3}+\frac{2 \left(a+c x^2\right)^{5/2} (d+e x)^{3/2}}{13 e}-\frac{20 d \left(a+c x^2\right)^{5/2} \sqrt{d+e x}}{143 e}",1,"(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(a^2*e^4*(971*d + 2387*e*x) + 2*a*c*e^2*(266*d^3 - 197*d^2*e*x + 163*d*e^2*x^2 + 1078*e^3*x^3) + c^2*(128*d^5 - 96*d^4*e*x + 80*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + (8*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*c*d^2*e^4 - 231*a^3*e^6)*(a + c*x^2)) + Sqrt[c]*((32*I)*c^(7/2)*d^7 - 32*Sqrt[a]*c^3*d^6*e + (137*I)*a*c^(5/2)*d^5*e^2 - 137*a^(3/2)*c^2*d^4*e^3 + (258*I)*a^2*c^(3/2)*d^3*e^4 - 258*a^(5/2)*c*d^2*e^5 - (231*I)*a^3*Sqrt[c]*d*e^6 + 231*a^(7/2)*e^7)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(32*c^3*d^6 + (8*I)*Sqrt[a]*c^(5/2)*d^5*e + 137*a*c^2*d^4*e^2 + (32*I)*a^(3/2)*c^(3/2)*d^3*e^3 + 258*a^2*c*d^2*e^4 + (408*I)*a^(5/2)*Sqrt[c]*d*e^5 - 231*a^3*e^6)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(9009*e^7*Sqrt[a + c*x^2])","C",1
671,1,634,494,3.6195498,"\int \frac{\left(a+c x^2\right)^{5/2}}{\sqrt{d+e x}} \, dx","Integrate[(a + c*x^2)^(5/2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(-\frac{8 d e^2 \left(a+c x^2\right) \left(93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right)}{d+e x}-8 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \left(a+c x^2\right) \left(333 a^2 e^4+2 a c e^2 \left(178 d^2-131 d e x+108 e^2 x^2\right)+c^2 \left(128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right)\right)+\frac{8 \sqrt{a} e \sqrt{d+e x} \left(21 i a^{3/2} c d^2 e^3+45 i a^{5/2} e^5+93 a^2 \sqrt{c} d e^4+93 a c^{3/2} d^3 e^2+8 i \sqrt{a} c^2 d^4 e+32 c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{693 e^7 \sqrt{a+c x^2}}","-\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{693 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{693 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left(45 a^2 e^4-24 c d e x \left(2 a e^2+c d^2\right)+69 a c d^2 e^2+32 c^2 d^4\right)}{693 e^5}+\frac{20 \left(a+c x^2\right)^{3/2} \sqrt{d+e x} \left(9 a e^2+8 c d^2-7 c d e x\right)}{693 e^3}+\frac{2 \left(a+c x^2\right)^{5/2} \sqrt{d+e x}}{11 e}",1,"(2*Sqrt[d + e*x]*((-8*d*e^2*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(a + c*x^2))/(d + e*x) + e^2*(a + c*x^2)*(333*a^2*e^4 + 2*a*c*e^2*(178*d^2 - 131*d*e*x + 108*e^2*x^2) + c^2*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 - 70*d*e^3*x^3 + 63*e^4*x^4)) - (8*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + (8*Sqrt[a]*e*(32*c^(5/2)*d^5 + (8*I)*Sqrt[a]*c^2*d^4*e + 93*a*c^(3/2)*d^3*e^2 + (21*I)*a^(3/2)*c*d^2*e^3 + 93*a^2*Sqrt[c]*d*e^4 + (45*I)*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(693*e^7*Sqrt[a + c*x^2])","C",1
672,1,684,457,3.8626566,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(3/2),x]","\frac{\sqrt{d+e x} \left(-\frac{2 \left(a+c x^2\right) \left(63 a^2 e^4+2 a c e^2 \left(106 d^2+29 d e x-14 e^2 x^2\right)+c^2 \left(128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right)\right)}{e^5 (d+e x)}+\frac{16 \left(e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(12 i a^{3/2} \sqrt{c} d e^3+21 a^2 e^4+8 i \sqrt{a} c^{3/2} d^3 e+57 a c d^2 e^2+32 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(57 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt{c} d e^4-57 i a c^{3/2} d^3 e^2+32 \sqrt{a} c^2 d^4 e-32 i c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{63 \sqrt{a+c x^2}}","-\frac{16 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{63 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(33 a e^2+32 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{8 c \sqrt{a+c x^2} \sqrt{d+e x} \left(d \left(33 a e^2+32 c d^2\right)-3 e x \left(7 a e^2+8 c d^2\right)\right)}{63 e^5}-\frac{20 c \left(a+c x^2\right)^{3/2} (8 d-7 e x) \sqrt{d+e x}}{63 e^3}-\frac{2 \left(a+c x^2\right)^{5/2}}{e \sqrt{d+e x}}",1,"(Sqrt[d + e*x]*((-2*(a + c*x^2)*(63*a^2*e^4 + 2*a*c*e^2*(106*d^2 + 29*d*e*x - 14*e^2*x^2) + c^2*(128*d^4 + 32*d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)))/(e^5*(d + e*x)) + (16*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + Sqrt[c]*((-32*I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^4*e - (57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^3 + 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(63*Sqrt[a + c*x^2])","C",1
673,1,637,430,3.4929326,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) \left(-7 a^2 e^4+2 a c e^2 \left(50 d^2+65 d e x+8 e^2 x^2\right)+c^2 \left(128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right)\right)}{e^5 (d+e x)^2}-\frac{16 c \left(-\sqrt{a} e (d+e x)^{3/2} \left(5 i a^{3/2} e^3+8 i \sqrt{a} c d^2 e+29 a \sqrt{c} d e^2+32 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(29 a^{3/2} e^3+32 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(29 a^2 e^2+a c \left(32 d^2+29 e^2 x^2\right)+32 c^2 d^2 x^2\right)\right)}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{21 \sqrt{a+c x^2}}","\frac{16 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(29 a e^2+32 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{21 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{16 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(5 a e^2+32 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 c \sqrt{a+c x^2} \sqrt{d+e x} \left(5 a e^2+32 c d^2-24 c d e x\right)}{21 e^5}+\frac{20 c \left(a+c x^2\right)^{3/2} (8 d+e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left(a+c x^2\right)^{5/2}}{3 e (d+e x)^{3/2}}",1,"(Sqrt[d + e*x]*((2*(a + c*x^2)*(-7*a^2*e^4 + 2*a*c*e^2*(50*d^2 + 65*d*e*x + 8*e^2*x^2) + c^2*(128*d^4 + 160*d^3*e*x + 16*d^2*e^2*x^2 - 6*d*e^3*x^3 + 3*e^4*x^4)))/(e^5*(d + e*x)^2) - (16*c*(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(29*a^2*e^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 29*e^2*x^2)) + Sqrt[c]*d*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(32*c^(3/2)*d^3 + (8*I)*Sqrt[a]*c*d^2*e + 29*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(21*Sqrt[a + c*x^2])","C",1
674,1,613,420,3.3341591,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(7/2),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(3 a^2 e^4+2 a c e^2 \left(10 d^2+25 d e x+18 e^2 x^2\right)+c^2 \left(128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right)\right)+\frac{8 c (d+e x)^2 \left(\sqrt{c} (d+e x)^{3/2} \left(9 a^{3/2} e^3+32 \sqrt{a} c d^2 e-9 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(9 a^2 e^2+a c \left(32 d^2+9 e^2 x^2\right)+32 c^2 d^2 x^2\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(8 i \sqrt{a} \sqrt{c} d e+9 a e^2+32 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{15 e^7 \sqrt{a+c x^2} (d+e x)^{5/2}}","\frac{16 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \left(17 a e^2+32 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(9 a e^2+32 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 c \sqrt{a+c x^2} \left(9 a e^2+32 c d^2+8 c d e x\right)}{15 e^5 \sqrt{d+e x}}+\frac{4 c \left(a+c x^2\right)^{3/2} (8 d+3 e x)}{15 e^3 (d+e x)^{3/2}}-\frac{2 \left(a+c x^2\right)^{5/2}}{5 e (d+e x)^{5/2}}",1,"(2*(-(e^2*(a + c*x^2)*(3*a^2*e^4 + 2*a*c*e^2*(10*d^2 + 25*d*e*x + 18*e^2*x^2) + c^2*(128*d^4 + 288*d^3*e*x + 176*d^2*e^2*x^2 + 10*d*e^3*x^3 - 3*e^4*x^4))) + (8*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(9*a^2*e^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 9*e^2*x^2)) + Sqrt[c]*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (9*I)*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(32*c*d^2 + (8*I)*Sqrt[a]*Sqrt[c]*d*e + 9*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(15*e^7*(d + e*x)^(5/2)*Sqrt[a + c*x^2])","C",1
675,1,677,498,3.2797566,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(9/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a+c x^2\right) \left(\frac{2 c^2 d \left(67 a e^2+79 c d^2\right)}{(d+e x) \left(a e^2+c d^2\right)}-\frac{4 c \left(4 a e^2+13 c d^2\right)}{(d+e x)^2}+\frac{18 c d \left(a e^2+c d^2\right)}{(d+e x)^3}-\frac{3 \left(a e^2+c d^2\right)^2}{(d+e x)^4}+7 c^2\right)}{e^5}-\frac{16 c^2 \left(-\sqrt{a} e (d+e x)^{3/2} \left(5 i a^{3/2} e^3+8 i \sqrt{a} c d^2 e+29 a \sqrt{c} d e^2+32 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(29 a^{3/2} e^3+32 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(29 a^2 e^2+a c \left(32 d^2+29 e^2 x^2\right)+32 c^2 d^2 x^2\right)\right)}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(a e^2+c d^2\right)}\right)}{21 \sqrt{a+c x^2}}","-\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left(5 a e^2+32 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(29 a e^2+32 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{21 e^6 \sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 c^2 \sqrt{a+c x^2} \left(e x \left(5 a e^2+8 c d^2\right)+d \left(29 a e^2+32 c d^2\right)\right)}{21 e^5 \sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{4 c \left(a+c x^2\right)^{3/2} \left(e x \left(5 a e^2+11 c d^2\right)+2 d \left(a e^2+4 c d^2\right)\right)}{21 e^3 (d+e x)^{5/2} \left(a e^2+c d^2\right)}-\frac{2 \left(a+c x^2\right)^{5/2}}{7 e (d+e x)^{7/2}}",1,"(Sqrt[d + e*x]*((2*(a + c*x^2)*(7*c^2 - (3*(c*d^2 + a*e^2)^2)/(d + e*x)^4 + (18*c*d*(c*d^2 + a*e^2))/(d + e*x)^3 - (4*c*(13*c*d^2 + 4*a*e^2))/(d + e*x)^2 + (2*c^2*d*(79*c*d^2 + 67*a*e^2))/((c*d^2 + a*e^2)*(d + e*x))))/e^5 - (16*c^2*(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(29*a^2*e^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 29*e^2*x^2)) + Sqrt[c]*d*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(32*c^(3/2)*d^3 + (8*I)*Sqrt[a]*c*d^2*e + 29*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)*(d + e*x))))/(21*Sqrt[a + c*x^2])","C",1
676,1,762,553,3.7281023,"\int \frac{\left(a+c x^2\right)^{5/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(11/2),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(c^2 (d+e x)^4 \left(105 a^2 e^4+330 a c d^2 e^2+193 c^2 d^4\right)-2 c^2 d (d+e x)^3 \left(57 a e^2+61 c d^2\right) \left(a e^2+c d^2\right)-38 c d (d+e x) \left(a e^2+c d^2\right)^3+4 c (d+e x)^2 \left(7 a e^2+22 c d^2\right) \left(a e^2+c d^2\right)^2+7 \left(a e^2+c d^2\right)^4\right)+\frac{8 c^2 (d+e x)^4 \left(e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(12 i a^{3/2} \sqrt{c} d e^3+21 a^2 e^4+8 i \sqrt{a} c^{3/2} d^3 e+57 a c d^2 e^2+32 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(57 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt{c} d e^4-57 i a c^{3/2} d^3 e^2+32 \sqrt{a} c^2 d^4 e-32 i c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{63 e^7 \sqrt{a+c x^2} (d+e x)^{9/2} \left(a e^2+c d^2\right)^2}","-\frac{8 c^2 \sqrt{a+c x^2} \left(e x \left(21 a^2 e^4+69 a c d^2 e^2+40 c^2 d^4\right)+d \left(9 a^2 e^4+49 a c d^2 e^2+32 c^2 d^4\right)\right)}{63 e^5 (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}-\frac{16 \sqrt{-a} c^{5/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{63 e^6 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \left(33 a e^2+32 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{4 c \left(a+c x^2\right)^{3/2} \left(e x \left(7 a e^2+13 c d^2\right)+2 d \left(a e^2+4 c d^2\right)\right)}{63 e^3 (d+e x)^{7/2} \left(a e^2+c d^2\right)}-\frac{2 \left(a+c x^2\right)^{5/2}}{9 e (d+e x)^{9/2}}",1,"(2*(-(e^2*(a + c*x^2)*(7*(c*d^2 + a*e^2)^4 - 38*c*d*(c*d^2 + a*e^2)^3*(d + e*x) + 4*c*(c*d^2 + a*e^2)^2*(22*c*d^2 + 7*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)*(61*c*d^2 + 57*a*e^2)*(d + e*x)^3 + c^2*(193*c^2*d^4 + 330*a*c*d^2*e^2 + 105*a^2*e^4)*(d + e*x)^4)) + (8*c^2*(d + e*x)^4*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + Sqrt[c]*((-32*I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^4*e - (57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^3 + 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(63*e^7*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2)*Sqrt[a + c*x^2])","C",1
677,1,548,413,3.0835608,"\int \frac{(d+e x)^{7/2}}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[a + c*x^2],x]","\frac{2 \sqrt{d+e x} \left(\frac{16 d e \left(-13 a^2 e^2+a c \left(11 d^2-13 e^2 x^2\right)+11 c^2 d^2 x^2\right)}{d+e x}+\frac{\sqrt{d+e x} \left(208 a^{3/2} \sqrt{c} d e^3+25 i a^2 e^4-176 \sqrt{a} c^{3/2} d^3 e-254 i a c d^2 e^2+105 i c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{16 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(11 c d^2-13 a e^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e}+\left(a+c x^2\right) \left(c e \left(122 d^2+66 d e x+15 e^2 x^2\right)-25 a e^3\right)\right)}{105 c^2 \sqrt{a+c x^2}}","\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(71 c d^2-25 a e^2\right) \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{105 c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(11 c d^2-13 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{105 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x} \left(71 c d^2-25 a e^2\right)}{105 c^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{5/2}}{7 c}+\frac{24 d e \sqrt{a+c x^2} (d+e x)^{3/2}}{35 c}",1,"(2*Sqrt[d + e*x]*((16*d*e*(-13*a^2*e^2 + 11*c^2*d^2*x^2 + a*c*(11*d^2 - 13*e^2*x^2)))/(d + e*x) + (a + c*x^2)*(-25*a*e^3 + c*e*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + ((16*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(11*c*d^2 - 13*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (((105*I)*c^2*d^4 - 176*Sqrt[a]*c^(3/2)*d^3*e - (254*I)*a*c*d^2*e^2 + 208*a^(3/2)*Sqrt[c]*d*e^3 + (25*I)*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(105*c^2*Sqrt[a + c*x^2])","C",1
678,1,557,359,2.9802652,"\int \frac{(d+e x)^{5/2}}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[a + c*x^2],x]","\frac{\sqrt{d+e x} \left(\frac{2 e \left(a+c x^2\right) (11 d+3 e x)}{c}+\frac{2 \left(\sqrt{c} (d+e x)^{3/2} \left(9 a^{3/2} e^3-23 \sqrt{a} c d^2 e-17 i a \sqrt{c} d e^2+15 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(-9 a^{3/2} e^3+23 \sqrt{a} c d^2 e+9 i a \sqrt{c} d e^2-23 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-9 a^2 e^2+a c \left(23 d^2-9 e^2 x^2\right)+23 c^2 d^2 x^2\right)\right)}{c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{15 \sqrt{a+c x^2}}","\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(23 c d^2-9 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c}+\frac{16 d e \sqrt{a+c x^2} \sqrt{d+e x}}{15 c}",1,"(Sqrt[d + e*x]*((2*e*(11*d + 3*e*x)*(a + c*x^2))/c + (2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-9*a^2*e^2 + 23*c^2*d^2*x^2 + a*c*(23*d^2 - 9*e^2*x^2)) + Sqrt[c]*((-23*I)*c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (9*I)*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[c]*((15*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (17*I)*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])","C",1
679,1,445,317,2.1505406,"\int \frac{(d+e x)^{3/2}}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[a + c*x^2],x]","\frac{2 \sqrt{d+e x} \left(\frac{i \sqrt{d+e x} \left(4 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{4 d e^2 \left(a+c x^2\right)}{d+e x}+4 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \left(a+c x^2\right)\right)}{3 c e \sqrt{a+c x^2}}","\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x}}{3 c}-\frac{8 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(2*Sqrt[d + e*x]*(e^2*(a + c*x^2) + (4*d*e^2*(a + c*x^2))/(d + e*x) + (4*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + (I*(3*c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*c*e*Sqrt[a + c*x^2])","C",1
680,1,294,136,0.4932588,"\int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[a + c*x^2],x]","\frac{2 i \sqrt{d+e x} \left(\sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(\sqrt{a}+i \sqrt{c} x\right)}{\sqrt{a} e-i \sqrt{c} d}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{e \left(\sqrt{c} x+i \sqrt{a}\right)}}}","-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"((2*I)*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[d + e*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt[a + c*x^2])","C",1
681,1,186,136,0.2235915,"\int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[a + c*x^2]),x]","\frac{2 i (d+e x) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e \sqrt{a+c x^2} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}","-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}",1,"((2*I)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[a + c*x^2])","C",1
682,1,331,186,0.3965772,"\int \frac{1}{(d+e x)^{3/2} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[a + c*x^2]),x]","-\frac{2 e \sqrt{a+c x^2}}{\sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{2 \sqrt{c} \sqrt{d+e x} \sqrt{\frac{e \left(\sqrt{a}+i \sqrt{c} x\right)}{\sqrt{a} e-i \sqrt{c} d}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{e \sqrt{a+c x^2} \left(\sqrt{a} e+i \sqrt{c} d\right) \sqrt{\frac{\sqrt{c} (d+e x)}{e \left(\sqrt{c} x+i \sqrt{a}\right)}}}","-\frac{2 e \sqrt{a+c x^2}}{\sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(-2*e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*Sqrt[c]*Sqrt[(e*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[d + e*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e*(I*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(Sqrt[c]*(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt[a + c*x^2])","C",1
683,1,494,382,1.5169568,"\int \frac{1}{(d+e x)^{5/2} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[a + c*x^2]),x]","\frac{2 \left(-e^2 \left(a+c x^2\right) \left(a e^2+c d (5 d+4 e x)\right)+\frac{c (d+e x) \left(i (d+e x)^{3/2} \left(4 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}+4 \sqrt{c} d (d+e x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{3 e \sqrt{a+c x^2} (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}","-\frac{8 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 c d e \sqrt{a+c x^2}}{3 \sqrt{d+e x} \left(a e^2+c d^2\right)^2}-\frac{2 e \sqrt{a+c x^2}}{3 (d+e x)^{3/2} \left(a e^2+c d^2\right)}+\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}",1,"(2*(-(e^2*(a + c*x^2)*(a*e^2 + c*d*(5*d + 4*e*x))) + (c*(d + e*x)*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + I*(3*c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*e*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)*Sqrt[a + c*x^2])","C",1
684,1,618,447,2.4530571,"\int \frac{1}{(d+e x)^{7/2} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^(7/2)*Sqrt[a + c*x^2]),x]","\frac{2 \left(-\left(e^2 \left(a+c x^2\right) \left(8 c d (d+e x) \left(a e^2+c d^2\right)+c (d+e x)^2 \left(23 c d^2-9 a e^2\right)+3 \left(a e^2+c d^2\right)^2\right)\right)-\frac{c (d+e x)^2 \left(\sqrt{c} (d+e x)^{3/2} \left(-9 a^{3/2} e^3+23 \sqrt{a} c d^2 e+17 i a \sqrt{c} d e^2-15 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(9 a^{3/2} e^3-23 \sqrt{a} c d^2 e-9 i a \sqrt{c} d e^2+23 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-9 a^2 e^2+a c \left(23 d^2-9 e^2 x^2\right)+23 c^2 d^2 x^2\right)\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{15 e \sqrt{a+c x^2} (d+e x)^{5/2} \left(a e^2+c d^2\right)^3}","\frac{16 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)^2}-\frac{2 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(23 c d^2-9 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{15 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 c e \sqrt{a+c x^2} \left(23 c d^2-9 a e^2\right)}{15 \sqrt{d+e x} \left(a e^2+c d^2\right)^3}-\frac{16 c d e \sqrt{a+c x^2}}{15 (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}-\frac{2 e \sqrt{a+c x^2}}{5 (d+e x)^{5/2} \left(a e^2+c d^2\right)}",1,"(2*(-(e^2*(a + c*x^2)*(3*(c*d^2 + a*e^2)^2 + 8*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(23*c*d^2 - 9*a*e^2)*(d + e*x)^2)) - (c*(d + e*x)^2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-9*a^2*e^2 + 23*c^2*d^2*x^2 + a*c*(23*d^2 - 9*e^2*x^2))) + Sqrt[c]*((23*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (9*I)*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[c]*((-15*I)*c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (17*I)*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(15*e*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2)*Sqrt[a + c*x^2])","C",1
685,1,586,426,3.4240284,"\int \frac{(d+e x)^{7/2}}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a + c*x^2)^(3/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(\sqrt{a} e (d+e x)^{3/2} \left(-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-29 a^2 e^2+a c \left(3 d^2-29 e^2 x^2\right)+3 c^2 d^2 x^2\right)\right)}{a c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{10 a e^3}{c^2}+\frac{6 d^3 x}{a}+\frac{2 e \left(-9 d^2-9 d e x+2 e^2 x^2\right)}{c}\right)}{6 \sqrt{a+c x^2}}","\frac{\sqrt{\frac{c x^2}{a}+1} \left(3 c d^2-5 a e^2\right) \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{-a} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 c d^2-29 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{e \sqrt{a+c x^2} \sqrt{d+e x} \left(3 c d^2-5 a e^2\right)}{3 a c^2}-\frac{(d+e x)^{5/2} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^{3/2}}{a c}",1,"(Sqrt[d + e*x]*((10*a*e^3)/c^2 + (6*d^3*x)/a + (2*e*(-9*d^2 - 9*d*e*x + 2*e^2*x^2))/c + (2*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^2*x^2 + a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29*a*Sqrt[c]*d*e^2 - (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(a*c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(6*Sqrt[a + c*x^2])","C",1
686,1,495,363,2.8493939,"\int \frac{(d+e x)^{5/2}}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a + c*x^2)^(3/2),x]","\frac{\sqrt{d+e x} \left(-\frac{e \left(-3 a^2 e^2+a c \left(d^2-3 e^2 x^2\right)+c^2 d^2 x^2\right)}{d+e x}+\frac{\sqrt{a} \sqrt{c} \sqrt{d+e x} \left(4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{i c \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(c d^2-3 a e^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e}+c \left(c d^2 x-a e (2 d+e x)\right)\right)}{a c^2 \sqrt{a+c x^2}}","\frac{d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(c d^2-3 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{3/2} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} \sqrt{d+e x}}{a c}",1,"(Sqrt[d + e*x]*(c*(c*d^2*x - a*e*(2*d + e*x)) - (e*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3*e^2*x^2)))/(d + e*x) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 - 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[c]*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(a*c^2*Sqrt[a + c*x^2])","C",1
687,1,414,321,4.9158071,"\int \frac{(d+e x)^{3/2}}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a + c*x^2)^(3/2),x]","\frac{e \left(c d^2 x-a e (2 d+e x)\right)-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-i c d (d+e x)^{3/2} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{a c e \sqrt{a+c x^2} \sqrt{d+e x}}","\frac{\sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(e*(c*d^2*x - a*e*(2*d + e*x)) - I*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*Sqrt[c]*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a*c*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
688,1,408,298,2.0606524,"\int \frac{\sqrt{d+e x}}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(a + c*x^2)^(3/2),x]","\frac{\sqrt{d+e x} \left(-\frac{e \left(a+c x^2\right)}{c (d+e x)}+\frac{\sqrt{a} \sqrt{d+e x} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{c} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{i \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e}+x\right)}{a \sqrt{a+c x^2}}","\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}+\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(Sqrt[d + e*x]*(x - (e*(a + c*x^2))/(c*(d + e*x)) - (I*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[c]*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(a*Sqrt[a + c*x^2])","C",1
689,1,430,331,0.9621714,"\int \frac{1}{\sqrt{d+e x} \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)^(3/2)),x]","\frac{e x \left(\sqrt{c} d-i \sqrt{a} e\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}+\sqrt{a} e (d+e x)^{3/2} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+i \sqrt{c} d (d+e x)^{3/2} \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{a e \sqrt{a+c x^2} \sqrt{d+e x} \left(\sqrt{c} d-i \sqrt{a} e\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}","\frac{\sqrt{d+e x} (a e+c d x)}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}",1,"(e*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*x + I*Sqrt[c]*d*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a*e*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
690,1,583,406,1.3195058,"\int \frac{1}{(d+e x)^{3/2} \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(3/2)),x]","\frac{\sqrt{c} (d+e x)^{3/2} \left(3 a^{3/2} e^3-\sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2+i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+e^2 \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-3 a^2 e^2+a c \left(d^2-3 e^2 x^2\right)+c^2 d^2 x^2\right)+e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(c (d+e x) \left(a e (2 d-e x)+c d^2 x\right)-2 a e^3 \left(a+c x^2\right)\right)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{a e \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(a e^2+c d^2\right)^2}","\frac{e \sqrt{a+c x^2} \left(c d^2-3 a e^2\right)}{a \sqrt{d+e x} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(c d^2-3 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{\sqrt{-a} \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3*e^2*x^2))) + e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-2*a*e^3*(a + c*x^2) + c*(d + e*x)*(c*d^2*x + a*e*(2*d - e*x))) + Sqrt[c]*(I*c^(3/2)*d^3 - Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
691,1,634,485,3.3513417,"\int \frac{1}{(d+e x)^{5/2} \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^(3/2)),x]","\frac{3 c (d+e x)^2 \left(-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right)+\frac{c (d+e x) \left(\sqrt{a} e (d+e x)^{3/2} \left(-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} d (d+e x)^{3/2} \left(29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-29 a^2 e^2+a c \left(3 d^2-29 e^2 x^2\right)+3 c^2 d^2 x^2\right)\right)}{e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-2 a e^3 \left(a+c x^2\right) \left(a e^2+c d^2\right)-20 a c d e^3 \left(a+c x^2\right) (d+e x)}{3 a \sqrt{a+c x^2} (d+e x)^{3/2} \left(a e^2+c d^2\right)^3}","-\frac{c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 c d^2-29 a e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{-a} \sqrt{a+c x^2} \left(a e^2+c d^2\right)^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{c d e \sqrt{a+c x^2} \left(3 c d^2-29 a e^2\right)}{3 a \sqrt{d+e x} \left(a e^2+c d^2\right)^3}+\frac{e \sqrt{a+c x^2} \left(3 c d^2-5 a e^2\right)}{3 a (d+e x)^{3/2} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^{3/2} \left(a e^2+c d^2\right)}+\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left(3 c d^2-5 a e^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 \sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)^2}",1,"(-2*a*e^3*(c*d^2 + a*e^2)*(a + c*x^2) - 20*a*c*d*e^3*(d + e*x)*(a + c*x^2) + 3*c*(d + e*x)^2*(-(a^2*e^3) + c^2*d^3*x + 3*a*c*d*e*(d - e*x)) + (c*(d + e*x)*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^2*x^2 + a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29*a*Sqrt[c]*d*e^2 - (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*a*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*Sqrt[a + c*x^2])","C",1
692,1,700,475,3.4452163,"\int \frac{(d+e x)^{9/2}}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(9/2)/(a + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{-2 a^3 e^3 (19 d+7 e x)-2 a^2 c e \left(7 d^3-3 d^2 e x+27 d e^2 x^2+9 e^3 x^3\right)+2 a c^2 d^2 x \left(6 d^2+d e x+15 e^2 x^2\right)+8 c^3 d^4 x^3}{a^2 c^2 \left(a+c x^2\right)}+\frac{2 \left(e^2 \left(a+c x^2\right) \left(-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right) \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left(33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+\sqrt{c} (d+e x)^{3/2} \left(-15 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt{c} d e^4+15 i a c^{3/2} d^3 e^2-4 \sqrt{a} c^2 d^4 e+4 i c^{5/2} d^5\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)\right)}{a^2 c^3 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{12 \sqrt{a+c x^2}}","-\frac{(d+e x)^{3/2} \left(a e \left(7 a e^2+c d^2\right)-2 c d x \left(5 a e^2+2 c d^2\right)\right)}{6 a^2 c^2 \sqrt{a+c x^2}}-\frac{2 d e \sqrt{a+c x^2} \sqrt{d+e x} \left(3 a e^2+c d^2\right)}{3 a^2 c^2}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} c^{5/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(3 a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{(d+e x)^{7/2} (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(Sqrt[d + e*x]*((8*c^3*d^4*x^3 - 2*a^3*e^3*(19*d + 7*e*x) + 2*a*c^2*d^2*x*(6*d^2 + d*e*x + 15*e^2*x^2) - 2*a^2*c*e*(7*d^3 - 3*d^2*e*x + 27*d*e^2*x^2 + 9*e^3*x^3))/(a^2*c^2*(a + c*x^2)) + (2*(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*(a + c*x^2)) + Sqrt[c]*((4*I)*c^(5/2)*d^5 - 4*Sqrt[a]*c^2*d^4*e + (15*I)*a*c^(3/2)*d^3*e^2 - 15*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(a^2*c^3*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(12*Sqrt[a + c*x^2])","C",1
693,1,627,418,3.186281,"\int \frac{(d+e x)^{7/2}}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{-5 a^3 e^3+a^2 c e \left(-5 d^2+2 d e x-7 e^2 x^2\right)+a c^2 d x \left(6 d^2+d e x+8 e^2 x^2\right)+4 c^3 d^3 x^3}{a^2 c^2 \left(a+c x^2\right)}-\frac{-\sqrt{a} e (d+e x)^{3/2} \left(5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 \sqrt{c} d (d+e x)^{3/2} \left(2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(2 a^2 e^2+a c \left(d^2+2 e^2 x^2\right)+c^2 d^2 x^2\right)}{a^2 c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{6 \sqrt{a+c x^2}}","-\frac{\sqrt{d+e x} \left(a e \left(5 a e^2+3 c d^2\right)-2 c d x \left(3 a e^2+2 c d^2\right)\right)}{6 a^2 c^2 \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \left(5 a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(2 a e^2+c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{5/2} (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(Sqrt[d + e*x]*((-5*a^3*e^3 + 4*c^3*d^3*x^3 + a^2*c*e*(-5*d^2 + 2*d*e*x - 7*e^2*x^2) + a*c^2*d*x*(6*d^2 + d*e*x + 8*e^2*x^2))/(a^2*c^2*(a + c*x^2)) - (4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a^2*c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(6*Sqrt[a + c*x^2])","C",1
694,1,597,392,3.4144349,"\int \frac{(d+e x)^{5/2}}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a^2 e (e x-3 d)+a c x \left(6 d^2+d e x+3 e^2 x^2\right)+4 c^2 d^2 x^3\right)}{a^2 c \left(a+c x^2\right)}+\frac{(d+e x) \left(\frac{2 i \sqrt{c} \left(3 i a^{3/2} e^3+4 i \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}-\frac{2 e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(3 a^2 e^2+a c \left(4 d^2+3 e^2 x^2\right)+4 c^2 d^2 x^2\right)}{(d+e x)^2}+\frac{2 \sqrt{a} \sqrt{c} e \left(i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}\right)}{a^2 c^2 e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{12 \sqrt{a+c x^2}}","-\frac{\sqrt{d+e x} \left(a d e-x \left(3 a e^2+4 c d^2\right)\right)}{6 a^2 c \sqrt{a+c x^2}}-\frac{2 d \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 a e^2+4 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{3/2} (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(Sqrt[d + e*x]*((2*(4*c^2*d^2*x^3 + a^2*e*(-3*d + e*x) + a*c*x*(6*d^2 + d*e*x + 3*e^2*x^2)))/(a^2*c*(a + c*x^2)) + ((d + e*x)*((-2*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)))/(d + e*x)^2 + ((2*I)*Sqrt[c]*(4*c^(3/2)*d^3 + (4*I)*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + (3*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x] + (2*Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(a^2*c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(12*Sqrt[a + c*x^2])","C",1
695,1,504,368,3.5392948,"\int \frac{(d+e x)^{3/2}}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{-2 a^2 e+2 a c x (6 d+e x)+8 c^2 d x^3}{a^2 c \left(a+c x^2\right)}+\frac{(d+e x) \left(-\frac{8 d e^2 \left(a+c x^2\right) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{(d+e x)^2}+\frac{2 \sqrt{a} e \left(4 \sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}+\frac{8 i \sqrt{c} d \left(\sqrt{c} d+i \sqrt{a} e\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}\right)}{a^2 c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right)}{12 \sqrt{a+c x^2}}","\frac{\sqrt{d+e x} (a e+4 c d x)}{6 a^2 c \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left(a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{3 a c \left(a+c x^2\right)^{3/2}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(Sqrt[d + e*x]*((-2*a^2*e + 8*c^2*d*x^3 + 2*a*c*x*(6*d + e*x))/(a^2*c*(a + c*x^2)) + ((d + e*x)*((-8*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2))/(d + e*x)^2 + ((8*I)*Sqrt[c]*d*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x] + (2*Sqrt[a]*e*(4*Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(a^2*c*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(12*Sqrt[a + c*x^2])","C",1
696,1,619,392,1.9212572,"\int \frac{\sqrt{d+e x}}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[d + e*x]/(a + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(a^2 e (d+5 e x)+a c x \left(6 d^2+d e x+3 e^2 x^2\right)+4 c^2 d^2 x^3\right)}{a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{(d+e x) \left(\frac{2 i \sqrt{c} \left(3 i a^{3/2} e^3+4 i \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}-\frac{2 e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(3 a^2 e^2+a c \left(4 d^2+3 e^2 x^2\right)+4 c^2 d^2 x^2\right)}{(d+e x)^2}+\frac{2 \sqrt{a} \sqrt{c} e \left(i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{d+e x}}\right)}{a^2 c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(a e^2+c d^2\right)}\right)}{12 \sqrt{a+c x^2}}","\frac{\sqrt{d+e x} \left(x \left(3 a e^2+4 c d^2\right)+a d e\right)}{6 a^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(3 a e^2+4 c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{x \sqrt{d+e x}}{3 a \left(a+c x^2\right)^{3/2}}-\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}",1,"(Sqrt[d + e*x]*((2*(4*c^2*d^2*x^3 + a^2*e*(d + 5*e*x) + a*c*x*(6*d^2 + d*e*x + 3*e^2*x^2)))/(a^2*(c*d^2 + a*e^2)*(a + c*x^2)) + ((d + e*x)*((-2*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)))/(d + e*x)^2 + ((2*I)*Sqrt[c]*(4*c^(3/2)*d^3 + (4*I)*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + (3*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x] + (2*Sqrt[a]*Sqrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(a^2*c*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2))))/(12*Sqrt[a + c*x^2])","C",1
697,1,570,450,2.1408107,"\int \frac{1}{\sqrt{d+e x} \left(a+c x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)^(5/2)),x]","\frac{\sqrt{d+e x} \left(\frac{\sqrt{a} \sqrt{d+e x} \left(5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{4 d e \left(2 a^2 e^2+a c \left(d^2+2 e^2 x^2\right)+c^2 d^2 x^2\right)}{d+e x}+5 a^2 e^3+\frac{2 a \left(a e^2+c d^2\right) (a e+c d x)}{a+c x^2}-\frac{4 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(2 a e^2+c d^2\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e}+a c d^2 e+8 a c d e^2 x+4 c^2 d^3 x\right)}{6 a^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2}","\frac{\sqrt{d+e x} \left(4 c d x \left(2 a e^2+c d^2\right)+a e \left(5 a e^2+c d^2\right)\right)}{6 a^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2}+\frac{\sqrt{d+e x} (a e+c d x)}{3 a \left(a+c x^2\right)^{3/2} \left(a e^2+c d^2\right)}-\frac{\sqrt{\frac{c x^2}{a}+1} \left(5 a e^2+4 c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(2 a e^2+c d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} \sqrt{a+c x^2} \left(a e^2+c d^2\right)^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}",1,"(Sqrt[d + e*x]*(a*c*d^2*e + 5*a^2*e^3 + 4*c^2*d^3*x + 8*a*c*d*e^2*x + (2*a*(c*d^2 + a*e^2)*(a*e + c*d*x))/(a + c*x^2) - (4*d*e*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)))/(d + e*x) - ((4*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + 2*a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2])","C",1
698,1,669,532,4.0584568,"\int \frac{1}{(d+e x)^{3/2} \left(a+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]","\frac{21 a^3 e^5+c (d+e x) \left(3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)+4 c^2 d^4 x\right)-\frac{i c (d+e x)^{3/2} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{e}+3 a^2 c e^3 \left(7 e^2 x^2-5 d^2\right)-12 a^2 e^5 \left(a+c x^2\right)+\frac{\sqrt{a} \sqrt{c} (d+e x)^{3/2} \left(33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right) \sqrt{\frac{e \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right)|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right)}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-a c^2 d^2 e \left(4 d^2+15 e^2 x^2\right)+\frac{2 a c (d+e x) \left(a e^2+c d^2\right) \left(a e (2 d-e x)+c d^2 x\right)}{a+c x^2}-4 c^3 d^4 e x^2}{6 a^2 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)^3}","\frac{e \sqrt{a+c x^2} \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right)}{6 a^2 \sqrt{d+e x} \left(a e^2+c d^2\right)^3}+\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left(-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{6 (-a)^{3/2} \sqrt{a+c x^2} \left(a e^2+c d^2\right)^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{a e \left(c d^2-7 a e^2\right)-4 c d x \left(3 a e^2+c d^2\right)}{6 a^2 \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)^2}+\frac{a e+c d x}{3 a \left(a+c x^2\right)^{3/2} \sqrt{d+e x} \left(a e^2+c d^2\right)}-\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left(3 a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right)}{3 (-a)^{3/2} \sqrt{a+c x^2} \sqrt{d+e x} \left(a e^2+c d^2\right)^2}",1,"(21*a^3*e^5 - 4*c^3*d^4*e*x^2 - 12*a^2*e^5*(a + c*x^2) + 3*a^2*c*e^3*(-5*d^2 + 7*e^2*x^2) - a*c^2*d^2*e*(4*d^2 + 15*e^2*x^2) + (2*a*c*(c*d^2 + a*e^2)*(d + e*x)*(c*d^2*x + a*e*(2*d - e*x)))/(a + c*x^2) + c*(d + e*x)*(4*c^2*d^4*x + 3*a^2*e^3*(7*d - 3*e*x) + a*c*d^2*e*(d + 15*e*x)) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[c]*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])","C",1
699,1,176,151,0.1456287,"\int \frac{1}{(d+e x) \sqrt[3]{d^2+3 e^2 x^2}} \, dx","Integrate[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{e \left(\sqrt{3} \sqrt{-\frac{d^2}{e^2}}+3 x\right)}{d+e x}} \sqrt[3]{\frac{e \left(9 x-3 \sqrt{3} \sqrt{-\frac{d^2}{e^2}}\right)}{d+e x}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{3} \sqrt{-\frac{d^2}{e^2}} e}{3 d+3 e x}\right)}{2 e \sqrt[3]{d^2+3 e^2 x^2}}","-\frac{\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (d-e x)}{\sqrt{3} \sqrt[3]{d} \sqrt[3]{d^2+3 e^2 x^2}}+\frac{1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} d^{2/3} e}+\frac{\log \left(-3 \sqrt[3]{2} \sqrt[3]{d} e^2 \sqrt[3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right)}{2\ 2^{2/3} d^{2/3} e}",1,"-1/2*(((e*(Sqrt[3]*Sqrt[-(d^2/e^2)] + 3*x))/(d + e*x))^(1/3)*((e*(-3*Sqrt[3]*Sqrt[-(d^2/e^2)] + 9*x))/(d + e*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3*d - Sqrt[3]*Sqrt[-(d^2/e^2)]*e)/(3*d + 3*e*x), (3*d + Sqrt[3]*Sqrt[-(d^2/e^2)]*e)/(3*d + 3*e*x)])/(e*(d^2 + 3*e^2*x^2)^(1/3))","C",0
700,1,53,558,0.0365132,"\int \frac{(2+3 x)^3}{\sqrt[3]{4+27 x^2}} \, dx","Integrate[(2 + 3*x)^3/(4 + 27*x^2)^(1/3),x]","\frac{16}{7} \sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27 x^2}{4}\right)+\frac{1}{210} \left(27 x^2+4\right)^{2/3} \left(63 x^2+180 x+196\right)","\frac{1}{30} \left(27 x^2+4\right)^{2/3} (3 x+2)^2+\frac{4}{35} (4 x+7) \left(27 x^2+4\right)^{2/3}-\frac{96 x}{7 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)}-\frac{32\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{63 \sqrt[4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}}}+\frac{16 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}}}",1,"((4 + 27*x^2)^(2/3)*(196 + 180*x + 63*x^2))/210 + (16*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4])/7","C",1
701,1,47,551,0.0279367,"\int \frac{(2+3 x)^2}{\sqrt[3]{4+27 x^2}} \, dx","Integrate[(2 + 3*x)^2/(4 + 27*x^2)^(1/3),x]","\frac{1}{21} \left(36 \sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27 x^2}{4}\right)+\left(27 x^2+4\right)^{2/3} (3 x+7)\right)","-\frac{72 x}{7 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)}+\frac{1}{21} (3 x+2) \left(27 x^2+4\right)^{2/3}+\frac{5}{21} \left(27 x^2+4\right)^{2/3}-\frac{8\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{21 \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}+\frac{4 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{7\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}",1,"((7 + 3*x)*(4 + 27*x^2)^(2/3) + 36*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4])/21","C",1
702,1,40,529,0.0168098,"\int \frac{2+3 x}{\sqrt[3]{4+27 x^2}} \, dx","Integrate[(2 + 3*x)/(4 + 27*x^2)^(1/3),x]","\sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27 x^2}{4}\right)+\frac{1}{12} \left(27 x^2+4\right)^{2/3}","-\frac{6 x}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}+\frac{1}{12} \left(27 x^2+4\right)^{2/3}-\frac{2\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{9 \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}+\frac{\sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}",1,"(4 + 27*x^2)^(2/3)/12 + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4]","C",1
703,1,121,97,0.0821835,"\int \frac{1}{(2+3 x) \sqrt[3]{4+27 x^2}} \, dx","Integrate[1/((2 + 3*x)*(4 + 27*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt[3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right)}{2\ 3^{2/3} \sqrt[3]{27 x^2+4}}","\frac{\log \left(-27\ 2^{2/3} \sqrt[3]{27 x^2+4}-81 x+54\right)}{12 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} (2-3 x)}{\sqrt{3} \sqrt[3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right)}{6 \sqrt[3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt[3]{2}}",1,"-1/2*((((-2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)])/(3^(2/3)*(4 + 27*x^2)^(1/3))","C",0
704,1,211,634,0.2516029,"\int \frac{1}{(2+3 x)^2 \sqrt[3]{4+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^2*(4 + 27*x^2)^(1/3)),x]","\frac{-8 \sqrt[3]{3} (3 x+2) \sqrt[3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt[3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right)+\sqrt[3]{6} \sqrt[3]{2 \sqrt{3}-9 i x} (3 x+2) \left(3 \sqrt{3} x-2 i\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right)-4 \left(27 x^2+4\right)}{192 (3 x+2) \sqrt[3]{27 x^2+4}}","-\frac{3 x}{16 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)}-\frac{\left(27 x^2+4\right)^{2/3}}{48 (3 x+2)}+\frac{\log \left(-27\ 2^{2/3} \sqrt[3]{27 x^2+4}-81 x+54\right)}{48 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} (2-3 x)}{\sqrt{3} \sqrt[3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right)}{24 \sqrt[3]{2} \sqrt{3}}-\frac{\left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{72 \sqrt[6]{2} \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}-\frac{\log (3 x+2)}{48 \sqrt[3]{2}}",1,"(-4*(4 + 27*x^2) - 8*3^(1/3)*(2 + 3*x)*(((-2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)] + 6^(1/3)*(2*Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)*(-2*I + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, 1/2 + ((3*I)/4)*Sqrt[3]*x])/(192*(2 + 3*x)*(4 + 27*x^2)^(1/3))","C",0
705,1,223,656,0.2655948,"\int \frac{1}{(2+3 x)^3 \sqrt[3]{4+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^3*(4 + 27*x^2)^(1/3)),x]","\frac{-4 \sqrt[3]{3} \sqrt[3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt[3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} (3 x+2)^2 F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right)+\sqrt[3]{6} \sqrt[3]{2 \sqrt{3}-9 i x} \left(3 \sqrt{3} x-2 i\right) (3 x+2)^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right)-12 \left(27 x^3+27 x^2+4 x+4\right)}{384 (3 x+2)^2 \sqrt[3]{27 x^2+4}}","-\frac{3 x}{32 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)}-\frac{\left(27 x^2+4\right)^{2/3}}{96 (3 x+2)}-\frac{\left(27 x^2+4\right)^{2/3}}{96 (3 x+2)^2}+\frac{\log \left(-27\ 2^{2/3} \sqrt[3]{27 x^2+4}-81 x+54\right)}{192 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} (2-3 x)}{\sqrt{3} \sqrt[3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right)}{96 \sqrt[3]{2} \sqrt{3}}-\frac{\left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{144 \sqrt[6]{2} \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{27 x^2+4}\right) \sqrt{\frac{\left(27 x^2+4\right)^{2/3}+2^{2/3} \sqrt[3]{27 x^2+4}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}}\right)|-7+4 \sqrt{3}\right)}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{27 x^2+4}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{27 x^2+4}\right)^2}} x}-\frac{\log (3 x+2)}{192 \sqrt[3]{2}}",1,"(-12*(4 + 4*x + 27*x^2 + 27*x^3) - 4*3^(1/3)*(2 + 3*x)^2*(((-2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)] + 6^(1/3)*(2*Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)^2*(-2*I + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, 1/2 + ((3*I)/4)*Sqrt[3]*x])/(384*(2 + 3*x)^2*(4 + 27*x^2)^(1/3))","C",0
706,1,60,564,0.0380314,"\int \frac{(2+3 i x)^3}{\sqrt[3]{4-27 x^2}} \, dx","Integrate[(2 + (3*I)*x)^3/(4 - 27*x^2)^(1/3),x]","\frac{16}{7} \sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right)+\left(4-27 x^2\right)^{2/3} \left(\frac{3 i x^2}{10}+\frac{6 x}{7}-\frac{14 i}{15}\right)","-\frac{1}{30} i \left(4-27 x^2\right)^{2/3} (2+3 i x)^2-\frac{4}{35} (-4 x+7 i) \left(4-27 x^2\right)^{2/3}-\frac{96 x}{7 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)}+\frac{32\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{63 \sqrt[4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}}}-\frac{16 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}}}",1,"(4 - 27*x^2)^(2/3)*((-14*I)/15 + (6*x)/7 + ((3*I)/10)*x^2) + (16*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (27*x^2)/4])/7","C",1
707,1,51,557,0.0347414,"\int \frac{(2+3 i x)^2}{\sqrt[3]{4-27 x^2}} \, dx","Integrate[(2 + (3*I)*x)^2/(4 - 27*x^2)^(1/3),x]","\frac{12}{7} \sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right)+\left(4-27 x^2\right)^{2/3} \left(\frac{x}{7}-\frac{i}{3}\right)","-\frac{1}{21} i \left(4-27 x^2\right)^{2/3} (2+3 i x)-\frac{5}{21} i \left(4-27 x^2\right)^{2/3}-\frac{72 x}{7 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)}+\frac{8\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{21 \sqrt[4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}}}-\frac{4 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{7\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}}}",1,"(-1/3*I + x/7)*(4 - 27*x^2)^(2/3) + (12*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (27*x^2)/4])/7","C",1
708,1,42,531,0.0187989,"\int \frac{2+3 i x}{\sqrt[3]{4-27 x^2}} \, dx","Integrate[(2 + (3*I)*x)/(4 - 27*x^2)^(1/3),x]","\sqrt[3]{2} x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right)-\frac{1}{12} i \left(4-27 x^2\right)^{2/3}","-\frac{6 x}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}-\frac{1}{12} i \left(4-27 x^2\right)^{2/3}+\frac{2\ 2^{5/6} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{9 \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}-\frac{\sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}",1,"(-1/12*I)*(4 - 27*x^2)^(2/3) + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (27*x^2)/4]","C",1
709,1,125,109,0.0862968,"\int \frac{1}{(2+3 i x) \sqrt[3]{4-27 x^2}} \, dx","Integrate[1/((2 + (3*I)*x)*(4 - 27*x^2)^(1/3)),x]","\frac{i \sqrt[3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt[3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 \left(3 i+\sqrt{3}\right)}{6 i-9 x},\frac{2 \left(-3 i+\sqrt{3}\right)}{9 x-6 i}\right)}{2\ 3^{2/3} \sqrt[3]{4-27 x^2}}","-\frac{i \log \left(27\ 2^{2/3} \sqrt[3]{4-27 x^2}+81 i x-54\right)}{12 \sqrt[3]{2}}+\frac{i \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{2} (2-3 i x)}{\sqrt{3} \sqrt[3]{4-27 x^2}}\right)}{6 \sqrt[3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{12 \sqrt[3]{2}}",1,"((I/2)*((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(3^(2/3)*(4 - 27*x^2)^(1/3))","C",0
710,1,132,650,0.112684,"\int \frac{1}{(2+3 i x)^2 \sqrt[3]{4-27 x^2}} \, dx","Integrate[1/((2 + (3*I)*x)^2*(4 - 27*x^2)^(1/3)),x]","\frac{\sqrt[3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt[3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left(\frac{5}{3};\frac{1}{3},\frac{1}{3};\frac{8}{3};\frac{2 \left(3 i+\sqrt{3}\right)}{6 i-9 x},\frac{2 \left(-3 i+\sqrt{3}\right)}{9 x-6 i}\right)}{5\ 3^{2/3} (3 x-2 i) \sqrt[3]{4-27 x^2}}","-\frac{3 x}{16 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)}+\frac{i \left(4-27 x^2\right)^{2/3}}{48 (2+3 i x)}-\frac{i \log \left(27\ 2^{2/3} \sqrt[3]{4-27 x^2}+81 i x-54\right)}{48 \sqrt[3]{2}}+\frac{i \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{2} (2-3 i x)}{\sqrt{3} \sqrt[3]{4-27 x^2}}\right)}{24 \sqrt[3]{2} \sqrt{3}}+\frac{\left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{72 \sqrt[6]{2} \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}+\frac{i \log (2+3 i x)}{48 \sqrt[3]{2}}",1,"(((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*AppellF1[5/3, 1/3, 1/3, 8/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(5*3^(2/3)*(-2*I + 3*x)*(4 - 27*x^2)^(1/3))","C",0
711,1,134,676,0.0988485,"\int \frac{1}{(2+3 i x)^3 \sqrt[3]{4-27 x^2}} \, dx","Integrate[1/((2 + (3*I)*x)^3*(4 - 27*x^2)^(1/3)),x]","-\frac{i \sqrt[3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt[3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left(\frac{8}{3};\frac{1}{3},\frac{1}{3};\frac{11}{3};\frac{2 \left(3 i+\sqrt{3}\right)}{6 i-9 x},\frac{2 \left(-3 i+\sqrt{3}\right)}{9 x-6 i}\right)}{8\ 3^{2/3} (-3 x+2 i)^2 \sqrt[3]{4-27 x^2}}","-\frac{3 x}{32 \left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)}+\frac{i \left(4-27 x^2\right)^{2/3}}{96 (2+3 i x)}+\frac{i \left(4-27 x^2\right)^{2/3}}{96 (2+3 i x)^2}-\frac{i \log \left(27\ 2^{2/3} \sqrt[3]{4-27 x^2}+81 i x-54\right)}{192 \sqrt[3]{2}}+\frac{i \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{2} (2-3 i x)}{\sqrt{3} \sqrt[3]{4-27 x^2}}\right)}{96 \sqrt[3]{2} \sqrt{3}}+\frac{\left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{144 \sqrt[6]{2} \sqrt[4]{3} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left(2^{2/3}-\sqrt[3]{4-27 x^2}\right) \sqrt{\frac{\left(4-27 x^2\right)^{2/3}+2^{2/3} \sqrt[3]{4-27 x^2}+2 \sqrt[3]{2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{2^{2/3} \left(1+\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}{2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}}\right)|-7+4 \sqrt{3}\right)}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt[3]{4-27 x^2}}{\left(2^{2/3} \left(1-\sqrt{3}\right)-\sqrt[3]{4-27 x^2}\right)^2}} x}+\frac{i \log (2+3 i x)}{192 \sqrt[3]{2}}",1,"((-1/8*I)*((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*AppellF1[8/3, 1/3, 1/3, 11/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(3^(2/3)*(2*I - 3*x)^2*(4 - 27*x^2)^(1/3))","C",0
712,1,102,104,0.0742159,"\int \frac{1}{\left(\sqrt{3}+x\right) \sqrt[3]{1+x^2}} \, dx","Integrate[1/((Sqrt[3] + x)*(1 + x^2)^(1/3)),x]","-\frac{3 \sqrt[3]{\frac{x-i}{x+\sqrt{3}}} \sqrt[3]{\frac{x+i}{x+\sqrt{3}}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{x+\sqrt{3}},\frac{i+\sqrt{3}}{x+\sqrt{3}}\right)}{2 \sqrt[3]{x^2+1}}","\frac{\log \left(-\sqrt[3]{2} \sqrt{3} \sqrt[3]{x^2+1}-x+\sqrt{3}\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \left(\sqrt{3}-x\right)}{3 \sqrt[3]{x^2+1}}+\frac{1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(x+\sqrt{3}\right)}{2\ 2^{2/3}}",1,"(-3*((-I + x)/(Sqrt[3] + x))^(1/3)*((I + x)/(Sqrt[3] + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I + Sqrt[3])/(Sqrt[3] + x), (I + Sqrt[3])/(Sqrt[3] + x)])/(2*(1 + x^2)^(1/3))","C",0
713,1,110,101,0.0778706,"\int \frac{1}{\left(\sqrt{3}-x\right) \sqrt[3]{1+x^2}} \, dx","Integrate[1/((Sqrt[3] - x)*(1 + x^2)^(1/3)),x]","\frac{3 \sqrt[3]{\frac{x-i}{x-\sqrt{3}}} \sqrt[3]{\frac{x+i}{x-\sqrt{3}}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{-i+\sqrt{3}}{\sqrt{3}-x},\frac{i+\sqrt{3}}{\sqrt{3}-x}\right)}{2 \sqrt[3]{x^2+1}}","-\frac{\log \left(-\sqrt[3]{2} \sqrt{3} \sqrt[3]{x^2+1}+x+\sqrt{3}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \left(x+\sqrt{3}\right)}{3 \sqrt[3]{x^2+1}}+\frac{1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(\sqrt{3}-x\right)}{2\ 2^{2/3}}",1,"(3*((-I + x)/(-Sqrt[3] + x))^(1/3)*((I + x)/(-Sqrt[3] + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I + Sqrt[3])/(Sqrt[3] - x), (I + Sqrt[3])/(Sqrt[3] - x)])/(2*(1 + x^2)^(1/3))","C",0
714,1,68,78,0.0569882,"\int \frac{1}{(3-x) \sqrt[3]{1-x^2}} \, dx","Integrate[1/((3 - x)*(1 - x^2)^(1/3)),x]","\frac{3 \sqrt[3]{\frac{x-1}{x-3}} \sqrt[3]{\frac{x+1}{x-3}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right)}{2 \sqrt[3]{1-x^2}}","-\frac{1}{4} \log (3-x)+\frac{3}{8} \log \left(-\frac{1}{2} (x+1)^{2/3}-\sqrt[3]{1-x}\right)-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{(x+1)^{2/3}}{\sqrt{3} \sqrt[3]{1-x}}\right)",1,"(3*((-1 + x)/(-3 + x))^(1/3)*((1 + x)/(-3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -4/(-3 + x), -2/(-3 + x)])/(2*(1 - x^2)^(1/3))","C",0
715,1,68,76,0.0380379,"\int \frac{1}{(3+x) \sqrt[3]{1-x^2}} \, dx","Integrate[1/((3 + x)*(1 - x^2)^(1/3)),x]","-\frac{3 \sqrt[3]{\frac{x-1}{x+3}} \sqrt[3]{\frac{x+1}{x+3}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4}{x+3},\frac{2}{x+3}\right)}{2 \sqrt[3]{1-x^2}}","\frac{1}{4} \log (x+3)-\frac{3}{8} \log \left(-\frac{1}{2} (1-x)^{2/3}-\sqrt[3]{x+1}\right)+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt[3]{x+1}}\right)",1,"(-3*((-1 + x)/(3 + x))^(1/3)*((1 + x)/(3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, 4/(3 + x), 2/(3 + x)])/(2*(1 - x^2)^(1/3))","C",0
716,1,155,206,0.1071701,"\int \frac{1}{(d+e x) \sqrt[3]{d^2-9 e^2 x^2}} \, dx","Integrate[1/((d + e*x)*(d^2 - 9*e^2*x^2)^(1/3)),x]","-\frac{\sqrt[3]{3} \sqrt[3]{-\frac{e \left(\sqrt{\frac{d^2}{e^2}}-3 x\right)}{d+e x}} \sqrt[3]{\frac{e \left(\sqrt{\frac{d^2}{e^2}}+3 x\right)}{d+e x}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{3 d-\sqrt{\frac{d^2}{e^2}} e}{3 d+3 e x},\frac{3 d+\sqrt{\frac{d^2}{e^2}} e}{3 d+3 e x}\right)}{2 e \sqrt[3]{d^2-9 e^2 x^2}}","\frac{\sqrt[3]{1-\frac{9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt[3]{d^2-9 e^2 x^2}}-\frac{3 \sqrt[3]{1-\frac{9 e^2 x^2}{d^2}} \log \left(-\frac{1}{2} \left(1-\frac{3 e x}{d}\right)^{2/3}-\sqrt[3]{\frac{3 e x}{d}+1}\right)}{8 e \sqrt[3]{d^2-9 e^2 x^2}}+\frac{\sqrt{3} \sqrt[3]{1-\frac{9 e^2 x^2}{d^2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\left(1-\frac{3 e x}{d}\right)^{2/3}}{\sqrt{3} \sqrt[3]{\frac{3 e x}{d}+1}}\right)}{4 e \sqrt[3]{d^2-9 e^2 x^2}}",1,"-1/2*(3^(1/3)*(-((e*(Sqrt[d^2/e^2] - 3*x))/(d + e*x)))^(1/3)*((e*(Sqrt[d^2/e^2] + 3*x))/(d + e*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (3*d - Sqrt[d^2/e^2]*e)/(3*d + 3*e*x), (3*d + Sqrt[d^2/e^2]*e)/(3*d + 3*e*x)])/(e*(d^2 - 9*e^2*x^2)^(1/3))","C",0
717,1,126,278,0.1076252,"\int \frac{1}{(a+b x) \sqrt[4]{c+d x^2}} \, dx","Integrate[1/((a + b*x)*(c + d*x^2)^(1/4)),x]","-\frac{2 \sqrt[4]{\frac{b \left(x-\sqrt{-\frac{c}{d}}\right)}{a+b x}} \sqrt[4]{\frac{b \left(\sqrt{-\frac{c}{d}}+x\right)}{a+b x}} F_1\left(\frac{1}{2};\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{a-b \sqrt{-\frac{c}{d}}}{a+b x},\frac{a+b \sqrt{-\frac{c}{d}}}{a+b x}\right)}{b \sqrt[4]{c+d x^2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)}{\sqrt{b} \sqrt[4]{a^2 d+b^2 c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)}{\sqrt{b} \sqrt[4]{a^2 d+b^2 c}}-\frac{a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)}{b x \sqrt{a^2 d+b^2 c}}+\frac{a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)}{b x \sqrt{a^2 d+b^2 c}}",1,"(-2*((b*(-Sqrt[-(c/d)] + x))/(a + b*x))^(1/4)*((b*(Sqrt[-(c/d)] + x))/(a + b*x))^(1/4)*AppellF1[1/2, 1/4, 1/4, 3/2, (a - b*Sqrt[-(c/d)])/(a + b*x), (a + b*Sqrt[-(c/d)])/(a + b*x)])/(b*(c + d*x^2)^(1/4))","C",0
718,1,233,268,0.1672614,"\int \frac{1}{(a+b x) \left(c+d x^2\right)^{3/4}} \, dx","Integrate[1/((a + b*x)*(c + d*x^2)^(3/4)),x]","\frac{-\sqrt{b} x \sqrt[4]{a^2 d+b^2 c} \left(\tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)\right)+a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)+a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)}{x \left(a^2 d+b^2 c\right)}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)}{\left(a^2 d+b^2 c\right)^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt[4]{c+d x^2}}{\sqrt[4]{a^2 d+b^2 c}}\right)}{\left(a^2 d+b^2 c\right)^{3/4}}+\frac{a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)}{x \left(a^2 d+b^2 c\right)}+\frac{a \sqrt[4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left(\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{d x^2+c}}{\sqrt[4]{c}}\right)\right|-1\right)}{x \left(a^2 d+b^2 c\right)}",1,"(-(Sqrt[b]*(b^2*c + a^2*d)^(1/4)*x*(ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)] + ArcTanh[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])) + a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[-((b*Sqrt[c])/Sqrt[b^2*c + a^2*d]), ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1] + a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sqrt[b^2*c + a^2*d], ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x)","A",1
719,1,108,200,0.5069247,"\int \frac{1}{(d+e x)^{3/2} \sqrt[4]{a+c x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(1/4)),x]","\frac{\left(a+c x^2\right)^{3/4} (c d x-a e) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{(a e-c d x)^2}{a c (d+e x)^2}\right)}{a c (d+e x)^{5/2} \left(\frac{\left(a+c x^2\right) \left(a e^2+c d^2\right)}{a c (d+e x)^2}\right)^{3/4}}","-\frac{2 \left(\sqrt{-a}-\sqrt{c} x\right) \sqrt[4]{-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{\sqrt[4]{a+c x^2} \sqrt{d+e x} \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"((-(a*e) + c*d*x)*(a + c*x^2)^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((a*e - c*d*x)^2/(a*c*(d + e*x)^2))])/(a*c*(d + e*x)^(5/2)*(((c*d^2 + a*e^2)*(a + c*x^2))/(a*c*(d + e*x)^2))^(3/4))","A",1
720,1,72,203,0.0402962,"\int \frac{1}{(1+x) \sqrt[6]{1+x^2}} \, dx","Integrate[1/((1 + x)*(1 + x^2)^(1/6)),x]","-\frac{3 \sqrt[6]{\frac{x-i}{x+1}} \sqrt[6]{\frac{x+i}{x+1}} F_1\left(\frac{1}{3};\frac{1}{6},\frac{1}{6};\frac{4}{3};\frac{1-i}{x+1},\frac{1+i}{x+1}\right)}{\sqrt[6]{x^2+1}}","x F_1\left(\frac{1}{2};1,\frac{1}{6};\frac{3}{2};x^2,-x^2\right)+\frac{\log \left(\sqrt[3]{x^2+1}-\sqrt[6]{2} \sqrt[6]{x^2+1}+\sqrt[3]{2}\right)}{4 \sqrt[6]{2}}-\frac{\log \left(\sqrt[3]{x^2+1}+\sqrt[6]{2} \sqrt[6]{x^2+1}+\sqrt[3]{2}\right)}{4 \sqrt[6]{2}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-2^{5/6} \sqrt[6]{x^2+1}}{\sqrt{3}}\right)}{2 \sqrt[6]{2}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{2^{5/6} \sqrt[6]{x^2+1}+1}{\sqrt{3}}\right)}{2 \sqrt[6]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{x^2+1}}{\sqrt[6]{2}}\right)}{\sqrt[6]{2}}",1,"(-3*((-I + x)/(1 + x))^(1/6)*((I + x)/(1 + x))^(1/6)*AppellF1[1/3, 1/6, 1/6, 4/3, (1 - I)/(1 + x), (1 + I)/(1 + x)])/(1 + x^2)^(1/6)","C",0
721,1,379,223,0.6622403,"\int (d+e x)^m \left(a+c x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(a + c*x^2)^3,x]","\frac{(d+e x)^{m+1} \left(\frac{6 \left((m+6) \left(a e^2+c d^2\right) \left(4 (m+4) \left(a e^2+c d^2\right) \left(a e^2 \left(m^2+5 m+6\right)+c \left(2 d^2-2 d e (m+1) x+e^2 \left(m^2+3 m+2\right) x^2\right)\right)-4 c d (m+1) (d+e x) \left(a e^2 \left(m^2+7 m+12\right)+c \left(2 d^2-2 d e (m+2) x+e^2 \left(m^2+5 m+6\right) x^2\right)\right)+e^4 (m+1) (m+2) (m+3) (m+4) \left(a+c x^2\right)^2\right)-c d (m+1) (d+e x) \left(4 (m+5) \left(a e^2+c d^2\right) \left(a e^2 \left(m^2+7 m+12\right)+c \left(2 d^2-2 d e (m+2) x+e^2 \left(m^2+5 m+6\right) x^2\right)\right)-4 c d (m+2) (d+e x) \left(a e^2 \left(m^2+9 m+20\right)+c \left(2 d^2-2 d e (m+3) x+e^2 \left(m^2+7 m+12\right) x^2\right)\right)+e^4 (m+2) (m+3) (m+4) (m+5) \left(a+c x^2\right)^2\right)\right)}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)}+\left(a+c x^2\right)^3\right)}{e (m+7)}","-\frac{4 c^2 d \left(3 a e^2+5 c d^2\right) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c^2 \left(a e^2+5 c d^2\right) (d+e x)^{m+5}}{e^7 (m+5)}+\frac{\left(a e^2+c d^2\right)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 c d \left(a e^2+c d^2\right)^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{3 c \left(a e^2+c d^2\right) \left(a e^2+5 c d^2\right) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*((a + c*x^2)^3 + (6*((c*d^2 + a*e^2)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(4 + m)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) - 4*c*d*(1 + m)*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2))) - c*d*(1 + m)*(d + e*x)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))))/(e*(7 + m))","A",1
722,1,176,140,0.186097,"\int (d+e x)^m \left(a+c x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(a + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{4 \left(a e^2+c d^2\right) \left(a e^2 \left(m^2+5 m+6\right)+c \left(2 d^2-2 d e (m+1) x+e^2 \left(m^2+3 m+2\right) x^2\right)\right)}{e^4 (m+1) (m+2) (m+3)}-\frac{4 c d (d+e x) \left(a e^2 \left(m^2+7 m+12\right)+c \left(2 d^2-2 d e (m+2) x+e^2 \left(m^2+5 m+6\right) x^2\right)\right)}{e^4 (m+2) (m+3) (m+4)}+\left(a+c x^2\right)^2\right)}{e (m+5)}","\frac{\left(a e^2+c d^2\right)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 c d \left(a e^2+c d^2\right) (d+e x)^{m+2}}{e^5 (m+2)}+\frac{2 c \left(a e^2+3 c d^2\right) (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)}",1,"((d + e*x)^(1 + m)*((a + c*x^2)^2 + (4*(c*d^2 + a*e^2)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)))/(e^4*(1 + m)*(2 + m)*(3 + m)) - (4*c*d*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))/(e^4*(2 + m)*(3 + m)*(4 + m))))/(e*(5 + m))","A",1
723,1,59,70,0.0370967,"\int (d+e x)^m \left(a+c x^2\right) \, dx","Integrate[(d + e*x)^m*(a + c*x^2),x]","\frac{(d+e x)^{m+1} \left(\frac{a e^2+c d^2}{m+1}+\frac{c (d+e x)^2}{m+3}-\frac{2 c d (d+e x)}{m+2}\right)}{e^3}","\frac{\left(a e^2+c d^2\right) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 c d (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)}",1,"((d + e*x)^(1 + m)*((c*d^2 + a*e^2)/(1 + m) - (2*c*d*(d + e*x))/(2 + m) + (c*(d + e*x)^2)/(3 + m)))/e^3","A",1
724,1,145,167,0.1469817,"\int \frac{(d+e x)^m}{a+c x^2} \, dx","Integrate[(d + e*x)^m/(a + c*x^2),x]","\frac{(d+e x)^{m+1} \left(\frac{\, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}-\frac{\, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}\right)}{2 \sqrt{-a} (m+1)}","\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 \sqrt{-a} (m+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 \sqrt{-a} (m+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"((d + e*x)^(1 + m)*(Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[c]*d - Sqrt[-a]*e) - Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[c]*d + Sqrt[-a]*e)))/(2*Sqrt[-a]*(1 + m))","A",1
725,1,253,304,0.4312104,"\int \frac{(d+e x)^m}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(a + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{\left(\sqrt{-a} \sqrt{c} d e m-a e^2 (m-1)+c d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} (m+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\left(\sqrt{-a} \sqrt{c} d e m+a e^2 (m-1)-c d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} (m+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{2 (a e+c d x)}{a+c x^2}\right)}{4 a \left(a e^2+c d^2\right)}","-\frac{(d+e x)^{m+1} \left(\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 (-a)^{3/2} (m+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{m+1} \left(-\sqrt{-a} \sqrt{c} d e m+a e^2 (1-m)+c d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 (-a)^{3/2} (m+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{m+1} (a e+c d x)}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + m)*((2*(a*e + c*d*x))/(a + c*x^2) + ((c*d^2 - a*e^2*(-1 + m) + Sqrt[-a]*Sqrt[c]*d*e*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + m)) + ((-(c*d^2) + a*e^2*(-1 + m) + Sqrt[-a]*Sqrt[c]*d*e*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + m))))/(4*a*(c*d^2 + a*e^2))","A",1
726,1,396,472,1.0877283,"\int \frac{(d+e x)^m}{\left(a+c x^2\right)^3} \, dx","Integrate[(d + e*x)^m/(a + c*x^2)^3,x]","\frac{(d+e x)^{m+1} \left(\frac{2 \left(-a^2 e^3 (m-3)+a c d e (d (m+1)+e (5-2 m) x)+3 c^2 d^3 x\right)}{a+c x^2}+\frac{\frac{\left(\sqrt{-a} \left(-a^2 e^4 \left(m^2-4 m+3\right)+a c d^2 e^2 \left(m^2+2 m-6\right)-3 c^2 d^4\right)+a \sqrt{c} d e m \left(a e^2 (5-2 m)+3 c d^2\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}+\frac{\left(\sqrt{-a} \left(a^2 e^4 \left(m^2-4 m+3\right)-a c d^2 e^2 \left(m^2+2 m-6\right)+3 c^2 d^4\right)+a \sqrt{c} d e m \left(a e^2 (5-2 m)+3 c d^2\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}}{a (m+1)}+\frac{4 a \left(a e^2+c d^2\right) (a e+c d x)}{\left(a+c x^2\right)^2}\right)}{16 a^2 \left(a e^2+c d^2\right)^2}","\frac{(d+e x)^{m+1} \left(c d x \left(a e^2 (5-2 m)+3 c d^2\right)+a e \left(a e^2 (3-m)+c d^2 (m+1)\right)\right)}{8 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)^2}+\frac{(d+e x)^{m+1} \left(a \sqrt{c} d e m \left(a e^2 (5-2 m)+3 c d^2\right)-\sqrt{-a} \left(a^2 e^4 \left(m^2-4 m+3\right)+a c d^2 e^2 \left(-m^2-2 m+6\right)+3 c^2 d^4\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{16 a^3 (m+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)^2}+\frac{(d+e x)^{m+1} \left(\sqrt{-a} \left(a^2 e^4 \left(m^2-4 m+3\right)+a c d^2 e^2 \left(-m^2-2 m+6\right)+3 c^2 d^4\right)+a \sqrt{c} d e m \left(a e^2 (5-2 m)+3 c d^2\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{16 a^3 (m+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)^2}+\frac{(d+e x)^{m+1} (a e+c d x)}{4 a \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + m)*((4*a*(c*d^2 + a*e^2)*(a*e + c*d*x))/(a + c*x^2)^2 + (2*(-(a^2*e^3*(-3 + m)) + 3*c^2*d^3*x + a*c*d*e*(d*(1 + m) + e*(5 - 2*m)*x)))/(a + c*x^2) + (((a*Sqrt[c]*d*e*(3*c*d^2 + a*e^2*(5 - 2*m))*m + Sqrt[-a]*(-3*c^2*d^4 - a^2*e^4*(3 - 4*m + m^2) + a*c*d^2*e^2*(-6 + 2*m + m^2)))*Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) + ((a*Sqrt[c]*d*e*(3*c*d^2 + a*e^2*(5 - 2*m))*m + Sqrt[-a]*(3*c^2*d^4 + a^2*e^4*(3 - 4*m + m^2) - a*c*d^2*e^2*(-6 + 2*m + m^2)))*Hypergeometric2F1[1, 1 + m, 2 + m, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e))/(a*(1 + m))))/(16*a^2*(c*d^2 + a*e^2)^2)","A",1
727,0,0,154,0.0760566,"\int (d+e x)^m \left(a+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(a + c*x^2)^(3/2),x]","\int (d+e x)^m \left(a+c x^2\right)^{3/2} \, dx","\frac{\left(a+c x^2\right)^{3/2} (d+e x)^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (m+1) \left(1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right)^{3/2} \left(1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right)^{3/2}}",1,"Integrate[(d + e*x)^m*(a + c*x^2)^(3/2), x]","F",-1
728,1,159,154,0.1074091,"\int (d+e x)^m \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[a + c*x^2],x]","\frac{\sqrt{a+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{e (m+1) \sqrt{\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}} \sqrt{\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}}}","\frac{\sqrt{a+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (m+1) \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}}}",1,"((d + e*x)^(1 + m)*Sqrt[a + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*Sqrt[(e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e)]*Sqrt[(e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e)])","A",0
729,1,159,154,0.1090147,"\int \frac{(d+e x)^m}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[a + c*x^2],x]","\frac{(d+e x)^{m+1} \sqrt{\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}} \sqrt{\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{e (m+1) \sqrt{a+c x^2}}","\frac{(d+e x)^{m+1} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (m+1) \sqrt{a+c x^2}}",1,"(Sqrt[(e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e)]*Sqrt[(e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*Sqrt[a + c*x^2])","A",0
730,1,159,154,0.1819019,"\int \frac{(d+e x)^m}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(a + c*x^2)^(3/2),x]","\frac{(d+e x)^{m+1} \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}\right)^{3/2} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}\right)^{3/2} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{e (m+1) \left(a+c x^2\right)^{3/2}}","\frac{(d+e x)^{m+1} \left(1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right)^{3/2} \left(1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right)^{3/2} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (m+1) \left(a+c x^2\right)^{3/2}}",1,"(((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^(3/2)*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^(3/2)*(d + e*x)^(1 + m)*AppellF1[1 + m, 3/2, 3/2, 2 + m, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*(a + c*x^2)^(3/2))","A",0
731,1,157,152,0.1311954,"\int (d+e x)^m \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(a + c*x^2)^p,x]","\frac{\left(a+c x^2\right)^p (d+e x)^{m+1} \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{e (m+1)}","\frac{\left(a+c x^2\right)^p (d+e x)^{m+1} \left(1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right)^{-p} \left(1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (m+1)}",1,"((d + e*x)^(1 + m)*(a + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^p)","A",0
732,1,223,178,0.2320995,"\int (d+e x)^3 \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^p,x]","\frac{\left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(e \left(-a^2 e^2 \left(\left(\frac{c x^2}{a}+1\right)^p-1\right)+c^2 x^2 \left(\frac{c x^2}{a}+1\right)^p \left(3 d^2 (p+2)+e^2 (p+1) x^2\right)+2 c^2 d e \left(p^2+3 p+2\right) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right)+a c \left(3 d^2 (p+2) \left(\left(\frac{c x^2}{a}+1\right)^p-1\right)+e^2 p x^2 \left(\frac{c x^2}{a}+1\right)^p\right)\right)+2 c^2 d^3 \left(p^2+3 p+2\right) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)\right)}{2 c^2 (p+1) (p+2)}","-\frac{e \left(a+c x^2\right)^{p+1} \left((2 p+3) \left(a e^2-c d^2 (2 p+5)\right)-2 c d e (p+1) (p+3) x\right)}{2 c^2 (p+2) \left(2 p^2+5 p+3\right)}-\frac{d x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(3 a e^2-c d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)}{c (2 p+3)}+\frac{e (d+e x)^2 \left(a+c x^2\right)^{p+1}}{2 c (p+2)}",1,"((a + c*x^2)^p*(2*c^2*d^3*(2 + 3*p + p^2)*x*Hypergeometric2F1[1/2, -p, 3/2, -((c*x^2)/a)] + e*(c^2*x^2*(1 + (c*x^2)/a)^p*(3*d^2*(2 + p) + e^2*(1 + p)*x^2) - a^2*e^2*(-1 + (1 + (c*x^2)/a)^p) + a*c*(e^2*p*x^2*(1 + (c*x^2)/a)^p + 3*d^2*(2 + p)*(-1 + (1 + (c*x^2)/a)^p)) + 2*c^2*d*e*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((c*x^2)/a)])))/(2*c^2*(1 + p)*(2 + p)*(1 + (c*x^2)/a)^p)","A",1
733,1,133,133,0.1001203,"\int (d+e x)^2 \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^p,x]","\frac{\left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(3 c d^2 (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)+e \left(3 d \left(c x^2 \left(\frac{c x^2}{a}+1\right)^p+a \left(\left(\frac{c x^2}{a}+1\right)^p-1\right)\right)+c e (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right)\right)\right)}{3 c (p+1)}","-\frac{x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(a e^2-c d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)}{c (2 p+3)}+\frac{e (d+e x) \left(a+c x^2\right)^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left(a+c x^2\right)^{p+1}}{c (p+1) (2 p+3)}",1,"((a + c*x^2)^p*(3*c*d^2*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((c*x^2)/a)] + e*(3*d*(c*x^2*(1 + (c*x^2)/a)^p + a*(-1 + (1 + (c*x^2)/a)^p)) + c*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((c*x^2)/a)])))/(3*c*(1 + p)*(1 + (c*x^2)/a)^p)","A",1
734,1,98,61,0.0525741,"\int (d+e x) \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)*(a + c*x^2)^p,x]","\frac{\left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(2 c d (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)+c e x^2 \left(\frac{c x^2}{a}+1\right)^p+a e \left(\left(\frac{c x^2}{a}+1\right)^p-1\right)\right)}{2 c (p+1)}","\frac{d x \left(a+c x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a}\right)}{a}+\frac{e \left(a+c x^2\right)^{p+1}}{2 c (p+1)}",1,"((a + c*x^2)^p*(c*e*x^2*(1 + (c*x^2)/a)^p + a*e*(-1 + (1 + (c*x^2)/a)^p) + 2*c*d*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((c*x^2)/a)]))/(2*c*(1 + p)*(1 + (c*x^2)/a)^p)","A",1
735,1,44,35,0.0035362,"\int \left(a+c x^2\right)^p \, dx","Integrate[(a + c*x^2)^p,x]","x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right)","\frac{x \left(a+c x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a}\right)}{a}",1,"(x*(a + c*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((c*x^2)/a)])/(1 + (c*x^2)/a)^p","A",1
736,1,131,125,0.086182,"\int \frac{\left(a+c x^2\right)^p}{d+e x} \, dx","Integrate[(a + c*x^2)^p/(d + e*x),x]","\frac{\left(a+c x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{c}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{c}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{c}} e}{d+e x}\right)}{2 e p}","\frac{x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d}-\frac{e \left(a+c x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(c x^2+a\right)}{c d^2+a e^2}\right)}{2 (p+1) \left(a e^2+c d^2\right)}",1,"((a + c*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/c)]*e)/(d + e*x), (d + Sqrt[-(a/c)]*e)/(d + e*x)])/(2*e*p*((e*(-Sqrt[-(a/c)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/c)] + x))/(d + e*x))^p)","A",0
737,1,141,191,0.0785214,"\int \frac{\left(a+c x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(a + c*x^2)^p/(d + e*x)^2,x]","\frac{\left(a+c x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{c}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{c}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{c}} e}{d+e x}\right)}{e (2 p-1) (d+e x)}","\frac{x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^2}+\frac{e^2 x^3 \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d^4}-\frac{c d e \left(a+c x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(c x^2+a\right)}{c d^2+a e^2}\right)}{(p+1) \left(a e^2+c d^2\right)^2}",1,"((a + c*x^2)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/c)]*e)/(d + e*x), (d + Sqrt[-(a/c)]*e)/(d + e*x)])/(e*(-1 + 2*p)*((e*(-Sqrt[-(a/c)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/c)] + x))/(d + e*x))^p*(d + e*x))","A",0
738,1,142,322,0.1185013,"\int \frac{\left(a+c x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(a + c*x^2)^p/(d + e*x)^3,x]","\frac{\left(a+c x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{c}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{c}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{c}} e}{d+e x}\right)}{2 e (p-1) (d+e x)^2}","\frac{e^2 x^3 \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,3;\frac{5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5}+\frac{x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3}-\frac{3 c^2 d^2 e \left(a+c x^2\right)^{p+1} \, _2F_1\left(3,p+1;p+2;\frac{e^2 \left(c x^2+a\right)}{c d^2+a e^2}\right)}{2 (p+1) \left(a e^2+c d^2\right)^3}+\frac{c e \left(a+c x^2\right)^{p+1} \left(2 a e^2+c d^2 (p+1)\right) \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(c x^2+a\right)}{c d^2+a e^2}\right)}{4 (p+1) \left(a e^2+c d^2\right)^3}-\frac{d^2 e \left(a+c x^2\right)^{p+1}}{4 \left(d^2-e^2 x^2\right)^2 \left(a e^2+c d^2\right)}",1,"((a + c*x^2)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/c)]*e)/(d + e*x), (d + Sqrt[-(a/c)]*e)/(d + e*x)])/(2*e*(-1 + p)*((e*(-Sqrt[-(a/c)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/c)] + x))/(d + e*x))^p*(d + e*x)^2)","A",0
739,1,166,160,0.1247524,"\int (d+e x)^{-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(a + c*x^2)^p/(d + e*x)^(2*p),x]","-\frac{\left(a+c x^2\right)^p (d+e x)^{1-2 p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{e (2 p-1)}","\frac{\left(a+c x^2\right)^p (d+e x)^{1-2 p} \left(1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right)^{-p} \left(1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{e (1-2 p)}",1,"-(((d + e*x)^(1 - 2*p)*(a + c*x^2)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(-1 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^p))","A",0
740,1,160,155,0.0856481,"\int (d+e x)^{-1-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-1 - 2*p)*(a + c*x^2)^p,x]","-\frac{\left(a+c x^2\right)^p (d+e x)^{-2 p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{c}}+x\right)}{e \sqrt{-\frac{a}{c}}-d}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)}{2 e p}","-\frac{\left(a+c x^2\right)^p (d+e x)^{-2 p} \left(1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right)^{-p} \left(1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right)}{2 e p}",1,"-1/2*((a + c*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*p*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^p*(d + e*x)^(2*p))","A",0
741,1,200,209,0.135795,"\int (d+e x)^{-2-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-2 - 2*p)*(a + c*x^2)^p,x]","\frac{\left(\sqrt{c} x-\sqrt{-a}\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(1-\frac{c (d+e x)}{c d-\sqrt{-a} \sqrt{c} e}\right)^{-p} \left(1-\frac{c (d+e x)}{\sqrt{-a} \sqrt{c} e+c d}\right)^p \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}","-\frac{\left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"((-Sqrt[-a] + Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a + c*x^2)^p*(1 - (c*(d + e*x))/(c*d + Sqrt[-a]*Sqrt[c]*e))^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + 2*p)*(1 - (c*(d + e*x))/(c*d - Sqrt[-a]*Sqrt[c]*e))^p)","A",0
742,1,368,270,37.0806525,"\int (d+e x)^{-3-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]","\frac{2^{-2 p-3} \Gamma \left(-p-\frac{1}{2}\right) \left(a+c x^2\right)^p (d+e x)^{-2 (p+1)} \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{e \sqrt{-\frac{a}{c}}+d}\right)^{-p} \left(1-\frac{d+e x}{e \sqrt{-\frac{a}{c}}+d}\right)^{p+1} \left(\Gamma (1-2 p) \Gamma (-p) \left(e \sqrt{-\frac{a}{c}}+d\right) \left(e \left(2 p \sqrt{-\frac{a}{c}}+\sqrt{-\frac{a}{c}}+x\right)+2 d (p+1)\right) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right)+\frac{2 e \Gamma (1-p) \Gamma (-2 p) \left(c x \sqrt{-\frac{a}{c}}+a\right) (d+e x) \, _2F_1\left(2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right)}{c \left(\sqrt{-\frac{a}{c}}+x\right)}\right)}{\sqrt{\pi } e (p+1) \Gamma (1-2 p) \Gamma (-2 p) \left(e \sqrt{-\frac{a}{c}}+d\right)^2}","-\frac{c d \left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}-\frac{e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left(a e^2+c d^2\right)}",1,"(2^(-3 - 2*p)*(a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*Gamma[-1/2 - p]*((d + Sqrt[-(a/c)]*e)*(2*d*(1 + p) + e*(Sqrt[-(a/c)] + 2*Sqrt[-(a/c)]*p + x))*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (2*e*(a + Sqrt[-(a/c)]*c*x)*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))])/(c*(Sqrt[-(a/c)] + x))))/(e*(d + Sqrt[-(a/c)]*e)^2*(1 + p)*Sqrt[Pi]*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(d + e*x)^(2*(1 + p))*Gamma[1 - 2*p]*Gamma[-2*p])","A",0
743,1,1439,347,17.9418549,"\int (d+e x)^{-4-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]","-\frac{2 \left(\frac{e \left(\sqrt{-\frac{a}{c}}-x\right)}{d+\sqrt{-\frac{a}{c}} e}\right)^{-p} (d+e x)^{-2 p-3} \left(c x^2+a\right)^p \left(1-\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right)^{p+1} \Gamma (-2 (p+1)) \left(2 p^2 \left(x+\sqrt{-\frac{a}{c}}\right) \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^3+3 p \left(x+\sqrt{-\frac{a}{c}}\right) \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^3+\left(x+\sqrt{-\frac{a}{c}}\right) \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^3+2 p \left(x+\sqrt{-\frac{a}{c}}\right) (d+e x) \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^2+\left(x+\sqrt{-\frac{a}{c}}\right) (d+e x) \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^2-4 \sqrt{-\frac{a}{c}} p (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _2F_1\left(2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^2-3 \sqrt{-\frac{a}{c}} (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _2F_1\left(2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^2+\sqrt{-\frac{a}{c}} (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left(2,2,1-p;1,1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)^2+\left(x+\sqrt{-\frac{a}{c}}\right) (d+e x)^2 \Gamma (1-2 p) \Gamma (-p) \, _2F_1\left(1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)+4 \sqrt{-\frac{a}{c}} p (d+e x)^2 \Gamma (1-p) \Gamma (-2 p) \, _2F_1\left(2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)-2 \sqrt{-\frac{a}{c}} (d+e x)^2 \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left(2,2,1-p;1,1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right) \left(d+\sqrt{-\frac{a}{c}} e\right)+3 \sqrt{-\frac{a}{c}} (d+e x)^3 \Gamma (1-p) \Gamma (-2 p) \, _2F_1\left(2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right)+\sqrt{-\frac{a}{c}} (d+e x)^3 \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left(2,2,1-p;1,1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left(d+\sqrt{-\frac{a}{c}} e\right) \left(x+\sqrt{-\frac{a}{c}}\right)}\right)\right)}{e \left(d+\sqrt{-\frac{a}{c}} e\right)^3 (2 p+3) \left(x+\sqrt{-\frac{a}{c}}\right) \Gamma (1-2 p) \Gamma (-2 p) \Gamma (-p)}","\frac{c \left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(a e^2-c d^2 (2 p+3)\right) \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) (2 p+3) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)^2}-\frac{e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left(a e^2+c d^2\right)}-\frac{c d e (p+2) \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left(a e^2+c d^2\right)^2}",1,"(-2*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*Gamma[-2*(1 + p)]*((d + Sqrt[-(a/c)]*e)^3*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (d + Sqrt[-(a/c)]*e)^2*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*(d + Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 3*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 4*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 4*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 2*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]))/(e*(d + Sqrt[-(a/c)]*e)^3*(3 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-2*p]*Gamma[-p])","B",0
744,1,2500,436,83.7648106,"\int (d+e x)^{-5-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-5 - 2*p)*(a + c*x^2)^p,x]","\text{Result too large to show}","\frac{c^2 d \left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(3 a e^2-c d^2 (2 p+3)\right) \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) (2 p+3) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)^3}+\frac{c e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+1)} \left(a e^2 (2 p+3)-c d^2 \left(2 p^2+8 p+9\right)\right)}{2 (p+1) (p+2) (2 p+3) \left(a e^2+c d^2\right)^3}-\frac{c d e (p+3) \left(a+c x^2\right)^{p+1} (d+e x)^{-2 p-3}}{(p+2) (2 p+3) \left(a e^2+c d^2\right)^2}-\frac{e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left(a e^2+c d^2\right)}",1,"-1/4*((a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*(6*(d + Sqrt[-(a/c)]*e)^4*p*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 22*(d + Sqrt[-(a/c)]*e)^4*p^2*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 24*(d + Sqrt[-(a/c)]*e)^4*p^3*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 8*(d + Sqrt[-(a/c)]*e)^4*p^4*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*(d + Sqrt[-(a/c)]*e)^2*p^2*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)*p*(Sqrt[-(a/c)] + x)*(d + e*x)^3*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*(d + Sqrt[-(a/c)]*e)^3*p^3*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 48*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p^2*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 24*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p^3*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 33*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 22*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 54*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 36*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p^2*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p^2*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 2, 1 - p}, {1, 1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 2, 1 - p}, {1, 1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]))/(e*(d + Sqrt[-(a/c)]*e)^4*p*(1 + p)*(2 + p)*(1 + 2*p)*(3 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c)] + x)*(d + e*x)^(2*(2 + p))*Gamma[-p]*Gamma[-2*(1 + p)])","B",0
745,1,5685,559,36.2868595,"\int (d+e x)^{-6-2 p} \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-6 - 2*p)*(a + c*x^2)^p,x]","\text{Result too large to show}","-\frac{c^2 \left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (d+e x)^{-2 p-1} \left(3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left(4 p^2+16 p+15\right)\right) \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} e+\sqrt{c} d\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} d-\sqrt{-a} e\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{(2 p+1) (2 p+3) (2 p+5) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)^4}+\frac{c^2 d e (p+3) \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+1)} \left(a e^2 (5 p+8)-c d^2 \left(2 p^2+7 p+8\right)\right)}{(p+1) (p+2) (2 p+3) (2 p+5) \left(a e^2+c d^2\right)^4}+\frac{c e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 p-3} \left(3 a e^2 (p+2)-c d^2 \left(2 p^2+11 p+18\right)\right)}{(p+2) (2 p+3) (2 p+5) \left(a e^2+c d^2\right)^3}-\frac{e \left(a+c x^2\right)^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left(a e^2+c d^2\right)}-\frac{c d e (p+4) \left(a+c x^2\right)^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left(a e^2+c d^2\right)^2}",1,"Result too large to show","B",0
746,1,48,43,0.0382876,"\int \frac{(3-4 x)^n}{\sqrt{1-x^2}} \, dx","Integrate[(3 - 4*x)^n/Sqrt[1 - x^2],x]","-\frac{(3-4 x)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (3-4 x),4 x-3\right)}{\sqrt{7} (n+1)}","\sqrt{2} 7^n \sqrt{x+1} F_1\left(\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{4 (x+1)}{7},\frac{x+1}{2}\right)",1,"-(((3 - 4*x)^(1 + n)*AppellF1[1 + n, 1/2, 1/2, 2 + n, (3 - 4*x)/7, -3 + 4*x])/(Sqrt[7]*(1 + n)))","A",0
747,1,65,83,0.0076657,"\int \frac{(a+b x)^6}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^6/(a^2 - b^2*x^2),x]","-\frac{32 a^5 \log (a-b x)}{b}-31 a^4 x-13 a^3 b x^2-\frac{16}{3} a^2 b^2 x^3-\frac{3}{2} a b^3 x^4-\frac{b^4 x^5}{5}","-\frac{32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b}",1,"-31*a^4*x - 13*a^3*b*x^2 - (16*a^2*b^2*x^3)/3 - (3*a*b^3*x^4)/2 - (b^4*x^5)/5 - (32*a^5*Log[a - b*x])/b","A",1
748,1,54,66,0.0063972,"\int \frac{(a+b x)^5}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^5/(a^2 - b^2*x^2),x]","-\frac{16 a^4 \log (a-b x)}{b}-15 a^3 x-\frac{11}{2} a^2 b x^2-\frac{5}{3} a b^2 x^3-\frac{b^3 x^4}{4}","-\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a^2 (a+b x)^2}{b}-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b}",1,"-15*a^3*x - (11*a^2*b*x^2)/2 - (5*a*b^2*x^3)/3 - (b^3*x^4)/4 - (16*a^4*Log[a - b*x])/b","A",1
749,1,39,49,0.0060615,"\int \frac{(a+b x)^4}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^4/(a^2 - b^2*x^2),x]","-\frac{8 a^3 \log (a-b x)}{b}-7 a^2 x-2 a b x^2-\frac{b^2 x^3}{3}","-\frac{8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b}",1,"-7*a^2*x - 2*a*b*x^2 - (b^2*x^3)/3 - (8*a^3*Log[a - b*x])/b","A",1
750,1,28,28,0.0053268,"\int \frac{(a+b x)^3}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^3/(a^2 - b^2*x^2),x]","-\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2}","-\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2}",1,"-3*a*x - (b*x^2)/2 - (4*a^2*Log[a - b*x])/b","A",1
751,1,17,17,0.0039038,"\int \frac{(a+b x)^2}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^2/(a^2 - b^2*x^2),x]","-\frac{2 a \log (a-b x)}{b}-x","-\frac{2 a \log (a-b x)}{b}-x",1,"-x - (2*a*Log[a - b*x])/b","A",1
752,1,12,12,0.0012755,"\int \frac{a+b x}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)/(a^2 - b^2*x^2),x]","-\frac{\log (a-b x)}{b}","-\frac{\log (a-b x)}{b}",1,"-(Log[a - b*x]/b)","A",1
753,1,47,35,0.0128702,"\int \frac{1}{(a+b x) \left(a^2-b^2 x^2\right)} \, dx","Integrate[1/((a + b*x)*(a^2 - b^2*x^2)),x]","\frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b (a+b x)}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{2 a^2 b}-\frac{1}{2 a b (a+b x)}",1,"(-2*a - (a + b*x)*Log[a - b*x] + (a + b*x)*Log[a + b*x])/(4*a^2*b*(a + b*x))","A",1
754,1,58,52,0.0194055,"\int \frac{1}{(a+b x)^2 \left(a^2-b^2 x^2\right)} \, dx","Integrate[1/((a + b*x)^2*(a^2 - b^2*x^2)),x]","\frac{-2 a (2 a+b x)+(a+b x)^2 (-\log (a-b x))+(a+b x)^2 \log (a+b x)}{8 a^3 b (a+b x)^2}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{4 a^3 b}-\frac{1}{4 a^2 b (a+b x)}-\frac{1}{4 a b (a+b x)^2}",1,"(-2*a*(2*a + b*x) - (a + b*x)^2*Log[a - b*x] + (a + b*x)^2*Log[a + b*x])/(8*a^3*b*(a + b*x)^2)","A",1
755,1,71,69,0.0226596,"\int \frac{1}{(a+b x)^3 \left(a^2-b^2 x^2\right)} \, dx","Integrate[1/((a + b*x)^3*(a^2 - b^2*x^2)),x]","\frac{-2 a \left(10 a^2+9 a b x+3 b^2 x^2\right)-3 (a+b x)^3 \log (a-b x)+3 (a+b x)^3 \log (a+b x)}{48 a^4 b (a+b x)^3}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{8 a^4 b}-\frac{1}{8 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{6 a b (a+b x)^3}",1,"(-2*a*(10*a^2 + 9*a*b*x + 3*b^2*x^2) - 3*(a + b*x)^3*Log[a - b*x] + 3*(a + b*x)^3*Log[a + b*x])/(48*a^4*b*(a + b*x)^3)","A",1
756,1,82,86,0.0259194,"\int \frac{1}{(a+b x)^4 \left(a^2-b^2 x^2\right)} \, dx","Integrate[1/((a + b*x)^4*(a^2 - b^2*x^2)),x]","\frac{-2 a \left(16 a^3+19 a^2 b x+12 a b^2 x^2+3 b^3 x^3\right)-3 (a+b x)^4 \log (a-b x)+3 (a+b x)^4 \log (a+b x)}{96 a^5 b (a+b x)^4}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{16 a^5 b}-\frac{1}{16 a^4 b (a+b x)}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{8 a b (a+b x)^4}",1,"(-2*a*(16*a^3 + 19*a^2*b*x + 12*a*b^2*x^2 + 3*b^3*x^3) - 3*(a + b*x)^4*Log[a - b*x] + 3*(a + b*x)^4*Log[a + b*x])/(96*a^5*b*(a + b*x)^4)","A",1
757,1,71,70,0.0266683,"\int \frac{(a+b x)^7}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^7/(a^2 - b^2*x^2)^2,x]","-\frac{32 a^5}{b (b x-a)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4}","\frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4}",1,"49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 - (32*a^5)/(b*(-a + b*x)) + (80*a^4*Log[a - b*x])/b","A",1
758,1,56,55,0.028469,"\int \frac{(a+b x)^6}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^6/(a^2 - b^2*x^2)^2,x]","-\frac{16 a^4}{b (b x-a)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}","\frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}",1,"17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 - (16*a^4)/(b*(-a + b*x)) + (32*a^3*Log[a - b*x])/b","A",1
759,1,45,44,0.0204506,"\int \frac{(a+b x)^5}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^5/(a^2 - b^2*x^2)^2,x]","-\frac{8 a^3}{b (b x-a)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2}","\frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2}",1,"5*a*x + (b*x^2)/2 - (8*a^3)/(b*(-a + b*x)) + (12*a^2*Log[a - b*x])/b","A",1
760,1,32,31,0.0200157,"\int \frac{(a+b x)^4}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]","-\frac{4 a^2}{b (b x-a)}+\frac{4 a \log (a-b x)}{b}+x","\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x",1,"x - (4*a^2)/(b*(-a + b*x)) + (4*a*Log[a - b*x])/b","A",1
761,1,23,26,0.0075911,"\int \frac{(a+b x)^3}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]","\frac{\frac{2 a}{a-b x}+\log (a-b x)}{b}","\frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b}",1,"((2*a)/(a - b*x) + Log[a - b*x])/b","A",1
762,1,12,12,0.0020595,"\int \frac{(a+b x)^2}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^2/(a^2 - b^2*x^2)^2,x]","\frac{1}{b (a-b x)}","\frac{1}{b (a-b x)}",1,"1/(b*(a - b*x))","A",1
763,1,50,36,0.0101015,"\int \frac{a+b x}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)/(a^2 - b^2*x^2)^2,x]","\frac{(b x-a) \log (a-b x)+(a-b x) \log (a+b x)+2 a}{4 a^2 b (a-b x)}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{2 a^2 b}+\frac{1}{2 a b (a-b x)}",1,"(2*a + (-a + b*x)*Log[a - b*x] + (a - b*x)*Log[a + b*x])/(4*a^2*b*(a - b*x))","A",1
764,1,87,70,0.0223345,"\int \frac{1}{(a+b x) \left(a^2-b^2 x^2\right)^2} \, dx","Integrate[1/((a + b*x)*(a^2 - b^2*x^2)^2),x]","-\frac{3 \log (a-b x)}{16 a^4 b}+\frac{3 \log (a+b x)}{16 a^4 b}-\frac{1}{8 a^3 b (b x-a)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}","\frac{3 \tanh ^{-1}\left(\frac{b x}{a}\right)}{8 a^4 b}+\frac{1}{8 a^3 b (a-b x)}-\frac{1}{4 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}",1,"-1/8*1/(a^3*b*(-a + b*x)) - 1/(8*a^2*b*(a + b*x)^2) - 1/(4*a^3*b*(a + b*x)) - (3*Log[a - b*x])/(16*a^4*b) + (3*Log[a + b*x])/(16*a^4*b)","A",1
765,1,75,87,0.0415851,"\int \frac{1}{(a+b x)^2 \left(a^2-b^2 x^2\right)^2} \, dx","Integrate[1/((a + b*x)^2*(a^2 - b^2*x^2)^2),x]","\frac{\frac{2 a \left(-4 a^3+a^2 b x+6 a b^2 x^2+3 b^3 x^3\right)}{(a-b x) (a+b x)^3}-3 \log (a-b x)+3 \log (a+b x)}{24 a^5 b}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{4 a^5 b}+\frac{1}{16 a^4 b (a-b x)}-\frac{3}{16 a^4 b (a+b x)}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}",1,"((2*a*(-4*a^3 + a^2*b*x + 6*a*b^2*x^2 + 3*b^3*x^3))/((a - b*x)*(a + b*x)^3) - 3*Log[a - b*x] + 3*Log[a + b*x])/(24*a^5*b)","A",1
766,1,112,104,0.0298364,"\int \frac{1}{(a+b x)^3 \left(a^2-b^2 x^2\right)^2} \, dx","Integrate[1/((a + b*x)^3*(a^2 - b^2*x^2)^2),x]","\frac{-64 a^5-30 a^4 b x+70 a^3 b^2 x^2+90 a^2 b^3 x^3+30 a b^4 x^4-15 (a-b x) (a+b x)^4 \log (a-b x)+15 (a-b x) (a+b x)^4 \log (a+b x)}{192 a^6 b (a-b x) (a+b x)^4}","\frac{5 \tanh ^{-1}\left(\frac{b x}{a}\right)}{32 a^6 b}+\frac{1}{32 a^5 b (a-b x)}-\frac{1}{8 a^5 b (a+b x)}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{12 a^3 b (a+b x)^3}-\frac{1}{16 a^2 b (a+b x)^4}",1,"(-64*a^5 - 30*a^4*b*x + 70*a^3*b^2*x^2 + 90*a^2*b^3*x^3 + 30*a*b^4*x^4 - 15*(a - b*x)*(a + b*x)^4*Log[a - b*x] + 15*(a - b*x)*(a + b*x)^4*Log[a + b*x])/(192*a^6*b*(a - b*x)*(a + b*x)^4)","A",1
767,1,73,71,0.0375785,"\int \frac{(a+b x)^8}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^8/(a^2 - b^2*x^2)^3,x]","\frac{16 a^5}{b (b x-a)^2}+\frac{80 a^4}{b (b x-a)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3}","\frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3}",1,"-31*a^2*x - 4*a*b*x^2 - (b^2*x^3)/3 + (16*a^5)/(b*(-a + b*x)^2) + (80*a^4)/(b*(-a + b*x)) - (80*a^3*Log[a - b*x])/b","A",1
768,1,62,60,0.0282807,"\int \frac{(a+b x)^7}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^7/(a^2 - b^2*x^2)^3,x]","\frac{8 a^4}{b (b x-a)^2}+\frac{32 a^3}{b (b x-a)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2}","\frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2}",1,"-7*a*x - (b*x^2)/2 + (8*a^4)/(b*(-a + b*x)^2) + (32*a^3)/(b*(-a + b*x)) - (24*a^2*Log[a - b*x])/b","A",1
769,1,41,49,0.0428093,"\int \frac{(a+b x)^6}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^6/(a^2 - b^2*x^2)^3,x]","\frac{4 a^2 (3 b x-2 a)}{b (a-b x)^2}-\frac{6 a \log (a-b x)}{b}-x","\frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x",1,"-x + (4*a^2*(-2*a + 3*b*x))/(b*(a - b*x)^2) - (6*a*Log[a - b*x])/b","A",1
770,1,30,43,0.0180269,"\int \frac{(a+b x)^5}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]","-\frac{\frac{2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b}","\frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b}",1,"-(((2*a*(a - 2*b*x))/(a - b*x)^2 + Log[a - b*x])/b)","A",1
771,1,10,10,0.0041755,"\int \frac{(a+b x)^4}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^3,x]","\frac{x}{(a-b x)^2}","\frac{x}{(a-b x)^2}",1,"x/(a - b*x)^2","A",1
772,1,15,15,0.0020682,"\int \frac{(a+b x)^3}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^3/(a^2 - b^2*x^2)^3,x]","\frac{1}{2 b (a-b x)^2}","\frac{1}{2 b (a-b x)^2}",1,"1/(2*b*(a - b*x)^2)","A",1
773,1,62,54,0.0180187,"\int \frac{(a+b x)^2}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^2/(a^2 - b^2*x^2)^3,x]","\frac{2 a (2 a-b x)+(a-b x)^2 (-\log (a-b x))+(a-b x)^2 \log (a+b x)}{8 a^3 b (a-b x)^2}","\frac{\tanh ^{-1}\left(\frac{b x}{a}\right)}{4 a^3 b}+\frac{1}{4 a^2 b (a-b x)}+\frac{1}{4 a b (a-b x)^2}",1,"(2*a*(2*a - b*x) - (a - b*x)^2*Log[a - b*x] + (a - b*x)^2*Log[a + b*x])/(8*a^3*b*(a - b*x)^2)","A",1
774,1,65,71,0.0330948,"\int \frac{a+b x}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)/(a^2 - b^2*x^2)^3,x]","\frac{\frac{2 a \left(2 a^2+3 a b x-3 b^2 x^2\right)}{(a-b x)^2 (a+b x)}-3 \log (a-b x)+3 \log (a+b x)}{16 a^4 b}","\frac{3 \tanh ^{-1}\left(\frac{b x}{a}\right)}{8 a^4 b}+\frac{1}{4 a^3 b (a-b x)}-\frac{1}{8 a^3 b (a+b x)}+\frac{1}{8 a^2 b (a-b x)^2}",1,"((2*a*(2*a^2 + 3*a*b*x - 3*b^2*x^2))/((a - b*x)^2*(a + b*x)) - 3*Log[a - b*x] + 3*Log[a + b*x])/(16*a^4*b)","A",1
775,1,87,105,0.0449894,"\int \frac{1}{(a+b x) \left(a^2-b^2 x^2\right)^3} \, dx","Integrate[1/((a + b*x)*(a^2 - b^2*x^2)^3),x]","\frac{-\frac{2 a \left(8 a^4-25 a^3 b x-25 a^2 b^2 x^2+15 a b^3 x^3+15 b^4 x^4\right)}{(a-b x)^2 (a+b x)^3}-15 \log (a-b x)+15 \log (a+b x)}{96 a^6 b}","\frac{5 \tanh ^{-1}\left(\frac{b x}{a}\right)}{16 a^6 b}+\frac{1}{8 a^5 b (a-b x)}-\frac{3}{16 a^5 b (a+b x)}+\frac{1}{32 a^4 b (a-b x)^2}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{24 a^3 b (a+b x)^3}",1,"((-2*a*(8*a^4 - 25*a^3*b*x - 25*a^2*b^2*x^2 + 15*a*b^3*x^3 + 15*b^4*x^4))/((a - b*x)^2*(a + b*x)^3) - 15*Log[a - b*x] + 15*Log[a + b*x])/(96*a^6*b)","A",1
776,1,98,122,0.0539027,"\int \frac{1}{(a+b x)^2 \left(a^2-b^2 x^2\right)^3} \, dx","Integrate[1/((a + b*x)^2*(a^2 - b^2*x^2)^3),x]","\frac{\frac{2 a \left(-16 a^5+17 a^4 b x+50 a^3 b^2 x^2+10 a^2 b^3 x^3-30 a b^4 x^4-15 b^5 x^5\right)}{(a-b x)^2 (a+b x)^4}-15 \log (a-b x)+15 \log (a+b x)}{128 a^7 b}","\frac{15 \tanh ^{-1}\left(\frac{b x}{a}\right)}{64 a^7 b}+\frac{5}{64 a^6 b (a-b x)}-\frac{5}{32 a^6 b (a+b x)}+\frac{1}{64 a^5 b (a-b x)^2}-\frac{3}{32 a^5 b (a+b x)^2}-\frac{1}{16 a^4 b (a+b x)^3}-\frac{1}{32 a^3 b (a+b x)^4}",1,"((2*a*(-16*a^5 + 17*a^4*b*x + 50*a^3*b^2*x^2 + 10*a^2*b^3*x^3 - 30*a*b^4*x^4 - 15*b^5*x^5))/((a - b*x)^2*(a + b*x)^4) - 15*Log[a - b*x] + 15*Log[a + b*x])/(128*a^7*b)","A",1
777,1,123,173,0.2112293,"\int (a+b x)^4 \sqrt{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^4*Sqrt[a^2 - b^2*x^2],x]","\frac{\sqrt{a^2-b^2 x^2} \left(315 a^5 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-448 a^5-75 a^4 b x+256 a^3 b^2 x^2+350 a^2 b^3 x^3+192 a b^4 x^4+40 b^5 x^5\right)\right)}{240 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","-\frac{21 a^2 (a+b x) \left(a^2-b^2 x^2\right)^{3/2}}{40 b}-\frac{3 a (a+b x)^2 \left(a^2-b^2 x^2\right)^{3/2}}{10 b}-\frac{(a+b x)^3 \left(a^2-b^2 x^2\right)^{3/2}}{6 b}+\frac{21 a^6 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{16 b}+\frac{21}{16} a^4 x \sqrt{a^2-b^2 x^2}-\frac{7 a^3 \left(a^2-b^2 x^2\right)^{3/2}}{8 b}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-448*a^5 - 75*a^4*b*x + 256*a^3*b^2*x^2 + 350*a^2*b^3*x^3 + 192*a*b^4*x^4 + 40*b^5*x^5) + 315*a^5*ArcSin[(b*x)/a]))/(240*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
778,1,112,140,0.2164044,"\int (a+b x)^3 \sqrt{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^3*Sqrt[a^2 - b^2*x^2],x]","\frac{\sqrt{a^2-b^2 x^2} \left(105 a^4 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right)\right)}{120 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","-\frac{7 a^2 \left(a^2-b^2 x^2\right)^{3/2}}{12 b}-\frac{7 a (a+b x) \left(a^2-b^2 x^2\right)^{3/2}}{20 b}-\frac{(a+b x)^2 \left(a^2-b^2 x^2\right)^{3/2}}{5 b}+\frac{7 a^5 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{8 b}+\frac{7}{8} a^3 x \sqrt{a^2-b^2 x^2}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-136*a^4 + 15*a^3*b*x + 112*a^2*b^2*x^2 + 90*a*b^3*x^3 + 24*b^4*x^4) + 105*a^4*ArcSin[(b*x)/a]))/(120*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
779,1,101,107,0.1752391,"\int (a+b x)^2 \sqrt{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^2*Sqrt[a^2 - b^2*x^2],x]","\frac{\sqrt{a^2-b^2 x^2} \left(15 a^3 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right)\right)}{24 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left(a^2-b^2 x^2\right)^{3/2}}{12 b}-\frac{(a+b x) \left(a^2-b^2 x^2\right)^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{8 b}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-16*a^3 + 9*a^2*b*x + 16*a*b^2*x^2 + 6*b^3*x^3) + 15*a^3*ArcSin[(b*x)/a]))/(24*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
780,1,90,76,0.077118,"\int (a+b x) \sqrt{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)*Sqrt[a^2 - b^2*x^2],x]","\frac{\sqrt{a^2-b^2 x^2} \left(\left(-2 a^2+3 a b x+2 b^2 x^2\right) \sqrt{1-\frac{b^2 x^2}{a^2}}+3 a^2 \sin ^{-1}\left(\frac{b x}{a}\right)\right)}{6 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{1}{2} a x \sqrt{a^2-b^2 x^2}-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{3 b}+\frac{a^3 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{2 b}",1,"(Sqrt[a^2 - b^2*x^2]*((-2*a^2 + 3*a*b*x + 2*b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2] + 3*a^2*ArcSin[(b*x)/a]))/(6*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
781,1,43,46,0.0371582,"\int \frac{\sqrt{a^2-b^2 x^2}}{a+b x} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x),x]","\frac{\sqrt{a^2-b^2 x^2}+a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}","\frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}",1,"(Sqrt[a^2 - b^2*x^2] + a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b","A",1
782,1,51,54,0.0546024,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^2} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^2,x]","-\frac{\frac{2 \sqrt{a^2-b^2 x^2}}{a+b x}+\tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}","-\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{\tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}",1,"-(((2*Sqrt[a^2 - b^2*x^2])/(a + b*x) + ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b)","A",1
783,1,39,33,0.0331822,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]","-\frac{(a-b x) \sqrt{a^2-b^2 x^2}}{3 a b (a+b x)^2}","-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{3 a b (a+b x)^3}",1,"-1/3*((a - b*x)*Sqrt[a^2 - b^2*x^2])/(a*b*(a + b*x)^2)","A",1
784,1,51,67,0.0364159,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^4} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^4,x]","\frac{\sqrt{a^2-b^2 x^2} \left(-4 a^2+3 a b x+b^2 x^2\right)}{15 a^2 b (a+b x)^3}","-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{15 a^2 b (a+b x)^3}-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{5 a b (a+b x)^4}",1,"(Sqrt[a^2 - b^2*x^2]*(-4*a^2 + 3*a*b*x + b^2*x^2))/(15*a^2*b*(a + b*x)^3)","A",1
785,1,63,100,0.0414324,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^5} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]","\frac{\sqrt{a^2-b^2 x^2} \left(-23 a^3+13 a^2 b x+8 a b^2 x^2+2 b^3 x^3\right)}{105 a^3 b (a+b x)^4}","-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{35 a^2 b (a+b x)^4}-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{7 a b (a+b x)^5}-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{105 a^3 b (a+b x)^3}",1,"(Sqrt[a^2 - b^2*x^2]*(-23*a^3 + 13*a^2*b*x + 8*a*b^2*x^2 + 2*b^3*x^3))/(105*a^3*b*(a + b*x)^4)","A",1
786,1,74,133,0.0441009,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^6} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^6,x]","\frac{\sqrt{a^2-b^2 x^2} \left(-58 a^4+25 a^3 b x+21 a^2 b^2 x^2+10 a b^3 x^3+2 b^4 x^4\right)}{315 a^4 b (a+b x)^5}","-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{21 a^2 b (a+b x)^5}-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{9 a b (a+b x)^6}-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{315 a^4 b (a+b x)^3}-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{105 a^3 b (a+b x)^4}",1,"(Sqrt[a^2 - b^2*x^2]*(-58*a^4 + 25*a^3*b*x + 21*a^2*b^2*x^2 + 10*a*b^3*x^3 + 2*b^4*x^4))/(315*a^4*b*(a + b*x)^5)","A",1
787,1,85,166,0.0456389,"\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^7} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^7,x]","\frac{\sqrt{a^2-b^2 x^2} \left(-547 a^5+183 a^4 b x+184 a^3 b^2 x^2+124 a^2 b^3 x^3+48 a b^4 x^4+8 b^5 x^5\right)}{3465 a^5 b (a+b x)^6}","-\frac{4 \left(a^2-b^2 x^2\right)^{3/2}}{99 a^2 b (a+b x)^6}-\frac{\left(a^2-b^2 x^2\right)^{3/2}}{11 a b (a+b x)^7}-\frac{8 \left(a^2-b^2 x^2\right)^{3/2}}{3465 a^5 b (a+b x)^3}-\frac{8 \left(a^2-b^2 x^2\right)^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{4 \left(a^2-b^2 x^2\right)^{3/2}}{231 a^3 b (a+b x)^5}",1,"(Sqrt[a^2 - b^2*x^2]*(-547*a^5 + 183*a^4*b*x + 184*a^3*b^2*x^2 + 124*a^2*b^3*x^3 + 48*a*b^4*x^4 + 8*b^5*x^5))/(3465*a^5*b*(a + b*x)^6)","A",1
788,1,134,164,0.213925,"\int (a+b x)^3 \left(a^2-b^2 x^2\right)^{3/2} \, dx","Integrate[(a + b*x)^3*(a^2 - b^2*x^2)^(3/2),x]","\frac{\sqrt{a^2-b^2 x^2} \left(315 a^6 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-368 a^6+245 a^5 b x+656 a^4 b^2 x^2+350 a^3 b^3 x^3-208 a^2 b^4 x^4-280 a b^5 x^5-80 b^6 x^6\right)\right)}{560 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","-\frac{3 a^2 \left(a^2-b^2 x^2\right)^{5/2}}{10 b}-\frac{3 a (a+b x) \left(a^2-b^2 x^2\right)^{5/2}}{14 b}-\frac{(a+b x)^2 \left(a^2-b^2 x^2\right)^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{16 b}+\frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left(a^2-b^2 x^2\right)^{3/2}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-368*a^6 + 245*a^5*b*x + 656*a^4*b^2*x^2 + 350*a^3*b^3*x^3 - 208*a^2*b^4*x^4 - 280*a*b^5*x^5 - 80*b^6*x^6) + 315*a^6*ArcSin[(b*x)/a]))/(560*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
789,1,123,131,0.1780557,"\int (a+b x)^2 \left(a^2-b^2 x^2\right)^{3/2} \, dx","Integrate[(a + b*x)^2*(a^2 - b^2*x^2)^(3/2),x]","\frac{\sqrt{a^2-b^2 x^2} \left(105 a^5 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-96 a^5+135 a^4 b x+192 a^3 b^2 x^2+10 a^2 b^3 x^3-96 a b^4 x^4-40 b^5 x^5\right)\right)}{240 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{7}{24} a^2 x \left(a^2-b^2 x^2\right)^{3/2}-\frac{7 a \left(a^2-b^2 x^2\right)^{5/2}}{30 b}-\frac{(a+b x) \left(a^2-b^2 x^2\right)^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{16 b}+\frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-96*a^5 + 135*a^4*b*x + 192*a^3*b^2*x^2 + 10*a^2*b^3*x^3 - 96*a*b^4*x^4 - 40*b^5*x^5) + 105*a^5*ArcSin[(b*x)/a]))/(240*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
790,1,112,100,0.1448537,"\int (a+b x) \left(a^2-b^2 x^2\right)^{3/2} \, dx","Integrate[(a + b*x)*(a^2 - b^2*x^2)^(3/2),x]","\frac{\sqrt{a^2-b^2 x^2} \left(15 a^4 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-8 a^4+25 a^3 b x+16 a^2 b^2 x^2-10 a b^3 x^3-8 b^4 x^4\right)\right)}{40 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{1}{4} a x \left(a^2-b^2 x^2\right)^{3/2}-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{8 b}+\frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2}",1,"(Sqrt[a^2 - b^2*x^2]*(Sqrt[1 - (b^2*x^2)/a^2]*(-8*a^4 + 25*a^3*b*x + 16*a^2*b^2*x^2 - 10*a*b^3*x^3 - 8*b^4*x^4) + 15*a^4*ArcSin[(b*x)/a]))/(40*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
791,1,90,76,0.0774131,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{a+b x} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x),x]","\frac{\sqrt{a^2-b^2 x^2} \left(\left(2 a^2+3 a b x-2 b^2 x^2\right) \sqrt{1-\frac{b^2 x^2}{a^2}}+3 a^2 \sin ^{-1}\left(\frac{b x}{a}\right)\right)}{6 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{1}{2} a x \sqrt{a^2-b^2 x^2}+\frac{\left(a^2-b^2 x^2\right)^{3/2}}{3 b}+\frac{a^3 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{2 b}",1,"(Sqrt[a^2 - b^2*x^2]*((2*a^2 + 3*a*b*x - 2*b^2*x^2)*Sqrt[1 - (b^2*x^2)/a^2] + 3*a^2*ArcSin[(b*x)/a]))/(6*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
792,1,60,85,0.0564172,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^2} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^2,x]","\left(\frac{2 a}{b}-\frac{x}{2}\right) \sqrt{a^2-b^2 x^2}+\frac{3 a^2 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{2 b}","\frac{3 a \sqrt{a^2-b^2 x^2}}{2 b}+\frac{\left(a^2-b^2 x^2\right)^{3/2}}{2 b (a+b x)}+\frac{3 a^2 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{2 b}",1,"((2*a)/b - x/2)*Sqrt[a^2 - b^2*x^2] + (3*a^2*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(2*b)","A",1
793,1,60,76,0.0840082,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^3} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^3,x]","-\frac{\frac{\sqrt{a^2-b^2 x^2} (5 a+b x)}{a+b x}+3 a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}","-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{b (a+b x)^2}-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{3 a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}",1,"-((((5*a + b*x)*Sqrt[a^2 - b^2*x^2])/(a + b*x) + 3*a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b)","A",1
794,1,61,83,0.0684973,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^4} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^4,x]","\frac{\frac{4 \sqrt{a^2-b^2 x^2} (a+2 b x)}{(a+b x)^2}+3 \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{3 b}","-\frac{2 \left(a^2-b^2 x^2\right)^{3/2}}{3 b (a+b x)^3}+\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}+\frac{\tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}",1,"((4*(a + 2*b*x)*Sqrt[a^2 - b^2*x^2])/(a + b*x)^2 + 3*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(3*b)","A",1
795,1,41,33,0.0446254,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^5} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^5,x]","-\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2}}{5 a b (a+b x)^3}","-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{5 a b (a+b x)^5}",1,"-1/5*((a - b*x)^2*Sqrt[a^2 - b^2*x^2])/(a*b*(a + b*x)^3)","A",1
796,1,48,67,0.0477656,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^6} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x]","-\frac{(a-b x)^2 (6 a+b x) \sqrt{a^2-b^2 x^2}}{35 a^2 b (a+b x)^4}","-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{35 a^2 b (a+b x)^5}-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{7 a b (a+b x)^6}",1,"-1/35*((a - b*x)^2*(6*a + b*x)*Sqrt[a^2 - b^2*x^2])/(a^2*b*(a + b*x)^4)","A",1
797,1,60,100,0.0491213,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^7} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^7,x]","-\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left(47 a^2+14 a b x+2 b^2 x^2\right)}{315 a^3 b (a+b x)^5}","-\frac{2 \left(a^2-b^2 x^2\right)^{5/2}}{63 a^2 b (a+b x)^6}-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{9 a b (a+b x)^7}-\frac{2 \left(a^2-b^2 x^2\right)^{5/2}}{315 a^3 b (a+b x)^5}",1,"-1/315*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(47*a^2 + 14*a*b*x + 2*b^2*x^2))/(a^3*b*(a + b*x)^5)","A",1
798,1,71,133,0.0566178,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^8} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x]","-\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left(152 a^3+61 a^2 b x+16 a b^2 x^2+2 b^3 x^3\right)}{1155 a^4 b (a+b x)^6}","-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{33 a^2 b (a+b x)^7}-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{11 a b (a+b x)^8}-\frac{2 \left(a^2-b^2 x^2\right)^{5/2}}{1155 a^4 b (a+b x)^5}-\frac{2 \left(a^2-b^2 x^2\right)^{5/2}}{231 a^3 b (a+b x)^6}",1,"-1/1155*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(152*a^3 + 61*a^2*b*x + 16*a*b^2*x^2 + 2*b^3*x^3))/(a^4*b*(a + b*x)^6)","A",1
799,1,82,166,0.0595083,"\int \frac{\left(a^2-b^2 x^2\right)^{3/2}}{(a+b x)^9} \, dx","Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]","-\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left(1763 a^4+852 a^3 b x+308 a^2 b^2 x^2+72 a b^3 x^3+8 b^4 x^4\right)}{15015 a^5 b (a+b x)^7}","-\frac{4 \left(a^2-b^2 x^2\right)^{5/2}}{143 a^2 b (a+b x)^8}-\frac{\left(a^2-b^2 x^2\right)^{5/2}}{13 a b (a+b x)^9}-\frac{8 \left(a^2-b^2 x^2\right)^{5/2}}{15015 a^5 b (a+b x)^5}-\frac{8 \left(a^2-b^2 x^2\right)^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{4 \left(a^2-b^2 x^2\right)^{5/2}}{429 a^3 b (a+b x)^7}",1,"-1/15015*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(1763*a^4 + 852*a^3*b*x + 308*a^2*b^2*x^2 + 72*a*b^3*x^3 + 8*b^4*x^4))/(a^5*b*(a + b*x)^7)","A",1
800,1,178,212,0.3093593,"\int (d+e x)^3 \left(d^2-e^2 x^2\right)^{7/2} \, dx","Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(45045 d^{10} \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-44800 d^{10}+81675 d^9 e x+167680 d^8 e^2 x^2+12210 d^7 e^3 x^3-222720 d^6 e^4 x^4-142296 d^5 e^5 x^5+110080 d^4 e^6 x^6+131472 d^3 e^7 x^7+1280 d^2 e^8 x^8-38016 d e^9 x^9-11520 e^{10} x^{10}\right)\right)}{126720 e \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{13 d^2 \left(d^2-e^2 x^2\right)^{9/2}}{90 e}-\frac{13 d (d+e x) \left(d^2-e^2 x^2\right)^{9/2}}{110 e}-\frac{(d+e x)^2 \left(d^2-e^2 x^2\right)^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{256 e}+\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left(d^2-e^2 x^2\right)^{3/2}+\frac{91}{480} d^5 x \left(d^2-e^2 x^2\right)^{5/2}+\frac{13}{80} d^3 x \left(d^2-e^2 x^2\right)^{7/2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-44800*d^10 + 81675*d^9*e*x + 167680*d^8*e^2*x^2 + 12210*d^7*e^3*x^3 - 222720*d^6*e^4*x^4 - 142296*d^5*e^5*x^5 + 110080*d^4*e^6*x^6 + 131472*d^3*e^7*x^7 + 1280*d^2*e^8*x^8 - 38016*d*e^9*x^9 - 11520*e^10*x^10) + 45045*d^10*ArcSin[(e*x)/d]))/(126720*e*Sqrt[1 - (e^2*x^2)/d^2])","A",1
801,1,118,179,0.3536988,"\int (d+e x)^2 \left(d^2-e^2 x^2\right)^{7/2} \, dx","Integrate[(d + e*x)^2*(d^2 - e^2*x^2)^(7/2),x]","\frac{\left(d^2-e^2 x^2\right)^{9/2} \left(\frac{33 d^2 \left(279 d^6 x-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left(\frac{e x}{d}\right)}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}-48 e^6 x^7\right)}{\left(d^2-e^2 x^2\right)^4}-\frac{2560 d}{e}-1152 x\right)}{11520}","\frac{11}{80} d^2 x \left(d^2-e^2 x^2\right)^{7/2}-\frac{11 d \left(d^2-e^2 x^2\right)^{9/2}}{90 e}-\frac{(d+e x) \left(d^2-e^2 x^2\right)^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{256 e}+\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left(d^2-e^2 x^2\right)^{3/2}+\frac{77}{480} d^4 x \left(d^2-e^2 x^2\right)^{5/2}",1,"((d^2 - e^2*x^2)^(9/2)*((-2560*d)/e - 1152*x + (33*d^2*(279*d^6*x - 326*d^4*e^2*x^3 + 200*d^2*e^4*x^5 - 48*e^6*x^7 + (105*d^7*ArcSin[(e*x)/d])/(e*Sqrt[1 - (e^2*x^2)/d^2])))/(d^2 - e^2*x^2)^4))/11520","A",1
802,1,114,148,0.2829392,"\int (d+e x) \left(d^2-e^2 x^2\right)^{7/2} \, dx","Integrate[(d + e*x)*(d^2 - e^2*x^2)^(7/2),x]","\frac{1}{384} d \sqrt{d^2-e^2 x^2} \left(279 d^6 x-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left(\frac{e x}{d}\right)}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}-48 e^6 x^7\right)-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{9 e}","\frac{1}{8} d x \left(d^2-e^2 x^2\right)^{7/2}-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e}+\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left(d^2-e^2 x^2\right)^{3/2}+\frac{7}{48} d^3 x \left(d^2-e^2 x^2\right)^{5/2}",1,"-1/9*(d^2 - e^2*x^2)^(9/2)/e + (d*Sqrt[d^2 - e^2*x^2]*(279*d^6*x - 326*d^4*e^2*x^3 + 200*d^2*e^4*x^5 - 48*e^6*x^7 + (105*d^7*ArcSin[(e*x)/d])/(e*Sqrt[1 - (e^2*x^2)/d^2])))/384","A",1
803,1,113,124,0.0968247,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{d+e x} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x),x]","\frac{105 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right)}{336 e}","\frac{1}{6} d x \left(d^2-e^2 x^2\right)^{5/2}+\frac{\left(d^2-e^2 x^2\right)^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e}+\frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left(d^2-e^2 x^2\right)^{3/2}",1,"(Sqrt[d^2 - e^2*x^2]*(48*d^6 + 231*d^5*e*x - 144*d^4*e^2*x^2 - 182*d^3*e^3*x^3 + 144*d^2*e^4*x^4 + 56*d*e^5*x^5 - 48*e^6*x^6) + 105*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(336*e)","A",1
804,1,102,132,0.07818,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^2,x]","\frac{105 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(96 d^5+135 d^4 e x-192 d^3 e^2 x^2+10 d^2 e^3 x^3+96 d e^4 x^4-40 e^5 x^5\right)}{240 e}","\frac{7}{24} d^2 x \left(d^2-e^2 x^2\right)^{3/2}+\frac{7 d \left(d^2-e^2 x^2\right)^{5/2}}{30 e}+\frac{(d-e x) \left(d^2-e^2 x^2\right)^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e}+\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(96*d^5 + 135*d^4*e*x - 192*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 96*d*e^4*x^4 - 40*e^5*x^5) + 105*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e)","A",1
805,1,91,142,0.0744028,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^3,x]","\frac{105 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(136 d^4+15 d^3 e x-112 d^2 e^2 x^2+90 d e^3 x^3-24 e^4 x^4\right)}{120 e}","\frac{7 d^2 \left(d^2-e^2 x^2\right)^{3/2}}{12 e}+\frac{7 d (d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{20 e}+\frac{(d-e x)^2 \left(d^2-e^2 x^2\right)^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}+\frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(136*d^4 + 15*d^3*e*x - 112*d^2*e^2*x^2 + 90*d*e^3*x^3 - 24*e^4*x^4) + 105*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e)","A",1
806,1,80,136,0.0766917,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^4,x]","\frac{105 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(160 d^3-81 d^2 e x+32 d e^2 x^2-6 e^3 x^3\right)}{24 e}","\frac{35}{8} d^2 x \sqrt{d^2-e^2 x^2}+\frac{35 d \left(d^2-e^2 x^2\right)^{3/2}}{12 e}+\frac{2 \left(d^2-e^2 x^2\right)^{7/2}}{e (d+e x)^3}+\frac{7 (d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}",1,"(Sqrt[d^2 - e^2*x^2]*(160*d^3 - 81*d^2*e*x + 32*d*e^2*x^2 - 6*e^3*x^3) + 105*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e)","A",1
807,1,85,132,0.0986097,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^5} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^5,x]","\frac{1}{6} \sqrt{d^2-e^2 x^2} \left(-\frac{96 d^3}{e (d+e x)}-\frac{70 d^2}{e}+15 d x-2 e x^2\right)-\frac{35 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}","-\frac{2 \left(d^2-e^2 x^2\right)^{7/2}}{e (d+e x)^4}-\frac{14 \left(d^2-e^2 x^2\right)^{5/2}}{e (d+e x)^2}-\frac{35 \left(d^2-e^2 x^2\right)^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"(Sqrt[d^2 - e^2*x^2]*((-70*d^2)/e + 15*d*x - 2*e*x^2 - (96*d^3)/(e*(d + e*x))))/6 - (35*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)","A",1
808,1,87,145,0.1045086,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^6} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^6,x]","\frac{105 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(164 d^3+229 d^2 e x+30 d e^2 x^2-3 e^3 x^3\right)}{(d+e x)^2}}{6 e}","-\frac{2 \left(d^2-e^2 x^2\right)^{7/2}}{3 e (d+e x)^5}+\frac{14 \left(d^2-e^2 x^2\right)^{5/2}}{3 e (d+e x)^3}+\frac{35 \left(d^2-e^2 x^2\right)^{3/2}}{6 e (d+e x)}+\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"((Sqrt[d^2 - e^2*x^2]*(164*d^3 + 229*d^2*e*x + 30*d*e^2*x^2 - 3*e^3*x^3))/(d + e*x)^2 + 105*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e)","A",1
809,1,87,138,0.0875283,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^7} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^7,x]","-\frac{7 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}-\frac{\sqrt{d^2-e^2 x^2} \left(167 d^3+381 d^2 e x+277 d e^2 x^2+15 e^3 x^3\right)}{15 e (d+e x)^3}","-\frac{2 \left(d^2-e^2 x^2\right)^{7/2}}{5 e (d+e x)^6}+\frac{14 \left(d^2-e^2 x^2\right)^{5/2}}{15 e (d+e x)^4}-\frac{14 \left(d^2-e^2 x^2\right)^{3/2}}{3 e (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(167*d^3 + 381*d^2*e*x + 277*d*e^2*x^2 + 15*e^3*x^3))/(e*(d + e*x)^3) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e","A",1
810,1,85,143,0.0941387,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^8} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]","\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}+\frac{8 \sqrt{d^2-e^2 x^2} \left(19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right)}{105 e (d+e x)^4}","-\frac{2 \left(d^2-e^2 x^2\right)^{7/2}}{7 e (d+e x)^7}+\frac{2 \left(d^2-e^2 x^2\right)^{5/2}}{5 e (d+e x)^5}-\frac{2 \left(d^2-e^2 x^2\right)^{3/2}}{3 e (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}+\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/(105*e*(d + e*x)^4) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e","A",1
811,1,41,33,0.0643757,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^9} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]","-\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}","-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{9 d e (d+e x)^9}",1,"-1/9*((d - e*x)^4*Sqrt[d^2 - e^2*x^2])/(d*e*(d + e*x)^5)","A",1
812,1,48,67,0.0687296,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^{10}} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]","-\frac{(d-e x)^4 (10 d+e x) \sqrt{d^2-e^2 x^2}}{99 d^2 e (d+e x)^6}","-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{99 d^2 e (d+e x)^9}-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{11 d e (d+e x)^{10}}",1,"-1/99*((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d^2*e*(d + e*x)^6)","A",1
813,1,60,100,0.0723677,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^{11}} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]","-\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left(119 d^2+22 d e x+2 e^2 x^2\right)}{1287 d^3 e (d+e x)^7}","-\frac{2 \left(d^2-e^2 x^2\right)^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{13 d e (d+e x)^{11}}-\frac{2 \left(d^2-e^2 x^2\right)^{9/2}}{1287 d^3 e (d+e x)^9}",1,"-1/1287*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(119*d^2 + 22*d*e*x + 2*e^2*x^2))/(d^3*e*(d + e*x)^7)","A",1
814,1,71,133,0.0760186,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^{12}} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]","-\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left(548 d^3+141 d^2 e x+24 d e^2 x^2+2 e^3 x^3\right)}{6435 d^4 e (d+e x)^8}","-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{15 d e (d+e x)^{12}}-\frac{2 \left(d^2-e^2 x^2\right)^{9/2}}{6435 d^4 e (d+e x)^9}-\frac{2 \left(d^2-e^2 x^2\right)^{9/2}}{715 d^3 e (d+e x)^{10}}",1,"-1/6435*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(548*d^3 + 141*d^2*e*x + 24*d*e^2*x^2 + 2*e^3*x^3))/(d^4*e*(d + e*x)^8)","A",1
815,1,82,166,0.0805437,"\int \frac{\left(d^2-e^2 x^2\right)^{7/2}}{(d+e x)^{13}} \, dx","Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]","-\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left(8627 d^4+2756 d^3 e x+660 d^2 e^2 x^2+104 d e^3 x^3+8 e^4 x^4\right)}{109395 d^5 e (d+e x)^9}","-\frac{4 \left(d^2-e^2 x^2\right)^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{\left(d^2-e^2 x^2\right)^{9/2}}{17 d e (d+e x)^{13}}-\frac{8 \left(d^2-e^2 x^2\right)^{9/2}}{109395 d^5 e (d+e x)^9}-\frac{8 \left(d^2-e^2 x^2\right)^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{4 \left(d^2-e^2 x^2\right)^{9/2}}{1105 d^3 e (d+e x)^{11}}",1,"-1/109395*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(8627*d^4 + 2756*d^3*e*x + 660*d^2*e^2*x^2 + 104*d*e^3*x^3 + 8*e^4*x^4))/(d^5*e*(d + e*x)^9)","A",1
816,1,47,47,0.0415465,"\int \frac{\sqrt{a^2-b^2 x^2}}{a-b x} \, dx","Integrate[Sqrt[a^2 - b^2*x^2]/(a - b*x),x]","\frac{a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}-\frac{\sqrt{a^2-b^2 x^2}}{b}","\frac{a \tan ^{-1}\left(\frac{b x}{\sqrt{a^2-b^2 x^2}}\right)}{b}-\frac{\sqrt{a^2-b^2 x^2}}{b}",1,"-(Sqrt[a^2 - b^2*x^2]/b) + (a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b","A",1
817,1,103,130,0.125787,"\int (a+b x)^2 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx","Integrate[(a + b*x)^2*Sqrt[-((a^2*c)/b^2) + c*x^2],x]","\frac{\sqrt{c \left(x^2-\frac{a^2}{b^2}\right)} \left(15 a^3 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right)\right)}{24 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{5}{8} a^2 x \sqrt{c x^2-\frac{a^2 c}{b^2}}+\frac{5 a b \left(c x^2-\frac{a^2 c}{b^2}\right)^{3/2}}{12 c}+\frac{b (a+b x) \left(c x^2-\frac{a^2 c}{b^2}\right)^{3/2}}{4 c}-\frac{5 a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right)}{8 b^2}",1,"(Sqrt[c*(-(a^2/b^2) + x^2)]*(Sqrt[1 - (b^2*x^2)/a^2]*(-16*a^3 + 9*a^2*b*x + 16*a*b^2*x^2 + 6*b^3*x^3) + 15*a^3*ArcSin[(b*x)/a]))/(24*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
818,1,114,167,0.1697954,"\int (a+b x)^3 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx","Integrate[(a + b*x)^3*Sqrt[-((a^2*c)/b^2) + c*x^2],x]","\frac{\sqrt{c \left(x^2-\frac{a^2}{b^2}\right)} \left(105 a^4 \sin ^{-1}\left(\frac{b x}{a}\right)+\sqrt{1-\frac{b^2 x^2}{a^2}} \left(-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right)\right)}{120 b \sqrt{1-\frac{b^2 x^2}{a^2}}}","\frac{7 a^2 b \left(c x^2-\frac{a^2 c}{b^2}\right)^{3/2}}{12 c}+\frac{7 a b (a+b x) \left(c x^2-\frac{a^2 c}{b^2}\right)^{3/2}}{20 c}+\frac{b (a+b x)^2 \left(c x^2-\frac{a^2 c}{b^2}\right)^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right)}{8 b^2}+\frac{7}{8} a^3 x \sqrt{c x^2-\frac{a^2 c}{b^2}}",1,"(Sqrt[c*(-(a^2/b^2) + x^2)]*(Sqrt[1 - (b^2*x^2)/a^2]*(-136*a^4 + 15*a^3*b*x + 112*a^2*b^2*x^2 + 90*a*b^3*x^3 + 24*b^4*x^4) + 105*a^4*ArcSin[(b*x)/a]))/(120*b*Sqrt[1 - (b^2*x^2)/a^2])","A",1
819,1,49,44,0.0283003,"\int (1+x) \sqrt{-1+x^2} \, dx","Integrate[(1 + x)*Sqrt[-1 + x^2],x]","\frac{\left(x^2-1\right) \left(\sqrt{1-x^2} \left(2 x^2+3 x-2\right)+3 \sin ^{-1}(x)\right)}{6 \sqrt{-\left(x^2-1\right)^2}}","\frac{1}{3} \left(x^2-1\right)^{3/2}+\frac{1}{2} x \sqrt{x^2-1}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2-1}}\right)",1,"((-1 + x^2)*(Sqrt[1 - x^2]*(-2 + 3*x + 2*x^2) + 3*ArcSin[x]))/(6*Sqrt[-(-1 + x^2)^2])","A",1
820,1,31,38,0.0085079,"\int (1+x) \sqrt{1-x^2} \, dx","Integrate[(1 + x)*Sqrt[1 - x^2],x]","\frac{1}{6} \left(\sqrt{1-x^2} \left(2 x^2+3 x-2\right)+3 \sin ^{-1}(x)\right)","-\frac{1}{3} \left(1-x^2\right)^{3/2}+\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x)",1,"(Sqrt[1 - x^2]*(-2 + 3*x + 2*x^2) + 3*ArcSin[x])/6","A",1
821,1,30,14,0.0515559,"\int \frac{\sqrt{1-x^2}}{1+x} \, dx","Integrate[Sqrt[1 - x^2]/(1 + x),x]","\sqrt{1-x^2}-2 \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)","\sqrt{1-x^2}+\sin ^{-1}(x)",1,"Sqrt[1 - x^2] - 2*ArcSin[Sqrt[1 - x]/Sqrt[2]]","B",1
822,1,32,38,0.0145104,"\int (1-x) \sqrt{1-x^2} \, dx","Integrate[(1 - x)*Sqrt[1 - x^2],x]","\frac{1}{6} \left(3 \sin ^{-1}(x)-\sqrt{1-x^2} \left(2 x^2-3 x-2\right)\right)","\frac{1}{3} \left(1-x^2\right)^{3/2}+\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x)",1,"(-(Sqrt[1 - x^2]*(-2 - 3*x + 2*x^2)) + 3*ArcSin[x])/6","A",1
823,1,44,16,0.0496285,"\int \frac{\sqrt{1-x^2}}{1-x} \, dx","Integrate[Sqrt[1 - x^2]/(1 - x),x]","\frac{x^2+2 \sqrt{1-x^2} \sin ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{2}}\right)-1}{\sqrt{1-x^2}}","\sin ^{-1}(x)-\sqrt{1-x^2}",1,"(-1 + x^2 + 2*Sqrt[1 - x^2]*ArcSin[Sqrt[1 + x]/Sqrt[2]])/Sqrt[1 - x^2]","B",1
824,1,50,25,0.0552901,"\int \frac{\sqrt{1-x^2}}{(1-x)^2} \, dx","Integrate[Sqrt[1 - x^2]/(1 - x)^2,x]","2 \sqrt{1-x^2} \left(\frac{1}{1-x}+\frac{\sinh ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{2}}\right)}{\sqrt{x-1} \sqrt{x+1}}\right)","\frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x)",1,"2*Sqrt[1 - x^2]*((1 - x)^(-1) + ArcSinh[Sqrt[-1 + x]/Sqrt[2]]/(Sqrt[-1 + x]*Sqrt[1 + x]))","A",1
825,1,23,22,0.0171587,"\int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx","Integrate[Sqrt[1 - x^2]/(1 - x)^3,x]","\frac{(x+1) \sqrt{1-x^2}}{3 (x-1)^2}","\frac{\left(1-x^2\right)^{3/2}}{3 (1-x)^3}",1,"((1 + x)*Sqrt[1 - x^2])/(3*(-1 + x)^2)","A",1
826,1,92,182,0.1162468,"\int \frac{(d+e x)^5}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^5/Sqrt[d^2 - e^2*x^2],x]","\frac{315 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(488 d^4+275 d^3 e x+144 d^2 e^2 x^2+50 d e^3 x^3+8 e^4 x^4\right)}{40 e}","-\frac{21 d^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{9 d (d+e x)^3 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{(d+e x)^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{63 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}-\frac{63 d^4 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{21 d^3 (d+e x) \sqrt{d^2-e^2 x^2}}{8 e}",1,"(-(Sqrt[d^2 - e^2*x^2]*(488*d^4 + 275*d^3*e*x + 144*d^2*e^2*x^2 + 50*d*e^3*x^3 + 8*e^4*x^4)) + 315*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40*e)","A",1
827,1,81,149,0.0795016,"\int \frac{(d+e x)^4}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^4/Sqrt[d^2 - e^2*x^2],x]","\frac{105 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(160 d^3+81 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)}{24 e}","-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}",1,"(-(Sqrt[d^2 - e^2*x^2]*(160*d^3 + 81*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 105*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e)","A",1
828,1,70,116,0.0597491,"\int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[d^2 - e^2*x^2],x]","\frac{15 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(22 d^2+9 d e x+2 e^2 x^2\right)}{6 e}","-\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"(-(Sqrt[d^2 - e^2*x^2]*(22*d^2 + 9*d*e*x + 2*e^2*x^2)) + 15*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e)","A",1
829,1,58,74,0.0363314,"\int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[d^2 - e^2*x^2],x]","\frac{3 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-(4 d+e x) \sqrt{d^2-e^2 x^2}}{2 e}","-\frac{2 d \sqrt{d^2-e^2 x^2}}{e}-\frac{1}{2} x \sqrt{d^2-e^2 x^2}+\frac{3 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"(-((4*d + e*x)*Sqrt[d^2 - e^2*x^2]) + 3*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)","A",1
830,1,47,47,0.0157055,"\int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)/Sqrt[d^2 - e^2*x^2],x]","\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e}","\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e}",1,"-(Sqrt[d^2 - e^2*x^2]/e) + (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e","A",1
831,1,32,31,0.0060775,"\int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2}}{d^2 e+d e^2 x}","-\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)}",1,"-(Sqrt[d^2 - e^2*x^2]/(d^2*e + d*e^2*x))","A",1
832,1,40,67,0.0400251,"\int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[d^2 - e^2*x^2]),x]","-\frac{(2 d+e x) \sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)^2}","-\frac{\sqrt{d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{3 d e (d+e x)^2}",1,"-1/3*((2*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d^2*e*(d + e*x)^2)","A",1
833,1,52,100,0.0383624,"\int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(7 d^2+6 d e x+2 e^2 x^2\right)}{15 d^3 e (d+e x)^3}","-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(7*d^2 + 6*d*e*x + 2*e^2*x^2))/(d^3*e*(d + e*x)^3)","A",1
834,1,63,133,0.040777,"\int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(12 d^3+13 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right)}{35 d^4 e (d+e x)^4}","-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}",1,"-1/35*(Sqrt[d^2 - e^2*x^2]*(12*d^3 + 13*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3))/(d^4*e*(d + e*x)^4)","A",1
835,1,74,166,0.0460068,"\int \frac{1}{(d+e x)^5 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(83 d^4+100 d^3 e x+84 d^2 e^2 x^2+40 d e^3 x^3+8 e^4 x^4\right)}{315 d^5 e (d+e x)^5}","-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}",1,"-1/315*(Sqrt[d^2 - e^2*x^2]*(83*d^4 + 100*d^3*e*x + 84*d^2*e^2*x^2 + 40*d*e^3*x^3 + 8*e^4*x^4))/(d^5*e*(d + e*x)^5)","A",1
836,1,121,143,0.2085765,"\int \frac{(d+e x)^6}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(5/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(164 d^3-229 d^2 e x+30 d e^2 x^2+3 e^3 x^3\right)-105 d (d-e x)^2 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{6 e (e x-d) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^5}{3 e \left(d^2-e^2 x^2\right)^{3/2}}-\frac{14 (d+e x)^3}{3 e \sqrt{d^2-e^2 x^2}}-\frac{35 \sqrt{d^2-e^2 x^2} (d+e x)}{6 e}-\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(164*d^3 - 229*d^2*e*x + 30*d*e^2*x^2 + 3*e^3*x^3) - 105*d*(d - e*x)^2*ArcSin[(e*x)/d]))/(6*e*(-d + e*x)*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
837,1,109,108,0.1568681,"\int \frac{(d+e x)^5}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(5/2),x]","\frac{(d+e x) \left(\left(23 d^2-34 d e x+3 e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}}-15 (d-e x)^2 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{3 e (e x-d) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^4}{3 e \left(d^2-e^2 x^2\right)^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"((d + e*x)*((23*d^2 - 34*d*e*x + 3*e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2] - 15*(d - e*x)^2*ArcSin[(e*x)/d]))/(3*e*(-d + e*x)*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
838,1,100,81,0.1225724,"\int \frac{(d+e x)^4}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x]","-\frac{(d+e x) \left(4 d (d-2 e x) \sqrt{1-\frac{e^2 x^2}{d^2}}-3 (d-e x)^2 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{3 d e (d-e x) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^3}{3 e \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"-1/3*((d + e*x)*(4*d*(d - 2*e*x)*Sqrt[1 - (e^2*x^2)/d^2] - 3*(d - e*x)^2*ArcSin[(e*x)/d]))/(d*e*(d - e*x)*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
839,1,41,33,0.0445778,"\int \frac{(d+e x)^3}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(5/2),x]","\frac{(d+e x)^2}{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^3}{3 d e \left(d^2-e^2 x^2\right)^{3/2}}",1,"(d + e*x)^2/(3*d*e*(d - e*x)*Sqrt[d^2 - e^2*x^2])","A",1
840,1,47,53,0.0285406,"\int \frac{(d+e x)^2}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(5/2),x]","\frac{(2 d-e x) (d+e x)}{3 d^2 e (d-e x) \sqrt{d^2-e^2 x^2}}","\frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left(d^2-e^2 x^2\right)^{3/2}}",1,"((2*d - e*x)*(d + e*x))/(3*d^2*e*(d - e*x)*Sqrt[d^2 - e^2*x^2])","A",1
841,1,51,56,0.0228345,"\int \frac{d+e x}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(d^2 - e^2*x^2)^(5/2),x]","\frac{d^2+2 d e x-2 e^2 x^2}{3 d^3 e (d-e x) \sqrt{d^2-e^2 x^2}}","\frac{d+e x}{3 d e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}",1,"(d^2 + 2*d*e*x - 2*e^2*x^2)/(3*d^3*e*(d - e*x)*Sqrt[d^2 - e^2*x^2])","A",1
842,1,82,82,0.0625998,"\int \frac{1}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(3 d^4-12 d^3 e x-12 d^2 e^2 x^2+8 d e^3 x^3+8 e^4 x^4\right)}{15 d^5 e (d-e x)^2 (d+e x)^3}","-\frac{1}{5 d e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(3*d^4 - 12*d^3*e*x - 12*d^2*e^2*x^2 + 8*d*e^3*x^3 + 8*e^4*x^4))/(d^5*e*(d - e*x)^2*(d + e*x)^3)","A",1
843,1,93,115,0.0625884,"\int \frac{1}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(5/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right)}{21 d^6 e (d-e x)^2 (d+e x)^4}","-\frac{1}{7 d^2 e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-6*d^5 + 9*d^4*e*x + 24*d^3*e^2*x^2 + 4*d^2*e^3*x^3 - 16*d*e^4*x^4 - 8*e^5*x^5))/(21*d^6*e*(d - e*x)^2*(d + e*x)^4)","A",1
844,1,104,148,0.0683371,"\int \frac{1}{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(19 d^6-6 d^5 e x-66 d^4 e^2 x^2-56 d^3 e^3 x^3+24 d^2 e^4 x^4+48 d e^5 x^5+16 e^6 x^6\right)}{63 d^7 e (d-e x)^2 (d+e x)^5}","-\frac{2}{21 d^2 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{63 d^5 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/63*(Sqrt[d^2 - e^2*x^2]*(19*d^6 - 6*d^5*e*x - 66*d^4*e^2*x^2 - 56*d^3*e^3*x^3 + 24*d^2*e^4*x^4 + 48*d*e^5*x^5 + 16*e^6*x^6))/(d^7*e*(d - e*x)^2*(d + e*x)^5)","A",1
845,1,115,181,0.0683077,"\int \frac{1}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(28 d^7+13 d^6 e x-72 d^5 e^2 x^2-122 d^4 e^3 x^3-32 d^3 e^4 x^4+72 d^2 e^5 x^5+64 d e^6 x^6+16 e^7 x^7\right)}{99 d^8 e (d-e x)^2 (d+e x)^6}","-\frac{7}{99 d^2 e (d+e x)^3 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{99 d^6 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/99*(Sqrt[d^2 - e^2*x^2]*(28*d^7 + 13*d^6*e*x - 72*d^5*e^2*x^2 - 122*d^4*e^3*x^3 - 32*d^3*e^4*x^4 + 72*d^2*e^5*x^5 + 64*d*e^6*x^6 + 16*e^7*x^7))/(d^8*e*(d - e*x)^2*(d + e*x)^6)","A",1
846,1,144,206,0.3426218,"\int \frac{(d+e x)^9}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(5446 d^5-12843 d^4 e x+8711 d^3 e^2 x^2-815 d^2 e^3 x^3-105 d e^4 x^4-10 e^5 x^5\right)-3465 d^2 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{30 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^8}{5 e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{22 (d+e x)^6}{15 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(5446*d^5 - 12843*d^4*e*x + 8711*d^3*e^2*x^2 - 815*d^2*e^3*x^3 - 105*d*e^4*x^4 - 10*e^5*x^5) - 3465*d^2*(d - e*x)^3*ArcSin[(e*x)/d]))/(30*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
847,1,131,173,0.2790283,"\int \frac{(d+e x)^8}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(496 d^4-1163 d^3 e x+801 d^2 e^2 x^2-65 d e^3 x^3-5 e^4 x^4\right)-315 d (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{10 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^7}{5 e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{6 (d+e x)^5}{5 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(496*d^4 - 1163*d^3*e*x + 801*d^2*e^2*x^2 - 65*d*e^3*x^3 - 5*e^4*x^4) - 315*d*(d - e*x)^3*ArcSin[(e*x)/d]))/(10*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
848,1,119,138,0.2337833,"\int \frac{(d+e x)^7}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^7/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(167 d^3-381 d^2 e x+277 d e^2 x^2-15 e^3 x^3\right)-105 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{15 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^6}{5 e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{14 (d+e x)^4}{15 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(167*d^3 - 381*d^2*e*x + 277*d*e^2*x^2 - 15*e^3*x^3) - 105*(d - e*x)^3*ArcSin[(e*x)/d]))/(15*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
849,1,113,112,0.186331,"\int \frac{(d+e x)^6}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(2 d \left(13 d^2-24 d e x+23 e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}}-15 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{15 d e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{2 (d+e x)^5}{5 e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 (d+e x)^3}{3 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"((d + e*x)*(2*d*(13*d^2 - 24*d*e*x + 23*e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2] - 15*(d - e*x)^3*ArcSin[(e*x)/d]))/(15*d*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
850,1,41,33,0.0911314,"\int \frac{(d+e x)^5}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x)^3}{5 d e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^5}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}",1,"(d + e*x)^3/(5*d*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
851,1,49,67,0.0672332,"\int \frac{(d+e x)^4}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(7/2),x]","\frac{(4 d-e x) (d+e x)^2}{15 d^2 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^4}{3 d e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left(d^2-e^2 x^2\right)^{5/2}}",1,"((4*d - e*x)*(d + e*x)^2)/(15*d^2*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
852,1,58,103,0.0600048,"\int \frac{(d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(7 d^2-6 d e x+2 e^2 x^2\right)}{15 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}",1,"((d + e*x)*(7*d^2 - 6*d*e*x + 2*e^2*x^2))/(15*d^3*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
853,1,63,77,0.0450169,"\int \frac{(d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(7/2),x]","\frac{2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3}{5 d^4 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{x}{5 d^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 (d+e x)}{5 e \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}",1,"(2*d^3 + d^2*e*x - 4*d*e^2*x^2 + 2*e^3*x^3)/(5*d^4*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
854,1,82,80,0.0308329,"\int \frac{d+e x}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(d^2 - e^2*x^2)^(7/2),x]","\frac{3 d^4+12 d^3 e x-12 d^2 e^2 x^2-8 d e^3 x^3+8 e^4 x^4}{15 d^5 e (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{d+e x}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(3*d^4 + 12*d^3*e*x - 12*d^2*e^2*x^2 - 8*d*e^3*x^3 + 8*e^4*x^4)/(15*d^5*e*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
855,1,104,106,0.0785335,"\int \frac{1}{(d+e x) \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-5 d^6+30 d^5 e x+30 d^4 e^2 x^2-40 d^3 e^3 x^3-40 d^2 e^4 x^4+16 d e^5 x^5+16 e^6 x^6\right)}{35 d^7 e (d-e x)^3 (d+e x)^4}","-\frac{1}{7 d e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}+\frac{16 x}{35 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{35 d^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{6 x}{35 d^3 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-5*d^6 + 30*d^5*e*x + 30*d^4*e^2*x^2 - 40*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 16*d*e^5*x^5 + 16*e^6*x^6))/(35*d^7*e*(d - e*x)^3*(d + e*x)^4)","A",1
856,1,115,139,0.07324,"\int \frac{1}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-10 d^7+25 d^6 e x+60 d^5 e^2 x^2-10 d^4 e^3 x^3-80 d^3 e^4 x^4-24 d^2 e^5 x^5+32 d e^6 x^6+16 e^7 x^7\right)}{45 d^8 e (d-e x)^3 (d+e x)^5}","-\frac{1}{9 d^2 e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{45 d^6 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 x}{15 d^4 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-10*d^7 + 25*d^6*e*x + 60*d^5*e^2*x^2 - 10*d^4*e^3*x^3 - 80*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 32*d*e^6*x^6 + 16*e^7*x^7))/(45*d^8*e*(d - e*x)^3*(d + e*x)^5)","A",1
857,1,126,172,0.0790703,"\int \frac{1}{(d+e x)^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-125 d^8+120 d^7 e x+680 d^6 e^2 x^2+400 d^5 e^3 x^3-720 d^4 e^4 x^4-832 d^3 e^5 x^5+64 d^2 e^6 x^6+384 d e^7 x^7+128 e^8 x^8\right)}{495 d^9 e (d-e x)^3 (d+e x)^6}","-\frac{8}{99 d^2 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{16 x}{165 d^5 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-125*d^8 + 120*d^7*e*x + 680*d^6*e^2*x^2 + 400*d^5*e^3*x^3 - 720*d^4*e^4*x^4 - 832*d^3*e^5*x^5 + 64*d^2*e^6*x^6 + 384*d*e^7*x^7 + 128*e^8*x^8))/(495*d^9*e*(d - e*x)^3*(d + e*x)^6)","A",1
858,1,137,205,0.0854634,"\int \frac{1}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right)}{715 d^{10} e (d-e x)^3 (d+e x)^7}","-\frac{9}{143 d^2 e (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{48 x}{715 d^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7)","A",1
859,1,148,238,0.0895853,"\int \frac{1}{(d+e x)^5 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-503 d^{10}-370 d^9 e x+1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8+1280 d e^9 x^9+256 e^{10} x^{10}\right)}{2145 d^{11} e (d-e x)^3 (d+e x)^8}","-\frac{2}{39 d^2 e (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{128 x}{2145 d^9 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{32 x}{715 d^7 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-503*d^10 - 370*d^9*e*x + 1590*d^8*e^2*x^2 + 3760*d^7*e^3*x^3 + 1520*d^6*e^4*x^4 - 3744*d^5*e^5*x^5 - 4640*d^4*e^6*x^6 - 640*d^3*e^7*x^7 + 1920*d^2*e^8*x^8 + 1280*d*e^9*x^9 + 256*e^10*x^10))/(2145*d^11*e*(d - e*x)^3*(d + e*x)^8)","A",1
860,1,16,16,0.0061452,"\int \frac{1+x}{\sqrt{1-x^2}} \, dx","Integrate[(1 + x)/Sqrt[1 - x^2],x]","\sin ^{-1}(x)-\sqrt{1-x^2}","\sin ^{-1}(x)-\sqrt{1-x^2}",1,"-Sqrt[1 - x^2] + ArcSin[x]","A",1
861,1,14,14,0.0057945,"\int \frac{1-x}{\sqrt{1-x^2}} \, dx","Integrate[(1 - x)/Sqrt[1 - x^2],x]","\sqrt{1-x^2}+\sin ^{-1}(x)","\sqrt{1-x^2}+\sin ^{-1}(x)",1,"Sqrt[1 - x^2] + ArcSin[x]","A",1
862,1,75,160,0.0601229,"\int (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2} \, dx","Integrate[(d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 \left(319 d^4+2 d^3 e x-156 d^2 e^2 x^2-130 d e^3 x^3-35 e^4 x^4\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{315 e \sqrt{d+e x}}","-\frac{64 d^2 \left(c d^2-c e^2 x^2\right)^{3/2}}{105 c e \sqrt{d+e x}}-\frac{8 d \sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{3/2}}{21 c e}-\frac{2 (d+e x)^{3/2} \left(c d^2-c e^2 x^2\right)^{3/2}}{9 c e}-\frac{256 d^3 \left(c d^2-c e^2 x^2\right)^{3/2}}{315 c e (d+e x)^{3/2}}",1,"(-2*Sqrt[c*(d^2 - e^2*x^2)]*(319*d^4 + 2*d^3*e*x - 156*d^2*e^2*x^2 - 130*d*e^3*x^3 - 35*e^4*x^4))/(315*e*Sqrt[d + e*x])","A",1
863,1,64,119,0.0520469,"\int (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2],x]","\frac{2 \left(-71 d^3+17 d^2 e x+39 d e^2 x^2+15 e^3 x^3\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{105 e \sqrt{d+e x}}","-\frac{64 d^2 \left(c d^2-c e^2 x^2\right)^{3/2}}{105 c e (d+e x)^{3/2}}-\frac{16 d \left(c d^2-c e^2 x^2\right)^{3/2}}{35 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{3/2}}{7 c e}",1,"(2*Sqrt[c*(d^2 - e^2*x^2)]*(-71*d^3 + 17*d^2*e*x + 39*d*e^2*x^2 + 15*e^3*x^3))/(105*e*Sqrt[d + e*x])","A",1
864,1,53,78,0.0425896,"\int \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 \left(7 d^2-4 d e x-3 e^2 x^2\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{15 e \sqrt{d+e x}}","-\frac{2 \left(c d^2-c e^2 x^2\right)^{3/2}}{5 c e \sqrt{d+e x}}-\frac{8 d \left(c d^2-c e^2 x^2\right)^{3/2}}{15 c e (d+e x)^{3/2}}",1,"(-2*(7*d^2 - 4*d*e*x - 3*e^2*x^2)*Sqrt[c*(d^2 - e^2*x^2)])/(15*e*Sqrt[d + e*x])","A",1
865,1,40,38,0.0370313,"\int \frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[c*d^2 - c*e^2*x^2]/Sqrt[d + e*x],x]","-\frac{2 (d-e x) \sqrt{c \left(d^2-e^2 x^2\right)}}{3 e \sqrt{d+e x}}","-\frac{2 \left(c d^2-c e^2 x^2\right)^{3/2}}{3 c e (d+e x)^{3/2}}",1,"(-2*(d - e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(3*e*Sqrt[d + e*x])","A",1
866,1,98,99,0.0922405,"\int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(3/2),x]","\frac{2 \sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{1}{\sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d^2-e^2 x^2}}\right)}{e}","\frac{2 \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{2} \sqrt{c} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{e}",1,"(2*Sqrt[c*(d^2 - e^2*x^2)]*(1/Sqrt[d + e*x] - (Sqrt[2]*Sqrt[d]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/Sqrt[d^2 - e^2*x^2]))/e","A",1
867,1,101,98,0.1127031,"\int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(5/2),x]","\frac{\sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d} \sqrt{d^2-e^2 x^2}}-\frac{2}{(d+e x)^{3/2}}\right)}{2 e}","\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{2} \sqrt{d} e}-\frac{\sqrt{c d^2-c e^2 x^2}}{e (d+e x)^{3/2}}",1,"(Sqrt[c*(d^2 - e^2*x^2)]*(-2/(d + e*x)^(3/2) + (Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[d]*Sqrt[d^2 - e^2*x^2])))/(2*e)","A",1
868,1,111,141,0.1434708,"\int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(7/2),x]","\frac{\sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{d^{3/2} \sqrt{d^2-e^2 x^2}}+\frac{2 e x-6 d}{d (d+e x)^{5/2}}\right)}{16 e}","\frac{\sqrt{c d^2-c e^2 x^2}}{8 d e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{2 e (d+e x)^{5/2}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{8 \sqrt{2} d^{3/2} e}",1,"(Sqrt[c*(d^2 - e^2*x^2)]*((-6*d + 2*e*x)/(d*(d + e*x)^(5/2)) + (Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(d^(3/2)*Sqrt[d^2 - e^2*x^2])))/(16*e)","A",1
869,1,84,201,0.070385,"\int (d+e x)^{5/2} \left(c d^2-c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(5/2)*(c*d^2 - c*e^2*x^2)^(3/2),x]","-\frac{2 c (d-e x)^2 \left(9683 d^4+16700 d^3 e x+14210 d^2 e^2 x^2+6300 d e^3 x^3+1155 e^4 x^4\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{15015 e \sqrt{d+e x}}","-\frac{128 d^2 \left(c d^2-c e^2 x^2\right)^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left(c d^2-c e^2 x^2\right)^{5/2}}{13 c e}-\frac{4096 d^4 \left(c d^2-c e^2 x^2\right)^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left(c d^2-c e^2 x^2\right)^{5/2}}{3003 c e (d+e x)^{3/2}}",1,"(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(9683*d^4 + 16700*d^3*e*x + 14210*d^2*e^2*x^2 + 6300*d*e^3*x^3 + 1155*e^4*x^4))/(15015*e*Sqrt[d + e*x])","A",1
870,1,73,160,0.0583452,"\int (d+e x)^{3/2} \left(c d^2-c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(3/2),x]","-\frac{2 c (d-e x)^2 \left(533 d^3+755 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{1155 e \sqrt{d+e x}}","-\frac{64 d^2 \left(c d^2-c e^2 x^2\right)^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left(c d^2-c e^2 x^2\right)^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{5/2}}{11 c e}-\frac{256 d^3 \left(c d^2-c e^2 x^2\right)^{5/2}}{1155 c e (d+e x)^{5/2}}",1,"(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(533*d^3 + 755*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3))/(1155*e*Sqrt[d + e*x])","A",1
871,1,62,119,0.0504363,"\int \sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2),x]","-\frac{2 c (d-e x)^2 \left(107 d^2+110 d e x+35 e^2 x^2\right) \sqrt{c \left(d^2-e^2 x^2\right)}}{315 e \sqrt{d+e x}}","-\frac{2 \left(c d^2-c e^2 x^2\right)^{5/2}}{9 c e \sqrt{d+e x}}-\frac{16 d \left(c d^2-c e^2 x^2\right)^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{64 d^2 \left(c d^2-c e^2 x^2\right)^{5/2}}{315 c e (d+e x)^{5/2}}",1,"(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(107*d^2 + 110*d*e*x + 35*e^2*x^2))/(315*e*Sqrt[d + e*x])","A",1
872,1,51,78,0.047924,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/Sqrt[d + e*x],x]","-\frac{2 c (d-e x)^2 (9 d+5 e x) \sqrt{c \left(d^2-e^2 x^2\right)}}{35 e \sqrt{d+e x}}","-\frac{2 \left(c d^2-c e^2 x^2\right)^{5/2}}{7 c e (d+e x)^{3/2}}-\frac{8 d \left(c d^2-c e^2 x^2\right)^{5/2}}{35 c e (d+e x)^{5/2}}",1,"(-2*c*(d - e*x)^2*(9*d + 5*e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(35*e*Sqrt[d + e*x])","A",1
873,1,43,38,0.0492716,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(3/2),x]","-\frac{2 c (d-e x)^2 \sqrt{c \left(d^2-e^2 x^2\right)}}{5 e \sqrt{d+e x}}","-\frac{2 \left(c d^2-c e^2 x^2\right)^{5/2}}{5 c e (d+e x)^{5/2}}",1,"(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])/(5*e*Sqrt[d + e*x])","A",1
874,1,110,136,0.2129856,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(5/2),x]","\frac{2 c \sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{7 d-e x}{\sqrt{d+e x}}-\frac{6 \sqrt{2} d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d^2-e^2 x^2}}\right)}{3 e}","-\frac{4 \sqrt{2} c^{3/2} d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{e}+\frac{4 c d \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+\frac{2 \left(c d^2-c e^2 x^2\right)^{3/2}}{3 e (d+e x)^{3/2}}",1,"(2*c*Sqrt[c*(d^2 - e^2*x^2)]*((7*d - e*x)/Sqrt[d + e*x] - (6*Sqrt[2]*d^(3/2)*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/Sqrt[d^2 - e^2*x^2]))/(3*e)","A",1
875,1,107,133,0.194262,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]","\frac{c \sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{3 \sqrt{2} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d^2-e^2 x^2}}-\frac{2 (2 d+e x)}{(d+e x)^{3/2}}\right)}{e}","\frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{e (d+e x)^{5/2}}",1,"(c*Sqrt[c*(d^2 - e^2*x^2)]*((-2*(2*d + e*x))/(d + e*x)^(3/2) + (3*Sqrt[2]*Sqrt[d]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/Sqrt[d^2 - e^2*x^2]))/e","A",1
876,1,109,139,0.1666535,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(9/2),x]","\frac{c \sqrt{c \left(d^2-e^2 x^2\right)} \left(\frac{2 (d+5 e x)}{(d+e x)^{5/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d} \sqrt{d^2-e^2 x^2}}\right)}{8 e}","-\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{2 e (d+e x)^{7/2}}",1,"(c*Sqrt[c*(d^2 - e^2*x^2)]*((2*(d + 5*e*x))/(d + e*x)^(5/2) - (3*Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[d]*Sqrt[d^2 - e^2*x^2])))/(8*e)","A",1
877,1,134,178,0.2365238,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{11/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(11/2),x]","\frac{\left(c \left(d^2-e^2 x^2\right)\right)^{3/2} \left(-\frac{2 \sqrt{d} \left(7 d^2-22 d e x+3 e^2 x^2\right)}{(d-e x) (d+e x)^{9/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\left(d^2-e^2 x^2\right)^{3/2}}\right)}{96 d^{3/2} e}","-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{16 \sqrt{2} d^{3/2} e}-\frac{c \sqrt{c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{3 e (d+e x)^{9/2}}",1,"((c*(d^2 - e^2*x^2))^(3/2)*((-2*Sqrt[d]*(7*d^2 - 22*d*e*x + 3*e^2*x^2))/((d - e*x)*(d + e*x)^(9/2)) - (3*Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(d^2 - e^2*x^2)^(3/2)))/(96*d^(3/2)*e)","A",1
878,1,145,217,0.2647183,"\int \frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{(d+e x)^{13/2}} \, dx","Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(13/2),x]","\frac{\left(c \left(d^2-e^2 x^2\right)\right)^{3/2} \left(-\frac{3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 \sqrt{d} \left(39 d^3-79 d^2 e x+13 d e^2 x^2+3 e^3 x^3\right)}{(d-e x) (d+e x)^{11/2}}\right)}{512 d^{5/2} e}","-\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{256 \sqrt{2} d^{5/2} e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left(c d^2-c e^2 x^2\right)^{3/2}}{4 e (d+e x)^{11/2}}",1,"((c*(d^2 - e^2*x^2))^(3/2)*((-2*Sqrt[d]*(39*d^3 - 79*d^2*e*x + 13*d*e^2*x^2 + 3*e^3*x^3))/((d - e*x)*(d + e*x)^(11/2)) - (3*Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(d^2 - e^2*x^2)^(3/2)))/(512*d^(5/2)*e)","A",1
879,1,70,160,0.0701308,"\int \frac{(d+e x)^{7/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 (d-e x) \sqrt{d+e x} \left(177 d^3+71 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right)}{35 e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}",1,"(-2*(d - e*x)*Sqrt[d + e*x]*(177*d^3 + 71*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
880,1,59,119,0.0565387,"\int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 (d-e x) \sqrt{d+e x} \left(43 d^2+14 d e x+3 e^2 x^2\right)}{15 e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{64 d^2 \sqrt{c d^2-c e^2 x^2}}{15 c e \sqrt{d+e x}}-\frac{16 d \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{15 c e}-\frac{2 (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{5 c e}",1,"(-2*(d - e*x)*Sqrt[d + e*x]*(43*d^2 + 14*d*e*x + 3*e^2*x^2))/(15*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
881,1,47,78,0.0473521,"\int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 (d-e x) \sqrt{d+e x} (5 d+e x)}{3 e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e}",1,"(-2*(d - e*x)*Sqrt[d + e*x]*(5*d + e*x))/(3*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
882,1,35,36,0.0450286,"\int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]","-\frac{2 \sqrt{c \left(d^2-e^2 x^2\right)}}{c e \sqrt{d+e x}}","-\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}}",1,"(-2*Sqrt[c*(d^2 - e^2*x^2)])/(c*e*Sqrt[d + e*x])","A",1
883,1,86,65,0.0445568,"\int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]),x]","-\frac{\sqrt{2} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d} e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{c} \sqrt{d} e}",1,"-((Sqrt[2]*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[d]*e*Sqrt[c*(d^2 - e^2*x^2)]))","A",1
884,1,122,109,0.0783775,"\int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2]),x]","\frac{-\sqrt{2} \sqrt{d+e x} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)-2 \sqrt{d} (d-e x)}{4 d^{3/2} e \sqrt{d+e x} \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{\sqrt{c d^2-c e^2 x^2}}{2 c d e (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{2 \sqrt{2} \sqrt{c} d^{3/2} e}",1,"(-2*Sqrt[d]*(d - e*x) - Sqrt[2]*Sqrt[d + e*x]*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(4*d^(3/2)*e*Sqrt[d + e*x]*Sqrt[c*(d^2 - e^2*x^2)])","A",1
885,1,140,150,0.1047031,"\int \frac{1}{(d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2]),x]","\frac{2 \sqrt{d} \sqrt{d+e x} \left(-7 d^2+4 d e x+3 e^2 x^2\right)-3 \sqrt{2} (d+e x)^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{32 d^{5/2} e (d+e x)^2 \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{16 \sqrt{2} \sqrt{c} d^{5/2} e}",1,"(2*Sqrt[d]*Sqrt[d + e*x]*(-7*d^2 + 4*d*e*x + 3*e^2*x^2) - 3*Sqrt[2]*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(32*d^(5/2)*e*(d + e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])","A",1
886,1,66,160,0.0815833,"\int \frac{(d+e x)^{9/2}}{\left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} \left(-91 d^3+43 d^2 e x+7 d e^2 x^2+e^3 x^3\right)}{5 c e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}}",1,"(-2*Sqrt[d + e*x]*(-91*d^3 + 43*d^2*e*x + 7*d*e^2*x^2 + e^3*x^3))/(5*c*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
887,1,55,119,0.0586541,"\int \frac{(d+e x)^{7/2}}{\left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} \left(-23 d^2+10 d e x+e^2 x^2\right)}{3 c e \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}}",1,"(-2*Sqrt[d + e*x]*(-23*d^2 + 10*d*e*x + e^2*x^2))/(3*c*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
888,1,43,74,0.0536153,"\int \frac{(d+e x)^{5/2}}{\left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]","\frac{2 (3 d-e x) \sqrt{d+e x}}{c e \sqrt{c \left(d^2-e^2 x^2\right)}}","\frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}}",1,"(2*(3*d - e*x)*Sqrt[d + e*x])/(c*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
889,1,35,36,0.0440642,"\int \frac{(d+e x)^{3/2}}{\left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x}}{c e \sqrt{c \left(d^2-e^2 x^2\right)}}","\frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}",1,"(2*Sqrt[d + e*x])/(c*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
890,1,110,104,0.0536493,"\int \frac{\sqrt{d+e x}}{\left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(c*d^2 - c*e^2*x^2)^(3/2),x]","\frac{2 \sqrt{d} \sqrt{d+e x}-\sqrt{2} \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{2 c d^{3/2} e \sqrt{c \left(d^2-e^2 x^2\right)}}","\frac{\sqrt{d+e x}}{c d e \sqrt{c d^2-c e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{2} c^{3/2} d^{3/2} e}",1,"(2*Sqrt[d]*Sqrt[d + e*x] - Sqrt[2]*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(2*c*d^(3/2)*e*Sqrt[c*(d^2 - e^2*x^2)])","A",1
891,1,128,150,0.0865896,"\int \frac{1}{\sqrt{d+e x} \left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2)),x]","\frac{2 \sqrt{d} \sqrt{d+e x} (d+3 e x)-3 \sqrt{2} (d+e x) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{8 c d^{5/2} e (d+e x) \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{4 \sqrt{2} c^{3/2} d^{5/2} e}+\frac{3 \sqrt{d+e x}}{4 c d^2 e \sqrt{c d^2-c e^2 x^2}}-\frac{1}{2 c d e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}",1,"(2*Sqrt[d]*Sqrt[d + e*x]*(d + 3*e*x) - 3*Sqrt[2]*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(8*c*d^(5/2)*e*(d + e*x)*Sqrt[c*(d^2 - e^2*x^2)])","A",1
892,1,143,191,0.1107706,"\int \frac{1}{(d+e x)^{3/2} \left(c d^2-c e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(3/2)),x]","\frac{2 \sqrt{d} \sqrt{d+e x} \left(-3 d^2+20 d e x+15 e^2 x^2\right)-15 \sqrt{2} (d+e x)^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{64 c d^{7/2} e (d+e x)^2 \sqrt{c \left(d^2-e^2 x^2\right)}}","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right)}{32 \sqrt{2} c^{3/2} d^{7/2} e}-\frac{5}{16 c d^2 e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}-\frac{1}{4 c d e (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}+\frac{15 \sqrt{d+e x}}{32 c d^3 e \sqrt{c d^2-c e^2 x^2}}",1,"(2*Sqrt[d]*Sqrt[d + e*x]*(-3*d^2 + 20*d*e*x + 15*e^2*x^2) - 15*Sqrt[2]*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(64*c*d^(7/2)*e*(d + e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])","A",1
893,1,46,31,0.0299897,"\int \frac{1}{\sqrt{-1+x} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[-1 + x]*Sqrt[1 - x^2]),x]","-\frac{\sqrt{2} \sqrt{x-1} \sqrt{x+1} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{2}}\right)}{\sqrt{1-x^2}}","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{1-x^2}}{\sqrt{2} \sqrt{x-1}}\right)",1,"-((Sqrt[2]*Sqrt[-1 + x]*Sqrt[1 + x]*ArcTanh[Sqrt[1 + x]/Sqrt[2]])/Sqrt[1 - x^2])","A",1
894,1,58,87,0.0741233,"\int (2+e x)^{5/2} \sqrt{12-3 e^2 x^2} \, dx","Integrate[(2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{4-e^2 x^2} \left(35 e^3 x^3+330 e^2 x^2+1284 e x+2552\right)}{105 e \sqrt{3 e x+6}}","\frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(2552 + 1284*e*x + 330*e^2*x^2 + 35*e^3*x^3))/(105*e*Sqrt[6 + 3*e*x])","A",1
895,1,50,65,0.0576231,"\int (2+e x)^{3/2} \sqrt{12-3 e^2 x^2} \, dx","Integrate[(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{4-e^2 x^2} \left(15 e^2 x^2+108 e x+284\right)}{35 e \sqrt{3 e x+6}}","-\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(284 + 108*e*x + 15*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])","A",1
896,1,42,43,0.0439707,"\int \sqrt{2+e x} \sqrt{12-3 e^2 x^2} \, dx","Integrate[Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) (3 e x+14) \sqrt{4-e^2 x^2}}{5 e \sqrt{3 e x+6}}","\frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*(14 + 3*e*x)*Sqrt[4 - e^2*x^2])/(5*e*Sqrt[6 + 3*e*x])","A",1
897,1,34,20,0.0365804,"\int \frac{\sqrt{12-3 e^2 x^2}}{\sqrt{2+e x}} \, dx","Integrate[Sqrt[12 - 3*e^2*x^2]/Sqrt[2 + e*x],x]","\frac{2 (e x-2) \sqrt{4-e^2 x^2}}{e \sqrt{3 e x+6}}","-\frac{2 (2-e x)^{3/2}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[4 - e^2*x^2])/(e*Sqrt[6 + 3*e*x])","A",1
898,1,63,46,0.0585902,"\int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx","Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(3/2),x]","\frac{2 \sqrt{12-3 e^2 x^2} \left(\sqrt{e x-2}-2 \tan ^{-1}\left(\frac{1}{2} \sqrt{e x-2}\right)\right)}{e \sqrt{e x-2} \sqrt{e x+2}}","\frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{e}",1,"(2*Sqrt[12 - 3*e^2*x^2]*(Sqrt[-2 + e*x] - 2*ArcTan[Sqrt[-2 + e*x]/2]))/(e*Sqrt[-2 + e*x]*Sqrt[2 + e*x])","A",1
899,1,74,55,0.0898888,"\int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx","Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(5/2),x]","-\frac{\sqrt{12-3 e^2 x^2} \left(2 e x+\sqrt{2-e x} (e x+2) \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)-4\right)}{2 e (e x-2) (e x+2)^{3/2}}","\frac{\sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{2 e}-\frac{\sqrt{3} \sqrt{2-e x}}{e (e x+2)}",1,"-1/2*(Sqrt[12 - 3*e^2*x^2]*(-4 + 2*e*x + Sqrt[2 - e*x]*(2 + e*x)*ArcTanh[Sqrt[2 - e*x]/2]))/(e*(-2 + e*x)*(2 + e*x)^(3/2))","A",1
900,1,54,86,0.0630734,"\int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx","Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(7/2),x]","\frac{(e x-2) \sqrt{4-e^2 x^2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{1}{2}-\frac{e x}{4}\right)}{32 e \sqrt{3 e x+6}}","\frac{\sqrt{3} \sqrt{2-e x}}{16 e (e x+2)}-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (e x+2)^2}+\frac{\sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{32 e}",1,"((-2 + e*x)*Sqrt[4 - e^2*x^2]*Hypergeometric2F1[3/2, 3, 5/2, 1/2 - (e*x)/4])/(32*e*Sqrt[6 + 3*e*x])","C",1
901,1,67,109,0.067434,"\int (2+e x)^{5/2} \left(12-3 e^2 x^2\right)^{3/2} \, dx","Integrate[(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left(1155 e^4 x^4+12600 e^3 x^3+56840 e^2 x^2+133600 e x+154928\right)}{5005 e \sqrt{e x+2}}","-\frac{6 \sqrt{3} (2-e x)^{13/2}}{13 e}+\frac{96 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{64 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{1536 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{1536 \sqrt{3} (2-e x)^{5/2}}{5 e}",1,"(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(154928 + 133600*e*x + 56840*e^2*x^2 + 12600*e^3*x^3 + 1155*e^4*x^4))/(5005*e*Sqrt[2 + e*x])","A",1
902,1,59,87,0.0570788,"\int (2+e x)^{3/2} \left(12-3 e^2 x^2\right)^{3/2} \, dx","Integrate[(2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left(105 e^3 x^3+910 e^2 x^2+3020 e x+4264\right)}{385 e \sqrt{e x+2}}","\frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e}",1,"(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(4264 + 3020*e*x + 910*e^2*x^2 + 105*e^3*x^3))/(385*e*Sqrt[2 + e*x])","A",1
903,1,52,65,0.0449213,"\int \sqrt{2+e x} \left(12-3 e^2 x^2\right)^{3/2} \, dx","Integrate[Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 (e x-2)^2 \sqrt{4-e^2 x^2} \left(35 e^2 x^2+220 e x+428\right)}{35 e \sqrt{3 e x+6}}","-\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}",1,"(-2*(-2 + e*x)^2*Sqrt[4 - e^2*x^2]*(428 + 220*e*x + 35*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])","A",1
904,1,43,45,0.0499032,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{\sqrt{2+e x}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/Sqrt[2 + e*x],x]","-\frac{6 (e x-2)^2 (5 e x+18) \sqrt{12-3 e^2 x^2}}{35 e \sqrt{e x+2}}","\frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e}",1,"(-6*(-2 + e*x)^2*(18 + 5*e*x)*Sqrt[12 - 3*e^2*x^2])/(35*e*Sqrt[2 + e*x])","A",1
905,1,37,22,0.0479455,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{3/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(3/2),x]","-\frac{6 (e x-2)^2 \sqrt{12-3 e^2 x^2}}{5 e \sqrt{e x+2}}","-\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e}",1,"(-6*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2])/(5*e*Sqrt[2 + e*x])","A",1
906,1,69,66,0.0682852,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{5/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(5/2),x]","-\frac{2 \sqrt{12-3 e^2 x^2} \left(\sqrt{e x-2} (e x-14)+24 \tan ^{-1}\left(\frac{1}{2} \sqrt{e x-2}\right)\right)}{e \sqrt{e x-2} \sqrt{e x+2}}","\frac{2 \sqrt{3} (2-e x)^{3/2}}{e}+\frac{24 \sqrt{3} \sqrt{2-e x}}{e}-\frac{48 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{e}",1,"(-2*Sqrt[12 - 3*e^2*x^2]*((-14 + e*x)*Sqrt[-2 + e*x] + 24*ArcTan[Sqrt[-2 + e*x]/2]))/(e*Sqrt[-2 + e*x]*Sqrt[2 + e*x])","A",1
907,1,55,73,0.0569802,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{7/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(7/2),x]","-\frac{3 (e x-2)^2 \sqrt{12-3 e^2 x^2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{1}{2}-\frac{e x}{4}\right)}{40 e \sqrt{e x+2}}","-\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (e x+2)}-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}+\frac{18 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{e}",1,"(-3*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*Hypergeometric2F1[2, 5/2, 7/2, 1/2 - (e*x)/4])/(40*e*Sqrt[2 + e*x])","C",1
908,1,88,86,0.129093,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{9/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(9/2),x]","\frac{3 \sqrt{12-3 e^2 x^2} \left(2 \left(5 e^2 x^2-8 e x-4\right)+3 \sqrt{2-e x} (e x+2)^2 \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)\right)}{8 e (e x-2) (e x+2)^{5/2}}","-\frac{3 \sqrt{3} (2-e x)^{3/2}}{2 e (e x+2)^2}+\frac{9 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)}-\frac{9 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{8 e}",1,"(3*Sqrt[12 - 3*e^2*x^2]*(2*(-4 - 8*e*x + 5*e^2*x^2) + 3*Sqrt[2 - e*x]*(2 + e*x)^2*ArcTanh[Sqrt[2 - e*x]/2]))/(8*e*(-2 + e*x)*(2 + e*x)^(5/2))","A",1
909,1,55,113,0.0809457,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{11/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(11/2),x]","-\frac{3 (e x-2)^2 \sqrt{12-3 e^2 x^2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{1}{2}-\frac{e x}{4}\right)}{640 e \sqrt{e x+2}}","-\frac{\sqrt{3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (e x+2)}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)^2}-\frac{3 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{64 e}",1,"(-3*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*Hypergeometric2F1[5/2, 4, 7/2, 1/2 - (e*x)/4])/(640*e*Sqrt[2 + e*x])","C",1
910,1,55,144,0.0842759,"\int \frac{\left(12-3 e^2 x^2\right)^{3/2}}{(2+e x)^{13/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(13/2),x]","-\frac{3 (e x-2)^2 \sqrt{12-3 e^2 x^2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{1}{2}-\frac{e x}{4}\right)}{2560 e \sqrt{e x+2}}","-\frac{3 \sqrt{3} (2-e x)^{3/2}}{4 e (e x+2)^4}-\frac{9 \sqrt{3} \sqrt{2-e x}}{1024 e (e x+2)}-\frac{3 \sqrt{3} \sqrt{2-e x}}{128 e (e x+2)^2}+\frac{3 \sqrt{3} \sqrt{2-e x}}{8 e (e x+2)^3}-\frac{9 \sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{2048 e}",1,"(-3*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*Hypergeometric2F1[5/2, 5, 7/2, 1/2 - (e*x)/4])/(2560*e*Sqrt[2 + e*x])","C",1
911,1,57,85,0.0793521,"\int \frac{(2+e x)^{7/2}}{\sqrt{12-3 e^2 x^2}} \, dx","Integrate[(2 + e*x)^(7/2)/Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{e x+2} \left(5 e^3 x^3+54 e^2 x^2+284 e x+1416\right)}{35 e \sqrt{12-3 e^2 x^2}}","\frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[2 + e*x]*(1416 + 284*e*x + 54*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqrt[12 - 3*e^2*x^2])","A",1
912,1,49,65,0.0592369,"\int \frac{(2+e x)^{5/2}}{\sqrt{12-3 e^2 x^2}} \, dx","Integrate[(2 + e*x)^(5/2)/Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{e x+2} \left(3 e^2 x^2+28 e x+172\right)}{15 e \sqrt{12-3 e^2 x^2}}","-\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[2 + e*x]*(172 + 28*e*x + 3*e^2*x^2))/(15*e*Sqrt[12 - 3*e^2*x^2])","A",1
913,1,40,43,0.045408,"\int \frac{(2+e x)^{3/2}}{\sqrt{12-3 e^2 x^2}} \, dx","Integrate[(2 + e*x)^(3/2)/Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{e x+2} (e x+10)}{3 e \sqrt{12-3 e^2 x^2}}","\frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[2 + e*x]*(10 + e*x))/(3*e*Sqrt[12 - 3*e^2*x^2])","A",1
914,1,33,20,0.0346128,"\int \frac{\sqrt{2+e x}}{\sqrt{12-3 e^2 x^2}} \, dx","Integrate[Sqrt[2 + e*x]/Sqrt[12 - 3*e^2*x^2],x]","\frac{2 (e x-2) \sqrt{e x+2}}{e \sqrt{12-3 e^2 x^2}}","-\frac{2 \sqrt{2-e x}}{\sqrt{3} e}",1,"(2*(-2 + e*x)*Sqrt[2 + e*x])/(e*Sqrt[12 - 3*e^2*x^2])","A",1
915,1,50,25,0.0389313,"\int \frac{1}{\sqrt{2+e x} \sqrt{12-3 e^2 x^2}} \, dx","Integrate[1/(Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2]),x]","\frac{\sqrt{e x-2} \sqrt{e x+2} \tan ^{-1}\left(\frac{1}{2} \sqrt{e x-2}\right)}{e \sqrt{12-3 e^2 x^2}}","-\frac{\tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{\sqrt{3} e}",1,"(Sqrt[-2 + e*x]*Sqrt[2 + e*x]*ArcTan[Sqrt[-2 + e*x]/2])/(e*Sqrt[12 - 3*e^2*x^2])","A",1
916,1,54,57,0.0622294,"\int \frac{1}{(2+e x)^{3/2} \sqrt{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2]),x]","\frac{-2 \sqrt{2-e x}-\left((e x+2) \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)\right)}{8 \sqrt{3} e (e x+2)}","-\frac{\sqrt{2-e x}}{4 \sqrt{3} e (e x+2)}-\frac{\tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{8 \sqrt{3} e}",1,"(-2*Sqrt[2 - e*x] - (2 + e*x)*ArcTanh[Sqrt[2 - e*x]/2])/(8*Sqrt[3]*e*(2 + e*x))","A",1
917,1,53,86,0.056884,"\int \frac{1}{(2+e x)^{5/2} \sqrt{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2]),x]","\frac{(e x-2) \sqrt{e x+2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{1}{2}-\frac{e x}{4}\right)}{32 e \sqrt{12-3 e^2 x^2}}","-\frac{\sqrt{3} \sqrt{2-e x}}{64 e (e x+2)}-\frac{\sqrt{2-e x}}{8 \sqrt{3} e (e x+2)^2}-\frac{\sqrt{3} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{128 e}",1,"((-2 + e*x)*Sqrt[2 + e*x]*Hypergeometric2F1[1/2, 3, 3/2, 1/2 - (e*x)/4])/(32*e*Sqrt[12 - 3*e^2*x^2])","C",1
918,1,60,111,0.0885615,"\int \frac{(2+e x)^{11/2}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[(2 + e*x)^(11/2)/(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 \sqrt{e x+2} \left(5 e^4 x^4+72 e^3 x^3+568 e^2 x^2+5664 e x-23216\right)}{105 e \sqrt{12-3 e^2 x^2}}","-\frac{2 (2-e x)^{7/2}}{21 \sqrt{3} e}+\frac{32 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{64 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{512 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{512}{3 \sqrt{3} e \sqrt{2-e x}}",1,"(-2*Sqrt[2 + e*x]*(-23216 + 5664*e*x + 568*e^2*x^2 + 72*e^3*x^3 + 5*e^4*x^4))/(105*e*Sqrt[12 - 3*e^2*x^2])","A",1
919,1,51,87,0.0772067,"\int \frac{(2+e x)^{9/2}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[(2 + e*x)^(9/2)/(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 \sqrt{e x+2} \left(e^3 x^3+14 e^2 x^2+172 e x-728\right)}{15 e \sqrt{12-3 e^2 x^2}}","\frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{128}{3 \sqrt{3} e \sqrt{2-e x}}",1,"(-2*Sqrt[2 + e*x]*(-728 + 172*e*x + 14*e^2*x^2 + e^3*x^3))/(15*e*Sqrt[12 - 3*e^2*x^2])","A",1
920,1,43,67,0.0543493,"\int \frac{(2+e x)^{7/2}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[(2 + e*x)^(7/2)/(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 \sqrt{e x+2} \left(e^2 x^2+20 e x-92\right)}{9 e \sqrt{12-3 e^2 x^2}}","-\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}}",1,"(-2*Sqrt[2 + e*x]*(-92 + 20*e*x + e^2*x^2))/(9*e*Sqrt[12 - 3*e^2*x^2])","A",1
921,1,35,45,0.0523137,"\int \frac{(2+e x)^{5/2}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]","-\frac{2 (e x-6) \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}}","\frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{8}{3 \sqrt{3} e \sqrt{2-e x}}",1,"(-2*(-6 + e*x)*Sqrt[2 + e*x])/(3*e*Sqrt[12 - 3*e^2*x^2])","A",1
922,1,30,22,0.0414911,"\int \frac{(2+e x)^{3/2}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x]","\frac{2 \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}}","\frac{2}{3 \sqrt{3} e \sqrt{2-e x}}",1,"(2*Sqrt[2 + e*x])/(3*e*Sqrt[12 - 3*e^2*x^2])","A",1
923,1,48,50,0.0404684,"\int \frac{\sqrt{2+e x}}{\left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[Sqrt[2 + e*x]/(12 - 3*e^2*x^2)^(3/2),x]","\frac{\sqrt{e x+2} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right)}{6 e \sqrt{12-3 e^2 x^2}}","\frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{12 \sqrt{3} e}",1,"(Sqrt[2 + e*x]*Hypergeometric2F1[-1/2, 1, 1/2, 1/2 - (e*x)/4])/(6*e*Sqrt[12 - 3*e^2*x^2])","C",1
924,1,48,79,0.0456706,"\int \frac{1}{\sqrt{2+e x} \left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(3/2)),x]","\frac{\sqrt{e x+2} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right)}{24 e \sqrt{12-3 e^2 x^2}}","\frac{1}{16 \sqrt{3} e \sqrt{2-e x}}-\frac{1}{12 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{\tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{32 \sqrt{3} e}",1,"(Sqrt[2 + e*x]*Hypergeometric2F1[-1/2, 2, 1/2, 1/2 - (e*x)/4])/(24*e*Sqrt[12 - 3*e^2*x^2])","C",1
925,1,48,108,0.0490277,"\int \frac{1}{(2+e x)^{3/2} \left(12-3 e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(3/2)),x]","\frac{\sqrt{e x+2} \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right)}{96 e \sqrt{12-3 e^2 x^2}}","\frac{5}{256 \sqrt{3} e \sqrt{2-e x}}-\frac{5}{192 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{1}{24 \sqrt{3} e \sqrt{2-e x} (e x+2)^2}-\frac{5 \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-e x}\right)}{512 \sqrt{3} e}",1,"(Sqrt[2 + e*x]*Hypergeometric2F1[-1/2, 3, 1/2, 1/2 - (e*x)/4])/(96*e*Sqrt[12 - 3*e^2*x^2])","C",1
926,1,23,23,0.0062405,"\int \frac{1}{\sqrt{1-x} (1+x)} \, dx","Integrate[1/(Sqrt[1 - x]*(1 + x)),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)",1,"-(Sqrt[2]*ArcTanh[Sqrt[1 - x]/Sqrt[2]])","A",1
927,1,45,23,0.0286217,"\int \frac{1}{\sqrt{1+x} \sqrt{1-x^2}} \, dx","Integrate[1/(Sqrt[1 + x]*Sqrt[1 - x^2]),x]","\frac{\sqrt{2} \sqrt{x-1} \sqrt{x+1} \tan ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{2}}\right)}{\sqrt{1-x^2}}","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)",1,"(Sqrt[2]*Sqrt[-1 + x]*Sqrt[1 + x]*ArcTan[Sqrt[-1 + x]/Sqrt[2]])/Sqrt[1 - x^2]","A",1
928,1,27,27,0.0071625,"\int \frac{1}{\sqrt{1-a x} (1+a x)} \, dx","Integrate[1/(Sqrt[1 - a*x]*(1 + a*x)),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)}{a}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)}{a}",1,"-((Sqrt[2]*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/a)","A",1
929,1,53,27,0.04692,"\int \frac{1}{\sqrt{1+a x} \sqrt{1-a^2 x^2}} \, dx","Integrate[1/(Sqrt[1 + a*x]*Sqrt[1 - a^2*x^2]),x]","\frac{\sqrt{a x+1} \sqrt{2 a x-2} \tan ^{-1}\left(\frac{\sqrt{a x-1}}{\sqrt{2}}\right)}{a \sqrt{1-a^2 x^2}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)}{a}",1,"(Sqrt[1 + a*x]*Sqrt[-2 + 2*a*x]*ArcTan[Sqrt[-1 + a*x]/Sqrt[2]])/(a*Sqrt[1 - a^2*x^2])","A",1
930,1,60,309,0.0480929,"\int \sqrt{2+e x} \sqrt[4]{12-3 e^2 x^2} \, dx","Integrate[Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4),x]","\frac{8 \sqrt{2} (e x-2) \sqrt[4]{12-3 e^2 x^2} \, _2F_1\left(-\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right)}{5 e \sqrt[4]{e x+2}}","-\frac{\sqrt[4]{3} (e x+2)^{3/4} (2-e x)^{5/4}}{2 e}+\frac{3 \sqrt[4]{3} (e x+2)^{3/4} \sqrt[4]{2-e x}}{2 e}+\frac{3 \sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} e}-\frac{3 \sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} e}+\frac{3 \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{\sqrt{2} e}-\frac{3 \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{\sqrt{2} e}",1,"(8*Sqrt[2]*(-2 + e*x)*(12 - 3*e^2*x^2)^(1/4)*Hypergeometric2F1[-3/4, 5/4, 9/4, 1/2 - (e*x)/4])/(5*e*(2 + e*x)^(1/4))","C",1
931,1,60,269,0.0455991,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{\sqrt{2+e x}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/Sqrt[2 + e*x],x]","\frac{2 \sqrt{2} (e x-2) \sqrt[4]{12-3 e^2 x^2} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right)}{5 e \sqrt[4]{e x+2}}","\frac{\sqrt[4]{3} \sqrt[4]{2-e x} (e x+2)^{3/4}}{e}+\frac{\sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} e}-\frac{\sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} e}+\frac{\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{e}-\frac{\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{e}",1,"(2*Sqrt[2]*(-2 + e*x)*(12 - 3*e^2*x^2)^(1/4)*Hypergeometric2F1[1/4, 5/4, 9/4, 1/2 - (e*x)/4])/(5*e*(2 + e*x)^(1/4))","C",1
932,1,60,270,0.0471241,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(3/2),x]","\frac{(e x-2) \sqrt[4]{12-3 e^2 x^2} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right)}{5 \sqrt{2} e \sqrt[4]{e x+2}}","-\frac{4 \sqrt[4]{3} \sqrt[4]{2-e x}}{e \sqrt[4]{e x+2}}-\frac{\sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} e}+\frac{\sqrt[4]{3} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{e}+\frac{\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{e}",1,"((-2 + e*x)*(12 - 3*e^2*x^2)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, 1/2 - (e*x)/4])/(5*Sqrt[2]*e*(2 + e*x)^(1/4))","C",1
933,1,35,35,0.0400132,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(5/2),x]","\frac{(e x-2) \sqrt[4]{12-3 e^2 x^2}}{5 e (e x+2)^{3/2}}","-\frac{\sqrt[4]{3} \left(4-e^2 x^2\right)^{5/4}}{5 e (e x+2)^{5/2}}",1,"((-2 + e*x)*(12 - 3*e^2*x^2)^(1/4))/(5*e*(2 + e*x)^(3/2))","A",1
934,1,45,71,0.0528861,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(7/2),x]","\frac{(e x-2) (e x+7) \sqrt[4]{4-e^2 x^2}}{15\ 3^{3/4} e (e x+2)^{5/2}}","-\frac{\left(4-e^2 x^2\right)^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac{\left(4-e^2 x^2\right)^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}}",1,"((-2 + e*x)*(7 + e*x)*(4 - e^2*x^2)^(1/4))/(15*3^(3/4)*e*(2 + e*x)^(5/2))","A",1
935,1,57,106,0.0685536,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]","\frac{\sqrt[4]{4-e^2 x^2} \left(2 e^3 x^3+14 e^2 x^2+37 e x-146\right)}{195\ 3^{3/4} e (e x+2)^{7/2}}","-\frac{2 \left(4-e^2 x^2\right)^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left(4-e^2 x^2\right)^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac{\sqrt[4]{3} \left(4-e^2 x^2\right)^{5/4}}{13 e (e x+2)^{9/2}}",1,"((4 - e^2*x^2)^(1/4)*(-146 + 37*e*x + 14*e^2*x^2 + 2*e^3*x^3))/(195*3^(3/4)*e*(2 + e*x)^(7/2))","A",1
936,1,65,141,0.0699486,"\int \frac{\sqrt[4]{12-3 e^2 x^2}}{(2+e x)^{11/2}} \, dx","Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(11/2),x]","\frac{\sqrt[4]{4-e^2 x^2} \left(2 e^4 x^4+18 e^3 x^3+65 e^2 x^2+123 e x-682\right)}{1105\ 3^{3/4} e (e x+2)^{9/2}}","-\frac{2 \left(4-e^2 x^2\right)^{5/4}}{1105\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left(4-e^2 x^2\right)^{5/4}}{221\ 3^{3/4} e (e x+2)^{7/2}}-\frac{3 \sqrt[4]{3} \left(4-e^2 x^2\right)^{5/4}}{221 e (e x+2)^{9/2}}-\frac{\sqrt[4]{3} \left(4-e^2 x^2\right)^{5/4}}{17 e (e x+2)^{11/2}}",1,"((4 - e^2*x^2)^(1/4)*(-682 + 123*e*x + 65*e^2*x^2 + 18*e^3*x^3 + 2*e^4*x^4))/(1105*3^(3/4)*e*(2 + e*x)^(9/2))","A",1
937,1,60,340,0.0722671,"\int \frac{(2+e x)^{5/2}}{\sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(1/4),x]","\frac{64 \sqrt{2} (e x-2) \sqrt[4]{e x+2} \, _2F_1\left(-\frac{9}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{e x}{4}\right)}{3 e \sqrt[4]{12-3 e^2 x^2}}","-\frac{(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt[4]{3} e}-\frac{3^{3/4} (2-e x)^{3/4} (e x+2)^{5/4}}{2 e}-\frac{5\ 3^{3/4} (2-e x)^{3/4} \sqrt[4]{e x+2}}{2 e}-\frac{5\ 3^{3/4} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{\sqrt{2} e}-\frac{5\ 3^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{\sqrt{2} e}",1,"(64*Sqrt[2]*(-2 + e*x)*(2 + e*x)^(1/4)*Hypergeometric2F1[-9/4, 3/4, 7/4, 1/2 - (e*x)/4])/(3*e*(12 - 3*e^2*x^2)^(1/4))","C",1
938,1,60,309,0.0471468,"\int \frac{(2+e x)^{3/2}}{\sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(1/4),x]","\frac{16 \sqrt{2} (e x-2) \sqrt[4]{e x+2} \, _2F_1\left(-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{e x}{4}\right)}{3 e \sqrt[4]{12-3 e^2 x^2}}","-\frac{(2-e x)^{3/4} (e x+2)^{5/4}}{2 \sqrt[4]{3} e}-\frac{5 (2-e x)^{3/4} \sqrt[4]{e x+2}}{2 \sqrt[4]{3} e}-\frac{5 \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} \sqrt[4]{3} e}+\frac{5 \log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{2 \sqrt{2} \sqrt[4]{3} e}+\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{\sqrt{2} \sqrt[4]{3} e}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{\sqrt{2} \sqrt[4]{3} e}",1,"(16*Sqrt[2]*(-2 + e*x)*(2 + e*x)^(1/4)*Hypergeometric2F1[-5/4, 3/4, 7/4, 1/2 - (e*x)/4])/(3*e*(12 - 3*e^2*x^2)^(1/4))","C",1
939,1,60,270,0.0446314,"\int \frac{\sqrt{2+e x}}{\sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[Sqrt[2 + e*x]/(12 - 3*e^2*x^2)^(1/4),x]","\frac{4 \sqrt{2} (e x-2) \sqrt[4]{e x+2} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{e x}{4}\right)}{3 e \sqrt[4]{12-3 e^2 x^2}}","-\frac{(2-e x)^{3/4} \sqrt[4]{e x+2}}{\sqrt[4]{3} e}-\frac{\log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} \sqrt[4]{3} e}+\frac{\log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} \sqrt[4]{3} e}+\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{\sqrt[4]{3} e}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{\sqrt[4]{3} e}",1,"(4*Sqrt[2]*(-2 + e*x)*(2 + e*x)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, 1/2 - (e*x)/4])/(3*e*(12 - 3*e^2*x^2)^(1/4))","C",1
940,1,60,241,0.046127,"\int \frac{1}{\sqrt{2+e x} \sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[1/(Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4)),x]","\frac{\sqrt{2} (e x-2) \sqrt[4]{e x+2} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{e x}{4}\right)}{3 e \sqrt[4]{12-3 e^2 x^2}}","-\frac{\log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} \sqrt[4]{3} e}+\frac{\log \left(\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt[4]{2-e x} \sqrt[4]{e x+2}}{\sqrt{e x+2}}\right)}{\sqrt{2} \sqrt[4]{3} e}+\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}\right)}{\sqrt[4]{3} e}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2-e x}}{\sqrt[4]{e x+2}}+1\right)}{\sqrt[4]{3} e}",1,"(Sqrt[2]*(-2 + e*x)*(2 + e*x)^(1/4)*Hypergeometric2F1[3/4, 3/4, 7/4, 1/2 - (e*x)/4])/(3*e*(12 - 3*e^2*x^2)^(1/4))","C",1
941,1,35,35,0.043513,"\int \frac{1}{(2+e x)^{3/2} \sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]","\frac{e x-2}{3 e \sqrt{e x+2} \sqrt[4]{12-3 e^2 x^2}}","-\frac{\left(4-e^2 x^2\right)^{3/4}}{3 \sqrt[4]{3} e (e x+2)^{3/2}}",1,"(-2 + e*x)/(3*e*Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4))","A",1
942,1,40,71,0.0517658,"\int \frac{1}{(2+e x)^{5/2} \sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(1/4)),x]","\frac{(e x-2) (e x+5)}{21 e (e x+2)^{3/2} \sqrt[4]{12-3 e^2 x^2}}","-\frac{\left(4-e^2 x^2\right)^{3/4}}{21 \sqrt[4]{3} e (e x+2)^{3/2}}-\frac{\left(4-e^2 x^2\right)^{3/4}}{7 \sqrt[4]{3} e (e x+2)^{5/2}}",1,"((-2 + e*x)*(5 + e*x))/(21*e*(2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4))","A",1
943,1,49,106,0.063002,"\int \frac{1}{(2+e x)^{7/2} \sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(7/2)*(12 - 3*e^2*x^2)^(1/4)),x]","\frac{(e x-2) \left(2 e^2 x^2+14 e x+41\right)}{231 e (e x+2)^{5/2} \sqrt[4]{12-3 e^2 x^2}}","-\frac{2 \left(4-e^2 x^2\right)^{3/4}}{231 \sqrt[4]{3} e (e x+2)^{3/2}}-\frac{2 \left(4-e^2 x^2\right)^{3/4}}{77 \sqrt[4]{3} e (e x+2)^{5/2}}-\frac{\left(4-e^2 x^2\right)^{3/4}}{11 \sqrt[4]{3} e (e x+2)^{7/2}}",1,"((-2 + e*x)*(41 + 14*e*x + 2*e^2*x^2))/(231*e*(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(1/4))","A",1
944,1,57,141,0.0694857,"\int \frac{1}{(2+e x)^{9/2} \sqrt[4]{12-3 e^2 x^2}} \, dx","Integrate[1/((2 + e*x)^(9/2)*(12 - 3*e^2*x^2)^(1/4)),x]","\frac{(e x-2) \left(2 e^3 x^3+18 e^2 x^2+69 e x+159\right)}{1155 e (e x+2)^{7/2} \sqrt[4]{12-3 e^2 x^2}}","-\frac{2 \left(4-e^2 x^2\right)^{3/4}}{1155 \sqrt[4]{3} e (e x+2)^{3/2}}-\frac{2 \left(4-e^2 x^2\right)^{3/4}}{385 \sqrt[4]{3} e (e x+2)^{5/2}}-\frac{\left(4-e^2 x^2\right)^{3/4}}{55 \sqrt[4]{3} e (e x+2)^{7/2}}-\frac{\left(4-e^2 x^2\right)^{3/4}}{15 \sqrt[4]{3} e (e x+2)^{9/2}}",1,"((-2 + e*x)*(159 + 69*e*x + 18*e^2*x^2 + 2*e^3*x^3))/(1155*e*(2 + e*x)^(7/2)*(12 - 3*e^2*x^2)^(1/4))","A",1
945,1,68,84,0.0640697,"\int (a+b x)^m \left(a^2-b^2 x^2\right)^3 \, dx","Integrate[(a + b*x)^m*(a^2 - b^2*x^2)^3,x]","\frac{(a+b x)^{m+4} \left(\frac{8 a^3}{m+4}-\frac{12 a^2 (a+b x)}{m+5}+\frac{6 a (a+b x)^2}{m+6}-\frac{(a+b x)^3}{m+7}\right)}{b}","\frac{8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac{12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac{6 a (a+b x)^{m+6}}{b (m+6)}-\frac{(a+b x)^{m+7}}{b (m+7)}",1,"((a + b*x)^(4 + m)*((8*a^3)/(4 + m) - (12*a^2*(a + b*x))/(5 + m) + (6*a*(a + b*x)^2)/(6 + m) - (a + b*x)^3/(7 + m)))/b","A",1
946,1,50,61,0.0555469,"\int (a+b x)^m \left(a^2-b^2 x^2\right)^2 \, dx","Integrate[(a + b*x)^m*(a^2 - b^2*x^2)^2,x]","\frac{(a+b x)^{m+3} \left(\frac{4 a^2}{m+3}-\frac{4 a (a+b x)}{m+4}+\frac{(a+b x)^2}{m+5}\right)}{b}","\frac{4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac{4 a (a+b x)^{m+4}}{b (m+4)}+\frac{(a+b x)^{m+5}}{b (m+5)}",1,"((a + b*x)^(3 + m)*((4*a^2)/(3 + m) - (4*a*(a + b*x))/(4 + m) + (a + b*x)^2/(5 + m)))/b","A",1
947,1,36,40,0.0291209,"\int (a+b x)^m \left(a^2-b^2 x^2\right) \, dx","Integrate[(a + b*x)^m*(a^2 - b^2*x^2),x]","\frac{(a+b x)^{m+2} (a (m+4)-b (m+2) x)}{b (m+2) (m+3)}","\frac{2 a (a+b x)^{m+2}}{b (m+2)}-\frac{(a+b x)^{m+3}}{b (m+3)}",1,"((a + b*x)^(2 + m)*(a*(4 + m) - b*(2 + m)*x))/(b*(2 + m)*(3 + m))","A",1
948,1,59,38,0.0428093,"\int \frac{(a+b x)^m}{a^2-b^2 x^2} \, dx","Integrate[(a + b*x)^m/(a^2 - b^2*x^2),x]","\frac{(a+b x)^m \left(m (a+b x) \, _2F_1\left(1,m+1;m+2;\frac{a+b x}{2 a}\right)+2 a (m+1)\right)}{4 a^2 b m (m+1)}","\frac{(a+b x)^m \, _2F_1\left(1,m;m+1;\frac{a+b x}{2 a}\right)}{2 a b m}",1,"((a + b*x)^m*(2*a*(1 + m) + m*(a + b*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*x)/(2*a)]))/(4*a^2*b*m*(1 + m))","A",1
949,1,102,44,0.1565554,"\int \frac{(a+b x)^m}{\left(a^2-b^2 x^2\right)^2} \, dx","Integrate[(a + b*x)^m/(a^2 - b^2*x^2)^2,x]","\frac{(a+b x)^m \left(\frac{2 (a+b x) \, _2F_1\left(1,m+1;m+2;\frac{a+b x}{2 a}\right)}{m+1}+\frac{(a+b x) \, _2F_1\left(2,m+1;m+2;\frac{a+b x}{2 a}\right)}{m+1}+4 a \left(\frac{a}{(m-1) (a+b x)}+\frac{1}{m}\right)\right)}{16 a^4 b}","-\frac{(a+b x)^{m-1} \, _2F_1\left(2,m-1;m;\frac{a+b x}{2 a}\right)}{4 a^2 b (1-m)}",1,"((a + b*x)^m*(4*a*(m^(-1) + a/((-1 + m)*(a + b*x))) + (2*(a + b*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m) + ((a + b*x)*Hypergeometric2F1[2, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m)))/(16*a^4*b)","B",1
950,1,153,46,0.1929833,"\int \frac{(a+b x)^m}{\left(a^2-b^2 x^2\right)^3} \, dx","Integrate[(a + b*x)^m/(a^2 - b^2*x^2)^3,x]","\frac{(a+b x)^m \left(\frac{8 a^3}{(m-2) (a+b x)^2}+\frac{12 a^2}{(m-1) (a+b x)}+\frac{6 (a+b x) \, _2F_1\left(1,m+1;m+2;\frac{a+b x}{2 a}\right)}{m+1}+\frac{3 (a+b x) \, _2F_1\left(2,m+1;m+2;\frac{a+b x}{2 a}\right)}{m+1}+\frac{(a+b x) \, _2F_1\left(3,m+1;m+2;\frac{a+b x}{2 a}\right)}{m+1}+\frac{12 a}{m}\right)}{64 a^6 b}","-\frac{(a+b x)^{m-2} \, _2F_1\left(3,m-2;m-1;\frac{a+b x}{2 a}\right)}{8 a^3 b (2-m)}",1,"((a + b*x)^m*((12*a)/m + (8*a^3)/((-2 + m)*(a + b*x)^2) + (12*a^2)/((-1 + m)*(a + b*x)) + (6*(a + b*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m) + (3*(a + b*x)*Hypergeometric2F1[2, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m) + ((a + b*x)*Hypergeometric2F1[3, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m)))/(64*a^6*b)","B",1
951,1,347,59,0.492251,"\int (d+e x)^m \left(d^2-e^2 x^2\right)^{7/2} \, dx","Integrate[(d + e*x)^m*(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x)^m \left(\frac{e x}{d}+1\right)^{-m-\frac{1}{2}} \left(-105 d^4 e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} F_1\left(3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right)+63 d^2 e^5 x^5 \sqrt{d-e x} \sqrt{d+e x} F_1\left(5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right)-15 e^7 x^7 \sqrt{d-e x} \sqrt{d+e x} F_1\left(7;-\frac{1}{2},-m-\frac{1}{2};8;\frac{e x}{d},-\frac{e x}{d}\right)-35 d^7 2^{m+\frac{3}{2}} \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right)+35 d^6 e 2^{m+\frac{3}{2}} x \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right)\right)}{105 e \sqrt{1-\frac{e x}{d}}}","\frac{\left(d^2-e^2 x^2\right)^{9/2} (d+e x)^m \, _2F_1\left(1,m+9;m+\frac{11}{2};\frac{d+e x}{2 d}\right)}{d e (2 m+9)}",1,"((d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*(-105*d^4*e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[3, -1/2, -1/2 - m, 4, (e*x)/d, -((e*x)/d)] + 63*d^2*e^5*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[5, -1/2, -1/2 - m, 6, (e*x)/d, -((e*x)/d)] - 15*e^7*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[7, -1/2, -1/2 - m, 8, (e*x)/d, -((e*x)/d)] - 35*2^(3/2 + m)*d^7*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)] + 35*2^(3/2 + m)*d^6*e*x*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)]))/(105*e*Sqrt[1 - (e*x)/d])","C",0
952,1,227,83,0.388457,"\int (d+e x)^m \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^m*(d^2 - e^2*x^2)^(5/2),x]","\frac{(d+e x)^m \left(\frac{e x}{d}+1\right)^{-m-\frac{1}{2}} \left(-10 d^2 e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} F_1\left(3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right)+3 e^5 x^5 \sqrt{d-e x} \sqrt{d+e x} F_1\left(5;-\frac{1}{2},-m-\frac{1}{2};6;\frac{e x}{d},-\frac{e x}{d}\right)-5 d^4 2^{m+\frac{3}{2}} (d-e x) \sqrt{1-\frac{e x}{d}} \sqrt{d^2-e^2 x^2} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right)\right)}{15 e \sqrt{1-\frac{e x}{d}}}","-\frac{2^{m+\frac{7}{2}} \left(d^2-e^2 x^2\right)^{7/2} (d+e x)^m \left(\frac{e x}{d}+1\right)^{-m-\frac{7}{2}} \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{d-e x}{2 d}\right)}{7 d e}",1,"((d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*(-10*d^2*e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[3, -1/2, -1/2 - m, 4, (e*x)/d, -((e*x)/d)] + 3*e^5*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[5, -1/2, -1/2 - m, 6, (e*x)/d, -((e*x)/d)] - 5*2^(3/2 + m)*d^4*(d - e*x)*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)]))/(15*e*Sqrt[1 - (e*x)/d])","C",0
953,1,191,59,0.2237853,"\int (d+e x)^m \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(d^2 - e^2*x^2)^(3/2),x]","-\frac{2^m (d+e x)^m \left(\frac{e x}{d}+1\right)^{-2 m-\frac{1}{2}} \left(e^3 x^3 \sqrt{d-e x} \sqrt{d+e x} \left(\frac{e x}{2 d}+\frac{1}{2}\right)^m F_1\left(3;-\frac{1}{2},-m-\frac{1}{2};4;\frac{e x}{d},-\frac{e x}{d}\right)+2 d^2 (d-e x) \sqrt{2-\frac{2 e x}{d}} \sqrt{d^2-e^2 x^2} \left(\frac{e x}{d}+1\right)^m \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right)\right)}{3 e \sqrt{1-\frac{e x}{d}}}","\frac{\left(d^2-e^2 x^2\right)^{5/2} (d+e x)^m \, _2F_1\left(1,m+5;m+\frac{7}{2};\frac{d+e x}{2 d}\right)}{d e (2 m+5)}",1,"-1/3*(2^m*(d + e*x)^m*(1 + (e*x)/d)^(-1/2 - 2*m)*(e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*(1/2 + (e*x)/(2*d))^m*AppellF1[3, -1/2, -1/2 - m, 4, (e*x)/d, -((e*x)/d)] + 2*d^2*(d - e*x)*Sqrt[2 - (2*e*x)/d]*(1 + (e*x)/d)^m*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)]))/(e*Sqrt[1 - (e*x)/d])","C",0
954,1,86,67,0.0627253,"\int (d+e x)^m \sqrt{d^2-e^2 x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[d^2 - e^2*x^2],x]","-\frac{2^{m+\frac{3}{2}} (d-e x) \sqrt{d^2-e^2 x^2} (d+e x)^m \left(\frac{e x}{d}+1\right)^{-m-\frac{1}{2}} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right)}{3 e}","\frac{(d-e x) \sqrt{d^2-e^2 x^2} (d+e x)^{m+1} \, _2F_1\left(1,m+3;m+\frac{5}{2};\frac{d+e x}{2 d}\right)}{d e (2 m+3)}",1,"-1/3*(2^(3/2 + m)*(d - e*x)*(d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)])/e","A",1
955,1,84,67,0.0814027,"\int \frac{(d+e x)^m}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[d^2 - e^2*x^2],x]","-\frac{2^{m+\frac{1}{2}} (d-e x) (d+e x)^m \left(\frac{e x}{d}+1\right)^{\frac{1}{2}-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right)}{e \sqrt{d^2-e^2 x^2}}","\frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+\frac{3}{2};\frac{d+e x}{2 d}\right)}{d e (2 m+1) \sqrt{d^2-e^2 x^2}}",1,"-((2^(1/2 + m)*(d - e*x)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (d - e*x)/(2*d)])/(e*Sqrt[d^2 - e^2*x^2]))","A",1
956,1,80,80,0.0670714,"\int \frac{(d+e x)^m}{\left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(d^2 - e^2*x^2)^(3/2),x]","\frac{2^{m-\frac{1}{2}} (d+e x)^m \left(\frac{e x}{d}+1\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right)}{d e \sqrt{d^2-e^2 x^2}}","\frac{2^{m-\frac{1}{2}} (d+e x)^m \left(\frac{e x}{d}+1\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right)}{d e \sqrt{d^2-e^2 x^2}}",1,"(2^(-1/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, (d - e*x)/(2*d)])/(d*e*Sqrt[d^2 - e^2*x^2])","A",1
957,1,92,60,0.1047811,"\int \frac{(d+e x)^m}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^m/(d^2 - e^2*x^2)^(5/2),x]","\frac{2^{m-\frac{3}{2}} (d+e x)^m \left(\frac{e x}{d}+1\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{d-e x}{2 d}\right)}{\left(3 d^3 e-3 d^2 e^2 x\right) \sqrt{d^2-e^2 x^2}}","-\frac{(d+e x)^m \, _2F_1\left(1,m-3;m-\frac{1}{2};\frac{d+e x}{2 d}\right)}{d e (3-2 m) \left(d^2-e^2 x^2\right)^{3/2}}",1,"(2^(-3/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, (d - e*x)/(2*d)])/((3*d^3*e - 3*d^2*e^2*x)*Sqrt[d^2 - e^2*x^2])","A",1
958,1,91,68,0.0935842,"\int \frac{(d+e x)^m}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^m/(d^2 - e^2*x^2)^(7/2),x]","\frac{2^{m-\frac{5}{2}} (d+e x)^m \left(\frac{e x}{d}+1\right)^{\frac{1}{2}-m} \, _2F_1\left(-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right)}{5 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","-\frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left(1,m-5;m-\frac{3}{2};\frac{d+e x}{2 d}\right)}{d e (5-2 m) \left(d^2-e^2 x^2\right)^{7/2}}",1,"(2^(-5/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(1/2 - m)*Hypergeometric2F1[-5/2, 7/2 - m, -3/2, (d - e*x)/(2*d)])/(5*d^3*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
959,1,85,63,0.0863211,"\int (a+b x)^m \left(a^2-b^2 x^2\right)^p \, dx","Integrate[(a + b*x)^m*(a^2 - b^2*x^2)^p,x]","\frac{2^{m+p} (b x-a) (a+b x)^m \left(a^2-b^2 x^2\right)^p \left(\frac{b x}{a}+1\right)^{-m-p} \, _2F_1\left(-m-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{b (p+1)}","\frac{(a+b x)^m \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(1,m+2 p+2;m+p+2;\frac{a+b x}{2 a}\right)}{2 a b (m+p+1)}",1,"(2^(m + p)*(-a + b*x)*(a + b*x)^m*(1 + (b*x)/a)^(-m - p)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[-m - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p))","A",1
960,1,116,57,0.1131845,"\int (d+e x)^3 \left(1-\frac{e^2 x^2}{d^2}\right)^p \, dx","Integrate[(d + e*x)^3*(1 - (e^2*x^2)/d^2)^p,x]","d e^2 x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{2 d^4 \left(1-\frac{e^2 x^2}{d^2}\right)^{p+1}}{e (p+1)}+\frac{d^4 \left(1-\frac{e^2 x^2}{d^2}\right)^{p+2}}{2 e (p+2)}+d^3 x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)","-\frac{d^4 2^{p+3} \left(\frac{d-e x}{d}\right)^{p+1} \, _2F_1\left(-p-3,p+1;p+2;\frac{d-e x}{2 d}\right)}{e (p+1)}",1,"(-2*d^4*(1 - (e^2*x^2)/d^2)^(1 + p))/(e*(1 + p)) + (d^4*(1 - (e^2*x^2)/d^2)^(2 + p))/(2*e*(2 + p)) + d^3*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + d*e^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2]","B",1
961,1,86,57,0.0654476,"\int (d+e x)^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p \, dx","Integrate[(d + e*x)^2*(1 - (e^2*x^2)/d^2)^p,x]","d^2 x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+\frac{1}{3} e^2 x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{p+1}}{e (p+1)}","-\frac{d^3 2^{p+2} \left(\frac{d-e x}{d}\right)^{p+1} \, _2F_1\left(-p-2,p+1;p+2;\frac{d-e x}{2 d}\right)}{e (p+1)}",1,"-((d^3*(1 - (e^2*x^2)/d^2)^(1 + p))/(e*(1 + p))) + d^2*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (e^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/3","A",1
962,1,56,57,0.0284313,"\int (d+e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \, dx","Integrate[(d + e*x)*(1 - (e^2*x^2)/d^2)^p,x]","d x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{p+1}}{2 e (p+1)}","-\frac{d^2 2^{p+1} \left(\frac{d-e x}{d}\right)^{p+1} \, _2F_1\left(-p-1,p+1;p+2;\frac{d-e x}{2 d}\right)}{e (p+1)}",1,"-1/2*(d^2*(1 - (e^2*x^2)/d^2)^(1 + p))/(e*(1 + p)) + d*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2]","A",1
963,1,22,22,0.0018875,"\int \left(1-\frac{e^2 x^2}{d^2}\right)^p \, dx","Integrate[(1 - (e^2*x^2)/d^2)^p,x]","x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)","x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)",1,"x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2]","A",1
964,1,76,41,0.0834408,"\int \frac{\left(1-\frac{e^2 x^2}{d^2}\right)^p}{d+e x} \, dx","Integrate[(1 - (e^2*x^2)/d^2)^p/(d + e*x),x]","-\frac{2^{p-1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d e (p+1)}","\frac{2^p \left(\frac{d+e x}{d}\right)^p \, _2F_1\left(-p,p;p+1;\frac{d+e x}{2 d}\right)}{e p}",1,"-((2^(-1 + p)*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
965,1,76,57,0.0787883,"\int \frac{\left(1-\frac{e^2 x^2}{d^2}\right)^p}{(d+e x)^2} \, dx","Integrate[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^2,x]","-\frac{2^{p-2} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e (p+1)}","-\frac{2^{p-2} \left(\frac{d-e x}{d}\right)^{p+1} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d e (p+1)}",1,"-((2^(-2 + p)*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^2*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
966,1,76,57,0.076673,"\int \frac{\left(1-\frac{e^2 x^2}{d^2}\right)^p}{(d+e x)^3} \, dx","Integrate[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^3,x]","-\frac{2^{p-3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^3 e (p+1)}","-\frac{2^{p-3} \left(\frac{d-e x}{d}\right)^{p+1} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e (p+1)}",1,"-((2^(-3 + p)*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^3*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
967,1,155,60,0.172469,"\int (a+b x)^3 \left(a^2-b^2 x^2\right)^p \, dx","Integrate[(a + b*x)^3*(a^2 - b^2*x^2)^p,x]","\frac{1}{2} \left(a^2-b^2 x^2\right)^p \left(2 a b^2 x^3 \left(1-\frac{b^2 x^2}{a^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{b^2 x^2}{a^2}\right)+\frac{\left(b^2 x^2-a^2\right) \left(a^2 (3 p+7)+b^2 (p+1) x^2\right)}{b (p+1) (p+2)}+2 a^3 x \left(1-\frac{b^2 x^2}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right)\right)","\frac{(a+b x)^3 \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(1,2 p+5;p+5;\frac{a+b x}{2 a}\right)}{2 a b (p+4)}",1,"((a^2 - b^2*x^2)^p*(((-a^2 + b^2*x^2)*(a^2*(7 + 3*p) + b^2*(1 + p)*x^2))/(b*(1 + p)*(2 + p)) + (2*a^3*x*Hypergeometric2F1[1/2, -p, 3/2, (b^2*x^2)/a^2])/(1 - (b^2*x^2)/a^2)^p + (2*a*b^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, (b^2*x^2)/a^2])/(1 - (b^2*x^2)/a^2)^p))/2","B",1
968,1,134,60,0.0593126,"\int (a+b x)^2 \left(a^2-b^2 x^2\right)^p \, dx","Integrate[(a + b*x)^2*(a^2 - b^2*x^2)^p,x]","\frac{\left(a^2-b^2 x^2\right)^p \left(1-\frac{b^2 x^2}{a^2}\right)^{-p} \left(3 a^2 b (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right)-3 a \left(a^2-b^2 x^2\right) \left(1-\frac{b^2 x^2}{a^2}\right)^p+b^3 (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{b^2 x^2}{a^2}\right)\right)}{3 b (p+1)}","\frac{(a+b x)^2 \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(1,2 (p+2);p+4;\frac{a+b x}{2 a}\right)}{2 a b (p+3)}",1,"((a^2 - b^2*x^2)^p*(-3*a*(a^2 - b^2*x^2)*(1 - (b^2*x^2)/a^2)^p + 3*a^2*b*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (b^2*x^2)/a^2] + b^3*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (b^2*x^2)/a^2]))/(3*b*(1 + p)*(1 - (b^2*x^2)/a^2)^p)","B",1
969,1,83,83,0.0473428,"\int (a+b x) \left(a^2-b^2 x^2\right)^p \, dx","Integrate[(a + b*x)*(a^2 - b^2*x^2)^p,x]","a x \left(a^2-b^2 x^2\right)^p \left(1-\frac{b^2 x^2}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right)-\frac{\left(a^2-b^2 x^2\right)^{p+1}}{2 b (p+1)}","a x \left(a^2-b^2 x^2\right)^p \left(1-\frac{b^2 x^2}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right)-\frac{\left(a^2-b^2 x^2\right)^{p+1}}{2 b (p+1)}",1,"-1/2*(a^2 - b^2*x^2)^(1 + p)/(b*(1 + p)) + (a*x*(a^2 - b^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b^2*x^2)/a^2])/(1 - (b^2*x^2)/a^2)^p","A",1
970,1,75,55,0.0500186,"\int \frac{\left(a^2-b^2 x^2\right)^p}{a+b x} \, dx","Integrate[(a^2 - b^2*x^2)^p/(a + b*x),x]","-\frac{2^{p-1} (a-b x) \left(\frac{b x}{a}+1\right)^{-p} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a b (p+1)}","\frac{(a-b x) \left(a^2-b^2 x^2\right)^p \, _2F_1\left(1,2 p+1;p+1;\frac{a+b x}{2 a}\right)}{2 a b p}",1,"-((2^(-1 + p)*(a - b*x)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a*b*(1 + p)*(1 + (b*x)/a)^p))","A",1
971,1,75,58,0.0538839,"\int \frac{\left(a^2-b^2 x^2\right)^p}{(a+b x)^2} \, dx","Integrate[(a^2 - b^2*x^2)^p/(a + b*x)^2,x]","-\frac{2^{p-2} (a-b x) \left(\frac{b x}{a}+1\right)^{-p} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a^2 b (p+1)}","-\frac{\left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(1,2 p;p;\frac{a+b x}{2 a}\right)}{2 a b (1-p) (a+b x)^2}",1,"-((2^(-2 + p)*(a - b*x)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a^2*b*(1 + p)*(1 + (b*x)/a)^p))","A",1
972,1,75,62,0.055502,"\int \frac{\left(a^2-b^2 x^2\right)^p}{(a+b x)^3} \, dx","Integrate[(a^2 - b^2*x^2)^p/(a + b*x)^3,x]","-\frac{2^{p-3} (a-b x) \left(\frac{b x}{a}+1\right)^{-p} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a^3 b (p+1)}","-\frac{\left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(1,2 p-1;p-1;\frac{a+b x}{2 a}\right)}{2 a b (2-p) (a+b x)^3}",1,"-((2^(-3 + p)*(a - b*x)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a^3*b*(1 + p)*(1 + (b*x)/a)^p))","A",1
973,1,189,85,0.2596995,"\int (a+b x)^{3/2} \left(a^2-b^2 x^2\right)^p \, dx","Integrate[(a + b*x)^(3/2)*(a^2 - b^2*x^2)^p,x]","\frac{2^{p-1} \sqrt{a+b x} \left(1-\frac{b x}{a}\right)^{-p} \left(\frac{b x}{a}+1\right)^{-2 p-\frac{1}{2}} \left(b^2 (p+1) x^2 (a-b x)^p (a+b x)^p \left(\frac{b x}{2 a}+\frac{1}{2}\right)^p F_1\left(2;-p,-p-\frac{1}{2};3;\frac{b x}{a},-\frac{b x}{a}\right)-2 \sqrt{2} a (a-b x) \left(a^2-b^2 x^2\right)^p \left(1-\frac{b^2 x^2}{a^2}\right)^p \, _2F_1\left(-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right)\right)}{b (p+1)}","-\frac{2^{p+\frac{3}{2}} \sqrt{a+b x} \left(\frac{b x}{a}+1\right)^{-p-\frac{3}{2}} \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(-p-\frac{3}{2},p+1;p+2;\frac{a-b x}{2 a}\right)}{b (p+1)}",1,"(2^(-1 + p)*Sqrt[a + b*x]*(1 + (b*x)/a)^(-1/2 - 2*p)*(b^2*(1 + p)*x^2*(a - b*x)^p*(a + b*x)^p*(1/2 + (b*x)/(2*a))^p*AppellF1[2, -p, -1/2 - p, 3, (b*x)/a, -((b*x)/a)] - 2*Sqrt[2]*a*(a - b*x)*(a^2 - b^2*x^2)^p*(1 - (b^2*x^2)/a^2)^p*Hypergeometric2F1[-1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)]))/(b*(1 + p)*(1 - (b*x)/a)^p)","C",0
974,1,89,88,0.0918561,"\int \sqrt{a+b x} \left(a^2-b^2 x^2\right)^p \, dx","Integrate[Sqrt[a + b*x]*(a^2 - b^2*x^2)^p,x]","\frac{2^{p+\frac{1}{2}} (b x-a) \sqrt{a+b x} \left(\frac{b x}{a}+1\right)^{-p-\frac{1}{2}} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right)}{b (p+1)}","-\frac{2^{p+\frac{1}{2}} \sqrt{a+b x} \left(\frac{b x}{a}+1\right)^{-p-\frac{3}{2}} \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right)}{a b (p+1)}",1,"(2^(1/2 + p)*(-a + b*x)*Sqrt[a + b*x]*(1 + (b*x)/a)^(-1/2 - p)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[-1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p))","A",1
975,1,89,88,0.0941564,"\int \frac{\left(a^2-b^2 x^2\right)^p}{\sqrt{a+b x}} \, dx","Integrate[(a^2 - b^2*x^2)^p/Sqrt[a + b*x],x]","\frac{2^{p-\frac{1}{2}} (b x-a) \left(\frac{b x}{a}+1\right)^{\frac{1}{2}-p} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{b (p+1) \sqrt{a+b x}}","-\frac{2^{p-\frac{1}{2}} \left(\frac{b x}{a}+1\right)^{-p-\frac{1}{2}} \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a b (p+1) \sqrt{a+b x}}",1,"(2^(-1/2 + p)*(-a + b*x)*(1 + (b*x)/a)^(1/2 - p)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[1/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p)*Sqrt[a + b*x])","A",1
976,1,92,88,0.1096788,"\int \frac{\left(a^2-b^2 x^2\right)^p}{(a+b x)^{3/2}} \, dx","Integrate[(a^2 - b^2*x^2)^p/(a + b*x)^(3/2),x]","-\frac{2^{p-\frac{3}{2}} (a-b x) \left(\frac{b x}{a}+1\right)^{\frac{1}{2}-p} \left(a^2-b^2 x^2\right)^p \, _2F_1\left(\frac{3}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a b (p+1) \sqrt{a+b x}}","-\frac{2^{p-\frac{3}{2}} \left(\frac{b x}{a}+1\right)^{-p-\frac{1}{2}} \left(a^2-b^2 x^2\right)^{p+1} \, _2F_1\left(\frac{3}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right)}{a^2 b (p+1) \sqrt{a+b x}}",1,"-((2^(-3/2 + p)*(a - b*x)*(1 + (b*x)/a)^(1/2 - p)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[3/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a*b*(1 + p)*Sqrt[a + b*x]))","A",1
977,1,28,28,0.0051674,"\int \left(-a \left(a^2-b^2 x^2\right)^p+(a+b x) \left(a^2-b^2 x^2\right)^p\right) \, dx","Integrate[-(a*(a^2 - b^2*x^2)^p) + (a + b*x)*(a^2 - b^2*x^2)^p,x]","-\frac{\left(a^2-b^2 x^2\right)^{p+1}}{2 b (p+1)}","-\frac{\left(a^2-b^2 x^2\right)^{p+1}}{2 b (p+1)}",1,"-1/2*(a^2 - b^2*x^2)^(1 + p)/(b*(1 + p))","A",1
978,1,15,15,0.0025578,"\int (d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right) \, dx","Integrate[(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{c (d+e x)^5}{5 e}","\frac{c (d+e x)^5}{5 e}",1,"(c*(d + e*x)^5)/(5*e)","A",1
979,1,15,15,0.0014286,"\int (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right) \, dx","Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{c (d+e x)^4}{4 e}","\frac{c (d+e x)^4}{4 e}",1,"(c*(d + e*x)^4)/(4*e)","A",1
980,1,25,25,0.000075,"\int \left(c d^2+2 c d e x+c e^2 x^2\right) \, dx","Integrate[c*d^2 + 2*c*d*e*x + c*e^2*x^2,x]","c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3","c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3",1,"c*d^2*x + c*d*e*x^2 + (c*e^2*x^3)/3","A",1
981,1,14,14,0.0007085,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x),x]","c \left(d x+\frac{e x^2}{2}\right)","c d x+\frac{1}{2} c e x^2",1,"c*(d*x + (e*x^2)/2)","A",1
982,1,3,3,0.0004119,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^2,x]","c x","c x",1,"c*x","A",1
983,1,11,11,0.0013185,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^3,x]","\frac{c \log (d+e x)}{e}","\frac{c \log (d+e x)}{e}",1,"(c*Log[d + e*x])/e","A",1
984,1,13,13,0.0028011,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^4,x]","-\frac{c}{e (d+e x)}","-\frac{c}{e (d+e x)}",1,"-(c/(e*(d + e*x)))","A",1
985,1,15,15,0.003456,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^5,x]","-\frac{c}{2 e (d+e x)^2}","-\frac{c}{2 e (d+e x)^2}",1,"-1/2*c/(e*(d + e*x)^2)","A",1
986,1,15,15,0.003773,"\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^6} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^6,x]","-\frac{c}{3 e (d+e x)^3}","-\frac{c}{3 e (d+e x)^3}",1,"-1/3*c/(e*(d + e*x)^3)","A",1
987,1,17,17,0.0023974,"\int (d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{c^2 (d+e x)^7}{7 e}","\frac{c^2 (d+e x)^7}{7 e}",1,"(c^2*(d + e*x)^7)/(7*e)","A",1
988,1,17,17,0.001967,"\int (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^2 \, dx","Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{c^2 (d+e x)^6}{6 e}","\frac{c^2 (d+e x)^6}{6 e}",1,"(c^2*(d + e*x)^6)/(6*e)","A",1
989,1,17,17,0.0012106,"\int \left(c d^2+2 c d e x+c e^2 x^2\right)^2 \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{c^2 (d+e x)^5}{5 e}","\frac{c^2 (d+e x)^5}{5 e}",1,"(c^2*(d + e*x)^5)/(5*e)","A",1
990,1,17,17,0.0010786,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{d+e x} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x),x]","\frac{c^2 (d+e x)^4}{4 e}","\frac{c^2 (d+e x)^4}{4 e}",1,"(c^2*(d + e*x)^4)/(4*e)","A",1
991,1,17,17,0.0015227,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^2,x]","\frac{c^2 (d+e x)^3}{3 e}","\frac{c^2 (d+e x)^3}{3 e}",1,"(c^2*(d + e*x)^3)/(3*e)","A",1
992,1,16,17,0.0006812,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^3,x]","c^2 \left(d x+\frac{e x^2}{2}\right)","\frac{c^2 (d+e x)^2}{2 e}",1,"c^2*(d*x + (e*x^2)/2)","A",1
993,1,5,5,0.0002817,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]","c^2 x","c^2 x",1,"c^2*x","A",1
994,1,13,13,0.0013978,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^5,x]","\frac{c^2 \log (d+e x)}{e}","\frac{c^2 \log (d+e x)}{e}",1,"(c^2*Log[d + e*x])/e","A",1
995,1,15,15,0.0023156,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^6,x]","-\frac{c^2}{e (d+e x)}","-\frac{c^2}{e (d+e x)}",1,"-(c^2/(e*(d + e*x)))","A",1
996,1,17,17,0.0022307,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^7,x]","-\frac{c^2}{2 e (d+e x)^2}","-\frac{c^2}{2 e (d+e x)^2}",1,"-1/2*c^2/(e*(d + e*x)^2)","A",1
997,1,17,17,0.0024343,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^8,x]","-\frac{c^2}{3 e (d+e x)^3}","-\frac{c^2}{3 e (d+e x)^3}",1,"-1/3*c^2/(e*(d + e*x)^3)","A",1
998,1,17,17,0.0010074,"\int \frac{(d+e x)^5}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{(d+e x)^4}{4 c e}","\frac{(d+e x)^4}{4 c e}",1,"(d + e*x)^4/(4*c*e)","A",1
999,1,17,17,0.0008741,"\int \frac{(d+e x)^4}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{(d+e x)^3}{3 c e}","\frac{(d+e x)^3}{3 c e}",1,"(d + e*x)^3/(3*c*e)","A",1
1000,1,16,17,0.0006029,"\int \frac{(d+e x)^3}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{d x+\frac{e x^2}{2}}{c}","\frac{(d+e x)^2}{2 c e}",1,"(d*x + (e*x^2)/2)/c","A",1
1001,1,5,5,0.0003534,"\int \frac{(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{x}{c}","\frac{x}{c}",1,"x/c","A",1
1002,1,16,13,0.0018162,"\int \frac{d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{\log (c d+c e x)}{c e}","\frac{\log (d+e x)}{c e}",1,"Log[c*d + c*e*x]/(c*e)","A",1
1003,1,15,15,0.0018793,"\int \frac{1}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-1),x]","-\frac{1}{c e (d+e x)}","-\frac{1}{c e (d+e x)}",1,"-(1/(c*e*(d + e*x)))","A",1
1004,1,17,17,0.0020507,"\int \frac{1}{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)} \, dx","Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)),x]","-\frac{1}{2 c e (d+e x)^2}","-\frac{1}{2 c e (d+e x)^2}",1,"-1/2*1/(c*e*(d + e*x)^2)","A",1
1005,1,17,17,0.0022133,"\int \frac{1}{(d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)),x]","-\frac{1}{3 c e (d+e x)^3}","-\frac{1}{3 c e (d+e x)^3}",1,"-1/3*1/(c*e*(d + e*x)^3)","A",1
1006,1,17,17,0.0038693,"\int \frac{1}{(d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)),x]","-\frac{1}{4 c e (d+e x)^4}","-\frac{1}{4 c e (d+e x)^4}",1,"-1/4*1/(c*e*(d + e*x)^4)","A",1
1007,1,17,17,0.0011341,"\int \frac{(d+e x)^7}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{(d+e x)^4}{4 c^2 e}","\frac{(d+e x)^4}{4 c^2 e}",1,"(d + e*x)^4/(4*c^2*e)","A",1
1008,1,17,17,0.0009665,"\int \frac{(d+e x)^6}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{(d+e x)^3}{3 c^2 e}","\frac{(d+e x)^3}{3 c^2 e}",1,"(d + e*x)^3/(3*c^2*e)","A",1
1009,1,16,17,0.0006918,"\int \frac{(d+e x)^5}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{d x+\frac{e x^2}{2}}{c^2}","\frac{(d+e x)^2}{2 c^2 e}",1,"(d*x + (e*x^2)/2)/c^2","A",1
1010,1,5,5,0.0003675,"\int \frac{(d+e x)^4}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{x}{c^2}","\frac{x}{c^2}",1,"x/c^2","A",1
1011,1,13,13,0.0013858,"\int \frac{(d+e x)^3}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{\log (d+e x)}{c^2 e}","\frac{\log (d+e x)}{c^2 e}",1,"Log[d + e*x]/(c^2*e)","A",1
1012,1,15,15,0.0019958,"\int \frac{(d+e x)^2}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","-\frac{1}{c^2 e (d+e x)}","-\frac{1}{c^2 e (d+e x)}",1,"-(1/(c^2*e*(d + e*x)))","A",1
1013,1,17,17,0.0023269,"\int \frac{d+e x}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","-\frac{1}{2 c^2 e (d+e x)^2}","-\frac{1}{2 c^2 e (d+e x)^2}",1,"-1/2*1/(c^2*e*(d + e*x)^2)","A",1
1014,1,17,17,0.0020516,"\int \frac{1}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-2),x]","-\frac{1}{3 c^2 e (d+e x)^3}","-\frac{1}{3 c^2 e (d+e x)^3}",1,"-1/3*1/(c^2*e*(d + e*x)^3)","A",1
1015,1,17,17,0.0022197,"\int \frac{1}{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2),x]","-\frac{1}{4 c^2 e (d+e x)^4}","-\frac{1}{4 c^2 e (d+e x)^4}",1,"-1/4*1/(c^2*e*(d + e*x)^4)","A",1
1016,1,17,17,0.0051306,"\int \frac{1}{(d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2),x]","-\frac{1}{5 c^2 e (d+e x)^5}","-\frac{1}{5 c^2 e (d+e x)^5}",1,"-1/5*1/(c^2*e*(d + e*x)^5)","A",1
1017,1,17,17,0.0014244,"\int \frac{(d+e x)^9}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^9/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{(d+e x)^4}{4 c^3 e}","\frac{(d+e x)^4}{4 c^3 e}",1,"(d + e*x)^4/(4*c^3*e)","A",1
1018,1,17,17,0.0012438,"\int \frac{(d+e x)^8}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^8/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{(d+e x)^3}{3 c^3 e}","\frac{(d+e x)^3}{3 c^3 e}",1,"(d + e*x)^3/(3*c^3*e)","A",1
1019,1,16,17,0.0007273,"\int \frac{(d+e x)^7}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{d x+\frac{e x^2}{2}}{c^3}","\frac{(d+e x)^2}{2 c^3 e}",1,"(d*x + (e*x^2)/2)/c^3","A",1
1020,1,5,5,0.0004268,"\int \frac{(d+e x)^6}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{x}{c^3}","\frac{x}{c^3}",1,"x/c^3","A",1
1021,1,13,13,0.0020492,"\int \frac{(d+e x)^5}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{\log (d+e x)}{c^3 e}","\frac{\log (d+e x)}{c^3 e}",1,"Log[d + e*x]/(c^3*e)","A",1
1022,1,15,15,0.0026321,"\int \frac{(d+e x)^4}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","-\frac{1}{c^3 e (d+e x)}","-\frac{1}{c^3 e (d+e x)}",1,"-(1/(c^3*e*(d + e*x)))","A",1
1023,1,17,17,0.002963,"\int \frac{(d+e x)^3}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","-\frac{1}{2 c^3 e (d+e x)^2}","-\frac{1}{2 c^3 e (d+e x)^2}",1,"-1/2*1/(c^3*e*(d + e*x)^2)","A",1
1024,1,17,17,0.0031215,"\int \frac{(d+e x)^2}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","-\frac{1}{3 c^3 e (d+e x)^3}","-\frac{1}{3 c^3 e (d+e x)^3}",1,"-1/3*1/(c^3*e*(d + e*x)^3)","A",1
1025,1,17,17,0.0027313,"\int \frac{d+e x}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","-\frac{1}{4 c^3 e (d+e x)^4}","-\frac{1}{4 c^3 e (d+e x)^4}",1,"-1/4*1/(c^3*e*(d + e*x)^4)","A",1
1026,1,17,17,0.0027832,"\int \frac{1}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-3),x]","-\frac{1}{5 c^3 e (d+e x)^5}","-\frac{1}{5 c^3 e (d+e x)^5}",1,"-1/5*1/(c^3*e*(d + e*x)^5)","A",1
1027,1,17,17,0.0049506,"\int \frac{1}{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3),x]","-\frac{1}{6 c^3 e (d+e x)^6}","-\frac{1}{6 c^3 e (d+e x)^6}",1,"-1/6*1/(c^3*e*(d + e*x)^6)","A",1
1028,1,17,17,0.0043514,"\int \frac{1}{(d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3),x]","-\frac{1}{7 c^3 e (d+e x)^7}","-\frac{1}{7 c^3 e (d+e x)^7}",1,"-1/7*1/(c^3*e*(d + e*x)^7)","A",1
1029,1,27,34,0.0111622,"\int (d+e x)^3 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x)^4 \sqrt{c (d+e x)^2}}{5 e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{5 c^2 e}",1,"((d + e*x)^4*Sqrt[c*(d + e*x)^2])/(5*e)","A",1
1030,1,28,39,0.014022,"\int (d+e x)^2 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{3/2}}{4 c e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{4 c e}",1,"((d + e*x)*(c*(d + e*x)^2)^(3/2))/(4*c*e)","A",1
1031,1,23,34,0.0086647,"\int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{\left(c (d+e x)^2\right)^{3/2}}{3 c e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{3 c e}",1,"(c*(d + e*x)^2)^(3/2)/(3*c*e)","A",1
1032,1,31,36,0.0126882,"\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{c x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}","\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}",1,"(c*x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])","A",1
1033,1,21,28,0.0062138,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x),x]","\frac{c x (d+e x)}{\sqrt{c (d+e x)^2}}","\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e}",1,"(c*x*(d + e*x))/Sqrt[c*(d + e*x)^2]","A",1
1034,1,29,40,0.0080592,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^2,x]","\frac{c (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}}","\frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"(c*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])","A",1
1035,1,19,30,0.0080409,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^3,x]","-\frac{c}{e \sqrt{c (d+e x)^2}}","-\frac{c}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-(c/(e*Sqrt[c*(d + e*x)^2]))","A",1
1036,1,27,39,0.0063635,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^4,x]","-\frac{\sqrt{c (d+e x)^2}}{2 e (d+e x)^3}","-\frac{c}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*Sqrt[c*(d + e*x)^2]/(e*(d + e*x)^3)","A",1
1037,1,27,34,0.0074837,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^5,x]","-\frac{\sqrt{c (d+e x)^2}}{3 e (d+e x)^4}","-\frac{c^2}{3 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/3*Sqrt[c*(d + e*x)^2]/(e*(d + e*x)^4)","A",1
1038,1,27,41,0.0064101,"\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^6} \, dx","Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^6,x]","-\frac{\sqrt{c (d+e x)^2}}{4 e (d+e x)^5}","-\frac{c^2}{4 e (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/4*Sqrt[c*(d + e*x)^2]/(e*(d + e*x)^5)","A",1
1039,1,27,34,0.0170145,"\int (d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{(d+e x)^4 \left(c (d+e x)^2\right)^{3/2}}{7 e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{7/2}}{7 c^2 e}",1,"((d + e*x)^4*(c*(d + e*x)^2)^(3/2))/(7*e)","A",1
1040,1,28,39,0.0230624,"\int (d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{5/2}}{6 c e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{6 c e}",1,"((d + e*x)*(c*(d + e*x)^2)^(5/2))/(6*c*e)","A",1
1041,1,23,34,0.0131483,"\int (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{\left(c (d+e x)^2\right)^{5/2}}{5 c e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{5 c e}",1,"(c*(d + e*x)^2)^(5/2)/(5*c*e)","A",1
1042,1,25,36,0.0021684,"\int \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{3/2}}{4 e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{4 e}",1,"((d + e*x)*(c*(d + e*x)^2)^(3/2))/(4*e)","A",1
1043,1,20,31,0.00331,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x),x]","\frac{\left(c (d+e x)^2\right)^{3/2}}{3 e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{3 e}",1,"(c*(d + e*x)^2)^(3/2)/(3*e)","A",1
1044,1,33,37,0.0026994,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^2,x]","\frac{c^2 x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}","\frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}",1,"(c^2*x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])","A",1
1045,1,22,29,0.0094417,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^3,x]","\frac{x \left(c (d+e x)^2\right)^{3/2}}{(d+e x)^3}","\frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e}",1,"(x*(c*(d + e*x)^2)^(3/2))/(d + e*x)^3","A",1
1046,1,31,42,0.0039674,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^4} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^4,x]","\frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}}","\frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"(c^2*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])","A",1
1047,1,25,32,0.0094172,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^5} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^5,x]","-\frac{\left(c (d+e x)^2\right)^{3/2}}{e (d+e x)^4}","-\frac{c^2}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-((c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^4))","A",1
1048,1,27,41,0.0086898,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^6} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]","-\frac{\left(c (d+e x)^2\right)^{3/2}}{2 e (d+e x)^5}","-\frac{c^2}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*(c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^5)","A",1
1049,1,27,34,0.0088142,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{(d+e x)^7} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^7,x]","-\frac{\left(c (d+e x)^2\right)^{3/2}}{3 e (d+e x)^6}","-\frac{c^3}{3 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/3*(c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^6)","A",1
1050,1,27,34,0.0263164,"\int (d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{(d+e x)^4 \left(c (d+e x)^2\right)^{5/2}}{9 e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{9/2}}{9 c^2 e}",1,"((d + e*x)^4*(c*(d + e*x)^2)^(5/2))/(9*e)","A",1
1051,1,28,39,0.0206519,"\int (d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{7/2}}{8 c e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{7/2}}{8 c e}",1,"((d + e*x)*(c*(d + e*x)^2)^(7/2))/(8*c*e)","A",1
1052,1,23,34,0.0190617,"\int (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{\left(c (d+e x)^2\right)^{7/2}}{7 c e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{7/2}}{7 c e}",1,"(c*(d + e*x)^2)^(7/2)/(7*c*e)","A",1
1053,1,25,36,0.0018504,"\int \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{5/2}}{6 e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{6 e}",1,"((d + e*x)*(c*(d + e*x)^2)^(5/2))/(6*e)","A",1
1054,1,20,31,0.0056504,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x),x]","\frac{\left(c (d+e x)^2\right)^{5/2}}{5 e}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{5 e}",1,"(c*(d + e*x)^2)^(5/2)/(5*e)","A",1
1055,1,26,37,0.0054957,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^2,x]","\frac{c (d+e x) \left(c (d+e x)^2\right)^{3/2}}{4 e}","\frac{c (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{4 e}",1,"(c*(d + e*x)*(c*(d + e*x)^2)^(3/2))/(4*e)","A",1
1056,1,21,32,0.0033477,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]","\frac{c \left(c (d+e x)^2\right)^{3/2}}{3 e}","\frac{c \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{3 e}",1,"(c*(c*(d + e*x)^2)^(3/2))/(3*e)","A",1
1057,1,33,39,0.0023745,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^4,x]","\frac{c^3 x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}","\frac{c^2 (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}",1,"(c^3*x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])","A",1
1058,1,23,31,0.0061293,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^5} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^5,x]","\frac{c^3 x (d+e x)}{\sqrt{c (d+e x)^2}}","\frac{c^2 \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e}",1,"(c^3*x*(d + e*x))/Sqrt[c*(d + e*x)^2]","A",1
1059,1,31,42,0.0041059,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^6} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]","\frac{c^3 (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}}","\frac{c^3 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"(c^3*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])","A",1
1060,1,25,32,0.0132969,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^7} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^7,x]","-\frac{\left(c (d+e x)^2\right)^{5/2}}{e (d+e x)^6}","-\frac{c^3}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-((c*(d + e*x)^2)^(5/2)/(e*(d + e*x)^6))","A",1
1061,1,27,41,0.0110676,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}{(d+e x)^8} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^8,x]","-\frac{\left(c (d+e x)^2\right)^{5/2}}{2 e (d+e x)^7}","-\frac{c^3}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*(c*(d + e*x)^2)^(5/2)/(e*(d + e*x)^7)","A",1
1062,1,27,39,0.0082323,"\int \frac{(d+e x)^4}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[(d + e*x)^4/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x)^5}{4 e \sqrt{c (d+e x)^2}}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{4 c^2 e}",1,"(d + e*x)^5/(4*e*Sqrt[c*(d + e*x)^2])","A",1
1063,1,27,34,0.0060967,"\int \frac{(d+e x)^3}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x)^4}{3 e \sqrt{c (d+e x)^2}}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}{3 c^2 e}",1,"(d + e*x)^4/(3*e*Sqrt[c*(d + e*x)^2])","A",1
1064,1,30,39,0.0033828,"\int \frac{(d+e x)^2}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}","\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e}",1,"(x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])","A",1
1065,1,20,31,0.002851,"\int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[(d + e*x)/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{x (d+e x)}{\sqrt{c (d+e x)^2}}","\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e}",1,"(x*(d + e*x))/Sqrt[c*(d + e*x)^2]","A",1
1066,1,28,39,0.005498,"\int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[1/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}}","\frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"((d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])","A",1
1067,1,18,29,0.0113717,"\int \frac{1}{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]","-\frac{1}{e \sqrt{c (d+e x)^2}}","-\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-(1/(e*Sqrt[c*(d + e*x)^2]))","A",1
1068,1,26,38,0.0101638,"\int \frac{1}{(d+e x)^2 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]","-\frac{c (d+e x)}{2 e \left(c (d+e x)^2\right)^{3/2}}","-\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*(c*(d + e*x))/(e*(c*(d + e*x)^2)^(3/2))","A",1
1069,1,21,32,0.0154814,"\int \frac{1}{(d+e x)^3 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]","-\frac{c}{3 e \left(c (d+e x)^2\right)^{3/2}}","-\frac{c}{3 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/3*c/(e*(c*(d + e*x)^2)^(3/2))","A",1
1070,1,27,39,0.009679,"\int \frac{1}{(d+e x)^4 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]","-\frac{1}{4 e (d+e x)^3 \sqrt{c (d+e x)^2}}","-\frac{c}{4 e (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/4*1/(e*(d + e*x)^3*Sqrt[c*(d + e*x)^2])","A",1
1071,1,33,39,0.005316,"\int \frac{(d+e x)^4}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{x (d+e x) (2 d+e x)}{2 c \sqrt{c (d+e x)^2}}","\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e}",1,"(x*(d + e*x)*(2*d + e*x))/(2*c*Sqrt[c*(d + e*x)^2])","A",1
1072,1,22,31,0.0067345,"\int \frac{(d+e x)^3}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{x (d+e x)^3}{\left(c (d+e x)^2\right)^{3/2}}","\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e}",1,"(x*(d + e*x)^3)/(c*(d + e*x)^2)^(3/2)","A",1
1073,1,31,42,0.0033745,"\int \frac{(d+e x)^2}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{(d+e x) \log (d+e x)}{c e \sqrt{c (d+e x)^2}}","\frac{(d+e x) \log (d+e x)}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*(d + e*x)^2])","A",1
1074,1,21,32,0.0068821,"\int \frac{d+e x}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","-\frac{1}{c e \sqrt{c (d+e x)^2}}","-\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-(1/(c*e*Sqrt[c*(d + e*x)^2]))","A",1
1075,1,25,41,0.0031583,"\int \frac{1}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-3/2),x]","-\frac{d+e x}{2 e \left(c (d+e x)^2\right)^{3/2}}","-\frac{1}{2 c e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*(d + e*x)/(e*(c*(d + e*x)^2)^(3/2))","A",1
1076,1,20,31,0.0198861,"\int \frac{1}{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]","-\frac{1}{3 e \left(c (d+e x)^2\right)^{3/2}}","-\frac{1}{3 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/3*1/(e*(c*(d + e*x)^2)^(3/2))","A",1
1077,1,26,38,0.017494,"\int \frac{1}{(d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]","-\frac{c (d+e x)}{4 e \left(c (d+e x)^2\right)^{5/2}}","-\frac{1}{4 e (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/4*(c*(d + e*x))/(e*(c*(d + e*x)^2)^(5/2))","A",1
1078,1,21,32,0.0247904,"\int \frac{1}{(d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]","-\frac{c}{5 e \left(c (d+e x)^2\right)^{5/2}}","-\frac{c}{5 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}",1,"-1/5*c/(e*(c*(d + e*x)^2)^(5/2))","A",1
1079,1,33,39,0.0033288,"\int \frac{(d+e x)^6}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{x (d+e x) (2 d+e x)}{2 c^2 \sqrt{c (d+e x)^2}}","\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e}",1,"(x*(d + e*x)*(2*d + e*x))/(2*c^2*Sqrt[c*(d + e*x)^2])","A",1
1080,1,23,31,0.0027344,"\int \frac{(d+e x)^5}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{x (d+e x)}{c^2 \sqrt{c (d+e x)^2}}","\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e}",1,"(x*(d + e*x))/(c^2*Sqrt[c*(d + e*x)^2])","A",1
1081,1,31,42,0.005857,"\int \frac{(d+e x)^4}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","\frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c (d+e x)^2}}","\frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"((d + e*x)*Log[d + e*x])/(c^2*e*Sqrt[c*(d + e*x)^2])","A",1
1082,1,21,32,0.0121363,"\int \frac{(d+e x)^3}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","-\frac{1}{c^2 e \sqrt{c (d+e x)^2}}","-\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-(1/(c^2*e*Sqrt[c*(d + e*x)^2]))","A",1
1083,1,28,41,0.0071636,"\int \frac{(d+e x)^2}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","-\frac{d+e x}{2 c e \left(c (d+e x)^2\right)^{3/2}}","-\frac{1}{2 c^2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"-1/2*(d + e*x)/(c*e*(c*(d + e*x)^2)^(3/2))","A",1
1084,1,30,34,0.0109406,"\int \frac{d+e x}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]","-\frac{\sqrt{c (d+e x)^2}}{3 c^3 e (d+e x)^4}","-\frac{1}{3 c e \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/3*Sqrt[c*(d + e*x)^2]/(c^3*e*(d + e*x)^4)","A",1
1085,1,25,41,0.002876,"\int \frac{1}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-5/2),x]","-\frac{d+e x}{4 e \left(c (d+e x)^2\right)^{5/2}}","-\frac{1}{4 c e (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"-1/4*(d + e*x)/(e*(c*(d + e*x)^2)^(5/2))","A",1
1086,1,20,31,0.0246405,"\int \frac{1}{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]","-\frac{1}{5 e \left(c (d+e x)^2\right)^{5/2}}","-\frac{1}{5 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}",1,"-1/5*1/(e*(c*(d + e*x)^2)^(5/2))","A",1
1087,1,26,38,0.0169911,"\int \frac{1}{(d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]","-\frac{c (d+e x)}{6 e \left(c (d+e x)^2\right)^{7/2}}","-\frac{1}{6 e (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}}",1,"-1/6*(c*(d + e*x))/(e*(c*(d + e*x)^2)^(7/2))","A",1
1088,1,21,32,0.0276701,"\int \frac{1}{(d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]","-\frac{c}{7 e \left(c (d+e x)^2\right)^{7/2}}","-\frac{c}{7 e \left(c d^2+2 c d e x+c e^2 x^2\right)^{7/2}}",1,"-1/7*c/(e*(c*(d + e*x)^2)^(7/2))","A",1
1089,1,22,21,0.0165879,"\int (d+e x)^m \left(c d^2+2 c d e x+c e^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{c^2 (d+e x)^{m+5}}{e m+5 e}","\frac{c^2 (d+e x)^{m+5}}{e (m+5)}",1,"(c^2*(d + e*x)^(5 + m))/(5*e + e*m)","A",1
1090,1,20,19,0.0141836,"\int (d+e x)^m \left(c d^2+2 c d e x+c e^2 x^2\right) \, dx","Integrate[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{c (d+e x)^{m+3}}{e m+3 e}","\frac{c (d+e x)^{m+3}}{e (m+3)}",1,"(c*(d + e*x)^(3 + m))/(3*e + e*m)","A",1
1091,1,21,24,0.0143212,"\int \frac{(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]","\frac{(d+e x)^{m-1}}{c e (m-1)}","-\frac{(d+e x)^{m-1}}{c e (1-m)}",1,"(d + e*x)^(-1 + m)/(c*e*(-1 + m))","A",1
1092,1,21,24,0.0142131,"\int \frac{(d+e x)^m}{\left(c d^2+2 c d e x+c e^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]","\frac{(d+e x)^{m-3}}{c^2 e (m-3)}","-\frac{(d+e x)^{m-3}}{c^2 e (3-m)}",1,"(d + e*x)^(-3 + m)/(c^2*e*(-3 + m))","A",1
1093,1,21,24,0.0146747,"\int \frac{(d+e x)^m}{\left(c d^2+2 c d e x+c e^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]","\frac{(d+e x)^{m-5}}{c^3 e (m-5)}","-\frac{(d+e x)^{m-5}}{c^3 e (5-m)}",1,"(d + e*x)^(-5 + m)/(c^3*e*(-5 + m))","A",1
1094,1,31,42,0.0283627,"\int (d+e x)^m \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{\left(c (d+e x)^2\right)^{3/2} (d+e x)^{m+1}}{e (m+4)}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2} (d+e x)^{m+1}}{e (m+4)}",1,"((d + e*x)^(1 + m)*(c*(d + e*x)^2)^(3/2))/(e*(4 + m))","A",1
1095,1,31,42,0.0168091,"\int (d+e x)^m \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{\sqrt{c (d+e x)^2} (d+e x)^{m+1}}{e (m+2)}","\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+2)}",1,"((d + e*x)^(1 + m)*Sqrt[c*(d + e*x)^2])/(e*(2 + m))","A",1
1096,1,29,40,0.0123133,"\int \frac{(d+e x)^m}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]","\frac{(d+e x)^{m+1}}{e m \sqrt{c (d+e x)^2}}","\frac{(d+e x)^{m+1}}{e m \sqrt{c d^2+2 c d e x+c e^2 x^2}}",1,"(d + e*x)^(1 + m)/(e*m*Sqrt[c*(d + e*x)^2])","A",1
1097,1,31,45,0.0209843,"\int \frac{(d+e x)^m}{\left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]","\frac{(d+e x)^{m+1}}{e (m-2) \left(c (d+e x)^2\right)^{3/2}}","-\frac{(d+e x)^{m+1}}{e (2-m) \left(c d^2+2 c d e x+c e^2 x^2\right)^{3/2}}",1,"(d + e*x)^(1 + m)/(e*(-2 + m)*(c*(d + e*x)^2)^(3/2))","A",1
1098,1,32,43,0.0177783,"\int (d+e x)^m \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x)^{m+1} \left(c (d+e x)^2\right)^p}{e m+2 e p+e}","\frac{(d+e x)^{m+1} \left(c d^2+2 c d e x+c e^2 x^2\right)^p}{e (m+2 p+1)}",1,"((d + e*x)^(1 + m)*(c*(d + e*x)^2)^p)/(e + e*m + 2*e*p)","A",1
1099,1,31,44,0.0167571,"\int (d+e x)^p \left(c d^2+2 c d e x+c e^2 x^2\right)^{-p} \, dx","Integrate[(d + e*x)^p/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x)^{p+1} \left(c (d+e x)^2\right)^{-p}}{e-e p}","\frac{(d+e x)^{p+1} \left(c d^2+2 c d e x+c e^2 x^2\right)^{-p}}{e (1-p)}",1,"(d + e*x)^(1 + p)/((e - e*p)*(c*(d + e*x)^2)^p)","A",1
1100,1,30,39,0.0167197,"\int (d+e x)^3 \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x)^4 \left(c (d+e x)^2\right)^p}{2 e (p+2)}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{p+2}}{2 c^2 e (p+2)}",1,"((d + e*x)^4*(c*(d + e*x)^2)^p)/(2*e*(2 + p))","A",1
1101,1,32,43,0.019463,"\int (d+e x)^2 \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x) \left(c (d+e x)^2\right)^{p+1}}{c e (2 p+3)}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{p+1}}{c e (2 p+3)}",1,"((d + e*x)*(c*(d + e*x)^2)^(1 + p))/(c*e*(3 + 2*p))","A",1
1102,1,28,39,0.0145256,"\int (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{\left(c (d+e x)^2\right)^{p+1}}{2 c e (p+1)}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^{p+1}}{2 c e (p+1)}",1,"(c*(d + e*x)^2)^(1 + p)/(2*c*e*(1 + p))","A",1
1103,1,25,38,0.0102665,"\int \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x) \left(c (d+e x)^2\right)^p}{2 e p+e}","\frac{(d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^p}{e (2 p+1)}",1,"((d + e*x)*(c*(d + e*x)^2)^p)/(e + 2*e*p)","A",1
1104,1,21,32,0.0065956,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p/(d + e*x),x]","\frac{\left(c (d+e x)^2\right)^p}{2 e p}","\frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^p}{2 e p}",1,"(c*(d + e*x)^2)^p/(2*e*p)","A",1
1105,1,30,42,0.0171851,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p/(d + e*x)^2,x]","\frac{c (d+e x) \left(c (d+e x)^2\right)^{p-1}}{e (2 p-1)}","-\frac{c (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{p-1}}{e (1-2 p)}",1,"(c*(d + e*x)*(c*(d + e*x)^2)^(-1 + p))/(e*(-1 + 2*p))","A",1
1106,1,26,39,0.0127367,"\int \frac{\left(c d^2+2 c d e x+c e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p/(d + e*x)^3,x]","\frac{c \left(c (d+e x)^2\right)^{p-1}}{2 e (p-1)}","-\frac{c \left(c d^2+2 c d e x+c e^2 x^2\right)^{p-1}}{2 e (1-p)}",1,"(c*(c*(d + e*x)^2)^(-1 + p))/(2*e*(-1 + p))","A",1
1107,1,30,41,0.007032,"\int (d+e x)^{-1-2 p} \left(c d^2+2 c d e x+c e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^(-1 - 2*p)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x)^{-2 p} \log (d+e x) \left(c (d+e x)^2\right)^p}{e}","\frac{(d+e x)^{-2 p} \log (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^p}{e}",1,"((c*(d + e*x)^2)^p*Log[d + e*x])/(e*(d + e*x)^(2*p))","A",1
1108,1,32,43,0.006146,"\int (d+e x)^{-1+2 p} \left(c d^2+2 c d e x+c e^2 x^2\right)^{-p} \, dx","Integrate[(d + e*x)^(-1 + 2*p)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]","\frac{(d+e x)^{2 p} \log (d+e x) \left(c (d+e x)^2\right)^{-p}}{e}","\frac{(d+e x)^{2 p} \log (d+e x) \left(c d^2+2 c d e x+c e^2 x^2\right)^{-p}}{e}",1,"((d + e*x)^(2*p)*Log[d + e*x])/(e*(c*(d + e*x)^2)^p)","A",1
1109,1,102,45,0.0194744,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2),x]","d^4 \left(a b^4 x+\frac{8}{5} c^3 x^5 \left(2 a c+7 b^2\right)+8 b c^2 x^4 \left(a c+b^2\right)+b^2 c x^3 \left(8 a c+3 b^2\right)+\frac{1}{2} b^3 x^2 \left(8 a c+b^2\right)+8 b c^4 x^6+\frac{16 c^5 x^7}{7}\right)","\frac{d^4 (b+2 c x)^7}{56 c^2}-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^5}{40 c^2}",1,"d^4*(a*b^4*x + (b^3*(b^2 + 8*a*c)*x^2)/2 + b^2*c*(3*b^2 + 8*a*c)*x^3 + 8*b*c^2*(b^2 + a*c)*x^4 + (8*c^3*(7*b^2 + 2*a*c)*x^5)/5 + 8*b*c^4*x^6 + (16*c^5*x^7)/7)","B",1
1110,1,66,45,0.0157048,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2),x]","\frac{1}{6} d^3 x (b+c x) \left(6 a \left(b^2+2 b c x+2 c^2 x^2\right)+x \left(3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right)\right)","\frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left(b^2-4 a c\right) (b+2 c x)^4}{32 c^2}",1,"(d^3*x*(b + c*x)*(6*a*(b^2 + 2*b*c*x + 2*c^2*x^2) + x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/6","A",1
1111,1,64,45,0.0100938,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2),x]","d^2 \left(\frac{1}{3} c x^3 \left(4 a c+5 b^2\right)+\frac{1}{2} b x^2 \left(4 a c+b^2\right)+a b^2 x+2 b c^2 x^4+\frac{4 c^3 x^5}{5}\right)","\frac{d^2 (b+2 c x)^5}{40 c^2}-\frac{d^2 \left(b^2-4 a c\right) (b+2 c x)^3}{24 c^2}",1,"d^2*(a*b^2*x + (b*(b^2 + 4*a*c)*x^2)/2 + (c*(5*b^2 + 4*a*c)*x^3)/3 + 2*b*c^2*x^4 + (4*c^3*x^5)/5)","A",1
1112,1,22,17,0.0048226,"\int (b d+2 c d x) \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2),x]","\frac{1}{2} d x (b+c x) (2 a+x (b+c x))","\frac{1}{2} d \left(a+b x+c x^2\right)^2",1,"(d*x*(b + c*x)*(2*a + x*(b + c*x)))/2","A",1
1113,1,37,48,0.0134543,"\int \frac{a+b x+c x^2}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]","\frac{2 c x (b+c x)-\left(b^2-4 a c\right) \log (b+2 c x)}{8 c^2 d}","-\frac{\left(b^2-4 a c\right) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d}",1,"(2*c*x*(b + c*x) - (b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)","A",1
1114,1,41,38,0.0112968,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^2,x]","\frac{\frac{b^2-4 a c}{8 c^2 (b+2 c x)}+\frac{b+2 c x}{8 c^2}}{d^2}","\frac{b^2-4 a c}{8 c^2 d^2 (b+2 c x)}+\frac{x}{4 c d^2}",1,"((b^2 - 4*a*c)/(8*c^2*(b + 2*c*x)) + (b + 2*c*x)/(8*c^2))/d^2","A",1
1115,1,37,44,0.0160078,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^3,x]","\frac{\frac{b^2-4 a c}{(b+2 c x)^2}+2 \log (b+2 c x)}{16 c^2 d^3}","\frac{b^2-4 a c}{16 c^2 d^3 (b+2 c x)^2}+\frac{\log (b+2 c x)}{8 c^2 d^3}",1,"((b^2 - 4*a*c)/(b + 2*c*x)^2 + 2*Log[b + 2*c*x])/(16*c^2*d^3)","A",1
1116,1,38,45,0.0156243,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^4,x]","-\frac{2 c \left(a+3 c x^2\right)+b^2+6 b c x}{12 c^2 d^4 (b+2 c x)^3}","\frac{b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac{1}{8 c^2 d^4 (b+2 c x)}",1,"-1/12*(b^2 + 6*b*c*x + 2*c*(a + 3*c*x^2))/(c^2*d^4*(b + 2*c*x)^3)","A",1
1117,1,38,37,0.015078,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^5,x]","-\frac{4 c \left(a+2 c x^2\right)+b^2+8 b c x}{32 c^2 d^5 (b+2 c x)^4}","\frac{\left(a+b x+c x^2\right)^2}{2 d^5 \left(b^2-4 a c\right) (b+2 c x)^4}",1,"-1/32*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2))/(c^2*d^5*(b + 2*c*x)^4)","A",1
1118,1,43,45,0.0169209,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^6,x]","\frac{\frac{b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac{1}{24 c^2 (b+2 c x)^3}}{d^6}","\frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3}",1,"((b^2 - 4*a*c)/(40*c^2*(b + 2*c*x)^5) - 1/(24*c^2*(b + 2*c*x)^3))/d^6","A",1
1119,1,43,45,0.0146049,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^7} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^7,x]","\frac{\frac{b^2-4 a c}{48 c^2 (b+2 c x)^6}-\frac{1}{32 c^2 (b+2 c x)^4}}{d^7}","\frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4}",1,"((b^2 - 4*a*c)/(48*c^2*(b + 2*c*x)^6) - 1/(32*c^2*(b + 2*c*x)^4))/d^7","A",1
1120,1,39,45,0.0160529,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^8} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^8,x]","\frac{5 \left(b^2-4 a c\right)-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7}","\frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5}",1,"(5*(b^2 - 4*a*c) - 7*(b + 2*c*x)^2)/(280*c^2*d^8*(b + 2*c*x)^7)","A",1
1121,1,168,73,0.0598298,"\int (b d+2 c d x)^5 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]","\frac{1}{15} d^5 x (b+c x) \left(5 a^2 \left(3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right)+5 a x \left(3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right)+x^2 (b+c x)^2 \left(5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right)\right)","-\frac{d^5 \left(b^2-4 a c\right) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left(b^2-4 a c\right)^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3}",1,"(d^5*x*(b + c*x)*(5*a^2*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16*c^4*x^4) + x^2*(b + c*x)^2*(5*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + 5*a*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 + 88*b^2*c^3*x^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/15","B",1
1122,1,179,73,0.026451,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]","d^4 \left(a^2 b^4 x+\frac{1}{5} c^2 x^5 \left(16 a^2 c^2+112 a b^2 c+41 b^4\right)+\frac{1}{2} b c x^4 \left(16 a^2 c^2+32 a b^2 c+5 b^4\right)+\frac{1}{3} b^2 x^3 \left(24 a^2 c^2+18 a b^2 c+b^4\right)+\frac{8}{7} c^4 x^7 \left(4 a c+13 b^2\right)+\frac{4}{3} b c^3 x^6 \left(12 a c+11 b^2\right)+a b^3 x^2 \left(4 a c+b^2\right)+8 b c^5 x^8+\frac{16 c^6 x^9}{9}\right)","-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^7}{112 c^3}+\frac{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^5}{160 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3}",1,"d^4*(a^2*b^4*x + a*b^3*(b^2 + 4*a*c)*x^2 + (b^2*(b^4 + 18*a*b^2*c + 24*a^2*c^2)*x^3)/3 + (b*c*(5*b^4 + 32*a*b^2*c + 16*a^2*c^2)*x^4)/2 + (c^2*(41*b^4 + 112*a*b^2*c + 16*a^2*c^2)*x^5)/5 + (4*b*c^3*(11*b^2 + 12*a*c)*x^6)/3 + (8*c^4*(13*b^2 + 4*a*c)*x^7)/7 + 8*b*c^5*x^8 + (16*c^6*x^9)/9)","B",1
1123,1,97,55,0.0204159,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^2,x]","\frac{1}{3} d^3 x (b+c x) \left(3 a^2 \left(b^2+2 b c x+2 c^2 x^2\right)+a x \left(3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right)+x^2 (b+c x)^2 \left(b^2+3 b c x+3 c^2 x^2\right)\right)","\frac{1}{12} d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3+\frac{1}{4} d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^3",1,"(d^3*x*(b + c*x)*(3*a^2*(b^2 + 2*b*c*x + 2*c^2*x^2) + x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + a*x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/3","A",1
1124,1,111,73,0.0187307,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2,x]","d^2 \left(a^2 b^2 x+\frac{1}{3} x^3 \left(4 a^2 c^2+10 a b^2 c+b^4\right)+\frac{1}{5} c^2 x^5 \left(8 a c+13 b^2\right)+\frac{1}{2} b c x^4 \left(8 a c+3 b^2\right)+a b x^2 \left(2 a c+b^2\right)+2 b c^3 x^6+\frac{4 c^4 x^7}{7}\right)","-\frac{d^2 \left(b^2-4 a c\right) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left(b^2-4 a c\right)^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3}",1,"d^2*(a^2*b^2*x + a*b*(b^2 + 2*a*c)*x^2 + ((b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*c*(3*b^2 + 8*a*c)*x^4)/2 + (c^2*(13*b^2 + 8*a*c)*x^5)/5 + 2*b*c^3*x^6 + (4*c^4*x^7)/7)","A",1
1125,1,37,17,0.0095176,"\int (b d+2 c d x) \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^2,x]","\frac{1}{3} d x (b+c x) \left(3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right)","\frac{1}{3} d \left(a+b x+c x^2\right)^3",1,"(d*x*(b + c*x)*(3*a^2 + 3*a*x*(b + c*x) + x^2*(b + c*x)^2))/3","B",1
1126,1,61,72,0.0274965,"\int \frac{\left(a+b x+c x^2\right)^2}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x),x]","\frac{2 c x (b+c x) \left(2 c \left(4 a+c x^2\right)-b^2+2 b c x\right)+\left(b^2-4 a c\right)^2 \log (b+2 c x)}{32 c^3 d}","-\frac{\left(b^2-4 a c\right) (b+2 c x)^2}{32 c^3 d}+\frac{\left(b^2-4 a c\right)^2 \log (b+2 c x)}{32 c^3 d}+\frac{(b+2 c x)^4}{128 c^3 d}",1,"(2*c*x*(b + c*x)*(-b^2 + 2*b*c*x + 2*c*(4*a + c*x^2)) + (b^2 - 4*a*c)^2*Log[b + 2*c*x])/(32*c^3*d)","A",1
1127,1,59,72,0.0529535,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^2,x]","\frac{-\frac{3 \left(b^2-4 a c\right)^2}{c^3 (b+2 c x)}-\frac{6 x \left(b^2-8 a c\right)}{c^2}+\frac{12 b x^2}{c}+8 x^3}{96 d^2}","-\frac{\left(b^2-4 a c\right)^2}{32 c^3 d^2 (b+2 c x)}-\frac{x \left(b^2-8 a c\right)}{16 c^2 d^2}+\frac{b x^2}{8 c d^2}+\frac{x^3}{12 d^2}",1,"((-6*(b^2 - 8*a*c)*x)/c^2 + (12*b*x^2)/c + 8*x^3 - (3*(b^2 - 4*a*c)^2)/(c^3*(b + 2*c*x)))/(96*d^2)","A",1
1128,1,61,79,0.0407671,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^3,x]","\frac{-\frac{\left(b^2-4 a c\right)^2}{(b+2 c x)^2}-4 \left(b^2-4 a c\right) \log (b+2 c x)+4 b c x+4 c^2 x^2}{64 c^3 d^3}","-\frac{\left(b^2-4 a c\right)^2}{64 c^3 d^3 (b+2 c x)^2}-\frac{\left(b^2-4 a c\right) \log (b+2 c x)}{16 c^3 d^3}+\frac{b x}{16 c^2 d^3}+\frac{x^2}{16 c d^3}",1,"(4*b*c*x + 4*c^2*x^2 - (b^2 - 4*a*c)^2/(b + 2*c*x)^2 - 4*(b^2 - 4*a*c)*Log[b + 2*c*x])/(64*c^3*d^3)","A",1
1129,1,53,66,0.0555608,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^4,x]","\frac{-\frac{\left(b^2-4 a c\right)^2}{(b+2 c x)^3}+\frac{6 \left(b^2-4 a c\right)}{b+2 c x}+6 c x}{96 c^3 d^4}","-\frac{\left(b^2-4 a c\right)^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac{x}{16 c^2 d^4}",1,"(6*c*x - (b^2 - 4*a*c)^2/(b + 2*c*x)^3 + (6*(b^2 - 4*a*c))/(b + 2*c*x))/(96*c^3*d^4)","A",1
1130,1,59,72,0.0345656,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^5,x]","\frac{\frac{\left(b^2-4 a c\right) \left(4 c \left(a+4 c x^2\right)+3 b^2+16 b c x\right)}{(b+2 c x)^4}+4 \log (b+2 c x)}{128 c^3 d^5}","-\frac{\left(b^2-4 a c\right)^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5}",1,"(((b^2 - 4*a*c)*(3*b^2 + 16*b*c*x + 4*c*(a + 4*c*x^2)))/(b + 2*c*x)^4 + 4*Log[b + 2*c*x])/(128*c^3*d^5)","A",1
1131,1,59,73,0.0334673,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]","\frac{10 \left(b^2-4 a c\right) (b+2 c x)^2-3 \left(b^2-4 a c\right)^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5}","-\frac{\left(b^2-4 a c\right)^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)}",1,"(-3*(b^2 - 4*a*c)^2 + 10*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 15*(b + 2*c*x)^4)/(480*c^3*d^6*(b + 2*c*x)^5)","A",1
1132,1,65,37,0.0293266,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^7} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^7,x]","-\frac{16 a^2 c^2-3 \left(b^2-4 a c\right) (b+2 c x)^2-8 a b^2 c+b^4+3 (b+2 c x)^4}{192 c^3 d^7 (b+2 c x)^6}","\frac{\left(a+b x+c x^2\right)^3}{3 d^7 \left(b^2-4 a c\right) (b+2 c x)^6}",1,"-1/192*(b^4 - 8*a*b^2*c + 16*a^2*c^2 - 3*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 3*(b + 2*c*x)^4)/(c^3*d^7*(b + 2*c*x)^6)","A",1
1133,1,59,73,0.0317834,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^8} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]","\frac{42 \left(b^2-4 a c\right) (b+2 c x)^2-15 \left(b^2-4 a c\right)^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7}","-\frac{\left(b^2-4 a c\right)^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3}",1,"(-15*(b^2 - 4*a*c)^2 + 42*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 35*(b + 2*c*x)^4)/(3360*c^3*d^8*(b + 2*c*x)^7)","A",1
1134,1,59,73,0.0261766,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^9} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]","\frac{8 \left(b^2-4 a c\right) (b+2 c x)^2-3 \left(b^2-4 a c\right)^2-6 (b+2 c x)^4}{768 c^3 d^9 (b+2 c x)^8}","-\frac{\left(b^2-4 a c\right)^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4}",1,"(-3*(b^2 - 4*a*c)^2 + 8*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 6*(b + 2*c*x)^4)/(768*c^3*d^9*(b + 2*c*x)^8)","A",1
1135,1,59,73,0.0401268,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x]","\frac{90 \left(b^2-4 a c\right) (b+2 c x)^2-35 \left(b^2-4 a c\right)^2-63 (b+2 c x)^4}{10080 c^3 d^{10} (b+2 c x)^9}","-\frac{\left(b^2-4 a c\right)^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac{b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac{1}{160 c^3 d^{10} (b+2 c x)^5}",1,"(-35*(b^2 - 4*a*c)^2 + 90*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 63*(b + 2*c*x)^4)/(10080*c^3*d^10*(b + 2*c*x)^9)","A",1
1136,1,59,73,0.0324144,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{11}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^11,x]","\frac{15 \left(b^2-4 a c\right) (b+2 c x)^2-6 \left(b^2-4 a c\right)^2-10 (b+2 c x)^4}{1920 c^3 d^{11} (b+2 c x)^{10}}","-\frac{\left(b^2-4 a c\right)^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6}",1,"(-6*(b^2 - 4*a*c)^2 + 15*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 10*(b + 2*c*x)^4)/(1920*c^3*d^11*(b + 2*c*x)^10)","A",1
1137,1,224,101,0.0826445,"\int (b d+2 c d x)^5 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^3,x]","\frac{1}{60} d^5 x (b+c x) \left(20 a^3 \left(3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right)+30 a^2 x \left(3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right)+12 a x^2 (b+c x)^2 \left(5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right)+x^3 (b+c x)^3 \left(15 b^4+96 b^3 c x+256 b^2 c^2 x^2+320 b c^3 x^3+160 c^4 x^4\right)\right)","-\frac{3 d^5 \left(b^2-4 a c\right) (b+2 c x)^{10}}{1280 c^4}+\frac{3 d^5 \left(b^2-4 a c\right)^2 (b+2 c x)^8}{1024 c^4}-\frac{d^5 \left(b^2-4 a c\right)^3 (b+2 c x)^6}{768 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4}",1,"(d^5*x*(b + c*x)*(20*a^3*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16*c^4*x^4) + 12*a*x^2*(b + c*x)^2*(5*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + x^3*(b + c*x)^3*(15*b^4 + 96*b^3*c*x + 256*b^2*c^2*x^2 + 320*b*c^3*x^3 + 160*c^4*x^4) + 30*a^2*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 + 88*b^2*c^3*x^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/60","B",1
1138,1,259,101,0.0398409,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3,x]","d^4 \left(a^3 b^4 x+\frac{1}{2} b c^2 x^6 \left(48 a^2 c^2+88 a b^2 c+17 b^4\right)+a b^2 x^3 \left(8 a^2 c^2+9 a b^2 c+b^4\right)+\frac{3}{7} c^3 x^7 \left(16 a^2 c^2+104 a b^2 c+43 b^4\right)+\frac{1}{2} a^2 b^3 x^2 \left(8 a c+3 b^2\right)+\frac{1}{5} c x^5 \left(16 a^3 c^3+168 a^2 b^2 c^2+123 a b^4 c+11 b^6\right)+\frac{1}{4} b x^4 \left(32 a^3 c^3+96 a^2 b^2 c^2+30 a b^4 c+b^6\right)+\frac{8}{3} c^5 x^9 \left(2 a c+7 b^2\right)+24 b c^4 x^8 \left(a c+b^2\right)+8 b c^6 x^{10}+\frac{16 c^7 x^{11}}{11}\right)","-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^9}{384 c^4}+\frac{3 d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^7}{896 c^4}-\frac{d^4 \left(b^2-4 a c\right)^3 (b+2 c x)^5}{640 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4}",1,"d^4*(a^3*b^4*x + (a^2*b^3*(3*b^2 + 8*a*c)*x^2)/2 + a*b^2*(b^4 + 9*a*b^2*c + 8*a^2*c^2)*x^3 + (b*(b^6 + 30*a*b^4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*x^4)/4 + (c*(11*b^6 + 123*a*b^4*c + 168*a^2*b^2*c^2 + 16*a^3*c^3)*x^5)/5 + (b*c^2*(17*b^4 + 88*a*b^2*c + 48*a^2*c^2)*x^6)/2 + (3*c^3*(43*b^4 + 104*a*b^2*c + 16*a^2*c^2)*x^7)/7 + 24*b*c^4*(b^2 + a*c)*x^8 + (8*c^5*(7*b^2 + 2*a*c)*x^9)/3 + 8*b*c^6*x^10 + (16*c^7*x^11)/11)","B",1
1139,1,132,55,0.0322131,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^3,x]","\frac{1}{20} d^3 x (b+c x) \left(20 a^3 \left(b^2+2 b c x+2 c^2 x^2\right)+10 a^2 x \left(3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right)+20 a x^2 (b+c x)^2 \left(b^2+3 b c x+3 c^2 x^2\right)+x^3 (b+c x)^3 \left(5 b^2+16 b c x+16 c^2 x^2\right)\right)","\frac{1}{20} d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4+\frac{1}{5} d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^4",1,"(d^3*x*(b + c*x)*(20*a^3*(b^2 + 2*b*c*x + 2*c^2*x^2) + 20*a*x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + x^3*(b + c*x)^3*(5*b^2 + 16*b*c*x + 16*c^2*x^2) + 10*a^2*x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/20","B",1
1140,1,179,101,0.0304844,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3,x]","d^2 \left(a^3 b^2 x+\frac{1}{2} a^2 b x^2 \left(4 a c+3 b^2\right)+\frac{1}{5} c x^5 \left(12 a^2 c^2+39 a b^2 c+7 b^4\right)+\frac{1}{4} b x^4 \left(24 a^2 c^2+18 a b^2 c+b^4\right)+\frac{1}{3} a x^3 \left(4 a^2 c^2+15 a b^2 c+3 b^4\right)+\frac{1}{7} c^3 x^7 \left(12 a c+25 b^2\right)+\frac{1}{6} b c^2 x^6 \left(36 a c+19 b^2\right)+2 b c^4 x^8+\frac{4 c^5 x^9}{9}\right)","-\frac{3 d^2 \left(b^2-4 a c\right) (b+2 c x)^7}{896 c^4}+\frac{3 d^2 \left(b^2-4 a c\right)^2 (b+2 c x)^5}{640 c^4}-\frac{d^2 \left(b^2-4 a c\right)^3 (b+2 c x)^3}{384 c^4}+\frac{d^2 (b+2 c x)^9}{1152 c^4}",1,"d^2*(a^3*b^2*x + (a^2*b*(3*b^2 + 4*a*c)*x^2)/2 + (a*(3*b^4 + 15*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*(b^4 + 18*a*b^2*c + 24*a^2*c^2)*x^4)/4 + (c*(7*b^4 + 39*a*b^2*c + 12*a^2*c^2)*x^5)/5 + (b*c^2*(19*b^2 + 36*a*c)*x^6)/6 + (c^3*(25*b^2 + 12*a*c)*x^7)/7 + 2*b*c^4*x^8 + (4*c^5*x^9)/9)","A",1
1141,1,52,17,0.0171162,"\int (b d+2 c d x) \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^3,x]","\frac{1}{4} d x (b+c x) \left(4 a^3+6 a^2 x (b+c x)+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right)","\frac{1}{4} d \left(a+b x+c x^2\right)^4",1,"(d*x*(b + c*x)*(4*a^3 + 6*a^2*x*(b + c*x) + 4*a*x^2*(b + c*x)^2 + x^3*(b + c*x)^3))/4","B",1
1142,1,111,100,0.0462666,"\int \frac{\left(a+b x+c x^2\right)^3}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x),x]","\frac{2 c x (b+c x) \left(8 c^2 \left(18 a^2+9 a c x^2+2 c^2 x^4\right)+2 b^2 c \left(5 c x^2-18 a\right)+8 b c^2 x \left(9 a+4 c x^2\right)+3 b^4-6 b^3 c x\right)-3 \left(b^2-4 a c\right)^3 \log (b+2 c x)}{384 c^4 d}","-\frac{3 \left(b^2-4 a c\right) (b+2 c x)^4}{512 c^4 d}+\frac{3 \left(b^2-4 a c\right)^2 (b+2 c x)^2}{256 c^4 d}-\frac{\left(b^2-4 a c\right)^3 \log (b+2 c x)}{128 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d}",1,"(2*c*x*(b + c*x)*(3*b^4 - 6*b^3*c*x + 8*b*c^2*x*(9*a + 4*c*x^2) + 2*b^2*c*(-18*a + 5*c*x^2) + 8*c^2*(18*a^2 + 9*a*c*x^2 + 2*c^2*x^4)) - 3*(b^2 - 4*a*c)^3*Log[b + 2*c*x])/(384*c^4*d)","A",1
1143,1,101,94,0.0777625,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^2,x]","\frac{\frac{10 x \left(48 a^2 c^2-12 a b^2 c+b^4\right)}{c^3}+\frac{5 \left(b^2-4 a c\right)^3}{c^4 (b+2 c x)}-\frac{20 b x^2 \left(b^2-12 a c\right)}{c^2}+\frac{40 x^3 \left(4 a c+b^2\right)}{c}+80 b x^4+32 c x^5}{640 d^2}","-\frac{\left(b^2-4 a c\right) (b+2 c x)^3}{128 c^4 d^2}+\frac{\left(b^2-4 a c\right)^3}{128 c^4 d^2 (b+2 c x)}+\frac{3 x \left(b^2-4 a c\right)^2}{64 c^3 d^2}+\frac{(b+2 c x)^5}{640 c^4 d^2}",1,"((10*(b^4 - 12*a*b^2*c + 48*a^2*c^2)*x)/c^3 - (20*b*(b^2 - 12*a*c)*x^2)/c^2 + (40*(b^2 + 4*a*c)*x^3)/c + 80*b*x^4 + 32*c*x^5 + (5*(b^2 - 4*a*c)^3)/(c^4*(b + 2*c*x)))/(640*d^2)","A",1
1144,1,90,100,0.0946999,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^3,x]","\frac{\frac{\left(b^2-4 a c\right)^3}{c^4 (b+2 c x)^2}+\frac{6 \left(b^2-4 a c\right)^2 \log (b+2 c x)}{c^4}-\frac{8 b x \left(b^2-6 a c\right)}{c^3}+\frac{48 a x^2}{c}+\frac{16 b x^3}{c}+8 x^4}{256 d^3}","-\frac{3 \left(b^2-4 a c\right) (b+2 c x)^2}{256 c^4 d^3}+\frac{\left(b^2-4 a c\right)^3}{256 c^4 d^3 (b+2 c x)^2}+\frac{3 \left(b^2-4 a c\right)^2 \log (b+2 c x)}{128 c^4 d^3}+\frac{(b+2 c x)^4}{512 c^4 d^3}",1,"((-8*b*(b^2 - 6*a*c)*x)/c^3 + (48*a*x^2)/c + (16*b*x^3)/c + 8*x^4 + (b^2 - 4*a*c)^3/(c^4*(b + 2*c*x)^2) + (6*(b^2 - 4*a*c)^2*Log[b + 2*c*x])/c^4)/(256*d^3)","A",1
1145,1,81,103,0.0618597,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^4,x]","\frac{\frac{\left(b^2-4 a c\right)^3}{(b+2 c x)^3}-\frac{9 \left(b^2-4 a c\right)^2}{b+2 c x}+12 c x \left(6 a c-b^2\right)+12 b c^2 x^2+8 c^3 x^3}{384 c^4 d^4}","\frac{\left(b^2-4 a c\right)^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left(b^2-4 a c\right)^2}{128 c^4 d^4 (b+2 c x)}-\frac{x \left(b^2-6 a c\right)}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4}",1,"(12*c*(-b^2 + 6*a*c)*x + 12*b*c^2*x^2 + 8*c^3*x^3 + (b^2 - 4*a*c)^3/(b + 2*c*x)^3 - (9*(b^2 - 4*a*c)^2)/(b + 2*c*x))/(384*c^4*d^4)","A",1
1146,1,80,107,0.0441254,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x]","\frac{\frac{\left(b^2-4 a c\right)^3}{(b+2 c x)^4}-\frac{6 \left(b^2-4 a c\right)^2}{(b+2 c x)^2}-12 \left(b^2-4 a c\right) \log (b+2 c x)+8 b c x+8 c^2 x^2}{512 c^4 d^5}","\frac{\left(b^2-4 a c\right)^3}{512 c^4 d^5 (b+2 c x)^4}-\frac{3 \left(b^2-4 a c\right)^2}{256 c^4 d^5 (b+2 c x)^2}-\frac{3 \left(b^2-4 a c\right) \log (b+2 c x)}{128 c^4 d^5}+\frac{b x}{64 c^3 d^5}+\frac{x^2}{64 c^2 d^5}",1,"(8*b*c*x + 8*c^2*x^2 + (b^2 - 4*a*c)^3/(b + 2*c*x)^4 - (6*(b^2 - 4*a*c)^2)/(b + 2*c*x)^2 - 12*(b^2 - 4*a*c)*Log[b + 2*c*x])/(512*c^4*d^5)","A",1
1147,1,72,94,0.0625225,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]","\frac{\frac{\left(b^2-4 a c\right)^3}{(b+2 c x)^5}-\frac{5 \left(b^2-4 a c\right)^2}{(b+2 c x)^3}+\frac{15 \left(b^2-4 a c\right)}{b+2 c x}+10 c x}{640 c^4 d^6}","\frac{\left(b^2-4 a c\right)^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left(b^2-4 a c\right)^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left(b^2-4 a c\right)}{128 c^4 d^6 (b+2 c x)}+\frac{x}{64 c^3 d^6}",1,"(10*c*x + (b^2 - 4*a*c)^3/(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)^2)/(b + 2*c*x)^3 + (15*(b^2 - 4*a*c))/(b + 2*c*x))/(640*c^4*d^6)","A",1
1148,1,78,100,0.0420404,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^7} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]","\frac{\frac{2 \left(b^2-4 a c\right)^3}{(b+2 c x)^6}-\frac{9 \left(b^2-4 a c\right)^2}{(b+2 c x)^4}+\frac{18 \left(b^2-4 a c\right)}{(b+2 c x)^2}+12 \log (b+2 c x)}{1536 c^4 d^7}","\frac{\left(b^2-4 a c\right)^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left(b^2-4 a c\right)^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left(b^2-4 a c\right)}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7}",1,"((2*(b^2 - 4*a*c)^3)/(b + 2*c*x)^6 - (9*(b^2 - 4*a*c)^2)/(b + 2*c*x)^4 + (18*(b^2 - 4*a*c))/(b + 2*c*x)^2 + 12*Log[b + 2*c*x])/(1536*c^4*d^7)","A",1
1149,1,79,101,0.0492911,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^8} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x]","\frac{35 \left(b^2-4 a c\right) (b+2 c x)^4-21 \left(b^2-4 a c\right)^2 (b+2 c x)^2+5 \left(b^2-4 a c\right)^3-35 (b+2 c x)^6}{4480 c^4 d^8 (b+2 c x)^7}","\frac{\left(b^2-4 a c\right)^3}{896 c^4 d^8 (b+2 c x)^7}-\frac{3 \left(b^2-4 a c\right)^2}{640 c^4 d^8 (b+2 c x)^5}+\frac{b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac{1}{128 c^4 d^8 (b+2 c x)}",1,"(5*(b^2 - 4*a*c)^3 - 21*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 35*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 35*(b + 2*c*x)^6)/(4480*c^4*d^8*(b + 2*c*x)^7)","A",1
1150,1,96,37,0.0509522,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^9} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]","\frac{-64 a^3 c^3+48 a^2 b^2 c^2-12 a b^4 c+6 \left(b^2-4 a c\right) (b+2 c x)^4-4 \left(b^2-4 a c\right)^2 (b+2 c x)^2+b^6-4 (b+2 c x)^6}{1024 c^4 d^9 (b+2 c x)^8}","\frac{\left(a+b x+c x^2\right)^4}{4 d^9 \left(b^2-4 a c\right) (b+2 c x)^8}",1,"(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 - 4*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 6*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 4*(b + 2*c*x)^6)/(1024*c^4*d^9*(b + 2*c*x)^8)","B",1
1151,1,79,101,0.0568663,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x]","\frac{189 \left(b^2-4 a c\right) (b+2 c x)^4-135 \left(b^2-4 a c\right)^2 (b+2 c x)^2+35 \left(b^2-4 a c\right)^3-105 (b+2 c x)^6}{40320 c^4 d^{10} (b+2 c x)^9}","\frac{\left(b^2-4 a c\right)^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac{3 \left(b^2-4 a c\right)^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac{3 \left(b^2-4 a c\right)}{640 c^4 d^{10} (b+2 c x)^5}-\frac{1}{384 c^4 d^{10} (b+2 c x)^3}",1,"(35*(b^2 - 4*a*c)^3 - 135*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 189*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 105*(b + 2*c*x)^6)/(40320*c^4*d^10*(b + 2*c*x)^9)","A",1
1152,1,79,101,0.0568432,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{11}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^11,x]","\frac{20 \left(b^2-4 a c\right) (b+2 c x)^4-15 \left(b^2-4 a c\right)^2 (b+2 c x)^2+4 \left(b^2-4 a c\right)^3-10 (b+2 c x)^6}{5120 c^4 d^{11} (b+2 c x)^{10}}","\frac{\left(b^2-4 a c\right)^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left(b^2-4 a c\right)^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4}",1,"(4*(b^2 - 4*a*c)^3 - 15*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 20*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 10*(b + 2*c*x)^6)/(5120*c^4*d^11*(b + 2*c*x)^10)","A",1
1153,1,79,101,0.0631691,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{12}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^12,x]","\frac{495 \left(b^2-4 a c\right) (b+2 c x)^4-385 \left(b^2-4 a c\right)^2 (b+2 c x)^2+105 \left(b^2-4 a c\right)^3-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}}","\frac{\left(b^2-4 a c\right)^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac{\left(b^2-4 a c\right)^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac{3 \left(b^2-4 a c\right)}{896 c^4 d^{12} (b+2 c x)^7}-\frac{1}{640 c^4 d^{12} (b+2 c x)^5}",1,"(105*(b^2 - 4*a*c)^3 - 385*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 495*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 231*(b + 2*c*x)^6)/(147840*c^4*d^12*(b + 2*c*x)^11)","A",1
1154,1,188,122,0.1121041,"\int \frac{(b d+2 c d x)^8}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2),x]","d^8 \left(\frac{16}{105} c x \left(112 b^2 c^2 \left(15 a^2-10 a c x^2+12 c^2 x^4\right)+840 b c^3 x \left(a^2-a c x^2+c^2 x^4\right)+16 c^3 \left(-105 a^3+35 a^2 c x^2-21 a c^2 x^4+15 c^3 x^6\right)+70 b^4 c \left(11 c x^2-9 a\right)+420 b^3 c^2 x \left(3 c x^2-2 a\right)+105 b^6+315 b^5 c x\right)+2 \left(4 a c-b^2\right)^{7/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)","\frac{2}{5} d^8 \left(b^2-4 a c\right) (b+2 c x)^5+\frac{2}{3} d^8 \left(b^2-4 a c\right)^2 (b+2 c x)^3+2 d^8 \left(b^2-4 a c\right)^3 (b+2 c x)-2 d^8 \left(b^2-4 a c\right)^{7/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)+\frac{2}{7} d^8 (b+2 c x)^7",1,"d^8*((16*c*x*(105*b^6 + 315*b^5*c*x + 420*b^3*c^2*x*(-2*a + 3*c*x^2) + 70*b^4*c*(-9*a + 11*c*x^2) + 840*b*c^3*x*(a^2 - a*c*x^2 + c^2*x^4) + 112*b^2*c^2*(15*a^2 - 10*a*c*x^2 + 12*c^2*x^4) + 16*c^3*(-105*a^3 + 35*a^2*c*x^2 - 21*a*c^2*x^4 + 15*c^3*x^6)))/105 + 2*(-b^2 + 4*a*c)^(7/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1155,1,110,86,0.0550422,"\int \frac{(b d+2 c d x)^7}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2),x]","d^7 \left(\frac{4}{3} c x (b+c x) \left(8 c^2 \left(6 a^2-3 a c x^2+2 c^2 x^4\right)+b^2 \left(34 c^2 x^2-36 a c\right)+8 b c^2 x \left(4 c x^2-3 a\right)+9 b^4+18 b^3 c x\right)+\left(b^2-4 a c\right)^3 \log (a+x (b+c x))\right)","d^7 \left(b^2-4 a c\right)^3 \log \left(a+b x+c x^2\right)+\frac{1}{2} d^7 \left(b^2-4 a c\right) (b+2 c x)^4+d^7 \left(b^2-4 a c\right)^2 (b+2 c x)^2+\frac{1}{3} d^7 (b+2 c x)^6",1,"d^7*((4*c*x*(b + c*x)*(9*b^4 + 18*b^3*c*x + 8*b*c^2*x*(-3*a + 4*c*x^2) + b^2*(-36*a*c + 34*c^2*x^2) + 8*c^2*(6*a^2 - 3*a*c*x^2 + 2*c^2*x^4)))/3 + (b^2 - 4*a*c)^3*Log[a + x*(b + c*x)])","A",1
1156,1,120,97,0.0745314,"\int \frac{(b d+2 c d x)^6}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2),x]","d^6 \left(\frac{4}{15} c x \left(16 c^2 \left(15 a^2-5 a c x^2+3 c^2 x^4\right)+20 b^2 c \left(7 c x^2-9 a\right)+120 b c^2 x \left(c x^2-a\right)+45 b^4+90 b^3 c x\right)-2 \left(4 a c-b^2\right)^{5/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)","\frac{2}{3} d^6 \left(b^2-4 a c\right) (b+2 c x)^3+2 d^6 \left(b^2-4 a c\right)^2 (b+2 c x)-2 d^6 \left(b^2-4 a c\right)^{5/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)+\frac{2}{5} d^6 (b+2 c x)^5",1,"d^6*((4*c*x*(45*b^4 + 90*b^3*c*x + 120*b*c^2*x*(-a + c*x^2) + 20*b^2*c*(-9*a + 7*c*x^2) + 16*c^2*(15*a^2 - 5*a*c*x^2 + 3*c^2*x^4)))/15 - 2*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1157,1,54,61,0.0233154,"\int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2),x]","d^5 \left(8 c x (b+c x) \left(c \left(c x^2-2 a\right)+b^2+b c x\right)+\left(b^2-4 a c\right)^2 \log (a+x (b+c x))\right)","d^5 \left(b^2-4 a c\right)^2 \log \left(a+b x+c x^2\right)+d^5 \left(b^2-4 a c\right) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4",1,"d^5*(8*c*x*(b + c*x)*(b^2 + b*c*x + c*(-2*a + c*x^2)) + (b^2 - 4*a*c)^2*Log[a + x*(b + c*x)])","A",1
1158,1,72,72,0.0448322,"\int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2),x]","d^4 \left(\frac{8}{3} c x \left(2 c \left(c x^2-3 a\right)+3 b^2+3 b c x\right)+2 \left(4 a c-b^2\right)^{3/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)","2 d^4 \left(b^2-4 a c\right) (b+2 c x)-2 d^4 \left(b^2-4 a c\right)^{3/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)+\frac{2}{3} d^4 (b+2 c x)^3",1,"d^4*((8*c*x*(3*b^2 + 3*b*c*x + 2*c*(-3*a + c*x^2)))/3 + 2*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1159,1,33,36,0.012776,"\int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x]","d^3 \left(\left(b^2-4 a c\right) \log (a+x (b+c x))+4 c x (b+c x)\right)","d^3 \left(b^2-4 a c\right) \log \left(a+b x+c x^2\right)+d^3 (b+2 c x)^2",1,"d^3*(4*c*x*(b + c*x) + (b^2 - 4*a*c)*Log[a + x*(b + c*x)])","A",1
1160,1,47,49,0.0279563,"\int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2),x]","d^2 \left(4 c x-2 \sqrt{4 a c-b^2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)\right)","2 d^2 (b+2 c x)-2 d^2 \sqrt{b^2-4 a c} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)",1,"d^2*(4*c*x - 2*Sqrt[-b^2 + 4*a*c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1161,1,12,13,0.0021501,"\int \frac{b d+2 c d x}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2),x]","d \log (a+x (b+c x))","d \log \left(a+b x+c x^2\right)",1,"d*Log[a + x*(b + c*x)]","A",1
1162,1,34,48,0.0210634,"\int \frac{1}{(b d+2 c d x) \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)),x]","\frac{\log (a+x (b+c x))-2 \log (b+2 c x)}{d \left(b^2-4 a c\right)}","\frac{\log \left(a+b x+c x^2\right)}{d \left(b^2-4 a c\right)}-\frac{2 \log (b+2 c x)}{d \left(b^2-4 a c\right)}",1,"(-2*Log[b + 2*c*x] + Log[a + x*(b + c*x)])/((b^2 - 4*a*c)*d)","A",1
1163,1,63,61,0.0497293,"\int \frac{1}{(b d+2 c d x)^2 \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)),x]","\frac{\frac{2}{\left(b^2-4 a c\right) (b+2 c x)}-\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}}{d^2}","\frac{2}{d^2 \left(b^2-4 a c\right) (b+2 c x)}-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^2 \left(b^2-4 a c\right)^{3/2}}",1,"(2/((b^2 - 4*a*c)*(b + 2*c*x)) - (2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))/d^2","A",1
1164,1,65,70,0.0424773,"\int \frac{1}{(b d+2 c d x)^3 \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)),x]","\frac{\frac{\log \left(a+b x+c x^2\right)}{\left(b^2-4 a c\right)^2}+\frac{1}{\left(b^2-4 a c\right) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{\left(b^2-4 a c\right)^2}}{d^3}","\frac{\log \left(a+b x+c x^2\right)}{d^3 \left(b^2-4 a c\right)^2}+\frac{1}{d^3 \left(b^2-4 a c\right) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left(b^2-4 a c\right)^2}",1,"(1/((b^2 - 4*a*c)*(b + 2*c*x)^2) - (2*Log[b + 2*c*x])/(b^2 - 4*a*c)^2 + Log[a + b*x + c*x^2]/(b^2 - 4*a*c)^2)/d^3","A",1
1165,1,83,86,0.0511516,"\int \frac{1}{(b d+2 c d x)^4 \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)),x]","\frac{2 \left(\frac{b^2-4 a c}{(b+2 c x)^3}+\frac{3 \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}\right)}{3 d^4 \left(b^2-4 a c\right)^2}","\frac{2}{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)}+\frac{2}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3}-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^4 \left(b^2-4 a c\right)^{5/2}}",1,"(2*((b^2 - 4*a*c)/(b + 2*c*x)^3 + 3/(b + 2*c*x) + (3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]))/(3*(b^2 - 4*a*c)^2*d^4)","A",1
1166,1,155,126,0.0771967,"\int \frac{(b d+2 c d x)^8}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^2,x]","d^8 \left(32 c^2 x \left(24 a^2 c^2-16 a b^2 c+3 b^4\right)-\frac{512}{3} c^4 x^3 \left(a c-b^2\right)+128 b c^3 x^2 \left(b^2-2 a c\right)-\frac{\left(b^2-4 a c\right)^3 (b+2 c x)}{a+x (b+c x)}-28 c \left(4 a c-b^2\right)^{5/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+128 b c^5 x^4+\frac{256 c^6 x^5}{5}\right)","\frac{28}{3} c d^8 \left(b^2-4 a c\right) (b+2 c x)^3+28 c d^8 \left(b^2-4 a c\right)^2 (b+2 c x)-28 c d^8 \left(b^2-4 a c\right)^{5/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac{28}{5} c d^8 (b+2 c x)^5",1,"d^8*(32*c^2*(3*b^4 - 16*a*b^2*c + 24*a^2*c^2)*x + 128*b*c^3*(b^2 - 2*a*c)*x^2 - (512*c^4*(-b^2 + a*c)*x^3)/3 + 128*b*c^5*x^4 + (256*c^6*x^5)/5 - ((b^2 - 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x)) - 28*c*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1167,1,103,89,0.0831263,"\int \frac{(b d+2 c d x)^7}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^2,x]","d^7 \left(-16 c^3 x^2 \left(8 a c-5 b^2\right)+16 b c^2 x \left(3 b^2-8 a c\right)-\frac{\left(b^2-4 a c\right)^3}{a+x (b+c x)}+12 c \left(b^2-4 a c\right)^2 \log (a+x (b+c x))+64 b c^4 x^3+32 c^5 x^4\right)","12 c d^7 \left(b^2-4 a c\right)^2 \log \left(a+b x+c x^2\right)+12 c d^7 \left(b^2-4 a c\right) (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4",1,"d^7*(16*b*c^2*(3*b^2 - 8*a*c)*x - 16*c^3*(-5*b^2 + 8*a*c)*x^2 + 64*b*c^4*x^3 + 32*c^5*x^4 - (b^2 - 4*a*c)^3/(a + x*(b + c*x)) + 12*c*(b^2 - 4*a*c)^2*Log[a + x*(b + c*x)])","A",1
1168,1,108,100,0.0615907,"\int \frac{(b d+2 c d x)^6}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]","d^6 \left(-16 c^2 x \left(8 a c-3 b^2\right)-\frac{\left(b^2-4 a c\right)^2 (b+2 c x)}{a+x (b+c x)}+20 c \left(4 a c-b^2\right)^{3/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+32 b c^3 x^2+\frac{64 c^4 x^3}{3}\right)","20 c d^6 \left(b^2-4 a c\right) (b+2 c x)-20 c d^6 \left(b^2-4 a c\right)^{3/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac{20}{3} c d^6 (b+2 c x)^3",1,"d^6*(-16*c^2*(-3*b^2 + 8*a*c)*x + 32*b*c^3*x^2 + (64*c^4*x^3)/3 - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x)) + 20*c*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1169,1,64,65,0.0342467,"\int \frac{(b d+2 c d x)^5}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^2,x]","d^5 \left(-\frac{\left(b^2-4 a c\right)^2}{a+x (b+c x)}+8 c \left(b^2-4 a c\right) \log (a+x (b+c x))+16 b c^2 x+16 c^3 x^2\right)","8 c d^5 \left(b^2-4 a c\right) \log \left(a+b x+c x^2\right)-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2",1,"d^5*(16*b*c^2*x + 16*c^3*x^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x)) + 8*c*(b^2 - 4*a*c)*Log[a + x*(b + c*x)])","A",1
1170,1,77,76,0.0515827,"\int \frac{(b d+2 c d x)^4}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^2,x]","d^4 \left(-\frac{\left(b^2-4 a c\right) (b+2 c x)}{a+x (b+c x)}-12 c \sqrt{4 a c-b^2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+16 c^2 x\right)","-12 c d^4 \sqrt{b^2-4 a c} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x)",1,"d^4*(16*c^2*x - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)) - 12*c*Sqrt[-b^2 + 4*a*c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1171,1,42,43,0.021723,"\int \frac{(b d+2 c d x)^3}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^2,x]","d^3 \left(\frac{4 a c-b^2}{a+b x+c x^2}+4 c \log \left(a+b x+c x^2\right)\right)","4 c d^3 \log \left(a+b x+c x^2\right)-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2}",1,"d^3*((-b^2 + 4*a*c)/(a + b*x + c*x^2) + 4*c*Log[a + b*x + c*x^2])","A",1
1172,1,65,62,0.0358418,"\int \frac{(b d+2 c d x)^2}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^2,x]","d^2 \left(\frac{4 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{-b-2 c x}{a+b x+c x^2}\right)","-\frac{4 c d^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}-\frac{d^2 (b+2 c x)}{a+b x+c x^2}",1,"d^2*((-b - 2*c*x)/(a + b*x + c*x^2) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])","A",1
1173,1,14,15,0.0038937,"\int \frac{b d+2 c d x}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^2,x]","-\frac{d}{a+x (b+c x)}","-\frac{d}{a+b x+c x^2}",1,"-(d/(a + x*(b + c*x)))","A",1
1174,1,59,78,0.0558268,"\int \frac{1}{(b d+2 c d x) \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^2),x]","\frac{-\frac{b^2-4 a c}{a+x (b+c x)}-4 c \log (a+x (b+c x))+8 c \log (b+2 c x)}{d \left(b^2-4 a c\right)^2}","-\frac{1}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{4 c \log \left(a+b x+c x^2\right)}{d \left(b^2-4 a c\right)^2}+\frac{8 c \log (b+2 c x)}{d \left(b^2-4 a c\right)^2}",1,"(-((b^2 - 4*a*c)/(a + x*(b + c*x))) + 8*c*Log[b + 2*c*x] - 4*c*Log[a + x*(b + c*x)])/((b^2 - 4*a*c)^2*d)","A",1
1175,1,84,98,0.119796,"\int \frac{1}{(b d+2 c d x)^2 \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2),x]","-\frac{\frac{12 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}+\frac{8 c}{b+2 c x}}{d^2 \left(b^2-4 a c\right)^2}","-\frac{1}{d^2 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)}-\frac{12 c}{d^2 \left(b^2-4 a c\right)^2 (b+2 c x)}+\frac{12 c \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,"-(((8*c)/(b + 2*c*x) + (b + 2*c*x)/(a + x*(b + c*x)) + (12*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/((b^2 - 4*a*c)^2*d^2))","A",1
1176,1,79,110,0.0882905,"\int \frac{1}{(b d+2 c d x)^3 \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^2),x]","\frac{-\frac{4 c \left(b^2-4 a c\right)}{(b+2 c x)^2}+\frac{4 a c-b^2}{a+x (b+c x)}-8 c \log (a+x (b+c x))+16 c \log (b+2 c x)}{d^3 \left(b^2-4 a c\right)^3}","-\frac{1}{d^3 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)}-\frac{8 c \log \left(a+b x+c x^2\right)}{d^3 \left(b^2-4 a c\right)^3}-\frac{8 c}{d^3 \left(b^2-4 a c\right)^2 (b+2 c x)^2}+\frac{16 c \log (b+2 c x)}{d^3 \left(b^2-4 a c\right)^3}",1,"((-4*c*(b^2 - 4*a*c))/(b + 2*c*x)^2 + (-b^2 + 4*a*c)/(a + x*(b + c*x)) + 16*c*Log[b + 2*c*x] - 8*c*Log[a + x*(b + c*x)])/((b^2 - 4*a*c)^3*d^3)","A",1
1177,1,192,160,0.1120024,"\int \frac{(b d+2 c d x)^{10}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]","d^{10} \left(128 c^3 x \left(48 a^2 c^2-30 a b^2 c+5 b^4\right)-256 c^5 x^3 \left(4 a c-3 b^2\right)+128 b c^4 x^2 \left(5 b^2-12 a c\right)-252 c^2 \left(4 a c-b^2\right)^{5/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+\frac{17 c \left(4 a c-b^2\right)^3 (b+2 c x)}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^4 (b+2 c x)}{2 (a+x (b+c x))^2}+512 b c^6 x^4+\frac{1024 c^7 x^5}{5}\right)","84 c^2 d^{10} \left(b^2-4 a c\right) (b+2 c x)^3+252 c^2 d^{10} \left(b^2-4 a c\right)^2 (b+2 c x)-252 c^2 d^{10} \left(b^2-4 a c\right)^{5/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac{d^{10} (b+2 c x)^9}{2 \left(a+b x+c x^2\right)^2}+\frac{252}{5} c^2 d^{10} (b+2 c x)^5",1,"d^10*(128*c^3*(5*b^4 - 30*a*b^2*c + 48*a^2*c^2)*x + 128*b*c^4*(5*b^2 - 12*a*c)*x^2 - 256*c^5*(-3*b^2 + 4*a*c)*x^3 + 512*b*c^6*x^4 + (1024*c^7*x^5)/5 - ((b^2 - 4*a*c)^4*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (17*c*(-b^2 + 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x)) - 252*c^2*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1178,1,131,123,0.0576249,"\int \frac{(b d+2 c d x)^9}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x]","d^9 \left(-384 c^4 x^2 \left(2 a c-b^2\right)+256 b c^3 x \left(b^2-3 a c\right)+96 c^2 \left(b^2-4 a c\right)^2 \log (a+x (b+c x))+\frac{16 c \left(4 a c-b^2\right)^3}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^4}{2 (a+x (b+c x))^2}+256 b c^5 x^3+128 c^6 x^4\right)","96 c^2 d^9 \left(b^2-4 a c\right)^2 \log \left(a+b x+c x^2\right)+96 c^2 d^9 \left(b^2-4 a c\right) (b+2 c x)^2-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac{d^9 (b+2 c x)^8}{2 \left(a+b x+c x^2\right)^2}+48 c^2 d^9 (b+2 c x)^4",1,"d^9*(256*b*c^3*(b^2 - 3*a*c)*x - 384*c^4*(-b^2 + 2*a*c)*x^2 + 256*b*c^5*x^3 + 128*c^6*x^4 - (b^2 - 4*a*c)^4/(2*(a + x*(b + c*x))^2) + (16*c*(-b^2 + 4*a*c)^3)/(a + x*(b + c*x)) + 96*c^2*(b^2 - 4*a*c)^2*Log[a + x*(b + c*x)])","A",1
1179,1,142,134,0.0792958,"\int \frac{(b d+2 c d x)^8}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]","d^8 \left(-256 c^3 x \left(3 a c-b^2\right)+140 c^2 \left(4 a c-b^2\right)^{3/2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)-\frac{13 c \left(b^2-4 a c\right)^2 (b+2 c x)}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^3 (b+2 c x)}{2 (a+x (b+c x))^2}+128 b c^4 x^2+\frac{256 c^5 x^3}{3}\right)","140 c^2 d^8 \left(b^2-4 a c\right) (b+2 c x)-140 c^2 d^8 \left(b^2-4 a c\right)^{3/2} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left(a+b x+c x^2\right)^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3",1,"d^8*(-256*c^3*(-b^2 + 3*a*c)*x + 128*b*c^4*x^2 + (256*c^5*x^3)/3 - ((b^2 - 4*a*c)^3*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) - (13*c*(b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x)) + 140*c^2*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1180,1,92,97,0.035487,"\int \frac{(b d+2 c d x)^7}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^7/(a + b*x + c*x^2)^3,x]","d^7 \left(48 c^2 \left(b^2-4 a c\right) \log (a+x (b+c x))-\frac{12 c \left(b^2-4 a c\right)^2}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^3}{2 (a+x (b+c x))^2}+64 b c^3 x+64 c^4 x^2\right)","48 c^2 d^7 \left(b^2-4 a c\right) \log \left(a+b x+c x^2\right)-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac{d^7 (b+2 c x)^6}{2 \left(a+b x+c x^2\right)^2}+48 c^2 d^7 (b+2 c x)^2",1,"d^7*(64*b*c^3*x + 64*c^4*x^2 - (b^2 - 4*a*c)^3/(2*(a + x*(b + c*x))^2) - (12*c*(b^2 - 4*a*c)^2)/(a + x*(b + c*x)) + 48*c^2*(b^2 - 4*a*c)*Log[a + x*(b + c*x)])","A",1
1181,1,113,108,0.0630715,"\int \frac{(b d+2 c d x)^6}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^3,x]","d^6 \left(-60 c^2 \sqrt{4 a c-b^2} \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)+\frac{9 c \left(4 a c-b^2\right) (b+2 c x)}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^2 (b+2 c x)}{2 (a+x (b+c x))^2}+64 c^3 x\right)","-60 c^2 d^6 \sqrt{b^2-4 a c} \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)-\frac{5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\frac{d^6 (b+2 c x)^5}{2 \left(a+b x+c x^2\right)^2}+60 c^2 d^6 (b+2 c x)",1,"d^6*(64*c^3*x - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (9*c*(-b^2 + 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x)) - 60*c^2*Sqrt[-b^2 + 4*a*c]*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])","A",1
1182,1,65,73,0.041715,"\int \frac{(b d+2 c d x)^5}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^3,x]","d^5 \left(16 c^2 \log (a+x (b+c x))-\frac{\left(b^2-4 a c\right) \left(4 c \left(3 a+4 c x^2\right)+b^2+16 b c x\right)}{2 (a+x (b+c x))^2}\right)","16 c^2 d^5 \log \left(a+b x+c x^2\right)-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left(a+b x+c x^2\right)^2}",1,"d^5*(-1/2*((b^2 - 4*a*c)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(a + x*(b + c*x))^2 + 16*c^2*Log[a + x*(b + c*x)])","A",1
1183,1,89,92,0.0618166,"\int \frac{(b d+2 c d x)^4}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^3,x]","d^4 \left(\frac{12 c^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left(2 c \left(3 a+5 c x^2\right)+b^2+10 b c x\right)}{2 (a+x (b+c x))^2}\right)","-\frac{12 c^2 d^4 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left(a+b x+c x^2\right)^2}",1,"d^4*(-1/2*((b + 2*c*x)*(b^2 + 10*b*c*x + 2*c*(3*a + 5*c*x^2)))/(a + x*(b + c*x))^2 + (12*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])","A",1
1184,1,38,37,0.0222743,"\int \frac{(b d+2 c d x)^3}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^3,x]","-\frac{d^3 \left(4 c \left(a+2 c x^2\right)+b^2+8 b c x\right)}{2 (a+x (b+c x))^2}","-\frac{d^3 (b+2 c x)^4}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"-1/2*(d^3*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)))/(a + x*(b + c*x))^2","A",1
1185,1,98,100,0.0862644,"\int \frac{(b d+2 c d x)^2}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^3,x]","d^2 \left(\frac{4 c^2 \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{(b+2 c x) \left(2 c \left(c x^2-a\right)+b^2+2 b c x\right)}{2 \left(b^2-4 a c\right) (a+x (b+c x))^2}\right)","\frac{4 c^2 d^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{c d^2 (b+2 c x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{d^2 (b+2 c x)}{2 \left(a+b x+c x^2\right)^2}",1,"d^2*(-1/2*((b + 2*c*x)*(b^2 + 2*b*c*x + 2*c*(-a + c*x^2)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (4*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))","A",1
1186,1,16,17,0.0049512,"\int \frac{b d+2 c d x}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^3,x]","-\frac{d}{2 (a+x (b+c x))^2}","-\frac{d}{2 \left(a+b x+c x^2\right)^2}",1,"-1/2*d/(a + x*(b + c*x))^2","A",1
1187,1,90,112,0.0739285,"\int \frac{1}{(b d+2 c d x) \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^3),x]","\frac{\frac{8 c \left(b^2-4 a c\right)}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^2}{(a+x (b+c x))^2}+32 c^2 \log (a+x (b+c x))-64 c^2 \log (b+2 c x)}{2 d \left(b^2-4 a c\right)^3}","\frac{16 c^2 \log \left(a+b x+c x^2\right)}{d \left(b^2-4 a c\right)^3}-\frac{32 c^2 \log (b+2 c x)}{d \left(b^2-4 a c\right)^3}+\frac{4 c}{d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{1}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(-((b^2 - 4*a*c)^2/(a + x*(b + c*x))^2) + (8*c*(b^2 - 4*a*c))/(a + x*(b + c*x)) - 64*c^2*Log[b + 2*c*x] + 32*c^2*Log[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)^3*d)","A",1
1188,1,119,140,0.131049,"\int \frac{1}{(b d+2 c d x)^2 \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x]","\frac{\frac{120 c^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{\left(b^2-4 a c\right) (b+2 c x)}{(a+x (b+c x))^2}+\frac{14 c (b+2 c x)}{a+x (b+c x)}+\frac{64 c^2}{b+2 c x}}{2 d^2 \left(b^2-4 a c\right)^3}","\frac{60 c^2}{d^2 \left(b^2-4 a c\right)^3 (b+2 c x)}-\frac{60 c^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^2 \left(b^2-4 a c\right)^{7/2}}+\frac{5 c}{d^2 \left(b^2-4 a c\right)^2 (b+2 c x) \left(a+b x+c x^2\right)}-\frac{1}{2 d^2 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^2}",1,"((64*c^2)/(b + 2*c*x) - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (14*c*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^3*d^2)","A",1
1189,1,111,154,0.1358554,"\int \frac{1}{(b d+2 c d x)^3 \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^3),x]","\frac{\frac{32 c^2 \left(b^2-4 a c\right)}{(b+2 c x)^2}+\frac{16 c \left(b^2-4 a c\right)}{a+x (b+c x)}-\frac{\left(b^2-4 a c\right)^2}{(a+x (b+c x))^2}+96 c^2 \log (a+x (b+c x))-192 c^2 \log (b+2 c x)}{2 d^3 \left(b^2-4 a c\right)^4}","\frac{48 c^2 \log \left(a+b x+c x^2\right)}{d^3 \left(b^2-4 a c\right)^4}+\frac{48 c^2}{d^3 \left(b^2-4 a c\right)^3 (b+2 c x)^2}-\frac{96 c^2 \log (b+2 c x)}{d^3 \left(b^2-4 a c\right)^4}+\frac{6 c}{d^3 \left(b^2-4 a c\right)^2 (b+2 c x)^2 \left(a+b x+c x^2\right)}-\frac{1}{2 d^3 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)^2}",1,"((32*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x))^2 + (16*c*(b^2 - 4*a*c))/(a + x*(b + c*x)) - 192*c^2*Log[b + 2*c*x] + 96*c^2*Log[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)^4*d^3)","A",1
1190,1,140,168,0.2810259,"\int \frac{1}{(b d+2 c d x)^4 \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3),x]","\frac{\frac{64 c^2 \left(b^2-4 a c\right)}{(b+2 c x)^3}+\frac{840 c^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x)}{(a+x (b+c x))^2}+\frac{66 c (b+2 c x)}{a+x (b+c x)}+\frac{576 c^2}{b+2 c x}}{6 d^4 \left(b^2-4 a c\right)^4}","\frac{140 c^2}{d^4 \left(b^2-4 a c\right)^4 (b+2 c x)}+\frac{140 c^2}{3 d^4 \left(b^2-4 a c\right)^3 (b+2 c x)^3}-\frac{140 c^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{d^4 \left(b^2-4 a c\right)^{9/2}}+\frac{7 c}{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^3 \left(a+b x+c x^2\right)}-\frac{1}{2 d^4 \left(b^2-4 a c\right) (b+2 c x)^3 \left(a+b x+c x^2\right)^2}",1,"((64*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^3 + (576*c^2)/(b + 2*c*x) - (3*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (66*c*(b + 2*c*x))/(a + x*(b + c*x)) + (840*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(6*(b^2 - 4*a*c)^4*d^4)","A",1
1191,1,203,165,0.8691012,"\int (b d+2 c d x)^4 \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^4*Sqrt[a + b*x + c*x^2],x]","d^4 \left(\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 c \left(a+2 c x^2\right)+b^2+8 b c x\right)-\sqrt{c} \sqrt{4 a-\frac{b^2}{c}} \left(4 a c-b^2\right) \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)\right)}{64 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+\frac{1}{3} (b+2 c x)^3 (a+x (b+c x))^{3/2}\right)","-\frac{d^4 \left(b^2-4 a c\right)^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{64 c^{3/2}}-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}-\frac{d^4 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}",1,"d^4*(((b + 2*c*x)^3*(a + x*(b + c*x))^(3/2))/3 + ((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*(2*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)) - Sqrt[4*a - b^2/c]*Sqrt[c]*(-b^2 + 4*a*c)*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])]))/(64*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))","A",1
1192,1,44,59,0.0368676,"\int (b d+2 c d x)^3 \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^3*Sqrt[a + b*x + c*x^2],x]","\frac{2}{15} d^3 (a+x (b+c x))^{3/2} \left(4 c \left(3 c x^2-2 a\right)+5 b^2+12 b c x\right)","\frac{4}{15} d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^{3/2}",1,"(2*d^3*(a + x*(b + c*x))^(3/2)*(5*b^2 + 12*b*c*x + 4*c*(-2*a + 3*c*x^2)))/15","A",1
1193,1,140,123,0.3554142,"\int (b d+2 c d x)^2 \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^2*Sqrt[a + b*x + c*x^2],x]","\frac{d^2 \sqrt{a+x (b+c x)} \left(2 (b+2 c x) \left(4 c \left(a+2 c x^2\right)+b^2+8 b c x\right)-\frac{c^{3/2} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x)) \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{\left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/2}}\right)}{32 c}","-\frac{d^2 \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)}{32 c^{3/2}}-\frac{d^2 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}{16 c}+\frac{d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{8 c}",1,"(d^2*Sqrt[a + x*(b + c*x)]*(2*(b + 2*c*x)*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)) - (Sqrt[4*a - b^2/c]*c^(3/2)*(a + x*(b + c*x))*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/2)))/(32*c)","A",1
1194,1,18,19,0.0095278,"\int (b d+2 c d x) \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)*Sqrt[a + b*x + c*x^2],x]","\frac{2}{3} d (a+x (b+c x))^{3/2}","\frac{2}{3} d \left(a+b x+c x^2\right)^{3/2}",1,"(2*d*(a + x*(b + c*x))^(3/2))/3","A",1
1195,1,80,83,0.0543498,"\int \frac{\sqrt{a+b x+c x^2}}{b d+2 c d x} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x),x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}-\sqrt{b^2-4 a c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right)}{4 c^{3/2} d}","\frac{\sqrt{a+b x+c x^2}}{2 c d}-\frac{\sqrt{b^2-4 a c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{4 c^{3/2} d}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)] - Sqrt[b^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])/Sqrt[b^2 - 4*a*c]])/(4*c^(3/2)*d)","A",1
1196,1,114,75,0.2863849,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^2} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^2,x]","\frac{\sqrt{a+x (b+c x)} \left(\frac{\sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}-\frac{2 \sqrt{c}}{b+2 c x}\right)}{4 c^{3/2} d^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{4 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)}",1,"(Sqrt[a + x*(b + c*x)]*((-2*Sqrt[c])/(b + 2*c*x) + ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])]/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/(4*c^(3/2)*d^2)","A",1
1197,1,103,91,0.2022332,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^3} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^3,x]","\frac{-\sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)-\frac{2 c (a+x (b+c x))}{(b+2 c x)^2}}{8 c^2 d^3 \sqrt{a+x (b+c x)}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{8 c^{3/2} d^3 \sqrt{b^2-4 a c}}-\frac{\sqrt{a+b x+c x^2}}{4 c d^3 (b+2 c x)^2}",1,"((-2*c*(a + x*(b + c*x)))/(b + 2*c*x)^2 - Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]])/(8*c^2*d^3*Sqrt[a + x*(b + c*x)])","A",1
1198,1,38,39,0.0165735,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^4} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^4,x]","\frac{2 (a+x (b+c x))^{3/2}}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3}","\frac{2 \left(a+b x+c x^2\right)^{3/2}}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3}",1,"(2*(a + x*(b + c*x))^(3/2))/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3)","A",1
1199,1,62,133,0.0295553,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^5} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^5,x]","\frac{2 (a+x (b+c x))^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{3 d^5 \left(b^2-4 a c\right)^3}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{32 c^{3/2} d^5 \left(b^2-4 a c\right)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{16 c d^5 \left(b^2-4 a c\right) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}",1,"(2*(a + x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(3*(b^2 - 4*a*c)^3*d^5)","C",1
1200,1,62,79,0.0291865,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^6} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^6,x]","\frac{2 (a+x (b+c x))^{3/2} \left(4 c \left(2 c x^2-3 a\right)+5 b^2+8 b c x\right)}{15 d^6 \left(b^2-4 a c\right)^2 (b+2 c x)^5}","\frac{4 \left(a+b x+c x^2\right)^{3/2}}{15 d^6 \left(b^2-4 a c\right)^2 (b+2 c x)^3}+\frac{2 \left(a+b x+c x^2\right)^{3/2}}{5 d^6 \left(b^2-4 a c\right) (b+2 c x)^5}",1,"(2*(a + x*(b + c*x))^(3/2)*(5*b^2 + 8*b*c*x + 4*c*(-3*a + 2*c*x^2)))/(15*(b^2 - 4*a*c)^2*d^6*(b + 2*c*x)^5)","A",1
1201,1,62,175,0.0239442,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^7} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^7,x]","\frac{2 (a+x (b+c x))^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{3 d^7 \left(b^2-4 a c\right)^4}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{64 c^{3/2} d^7 \left(b^2-4 a c\right)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{32 c d^7 \left(b^2-4 a c\right)^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{48 c d^7 \left(b^2-4 a c\right) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{12 c d^7 (b+2 c x)^6}",1,"(2*(a + x*(b + c*x))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(3*(b^2 - 4*a*c)^4*d^7)","C",1
1202,1,92,98,0.0760283,"\int (b d+2 c d x)^5 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^(3/2),x]","\frac{2}{315} d^5 (a+x (b+c x))^{5/2} \left(16 c^2 \left(8 a^2-20 a c x^2+35 c^2 x^4\right)+8 b^2 c \left(115 c x^2-18 a\right)+160 b c^2 x \left(7 c x^2-2 a\right)+63 b^4+360 b^3 c x\right)","\frac{16}{315} d^5 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{5/2}+\frac{8}{63} d^5 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)^{5/2}+\frac{2}{9} d^5 (b+2 c x)^4 \left(a+b x+c x^2\right)^{5/2}",1,"(2*d^5*(a + x*(b + c*x))^(5/2)*(63*b^4 + 360*b^3*c*x + 160*b*c^2*x*(-2*a + 7*c*x^2) + 8*b^2*c*(-18*a + 115*c*x^2) + 16*c^2*(8*a^2 - 20*a*c*x^2 + 35*c^2*x^4)))/315","A",1
1203,1,249,207,2.001957,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2),x]","\frac{1}{4} d^4 \left((b+2 c x)^3 (a+x (b+c x))^{5/2}-2 c \left(a-\frac{b^2}{4 c}\right) (b+2 c x) \sqrt{a+x (b+c x)} \left((a+x (b+c x))^2-\frac{(a+x (b+c x)) \left(2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} \left(4 a c-b^2\right) \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)\right)}{256 c (b+2 c x) \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/2}}\right)\right)","\frac{3 d^4 \left(b^2-4 a c\right)^4 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2048 c^{5/2}}-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}+\frac{3 d^4 \left(b^2-4 a c\right)^3 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{d^4 (b+2 c x)^5 \left(a+b x+c x^2\right)^{3/2}}{16 c}",1,"(d^4*((b + 2*c*x)^3*(a + x*(b + c*x))^(5/2) - 2*(a - b^2/(4*c))*c*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(b + c*x))^2 - ((a + x*(b + c*x))*(2*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*Sqrt[4*a - b^2/c]*Sqrt[c]*(-b^2 + 4*a*c)*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])]))/(256*c*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/2)))))/4","A",1
1204,1,44,59,0.0456735,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2),x]","\frac{2}{35} d^3 (a+x (b+c x))^{5/2} \left(4 c \left(5 c x^2-2 a\right)+7 b^2+20 b c x\right)","\frac{4}{35} d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{5/2}+\frac{2}{7} d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^{5/2}",1,"(2*d^3*(a + x*(b + c*x))^(5/2)*(7*b^2 + 20*b*c*x + 4*c*(-2*a + 5*c*x^2)))/35","A",1
1205,1,211,165,0.427563,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2),x]","\frac{1}{3} d^2 (b+2 c x) \sqrt{a+x (b+c x)} \left((a+x (b+c x))^2-\frac{(a+x (b+c x)) \left(2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} \left(4 a c-b^2\right) \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)\right)}{256 c (b+2 c x) \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/2}}\right)","\frac{d^2 \left(b^2-4 a c\right)^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{5/2}}+\frac{d^2 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^2}-\frac{d^2 \left(b^2-4 a c\right) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{64 c^2}+\frac{d^2 (b+2 c x)^3 \left(a+b x+c x^2\right)^{3/2}}{12 c}",1,"(d^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(b + c*x))^2 - ((a + x*(b + c*x))*(2*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*Sqrt[4*a - b^2/c]*Sqrt[c]*(-b^2 + 4*a*c)*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])]))/(256*c*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/2))))/3","A",1
1206,1,18,19,0.0160692,"\int (b d+2 c d x) \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2),x]","\frac{2}{5} d (a+x (b+c x))^{5/2}","\frac{2}{5} d \left(a+b x+c x^2\right)^{5/2}",1,"(2*d*(a + x*(b + c*x))^(5/2))/5","A",1
1207,1,103,115,0.1034072,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x),x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left(4 c \left(4 a+c x^2\right)-3 b^2+4 b c x\right)+3 \left(b^2-4 a c\right)^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right)}{48 c^{5/2} d}","\frac{\left(b^2-4 a c\right)^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{16 c^{5/2} d}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left(a+b x+c x^2\right)^{3/2}}{6 c d}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 4*b*c*x + 4*c*(4*a + c*x^2)) + 3*(b^2 - 4*a*c)^(3/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])/Sqrt[b^2 - 4*a*c]])/(48*c^(5/2)*d)","A",1
1208,1,95,113,0.0450615,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^2,x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 c^2 d^2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{3 \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)}{32 c^{5/2} d^2}+\frac{3 (b+2 c x) \sqrt{a+b x+c x^2}}{16 c^2 d^2}-\frac{\left(a+b x+c x^2\right)^{3/2}}{2 c d^2 (b+2 c x)}",1,"((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -1/2, 1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*c^2*d^2*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1209,1,62,115,0.0314888,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^3,x]","\frac{2 (a+x (b+c x))^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{5 d^3 \left(b^2-4 a c\right)^2}","-\frac{3 \sqrt{b^2-4 a c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{32 c^{5/2} d^3}+\frac{3 \sqrt{a+b x+c x^2}}{16 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{3/2}}{4 c d^3 (b+2 c x)^2}",1,"(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^2*d^3)","C",1
1210,1,95,107,0.0452633,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^4,x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{48 c^2 d^4 (b+2 c x)^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{5/2} d^4}-\frac{\sqrt{a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac{\left(a+b x+c x^2\right)^{3/2}}{6 c d^4 (b+2 c x)^3}",1,"((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -3/2, -1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(48*c^2*d^4*(b + 2*c*x)^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1211,1,162,123,0.2456706,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^5,x]","\frac{-2 c \left(8 a^2 c+a \left(3 b^2+28 b c x+28 c^2 x^2\right)+x \left(3 b^3+23 b^2 c x+40 b c^2 x^2+20 c^3 x^3\right)\right)-3 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)}{128 c^3 d^5 (b+2 c x)^4 \sqrt{a+x (b+c x)}}","\frac{3 \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{128 c^{5/2} d^5 \sqrt{b^2-4 a c}}-\frac{3 \sqrt{a+b x+c x^2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left(a+b x+c x^2\right)^{3/2}}{8 c d^5 (b+2 c x)^4}",1,"(-2*c*(8*a^2*c + a*(3*b^2 + 28*b*c*x + 28*c^2*x^2) + x*(3*b^3 + 23*b^2*c*x + 40*b*c^2*x^2 + 20*c^3*x^3)) - 3*(b + 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]])/(128*c^3*d^5*(b + 2*c*x)^4*Sqrt[a + x*(b + c*x)])","A",1
1212,1,38,39,0.0242612,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]","\frac{2 (a+x (b+c x))^{5/2}}{5 d^6 \left(b^2-4 a c\right) (b+2 c x)^5}","\frac{2 \left(a+b x+c x^2\right)^{5/2}}{5 d^6 \left(b^2-4 a c\right) (b+2 c x)^5}",1,"(2*(a + x*(b + c*x))^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)","A",1
1213,1,62,165,0.0304549,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^7} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]","\frac{2 (a+x (b+c x))^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{5 d^7 \left(b^2-4 a c\right)^4}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{256 c^{5/2} d^7 \left(b^2-4 a c\right)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^7 \left(b^2-4 a c\right) (b+2 c x)^2}-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left(a+b x+c x^2\right)^{3/2}}{12 c d^7 (b+2 c x)^6}",1,"(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^4*d^7)","C",1
1214,1,62,79,0.0361926,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^8} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]","\frac{2 (a+x (b+c x))^{5/2} \left(4 c \left(2 c x^2-5 a\right)+7 b^2+8 b c x\right)}{35 d^8 \left(b^2-4 a c\right)^2 (b+2 c x)^7}","\frac{4 \left(a+b x+c x^2\right)^{5/2}}{35 d^8 \left(b^2-4 a c\right)^2 (b+2 c x)^5}+\frac{2 \left(a+b x+c x^2\right)^{5/2}}{7 d^8 \left(b^2-4 a c\right) (b+2 c x)^7}",1,"(2*(a + x*(b + c*x))^(5/2)*(7*b^2 + 8*b*c*x + 4*c*(-5*a + 2*c*x^2)))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^7)","A",1
1215,1,62,207,0.0296262,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^9} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^9,x]","\frac{2 (a+x (b+c x))^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{5 d^9 \left(b^2-4 a c\right)^5}","\frac{3 \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{2048 c^{5/2} d^9 \left(b^2-4 a c\right)^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 d^9 \left(b^2-4 a c\right)^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 d^9 \left(b^2-4 a c\right) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left(a+b x+c x^2\right)^{3/2}}{16 c d^9 (b+2 c x)^8}",1,"(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^5*d^9)","C",1
1216,1,110,118,0.0582934,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^10,x]","\frac{2 (a+x (b+c x))^{5/2} \left(16 c^2 \left(35 a^2-20 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(34 c x^2-45 a\right)+64 b c^2 x \left(4 c x^2-5 a\right)+63 b^4+144 b^3 c x\right)}{315 d^{10} \left(b^2-4 a c\right)^3 (b+2 c x)^9}","\frac{16 \left(a+b x+c x^2\right)^{5/2}}{315 d^{10} \left(b^2-4 a c\right)^3 (b+2 c x)^5}+\frac{8 \left(a+b x+c x^2\right)^{5/2}}{63 d^{10} \left(b^2-4 a c\right)^2 (b+2 c x)^7}+\frac{2 \left(a+b x+c x^2\right)^{5/2}}{9 d^{10} \left(b^2-4 a c\right) (b+2 c x)^9}",1,"(2*(a + x*(b + c*x))^(5/2)*(63*b^4 + 144*b^3*c*x + 64*b*c^2*x*(-5*a + 4*c*x^2) + 8*b^2*c*(-45*a + 34*c*x^2) + 16*c^2*(35*a^2 - 20*a*c*x^2 + 8*c^2*x^4)))/(315*(b^2 - 4*a*c)^3*d^10*(b + 2*c*x)^9)","A",1
1217,1,92,98,0.0930313,"\int (b d+2 c d x)^5 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^(5/2),x]","\frac{2}{693} d^5 (a+x (b+c x))^{7/2} \left(16 c^2 \left(8 a^2-28 a c x^2+63 c^2 x^4\right)+8 b^2 c \left(203 c x^2-22 a\right)+224 b c^2 x \left(9 c x^2-2 a\right)+99 b^4+616 b^3 c x\right)","\frac{16}{693} d^5 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{7/2}+\frac{8}{99} d^5 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)^{7/2}+\frac{2}{11} d^5 (b+2 c x)^4 \left(a+b x+c x^2\right)^{7/2}",1,"(2*d^5*(a + x*(b + c*x))^(7/2)*(99*b^4 + 616*b^3*c*x + 224*b*c^2*x*(-2*a + 9*c*x^2) + 8*b^2*c*(-22*a + 203*c*x^2) + 16*c^2*(8*a^2 - 28*a*c*x^2 + 63*c^2*x^4)))/693","A",1
1218,1,265,249,3.5359472,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]","\frac{1}{5} d^4 \left((b+2 c x)^3 (a+x (b+c x))^{7/2}-\frac{3}{2} c \left(a-\frac{b^2}{4 c}\right) (b+2 c x) \sqrt{a+x (b+c x)} \left(\frac{\left(b^2-4 a c\right) \left(16 c^2 \left(33 a^2+26 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(11 c x^2-20 a\right)+32 b c^2 x \left(13 a+8 c x^2\right)+15 b^4-40 b^3 c x\right)}{3072 c^3}-\frac{5 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{2048 (b+2 c x) \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{7/2}}+(a+x (b+c x))^3\right)\right)","-\frac{3 d^4 \left(b^2-4 a c\right)^5 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16384 c^{7/2}}-\frac{3 d^4 \left(b^2-4 a c\right)^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{d^4 \left(b^2-4 a c\right)^3 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{d^4 \left(b^2-4 a c\right) (b+2 c x)^5 \left(a+b x+c x^2\right)^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left(a+b x+c x^2\right)^{5/2}}{20 c}",1,"(d^4*((b + 2*c*x)^3*(a + x*(b + c*x))^(7/2) - (3*(a - b^2/(4*c))*c*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(b + c*x))^3 + ((b^2 - 4*a*c)*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)))/(3072*c^3) - (5*Sqrt[4*a - b^2/c]*Sqrt[c]*(a + x*(b + c*x))^3*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(2048*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(7/2))))/2))/5","A",1
1219,1,44,59,0.0556743,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2),x]","\frac{2}{63} d^3 (a+x (b+c x))^{7/2} \left(4 c \left(7 c x^2-2 a\right)+9 b^2+28 b c x\right)","\frac{4}{63} d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^{7/2}",1,"(2*d^3*(a + x*(b + c*x))^(7/2)*(9*b^2 + 28*b*c*x + 4*c*(-2*a + 7*c*x^2)))/63","A",1
1220,1,225,207,0.93254,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2),x]","\frac{1}{4} d^2 (b+2 c x) \sqrt{a+x (b+c x)} \left(\frac{\left(b^2-4 a c\right) \left(16 c^2 \left(33 a^2+26 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(11 c x^2-20 a\right)+32 b c^2 x \left(13 a+8 c x^2\right)+15 b^4-40 b^3 c x\right)}{3072 c^3}-\frac{5 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{2048 (b+2 c x) \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{7/2}}+(a+x (b+c x))^3\right)","-\frac{5 d^2 \left(b^2-4 a c\right)^4 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8192 c^{7/2}}-\frac{5 d^2 \left(b^2-4 a c\right)^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left(b^2-4 a c\right)^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left(b^2-4 a c\right) (b+2 c x)^3 \left(a+b x+c x^2\right)^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left(a+b x+c x^2\right)^{5/2}}{16 c}",1,"(d^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(b + c*x))^3 + ((b^2 - 4*a*c)*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)))/(3072*c^3) - (5*Sqrt[4*a - b^2/c]*Sqrt[c]*(a + x*(b + c*x))^3*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(2048*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(7/2))))/4","A",1
1221,1,18,19,0.0135699,"\int (b d+2 c d x) \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2),x]","\frac{2}{7} d (a+x (b+c x))^{7/2}","\frac{2}{7} d \left(a+b x+c x^2\right)^{7/2}",1,"(2*d*(a + x*(b + c*x))^(7/2))/7","A",1
1222,1,150,149,0.1665666,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x),x]","\frac{\frac{\sqrt{a+x (b+c x)} \left(16 c^2 \left(23 a^2+11 a c x^2+3 c^2 x^4\right)+28 b^2 c \left(c x^2-5 a\right)+16 b c^2 x \left(11 a+6 c x^2\right)+15 b^4-20 b^3 c x\right)}{480 c^3}-\frac{\left(b^2-4 a c\right)^{5/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right)}{64 c^{7/2}}}{d}","-\frac{\left(b^2-4 a c\right)^{5/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{64 c^{7/2} d}+\frac{\left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}{24 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2}}{10 c d}",1,"((Sqrt[a + x*(b + c*x)]*(15*b^4 - 20*b^3*c*x + 28*b^2*c*(-5*a + c*x^2) + 16*b*c^2*x*(11*a + 6*c*x^2) + 16*c^2*(23*a^2 + 11*a*c*x^2 + 3*c^2*x^4)))/(480*c^3) - ((b^2 - 4*a*c)^(5/2)*ArcTan[(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])/Sqrt[b^2 - 4*a*c]])/(64*c^(7/2)))/d","A",1
1223,1,97,153,0.0490914,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^2,x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64 c^3 d^2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{15 \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)}{512 c^{7/2} d^2}-\frac{15 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^3 d^2}+\frac{5 (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{32 c^2 d^2}-\frac{\left(a+b x+c x^2\right)^{5/2}}{2 c d^2 (b+2 c x)}",1,"-1/64*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -1/2, 1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^2*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1224,1,62,147,0.0376878,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^3,x]","\frac{2 (a+x (b+c x))^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{7 d^3 \left(b^2-4 a c\right)^2}","\frac{5 \left(b^2-4 a c\right)^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{128 c^{7/2} d^3}-\frac{5 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}{64 c^3 d^3}+\frac{5 \left(a+b x+c x^2\right)^{3/2}}{48 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{5/2}}{4 c d^3 (b+2 c x)^2}",1,"(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^2*d^3)","C",1
1225,1,97,145,0.0455161,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^4,x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{192 c^3 d^4 (b+2 c x)^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{5 \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)}{128 c^{7/2} d^4}+\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left(a+b x+c x^2\right)^{5/2}}{6 c d^4 (b+2 c x)^3}",1,"-1/192*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -3/2, -1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^4*(b + 2*c*x)^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1226,1,62,147,0.0378256,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^5,x]","\frac{2 (a+x (b+c x))^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{7 d^5 \left(b^2-4 a c\right)^3}","-\frac{15 \sqrt{b^2-4 a c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{512 c^{7/2} d^5}+\frac{15 \sqrt{a+b x+c x^2}}{256 c^3 d^5}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac{\left(a+b x+c x^2\right)^{5/2}}{8 c d^5 (b+2 c x)^4}",1,"(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^3*d^5)","C",1
1227,1,97,139,0.0492184,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{320 c^3 d^6 (b+2 c x)^5 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{64 c^{7/2} d^6}-\frac{\sqrt{a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac{\left(a+b x+c x^2\right)^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac{\left(a+b x+c x^2\right)^{5/2}}{10 c d^6 (b+2 c x)^5}",1,"-1/320*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -5/2, -3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1228,1,233,155,0.3705884,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^7} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]","\frac{-2 c \left(128 a^3 c^2+8 a^2 c \left(5 b^2+68 b c x+68 c^2 x^2\right)+a \left(15 b^4+200 b^3 c x+1144 b^2 c^2 x^2+1888 b c^3 x^3+944 c^4 x^4\right)+x \left(15 b^5+175 b^4 c x+848 b^3 c^2 x^2+1744 b^2 c^3 x^3+1584 b c^4 x^4+528 c^5 x^5\right)\right)-15 (b+2 c x)^6 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \tanh ^{-1}\left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)}{3072 c^4 d^7 (b+2 c x)^6 \sqrt{a+x (b+c x)}}","\frac{5 \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{1024 c^{7/2} d^7 \sqrt{b^2-4 a c}}-\frac{5 \sqrt{a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac{\left(a+b x+c x^2\right)^{5/2}}{12 c d^7 (b+2 c x)^6}",1,"(-2*c*(128*a^3*c^2 + 8*a^2*c*(5*b^2 + 68*b*c*x + 68*c^2*x^2) + a*(15*b^4 + 200*b^3*c*x + 1144*b^2*c^2*x^2 + 1888*b*c^3*x^3 + 944*c^4*x^4) + x*(15*b^5 + 175*b^4*c*x + 848*b^3*c^2*x^2 + 1744*b^2*c^3*x^3 + 1584*b*c^4*x^4 + 528*c^5*x^5)) - 15*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]])/(3072*c^4*d^7*(b + 2*c*x)^6*Sqrt[a + x*(b + c*x)])","A",1
1229,1,38,39,0.0269117,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^8} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]","\frac{2 (a+x (b+c x))^{7/2}}{7 d^8 \left(b^2-4 a c\right) (b+2 c x)^7}","\frac{2 \left(a+b x+c x^2\right)^{7/2}}{7 d^8 \left(b^2-4 a c\right) (b+2 c x)^7}",1,"(2*(a + x*(b + c*x))^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)","A",1
1230,1,62,197,0.0402362,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^9} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^9,x]","\frac{2 (a+x (b+c x))^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{7 d^9 \left(b^2-4 a c\right)^5}","\frac{5 \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{8192 c^{7/2} d^9 \left(b^2-4 a c\right)^{3/2}}+\frac{5 \sqrt{a+b x+c x^2}}{4096 c^3 d^9 \left(b^2-4 a c\right) (b+2 c x)^2}-\frac{5 \sqrt{a+b x+c x^2}}{2048 c^3 d^9 (b+2 c x)^4}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{384 c^2 d^9 (b+2 c x)^6}-\frac{\left(a+b x+c x^2\right)^{5/2}}{16 c d^9 (b+2 c x)^8}",1,"(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^5*d^9)","C",1
1231,1,62,79,0.0518084,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]","\frac{2 (a+x (b+c x))^{7/2} \left(4 c \left(2 c x^2-7 a\right)+9 b^2+8 b c x\right)}{63 d^{10} \left(b^2-4 a c\right)^2 (b+2 c x)^9}","\frac{4 \left(a+b x+c x^2\right)^{7/2}}{63 d^{10} \left(b^2-4 a c\right)^2 (b+2 c x)^7}+\frac{2 \left(a+b x+c x^2\right)^{7/2}}{9 d^{10} \left(b^2-4 a c\right) (b+2 c x)^9}",1,"(2*(a + x*(b + c*x))^(7/2)*(9*b^2 + 8*b*c*x + 4*c*(-7*a + 2*c*x^2)))/(63*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x)^9)","A",1
1232,1,62,239,0.0401416,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{11}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^11,x]","\frac{2 (a+x (b+c x))^{7/2} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{7 d^{11} \left(b^2-4 a c\right)^6}","\frac{3 \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{16384 c^{7/2} d^{11} \left(b^2-4 a c\right)^{5/2}}+\frac{3 \sqrt{a+b x+c x^2}}{8192 c^3 d^{11} \left(b^2-4 a c\right)^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{4096 c^3 d^{11} \left(b^2-4 a c\right) (b+2 c x)^4}-\frac{\sqrt{a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}-\frac{\left(a+b x+c x^2\right)^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac{\left(a+b x+c x^2\right)^{5/2}}{20 c d^{11} (b+2 c x)^{10}}",1,"(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^6*d^11)","C",1
1233,1,110,118,0.0769614,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{12}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^12,x]","\frac{2 (a+x (b+c x))^{7/2} \left(16 c^2 \left(63 a^2-28 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(38 c x^2-77 a\right)+64 b c^2 x \left(4 c x^2-7 a\right)+99 b^4+176 b^3 c x\right)}{693 d^{12} \left(b^2-4 a c\right)^3 (b+2 c x)^{11}}","\frac{16 \left(a+b x+c x^2\right)^{7/2}}{693 d^{12} \left(b^2-4 a c\right)^3 (b+2 c x)^7}+\frac{8 \left(a+b x+c x^2\right)^{7/2}}{99 d^{12} \left(b^2-4 a c\right)^2 (b+2 c x)^9}+\frac{2 \left(a+b x+c x^2\right)^{7/2}}{11 d^{12} \left(b^2-4 a c\right) (b+2 c x)^{11}}",1,"(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c*x^2) + 8*b^2*c*(-77*a + 38*c*x^2) + 16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)","A",1
1234,1,100,117,0.1053989,"\int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^4/Sqrt[a + b*x + c*x^2],x]","d^4 \left(\frac{1}{4} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(2 c x^2-3 a\right)+5 b^2+8 b c x\right)+\frac{3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{8 \sqrt{c}}\right)","\frac{3}{4} d^4 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}+\frac{3 d^4 \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)}{8 \sqrt{c}}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}",1,"d^4*(((b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(5*b^2 + 8*b*c*x + 4*c*(-3*a + 2*c*x^2)))/4 + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*Sqrt[c]))","A",1
1235,1,43,59,0.035746,"\int \frac{(b d+2 c d x)^3}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^3/Sqrt[a + b*x + c*x^2],x]","\frac{2}{3} d^3 \sqrt{a+x (b+c x)} \left(4 c \left(c x^2-2 a\right)+3 b^2+4 b c x\right)","\frac{4}{3} d^3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}",1,"(2*d^3*Sqrt[a + x*(b + c*x)]*(3*b^2 + 4*b*c*x + 4*c*(-2*a + c*x^2)))/3","A",1
1236,1,71,75,0.0549491,"\int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^2/Sqrt[a + b*x + c*x^2],x]","d^2 \left(\frac{\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}}+(b+2 c x) \sqrt{a+x (b+c x)}\right)","\frac{d^2 \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)}{2 \sqrt{c}}+d^2 (b+2 c x) \sqrt{a+b x+c x^2}",1,"d^2*((b + 2*c*x)*Sqrt[a + x*(b + c*x)] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[c]))","A",1
1237,1,16,17,0.0056736,"\int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)/Sqrt[a + b*x + c*x^2],x]","2 d \sqrt{a+x (b+c x)}","2 d \sqrt{a+b x+c x^2}",1,"2*d*Sqrt[a + x*(b + c*x)]","A",1
1238,1,54,55,0.0313026,"\int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right)}{\sqrt{c} d \sqrt{b^2-4 a c}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{\sqrt{c} d \sqrt{b^2-4 a c}}",1,"ArcTan[(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])/Sqrt[b^2 - 4*a*c]]/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*d)","A",1
1239,1,36,37,0.0151038,"\int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{a+x (b+c x)}}{d^2 \left(b^2-4 a c\right) (b+2 c x)}","\frac{2 \sqrt{a+b x+c x^2}}{d^2 \left(b^2-4 a c\right) (b+2 c x)}",1,"(2*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*d^2*(b + 2*c*x))","A",1
1240,1,107,95,0.2844278,"\int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*Sqrt[a + b*x + c*x^2]),x]","\frac{\sqrt{a+x (b+c x)} \left(\frac{2 \left(b^2-4 a c\right)}{(b+2 c x)^2}+\frac{\tanh ^{-1}\left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)}{\sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)}{2 d^3 \left(b^2-4 a c\right)^2}","\frac{\sqrt{a+b x+c x^2}}{d^3 \left(b^2-4 a c\right) (b+2 c x)^2}+\frac{\tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{2 \sqrt{c} d^3 \left(b^2-4 a c\right)^{3/2}}",1,"(Sqrt[a + x*(b + c*x)]*((2*(b^2 - 4*a*c))/(b + 2*c*x)^2 + ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]]/Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))/(2*(b^2 - 4*a*c)^2*d^3)","A",1
1241,1,60,79,0.0267781,"\int \frac{1}{(b d+2 c d x)^4 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^4*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{a+x (b+c x)} \left(-4 c \left(a-2 c x^2\right)+3 b^2+8 b c x\right)}{3 d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^3}","\frac{4 \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right)^2 (b+2 c x)}+\frac{2 \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3}",1,"(2*Sqrt[a + x*(b + c*x)]*(3*b^2 + 8*b*c*x - 4*c*(a - 2*c*x^2)))/(3*(b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^3)","A",1
1242,1,139,102,0.4001738,"\int \frac{(b d+2 c d x)^4}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x]","d^4 \left(-\frac{6 c^{3/2} (a+x (b+c x))^{3/2} \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{\sqrt{4 a-\frac{b^2}{c}} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/2}}-\frac{2 (b+2 c x) \left(-2 c \left(3 a+c x^2\right)+b^2-2 b c x\right)}{\sqrt{a+x (b+c x)}}\right)","6 \sqrt{c} d^4 \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)+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}}",1,"d^4*((-2*(b + 2*c*x)*(b^2 - 2*b*c*x - 2*c*(3*a + c*x^2)))/Sqrt[a + x*(b + c*x)] - (6*c^(3/2)*(a + x*(b + c*x))^(3/2)*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(Sqrt[4*a - b^2/c]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/2)))","A",1
1243,1,40,48,0.0274353,"\int \frac{(b d+2 c d x)^3}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(3/2),x]","\frac{d^3 \left(8 c \left(2 a+c x^2\right)-2 b^2+8 b c x\right)}{\sqrt{a+x (b+c x)}}","16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}}",1,"(d^3*(-2*b^2 + 8*b*c*x + 8*c*(2*a + c*x^2)))/Sqrt[a + x*(b + c*x)]","A",1
1244,1,118,66,0.189335,"\int \frac{(b d+2 c d x)^2}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^(3/2),x]","d^2 \left(\frac{4 \sqrt{c} \sqrt{a+x (b+c x)} \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}-\frac{2 (b+2 c x)}{\sqrt{a+x (b+c x)}}\right)","4 \sqrt{c} d^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)-\frac{2 d^2 (b+2 c x)}{\sqrt{a+b x+c x^2}}",1,"d^2*((-2*(b + 2*c*x))/Sqrt[a + x*(b + c*x)] + (4*Sqrt[c]*Sqrt[a + x*(b + c*x)]*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))","A",1
1245,1,16,17,0.0065599,"\int \frac{b d+2 c d x}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^(3/2),x]","-\frac{2 d}{\sqrt{a+x (b+c x)}}","-\frac{2 d}{\sqrt{a+b x+c x^2}}",1,"(-2*d)/Sqrt[a + x*(b + c*x)]","A",1
1246,1,60,86,0.0279114,"\int \frac{1}{(b d+2 c d x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{2}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{d \left(b^2-4 a c\right)^{3/2}}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)*d*Sqrt[a + x*(b + c*x)])","C",1
1247,1,56,76,0.0271429,"\int \frac{1}{(b d+2 c d x)^2 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(3/2)),x]","-\frac{2 \left(4 c \left(a+2 c x^2\right)+b^2+8 b c x\right)}{d^2 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+x (b+c x)}}","-\frac{16 c \sqrt{a+b x+c x^2}}{d^2 \left(b^2-4 a c\right)^2 (b+2 c x)}-\frac{2}{d^2 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}",1,"(-2*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)))/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])","A",1
1248,1,60,132,0.0269106,"\int \frac{1}{(b d+2 c d x)^3 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{d^3 \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)}}","-\frac{12 c \sqrt{a+b x+c x^2}}{d^3 \left(b^2-4 a c\right)^2 (b+2 c x)^2}-\frac{2}{d^3 \left(b^2-4 a c\right) (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{d^3 \left(b^2-4 a c\right)^{5/2}}",1,"(-2*Hypergeometric2F1[-1/2, 2, 1/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)^2*d^3*Sqrt[a + x*(b + c*x)])","C",1
1249,1,108,118,0.0488501,"\int \frac{1}{(b d+2 c d x)^4 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]","-\frac{2 \left(16 c^2 \left(-a^2+4 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(3 a+22 c x^2\right)+64 b c^2 x \left(a+4 c x^2\right)+3 b^4+48 b^3 c x\right)}{3 d^4 \left(b^2-4 a c\right)^3 (b+2 c x)^3 \sqrt{a+x (b+c x)}}","-\frac{64 c \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right)^3 (b+2 c x)}-\frac{32 c \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^3}-\frac{2}{d^4 \left(b^2-4 a c\right) (b+2 c x)^3 \sqrt{a+b x+c x^2}}",1,"(-2*(3*b^4 + 48*b^3*c*x + 64*b*c^2*x*(a + 4*c*x^2) + 8*b^2*c*(3*a + 22*c*x^2) + 16*c^2*(-a^2 + 4*a*c*x^2 + 8*c^2*x^4)))/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])","A",1
1250,1,204,136,1.4697819,"\int \frac{(b d+2 c d x)^6}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]","\frac{d^6 \left(3 (b+2 c x)^5-\frac{5 \left(b^2-4 a c\right) \left(\sqrt{4 a-\frac{b^2}{c}} (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 c \left(3 a+4 c x^2\right)+b^2+16 b c x\right)-24 c^{3/2} (a+x (b+c x))^2 \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)\right)}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)}{3 (a+x (b+c x))^{3/2}}","40 c^{3/2} d^6 \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)+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(d^6*(3*(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)*(Sqrt[4*a - b^2/c]*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)) - 24*c^(3/2)*(a + x*(b + c*x))^2*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])]))/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/(3*(a + x*(b + c*x))^(3/2))","A",1
1251,1,91,84,0.0528012,"\int \frac{(b d+2 c d x)^5}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^(5/2),x]","\frac{d^5 \left(32 c^2 \left(8 a^2+12 a c x^2+3 c^2 x^4\right)+16 b^2 c \left(3 c x^2-2 a\right)+192 b c^2 x \left(2 a+c x^2\right)-2 b^4-48 b^3 c x\right)}{3 (a+x (b+c x))^{3/2}}","\frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(d^5*(-2*b^4 - 48*b^3*c*x + 192*b*c^2*x*(2*a + c*x^2) + 16*b^2*c*(-2*a + 3*c*x^2) + 32*c^2*(8*a^2 + 12*a*c*x^2 + 3*c^2*x^4)))/(3*(a + x*(b + c*x))^(3/2))","A",1
1252,1,142,96,0.3345635,"\int \frac{(b d+2 c d x)^4}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x]","d^4 \left(\frac{16 c^{3/2} \sqrt{a+x (b+c x)} \sinh ^{-1}\left(\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right)}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}-\frac{2 (b+2 c x) \left(4 c \left(3 a+4 c x^2\right)+b^2+16 b c x\right)}{3 (a+x (b+c x))^{3/2}}\right)","16 c^{3/2} d^4 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}-\frac{2 d^4 (b+2 c x)^3}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"d^4*((-2*(b + 2*c*x)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(3*(a + x*(b + c*x))^(3/2)) + (16*c^(3/2)*Sqrt[a + x*(b + c*x)]*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))","A",1
1253,1,42,52,0.0289062,"\int \frac{(b d+2 c d x)^3}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 d^3 \left(4 c \left(2 a+3 c x^2\right)+b^2+12 b c x\right)}{3 (a+x (b+c x))^{3/2}}","-\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*d^3*(b^2 + 12*b*c*x + 4*c*(2*a + 3*c*x^2)))/(3*(a + x*(b + c*x))^(3/2))","A",1
1254,1,38,39,0.0206167,"\int \frac{(b d+2 c d x)^2}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 d^2 (b+2 c x)^3}{3 \left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}","-\frac{2 d^2 (b+2 c x)^3}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*d^2*(b + 2*c*x)^3)/(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2))","A",1
1255,1,18,19,0.0072105,"\int \frac{b d+2 c d x}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 d}{3 (a+x (b+c x))^{3/2}}","-\frac{2 d}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*d)/(3*(a + x*(b + c*x))^(3/2))","A",1
1256,1,62,118,0.030035,"\int \frac{1}{(b d+2 c d x) \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{3 d \left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}","\frac{16 c^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{d \left(b^2-4 a c\right)^{5/2}}+\frac{8 c}{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*Hypergeometric2F1[-3/2, 1, -1/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(3*(b^2 - 4*a*c)*d*(a + x*(b + c*x))^(3/2))","C",1
1257,1,108,122,0.0509395,"\int \frac{1}{(b d+2 c d x)^2 \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2)),x]","\frac{32 c^2 \left(3 a^2+12 a c x^2+8 c^2 x^4\right)+48 b^2 c \left(a+6 c x^2\right)+128 b c^2 x \left(3 a+4 c x^2\right)-2 b^4+32 b^3 c x}{3 d^2 \left(b^2-4 a c\right)^3 (b+2 c x) (a+x (b+c x))^{3/2}}","\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^2 \left(b^2-4 a c\right)^3 (b+2 c x)}+\frac{32 c}{3 d^2 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{2}{3 d^2 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*b^4 + 32*b^3*c*x + 128*b*c^2*x*(3*a + 4*c*x^2) + 48*b^2*c*(a + 6*c*x^2) + 32*c^2*(3*a^2 + 12*a*c*x^2 + 8*c^2*x^4))/(3*(b^2 - 4*a*c)^3*d^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))","A",1
1258,1,62,176,0.0354185,"\int \frac{1}{(b d+2 c d x)^3 \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(5/2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{3 d^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{40 c^{3/2} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{d^3 \left(b^2-4 a c\right)^{7/2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{d^3 \left(b^2-4 a c\right)^3 (b+2 c x)^2}+\frac{40 c}{3 d^3 \left(b^2-4 a c\right)^2 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^3 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*Hypergeometric2F1[-3/2, 2, -1/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(3*(b^2 - 4*a*c)^2*d^3*(a + x*(b + c*x))^(3/2))","C",1
1259,1,178,162,0.0817877,"\int \frac{1}{(b d+2 c d x)^4 \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]","\frac{2 \left(48 b^2 c^2 \left(3 a^2+44 a c x^2+72 c^2 x^4\right)+384 b c^3 x \left(a^2+8 a c x^2+8 c^2 x^4\right)+64 c^3 \left(-a^3+6 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right)+12 b^4 c \left(3 a+34 c x^2\right)+64 b^3 c^2 x \left(9 a+28 c x^2\right)-b^6+24 b^5 c x\right)}{3 d^4 \left(b^2-4 a c\right)^4 (b+2 c x)^3 (a+x (b+c x))^{3/2}}","\frac{512 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right)^4 (b+2 c x)}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^4 \left(b^2-4 a c\right)^3 (b+2 c x)^3}+\frac{16 c}{d^4 \left(b^2-4 a c\right)^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3 \left(a+b x+c x^2\right)^{3/2}}",1,"(2*(-b^6 + 24*b^5*c*x + 64*b^3*c^2*x*(9*a + 28*c*x^2) + 12*b^4*c*(3*a + 34*c*x^2) + 384*b*c^3*x*(a^2 + 8*a*c*x^2 + 8*c^2*x^4) + 48*b^2*c^2*(3*a^2 + 44*a*c*x^2 + 72*c^2*x^4) + 64*c^3*(-a^3 + 6*a^2*c*x^2 + 24*a*c^2*x^4 + 16*c^3*x^6)))/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)^3*(a + x*(b + c*x))^(3/2))","A",1
1260,1,19,27,0.0084713,"\int \frac{1}{(a+b x) \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{\tanh ^{-1}\left(\sqrt{(a+b x)^2+1}\right)}{b}","-\frac{\tanh ^{-1}\left(\sqrt{a^2+2 a b x+b^2 x^2+1}\right)}{b}",1,"-(ArcTanh[Sqrt[1 + (a + b*x)^2]]/b)","A",1
1261,1,45,55,0.0432956,"\int (b d+2 c d x)^{5/2} \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2),x]","\frac{\left(c \left(11 a+7 c x^2\right)-b^2+7 b c x\right) (d (b+2 c x))^{7/2}}{77 c^2 d}","\frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}{28 c^2 d}",1,"((d*(b + 2*c*x))^(7/2)*(-b^2 + 7*b*c*x + c*(11*a + 7*c*x^2)))/(77*c^2*d)","A",1
1262,1,45,55,0.0290413,"\int (b d+2 c d x)^{3/2} \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2),x]","\frac{\left(c \left(9 a+5 c x^2\right)-b^2+5 b c x\right) (d (b+2 c x))^{5/2}}{45 c^2 d}","\frac{(b d+2 c d x)^{9/2}}{36 c^2 d^3}-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{5/2}}{20 c^2 d}",1,"((d*(b + 2*c*x))^(5/2)*(-b^2 + 5*b*c*x + c*(9*a + 5*c*x^2)))/(45*c^2*d)","A",1
1263,1,45,55,0.0241443,"\int \sqrt{b d+2 c d x} \left(a+b x+c x^2\right) \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2),x]","\frac{\left(c \left(7 a+3 c x^2\right)-b^2+3 b c x\right) (d (b+2 c x))^{3/2}}{21 c^2 d}","\frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}{12 c^2 d}",1,"((d*(b + 2*c*x))^(3/2)*(-b^2 + 3*b*c*x + c*(7*a + 3*c*x^2)))/(21*c^2*d)","A",1
1264,1,43,55,0.0253123,"\int \frac{a+b x+c x^2}{\sqrt{b d+2 c d x}} \, dx","Integrate[(a + b*x + c*x^2)/Sqrt[b*d + 2*c*d*x],x]","\frac{\left(c \left(5 a+c x^2\right)-b^2+b c x\right) \sqrt{d (b+2 c x)}}{5 c^2 d}","\frac{(b d+2 c d x)^{5/2}}{20 c^2 d^3}-\frac{\left(b^2-4 a c\right) \sqrt{b d+2 c d x}}{4 c^2 d}",1,"(Sqrt[d*(b + 2*c*x)]*(-b^2 + b*c*x + c*(5*a + c*x^2)))/(5*c^2*d)","A",1
1265,1,41,55,0.0245684,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(3/2),x]","\frac{c \left(c x^2-3 a\right)+b^2+b c x}{3 c^2 d \sqrt{d (b+2 c x)}}","\frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3}",1,"(b^2 + b*c*x + c*(-3*a + c*x^2))/(3*c^2*d*Sqrt[d*(b + 2*c*x)])","A",1
1266,1,43,55,0.0266209,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(5/2),x]","\frac{c \left(3 c x^2-a\right)+b^2+3 b c x}{3 c^2 d (d (b+2 c x))^{3/2}}","\frac{b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{4 c^2 d^3}",1,"(b^2 + 3*b*c*x + c*(-a + 3*c*x^2))/(3*c^2*d*(d*(b + 2*c*x))^(3/2))","A",1
1267,1,44,55,0.0307905,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(7/2),x]","\frac{-c \left(a+5 c x^2\right)-b^2-5 b c x}{5 c^2 d (d (b+2 c x))^{5/2}}","\frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}}",1,"(-b^2 - 5*b*c*x - c*(a + 5*c*x^2))/(5*c^2*d*(d*(b + 2*c*x))^(5/2))","A",1
1268,1,51,55,0.03706,"\int \frac{a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(9/2),x]","-\frac{\left(c \left(3 a+7 c x^2\right)+b^2+7 b c x\right) \sqrt{d (b+2 c x)}}{21 c^2 d^5 (b+2 c x)^4}","\frac{b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac{1}{12 c^2 d^3 (b d+2 c d x)^{3/2}}",1,"-1/21*(Sqrt[d*(b + 2*c*x)]*(b^2 + 7*b*c*x + c*(3*a + 7*c*x^2)))/(c^2*d^5*(b + 2*c*x)^4)","A",1
1269,1,63,88,0.0602971,"\int (b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2,x]","\frac{\left(-130 \left(b^2-4 a c\right) (b+2 c x)^2+117 \left(b^2-4 a c\right)^2+45 (b+2 c x)^4\right) (d (b+2 c x))^{5/2}}{9360 c^3 d}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac{\left(b^2-4 a c\right)^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac{(b d+2 c d x)^{13/2}}{208 c^3 d^5}",1,"((d*(b + 2*c*x))^(5/2)*(117*(b^2 - 4*a*c)^2 - 130*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 45*(b + 2*c*x)^4))/(9360*c^3*d)","A",1
1270,1,92,88,0.0416887,"\int \sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^2 \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2,x]","\frac{\left(c^2 \left(77 a^2+66 a c x^2+21 c^2 x^4\right)+b^2 c \left(15 c x^2-22 a\right)+6 b c^2 x \left(11 a+7 c x^2\right)+2 b^4-6 b^3 c x\right) (d (b+2 c x))^{3/2}}{231 c^3 d}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{\left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}{48 c^3 d}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5}",1,"((d*(b + 2*c*x))^(3/2)*(2*b^4 - 6*b^3*c*x + 6*b*c^2*x*(11*a + 7*c*x^2) + b^2*c*(-22*a + 15*c*x^2) + c^2*(77*a^2 + 66*a*c*x^2 + 21*c^2*x^4)))/(231*c^3*d)","A",1
1271,1,92,88,0.041896,"\int \frac{\left(a+b x+c x^2\right)^2}{\sqrt{b d+2 c d x}} \, dx","Integrate[(a + b*x + c*x^2)^2/Sqrt[b*d + 2*c*d*x],x]","\frac{\left(c^2 \left(45 a^2+18 a c x^2+5 c^2 x^4\right)+3 b^2 c \left(c x^2-6 a\right)+2 b c^2 x \left(9 a+5 c x^2\right)+2 b^4-2 b^3 c x\right) \sqrt{d (b+2 c x)}}{45 c^3 d}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac{\left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}{16 c^3 d}+\frac{(b d+2 c d x)^{9/2}}{144 c^3 d^5}",1,"(Sqrt[d*(b + 2*c*x)]*(2*b^4 - 2*b^3*c*x + 3*b^2*c*(-6*a + c*x^2) + 2*b*c^2*x*(9*a + 5*c*x^2) + c^2*(45*a^2 + 18*a*c*x^2 + 5*c^2*x^4)))/(45*c^3*d)","A",1
1272,1,91,88,0.0376003,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(3/2),x]","\frac{c^2 \left(-21 a^2+14 a c x^2+3 c^2 x^4\right)+b^2 c \left(14 a+c x^2\right)+2 b c^2 x \left(7 a+3 c x^2\right)-2 b^4-2 b^3 c x}{21 c^3 d \sqrt{d (b+2 c x)}}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}{24 c^3 d^3}-\frac{\left(b^2-4 a c\right)^2}{16 c^3 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{7/2}}{112 c^3 d^5}",1,"(-2*b^4 - 2*b^3*c*x + b^2*c*(14*a + c*x^2) + 2*b*c^2*x*(7*a + 3*c*x^2) + c^2*(-21*a^2 + 14*a*c*x^2 + 3*c^2*x^4))/(21*c^3*d*Sqrt[d*(b + 2*c*x)])","A",1
1273,1,91,88,0.0388042,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(5/2),x]","\frac{c^2 \left(-5 a^2+30 a c x^2+3 c^2 x^4\right)+b^2 c \left(10 a-3 c x^2\right)+6 b c^2 x \left(5 a+c x^2\right)-2 b^4-6 b^3 c x}{15 c^3 d (d (b+2 c x))^{3/2}}","-\frac{\left(b^2-4 a c\right) \sqrt{b d+2 c d x}}{8 c^3 d^3}-\frac{\left(b^2-4 a c\right)^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5}",1,"(-2*b^4 - 6*b^3*c*x + b^2*c*(10*a - 3*c*x^2) + 6*b*c^2*x*(5*a + c*x^2) + c^2*(-5*a^2 + 30*a*c*x^2 + 3*c^2*x^4))/(15*c^3*d*(d*(b + 2*c*x))^(3/2))","A",1
1274,1,92,88,0.0413492,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(7/2),x]","\frac{c^2 \left(-3 a^2-30 a c x^2+5 c^2 x^4\right)+3 b^2 c \left(5 c x^2-2 a\right)+10 b c^2 x \left(c x^2-3 a\right)+2 b^4+10 b^3 c x}{15 c^3 d (d (b+2 c x))^{5/2}}","\frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left(b^2-4 a c\right)^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5}",1,"(2*b^4 + 10*b^3*c*x + 10*b*c^2*x*(-3*a + c*x^2) + 3*b^2*c*(-2*a + 5*c*x^2) + c^2*(-3*a^2 - 30*a*c*x^2 + 5*c^2*x^4))/(15*c^3*d*(d*(b + 2*c*x))^(5/2))","A",1
1275,1,63,88,0.0461355,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(9/2),x]","\frac{14 \left(b^2-4 a c\right) (b+2 c x)^2-3 \left(b^2-4 a c\right)^2+21 (b+2 c x)^4}{336 c^3 d (d (b+2 c x))^{7/2}}","\frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left(b^2-4 a c\right)^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5}",1,"(-3*(b^2 - 4*a*c)^2 + 14*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 21*(b + 2*c*x)^4)/(336*c^3*d*(d*(b + 2*c*x))^(7/2))","A",1
1276,1,63,88,0.049856,"\int \frac{\left(a+b x+c x^2\right)^2}{(b d+2 c d x)^{11/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x]","\frac{18 \left(b^2-4 a c\right) (b+2 c x)^2-5 \left(b^2-4 a c\right)^2-45 (b+2 c x)^4}{720 c^3 d (d (b+2 c x))^{9/2}}","\frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{\left(b^2-4 a c\right)^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}}",1,"(-5*(b^2 - 4*a*c)^2 + 18*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 45*(b + 2*c*x)^4)/(720*c^3*d*(d*(b + 2*c*x))^(9/2))","A",1
1277,1,83,121,0.0838525,"\int \sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^3 \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^3,x]","\frac{\left(-315 \left(b^2-4 a c\right) (b+2 c x)^4+495 \left(b^2-4 a c\right)^2 (b+2 c x)^2-385 \left(b^2-4 a c\right)^3+77 (b+2 c x)^6\right) (d (b+2 c x))^{3/2}}{73920 c^4 d}","-\frac{3 \left(b^2-4 a c\right) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{\left(b^2-4 a c\right)^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7}",1,"((d*(b + 2*c*x))^(3/2)*(-385*(b^2 - 4*a*c)^3 + 495*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 315*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 77*(b + 2*c*x)^6))/(73920*c^4*d)","A",1
1278,1,83,121,0.0750741,"\int \frac{\left(a+b x+c x^2\right)^3}{\sqrt{b d+2 c d x}} \, dx","Integrate[(a + b*x + c*x^2)^3/Sqrt[b*d + 2*c*d*x],x]","\frac{\left(-65 \left(b^2-4 a c\right) (b+2 c x)^4+117 \left(b^2-4 a c\right)^2 (b+2 c x)^2-195 \left(b^2-4 a c\right)^3+15 (b+2 c x)^6\right) \sqrt{d (b+2 c x)}}{12480 c^4 d}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left(b^2-4 a c\right)^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7}",1,"(Sqrt[d*(b + 2*c*x)]*(-195*(b^2 - 4*a*c)^3 + 117*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 65*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 15*(b + 2*c*x)^6))/(12480*c^4*d)","A",1
1279,1,83,121,0.0661456,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(3/2),x]","\frac{-33 \left(b^2-4 a c\right) (b+2 c x)^4+77 \left(b^2-4 a c\right)^2 (b+2 c x)^2+77 \left(b^2-4 a c\right)^3+7 (b+2 c x)^6}{4928 c^4 d \sqrt{d (b+2 c x)}}","-\frac{3 \left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac{\left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}+\frac{\left(b^2-4 a c\right)^3}{64 c^4 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{11/2}}{704 c^4 d^7}",1,"(77*(b^2 - 4*a*c)^3 + 77*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 33*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 7*(b + 2*c*x)^6)/(4928*c^4*d*Sqrt[d*(b + 2*c*x)])","A",1
1280,1,83,121,0.0664081,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(5/2),x]","\frac{-27 \left(b^2-4 a c\right) (b+2 c x)^4+135 \left(b^2-4 a c\right)^2 (b+2 c x)^2+15 \left(b^2-4 a c\right)^3+5 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{3/2}}","-\frac{3 \left(b^2-4 a c\right) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left(b^2-4 a c\right)^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7}",1,"(15*(b^2 - 4*a*c)^3 + 135*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 27*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 5*(b + 2*c*x)^6)/(2880*c^4*d*(d*(b + 2*c*x))^(3/2))","A",1
1281,1,83,121,0.0737378,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(7/2),x]","\frac{-35 \left(b^2-4 a c\right) (b+2 c x)^4-105 \left(b^2-4 a c\right)^2 (b+2 c x)^2+7 \left(b^2-4 a c\right)^3+5 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{5/2}}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}{64 c^4 d^5}-\frac{3 \left(b^2-4 a c\right)^2}{64 c^4 d^3 \sqrt{b d+2 c d x}}+\frac{\left(b^2-4 a c\right)^3}{320 c^4 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{7/2}}{448 c^4 d^7}",1,"(7*(b^2 - 4*a*c)^3 - 105*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 35*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 5*(b + 2*c*x)^6)/(2240*c^4*d*(d*(b + 2*c*x))^(5/2))","A",1
1282,1,83,121,0.0669611,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x]","\frac{-105 \left(b^2-4 a c\right) (b+2 c x)^4-35 \left(b^2-4 a c\right)^2 (b+2 c x)^2+5 \left(b^2-4 a c\right)^3+7 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{7/2}}","-\frac{3 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}{64 c^4 d^5}-\frac{\left(b^2-4 a c\right)^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}+\frac{\left(b^2-4 a c\right)^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7}",1,"(5*(b^2 - 4*a*c)^3 - 35*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 - 105*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 7*(b + 2*c*x)^6)/(2240*c^4*d*(d*(b + 2*c*x))^(7/2))","A",1
1283,1,83,121,0.0706431,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{11/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(11/2),x]","\frac{135 \left(b^2-4 a c\right) (b+2 c x)^4-27 \left(b^2-4 a c\right)^2 (b+2 c x)^2+5 \left(b^2-4 a c\right)^3+15 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{9/2}}","\frac{3 \left(b^2-4 a c\right)}{64 c^4 d^5 \sqrt{b d+2 c d x}}-\frac{3 \left(b^2-4 a c\right)^2}{320 c^4 d^3 (b d+2 c d x)^{5/2}}+\frac{\left(b^2-4 a c\right)^3}{576 c^4 d (b d+2 c d x)^{9/2}}+\frac{(b d+2 c d x)^{3/2}}{192 c^4 d^7}",1,"(5*(b^2 - 4*a*c)^3 - 27*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 135*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 15*(b + 2*c*x)^6)/(2880*c^4*d*(d*(b + 2*c*x))^(9/2))","A",1
1284,1,83,121,0.0738375,"\int \frac{\left(a+b x+c x^2\right)^3}{(b d+2 c d x)^{13/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]","\frac{77 \left(b^2-4 a c\right) (b+2 c x)^4-33 \left(b^2-4 a c\right)^2 (b+2 c x)^2+7 \left(b^2-4 a c\right)^3+77 (b+2 c x)^6}{4928 c^4 d (d (b+2 c x))^{11/2}}","\frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac{3 \left(b^2-4 a c\right)^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{\left(b^2-4 a c\right)^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7}",1,"(7*(b^2 - 4*a*c)^3 - 33*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 77*(b^2 - 4*a*c)*(b + 2*c*x)^4 + 77*(b + 2*c*x)^6)/(4928*c^4*d*(d*(b + 2*c*x))^(11/2))","A",1
1285,1,157,175,0.1977974,"\int \frac{(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2),x]","\frac{4 d (d (b+2 c x))^{9/2} \left(\frac{1}{9} (b+2 c x)^{9/2}-\left(4 a c-b^2\right) \left(\frac{1}{5} (b+2 c x)^{5/2}-\frac{1}{2} \left(4 a c-b^2\right) \left(2 \sqrt{b+2 c x}-\sqrt[4]{b^2-4 a c} \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)\right)\right)\right)}{(b+2 c x)^{9/2}}","-2 d^{11/2} \left(b^2-4 a c\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-2 d^{11/2} \left(b^2-4 a c\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+4 d^5 \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}+\frac{4}{5} d^3 \left(b^2-4 a c\right) (b d+2 c d x)^{5/2}+\frac{4}{9} d (b d+2 c d x)^{9/2}",1,"(4*d*(d*(b + 2*c*x))^(9/2)*((b + 2*c*x)^(9/2)/9 - (-b^2 + 4*a*c)*((b + 2*c*x)^(5/2)/5 - ((-b^2 + 4*a*c)*(2*Sqrt[b + 2*c*x] - (b^2 - 4*a*c)^(1/4)*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))/2)))/(b + 2*c*x)^(9/2)","A",1
1286,1,131,147,0.1421342,"\int \frac{(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]","\frac{4 (d (b+2 c x))^{9/2} \left(\frac{1}{7} (b+2 c x)^{7/2}-\frac{1}{6} \left(4 a c-b^2\right) \left(3 \left(b^2-4 a c\right)^{3/4} \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)-\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)+2 (b+2 c x)^{3/2}\right)\right)}{(b+2 c x)^{9/2}}","2 d^{9/2} \left(b^2-4 a c\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-2 d^{9/2} \left(b^2-4 a c\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+\frac{4}{3} d^3 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}",1,"(4*(d*(b + 2*c*x))^(9/2)*((b + 2*c*x)^(7/2)/7 - ((-b^2 + 4*a*c)*(2*(b + 2*c*x)^(3/2) + 3*(b^2 - 4*a*c)^(3/4)*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))/6))/(b + 2*c*x)^(9/2)","A",1
1287,1,141,145,0.1337782,"\int \frac{(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2),x]","\frac{2 d^3 \sqrt{d (b+2 c x)} \left(4 \sqrt{b+2 c x} \left(2 c \left(c x^2-5 a\right)+3 b^2+2 b c x\right)-5 \left(b^2-4 a c\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)-5 \left(b^2-4 a c\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{5 \sqrt{b+2 c x}}","-2 d^{7/2} \left(b^2-4 a c\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-2 d^{7/2} \left(b^2-4 a c\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+4 d^3 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}+\frac{4}{5} d (b d+2 c d x)^{5/2}",1,"(2*d^3*Sqrt[d*(b + 2*c*x)]*(4*Sqrt[b + 2*c*x]*(3*b^2 + 2*b*c*x + 2*c*(-5*a + c*x^2)) - 5*(b^2 - 4*a*c)^(5/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - 5*(b^2 - 4*a*c)^(5/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(5*Sqrt[b + 2*c*x])","A",1
1288,1,104,119,0.1175375,"\int \frac{(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2),x]","\frac{2 (d (b+2 c x))^{5/2} \left(3 \left(b^2-4 a c\right)^{3/4} \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)-\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)+2 (b+2 c x)^{3/2}\right)}{3 (b+2 c x)^{5/2}}","2 d^{5/2} \left(b^2-4 a c\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-2 d^{5/2} \left(b^2-4 a c\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+\frac{4}{3} d (b d+2 c d x)^{3/2}",1,"(2*(d*(b + 2*c*x))^(5/2)*(2*(b + 2*c*x)^(3/2) + 3*(b^2 - 4*a*c)^(3/4)*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))/(3*(b + 2*c*x)^(5/2))","A",1
1289,1,113,117,0.04602,"\int \frac{(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2),x]","\frac{2 (d (b+2 c x))^{3/2} \left(-\sqrt[4]{b^2-4 a c} \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)-\sqrt[4]{b^2-4 a c} \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+2 \sqrt{b+2 c x}\right)}{(b+2 c x)^{3/2}}","-2 d^{3/2} \sqrt[4]{b^2-4 a c} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-2 d^{3/2} \sqrt[4]{b^2-4 a c} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+4 d \sqrt{b d+2 c d x}",1,"(2*(d*(b + 2*c*x))^(3/2)*(2*Sqrt[b + 2*c*x] - (b^2 - 4*a*c)^(1/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - (b^2 - 4*a*c)^(1/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(b + 2*c*x)^(3/2)","A",1
1290,1,87,101,0.0381857,"\int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2),x]","\frac{2 \sqrt{d (b+2 c x)} \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)-\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{\sqrt[4]{b^2-4 a c} \sqrt{b+2 c x}}","\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}",1,"(2*Sqrt[d*(b + 2*c*x)]*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/((b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])","A",1
1291,1,85,101,0.048627,"\int \frac{1}{\sqrt{b d+2 c d x} \left(a+b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)),x]","-\frac{2 \sqrt{b+2 c x} \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{\left(b^2-4 a c\right)^{3/4} \sqrt{d (b+2 c x)}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{3/4}}",1,"(-2*Sqrt[b + 2*c*x]*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/((b^2 - 4*a*c)^(3/4)*Sqrt[d*(b + 2*c*x)])","A",1
1292,1,54,129,0.0464349,"\int \frac{1}{(b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)),x]","\frac{4 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d \left(b^2-4 a c\right) \sqrt{d (b+2 c x)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{5/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{5/4}}+\frac{4}{d \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}",1,"(4*Hypergeometric2F1[-1/4, 1, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*Sqrt[d*(b + 2*c*x)])","C",1
1293,1,56,131,0.0542589,"\int \frac{1}{(b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)),x]","\frac{4 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right) (d (b+2 c x))^{3/2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{7/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{7/4}}+\frac{4}{3 d \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}",1,"(4*Hypergeometric2F1[-3/4, 1, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(3/2))","C",1
1294,1,56,159,0.0597462,"\int \frac{1}{(b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)),x]","\frac{4 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 d \left(b^2-4 a c\right) (d (b+2 c x))^{5/2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{9/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{9/4}}+\frac{4}{d^3 \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}+\frac{4}{5 d \left(b^2-4 a c\right) (b d+2 c d x)^{5/2}}",1,"(4*Hypergeometric2F1[-5/4, 1, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(5/2))","C",1
1295,1,189,210,0.6036534,"\int \frac{(b d+2 c d x)^{15/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x]","-\frac{(d (b+2 c x))^{15/2} \left(-13 \left(b^2-4 a c\right) \left(3 \left(b^2-4 a c\right) \left(-30 \left(b^2-4 a c\right) \sqrt{b+2 c x}-60 c \sqrt[4]{b^2-4 a c} (a+x (b+c x)) \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)+24 (b+2 c x)^{5/2}\right)+8 (b+2 c x)^{9/2}\right)-40 (b+2 c x)^{13/2}\right)}{90 (b+2 c x)^{15/2} (a+x (b+c x))}","-26 c d^{15/2} \left(b^2-4 a c\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-26 c d^{15/2} \left(b^2-4 a c\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+52 c d^7 \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}+\frac{52}{5} c d^5 \left(b^2-4 a c\right) (b d+2 c d x)^{5/2}-\frac{d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac{52}{9} c d^3 (b d+2 c d x)^{9/2}",1,"-1/90*((d*(b + 2*c*x))^(15/2)*(-40*(b + 2*c*x)^(13/2) - 13*(b^2 - 4*a*c)*(8*(b + 2*c*x)^(9/2) + 3*(b^2 - 4*a*c)*(-30*(b^2 - 4*a*c)*Sqrt[b + 2*c*x] + 24*(b + 2*c*x)^(5/2) - 60*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))))/((b + 2*c*x)^(15/2)*(a + x*(b + c*x)))","A",1
1296,1,171,181,0.1733092,"\int \frac{(b d+2 c d x)^{13/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^2,x]","\frac{4 d^5 (d (b+2 c x))^{3/2} \left(308 c \left(4 a^2 c+a \left(-b^2+4 b c x+4 c^2 x^2\right)-b^2 x (b+c x)\right) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+16 c^2 \left(-77 a^2-11 a c x^2+3 c^2 x^4\right)+4 b^2 c \left(143 a+29 c x^2\right)+16 b c^2 x \left(6 c x^2-11 a\right)-63 b^4+68 b^3 c x\right)}{21 (a+x (b+c x))}","22 c d^{13/2} \left(b^2-4 a c\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-22 c d^{13/2} \left(b^2-4 a c\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+\frac{44}{3} c d^5 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2}",1,"(4*d^5*(d*(b + 2*c*x))^(3/2)*(-63*b^4 + 68*b^3*c*x + 16*b*c^2*x*(-11*a + 6*c*x^2) + 4*b^2*c*(143*a + 29*c*x^2) + 16*c^2*(-77*a^2 - 11*a*c*x^2 + 3*c^2*x^4) + 308*c*(4*a^2*c - b^2*x*(b + c*x) + a*(-b^2 + 4*b*c*x + 4*c^2*x^2))*Hypergeometric2F1[3/4, 2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(21*(a + x*(b + c*x)))","C",1
1297,1,167,179,0.3425845,"\int \frac{(b d+2 c d x)^{11/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^2,x]","-\frac{d (d (b+2 c x))^{9/2} \left(-3 \left(b^2-4 a c\right) \left(-30 \left(b^2-4 a c\right) \sqrt{b+2 c x}-60 c \sqrt[4]{b^2-4 a c} (a+x (b+c x)) \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)+24 (b+2 c x)^{5/2}\right)-8 (b+2 c x)^{9/2}\right)}{10 (b+2 c x)^{9/2} (a+x (b+c x))}","-18 c d^{11/2} \left(b^2-4 a c\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-18 c d^{11/2} \left(b^2-4 a c\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+36 c d^5 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}-\frac{d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac{36}{5} c d^3 (b d+2 c d x)^{5/2}",1,"-1/10*(d*(d*(b + 2*c*x))^(9/2)*(-8*(b + 2*c*x)^(9/2) - 3*(b^2 - 4*a*c)*(-30*(b^2 - 4*a*c)*Sqrt[b + 2*c*x] + 24*(b + 2*c*x)^(5/2) - 60*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))))/((b + 2*c*x)^(9/2)*(a + x*(b + c*x)))","A",1
1298,1,92,152,0.0902627,"\int \frac{(b d+2 c d x)^{9/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^2,x]","-\frac{8 d^3 (d (b+2 c x))^{3/2} \left(14 c (a+x (b+c x)) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-2 c \left(7 a+c x^2\right)+3 b^2-2 b c x\right)}{3 (a+x (b+c x))}","14 c d^{9/2} \left(b^2-4 a c\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-14 c d^{9/2} \left(b^2-4 a c\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-\frac{d (b d+2 c d x)^{7/2}}{a+b x+c x^2}+\frac{28}{3} c d^3 (b d+2 c d x)^{3/2}",1,"(-8*d^3*(d*(b + 2*c*x))^(3/2)*(3*b^2 - 2*b*c*x - 2*c*(7*a + c*x^2) + 14*c*(a + x*(b + c*x))*Hypergeometric2F1[3/4, 2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b + c*x)))","C",1
1299,1,168,150,0.1822813,"\int \frac{(b d+2 c d x)^{7/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^2,x]","-\frac{d^3 \sqrt{d (b+2 c x)} \left(\sqrt{b+2 c x} \left(-4 c \left(5 a+4 c x^2\right)+b^2-16 b c x\right)+10 c \sqrt[4]{b^2-4 a c} (a+x (b+c x)) \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+10 c \sqrt[4]{b^2-4 a c} (a+x (b+c x)) \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{\sqrt{b+2 c x} (a+x (b+c x))}","-10 c d^{7/2} \sqrt[4]{b^2-4 a c} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-10 c d^{7/2} \sqrt[4]{b^2-4 a c} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-\frac{d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+20 c d^3 \sqrt{b d+2 c d x}",1,"-((d^3*Sqrt[d*(b + 2*c*x)]*(Sqrt[b + 2*c*x]*(b^2 - 16*b*c*x - 4*c*(5*a + 4*c*x^2)) + 10*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 10*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(Sqrt[b + 2*c*x]*(a + x*(b + c*x))))","A",1
1300,1,83,131,0.0769008,"\int \frac{(b d+2 c d x)^{5/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^2,x]","-\frac{4 d (d (b+2 c x))^{3/2} \left(4 c (a+x (b+c x)) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-4 a c+b^2\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}","\frac{6 c d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}-\frac{6 c d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}-\frac{d (b d+2 c d x)^{3/2}}{a+b x+c x^2}",1,"(-4*d*(d*(b + 2*c*x))^(3/2)*(b^2 - 4*a*c + 4*c*(a + x*(b + c*x))*Hypergeometric2F1[3/4, 2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/((b^2 - 4*a*c)*(a + x*(b + c*x)))","C",1
1301,1,126,131,0.2340725,"\int \frac{(b d+2 c d x)^{3/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^2,x]","d \sqrt{d (b+2 c x)} \left(-\frac{2 c \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4} \sqrt{b+2 c x}}-\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4} \sqrt{b+2 c x}}-\frac{1}{a+x (b+c x)}\right)","-\frac{2 c d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4}}-\frac{2 c d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4}}-\frac{d \sqrt{b d+2 c d x}}{a+b x+c x^2}",1,"d*Sqrt[d*(b + 2*c*x)]*(-(a + x*(b + c*x))^(-1) - (2*c*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x]) - (2*c*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x]))","A",1
1302,1,57,143,0.0353666,"\int \frac{\sqrt{b d+2 c d x}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^2,x]","\frac{16 c (d (b+2 c x))^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^2}","-\frac{(b d+2 c d x)^{3/2}}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{2 c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/4}}+\frac{2 c \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/4}}",1,"(16*c*(d*(b + 2*c*x))^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^2*d)","C",1
1303,1,167,143,0.1637339,"\int \frac{1}{\sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^2),x]","\frac{-\left(\left(b^2-4 a c\right) (b+2 c x)\right)+6 c \sqrt[4]{b^2-4 a c} \sqrt{b+2 c x} (a+x (b+c x)) \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+6 c \sqrt[4]{b^2-4 a c} \sqrt{b+2 c x} (a+x (b+c x)) \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x)) \sqrt{d (b+2 c x)}}","-\frac{\sqrt{b d+2 c d x}}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{6 c \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{7/4}}+\frac{6 c \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{7/4}}",1,"(-((b^2 - 4*a*c)*(b + 2*c*x)) + 6*c*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x]*(a + x*(b + c*x))*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 6*c*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x]*(a + x*(b + c*x))*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*(a + x*(b + c*x)))","A",1
1304,1,55,172,0.0505461,"\int \frac{1}{(b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2),x]","-\frac{16 c \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d \left(b^2-4 a c\right)^2 \sqrt{d (b+2 c x)}}","-\frac{10 c \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{9/4}}+\frac{10 c \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{9/4}}-\frac{1}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right) \sqrt{b d+2 c d x}}-\frac{20 c}{d \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}",1,"(-16*c*Hypergeometric2F1[-1/4, 2, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^2*d*Sqrt[d*(b + 2*c*x)])","C",1
1305,1,57,174,0.0643748,"\int \frac{1}{(b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2),x]","-\frac{16 c \, _2F_1\left(-\frac{3}{4},2;\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^2 (d (b+2 c x))^{3/2}}","\frac{14 c \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{11/4}}+\frac{14 c \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{11/4}}-\frac{1}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right) (b d+2 c d x)^{3/2}}-\frac{28 c}{3 d \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}",1,"(-16*c*Hypergeometric2F1[-3/4, 2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^2*d*(d*(b + 2*c*x))^(3/2))","C",1
1306,1,57,203,0.0655228,"\int \frac{1}{(b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^2),x]","-\frac{16 c \, _2F_1\left(-\frac{5}{4},2;-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 d \left(b^2-4 a c\right)^2 (d (b+2 c x))^{5/2}}","-\frac{18 c \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{13/4}}+\frac{18 c \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{13/4}}-\frac{36 c}{d^3 \left(b^2-4 a c\right)^3 \sqrt{b d+2 c d x}}-\frac{1}{d \left(b^2-4 a c\right) \left(a+b x+c x^2\right) (b d+2 c d x)^{5/2}}-\frac{36 c}{5 d \left(b^2-4 a c\right)^2 (b d+2 c d x)^{5/2}}",1,"(-16*c*Hypergeometric2F1[-5/4, 2, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)^2*d*(d*(b + 2*c*x))^(5/2))","C",1
1307,1,138,222,0.3113728,"\int \frac{(b d+2 c d x)^{17/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x]","\frac{4 (d (b+2 c x))^{17/2} \left(\left(b^2-4 a c\right) \left(77 \left(\left(b^2-4 a c\right)^2-16 c^2 (a+x (b+c x))^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)-55 \left(b^2-4 a c\right) (b+2 c x)^2+5 (b+2 c x)^4\right)+(b+2 c x)^6\right)}{7 (b+2 c x)^7 (a+x (b+c x))^2}","165 c^2 d^{17/2} \left(b^2-4 a c\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-165 c^2 d^{17/2} \left(b^2-4 a c\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+110 c^2 d^7 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}-\frac{15 c d^3 (b d+2 c d x)^{11/2}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{15/2}}{2 \left(a+b x+c x^2\right)^2}+\frac{330}{7} c^2 d^5 (b d+2 c d x)^{7/2}",1,"(4*(d*(b + 2*c*x))^(17/2)*((b + 2*c*x)^6 + (b^2 - 4*a*c)*(-55*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 5*(b + 2*c*x)^4 + 77*((b^2 - 4*a*c)^2 - 16*c^2*(a + x*(b + c*x))^2*Hypergeometric2F1[3/4, 3, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))))/(7*(b + 2*c*x)^7*(a + x*(b + c*x))^2)","C",1
1308,1,225,222,0.7739544,"\int \frac{(b d+2 c d x)^{15/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]","\frac{(d (b+2 c x))^{15/2} \left(91 \left(b^2-4 a c\right) \left(-192 \left(b^2-4 a c\right) (b+2 c x)^{5/2}+120 \left(b^2-4 a c\right)^2 \sqrt{b+2 c x}+60 c \sqrt[4]{b^2-4 a c} (a+x (b+c x)) \left(2 \left(b^2-4 a c\right)^{3/4} \sqrt{b+2 c x}-12 c (a+x (b+c x)) \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)\right)+64 (b+2 c x)^{9/2}\right)+448 (b+2 c x)^{13/2}\right)}{560 (b+2 c x)^{15/2} (a+x (b+c x))^2}","-117 c^2 d^{15/2} \left(b^2-4 a c\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-117 c^2 d^{15/2} \left(b^2-4 a c\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)+234 c^2 d^7 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}-\frac{13 c d^3 (b d+2 c d x)^{9/2}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{13/2}}{2 \left(a+b x+c x^2\right)^2}+\frac{234}{5} c^2 d^5 (b d+2 c d x)^{5/2}",1,"((d*(b + 2*c*x))^(15/2)*(448*(b + 2*c*x)^(13/2) + 91*(b^2 - 4*a*c)*(120*(b^2 - 4*a*c)^2*Sqrt[b + 2*c*x] - 192*(b^2 - 4*a*c)*(b + 2*c*x)^(5/2) + 64*(b + 2*c*x)^(9/2) + 60*c*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))*(2*(b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x] - 12*c*(a + x*(b + c*x))*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))))/(560*(b + 2*c*x)^(15/2)*(a + x*(b + c*x))^2)","A",1
1309,1,145,193,0.1492842,"\int \frac{(b d+2 c d x)^{13/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^3,x]","-\frac{4 d^5 (d (b+2 c x))^{3/2} \left(-16 c^2 \left(77 a^2+55 a c x^2+5 c^2 x^4\right)+1232 c^2 (a+x (b+c x))^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+4 b^2 c \left(99 a+25 c x^2\right)-80 b c^2 x \left(11 a+2 c x^2\right)-27 b^4+180 b^3 c x\right)}{15 (a+x (b+c x))^2}","77 c^2 d^{13/2} \left(b^2-4 a c\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-77 c^2 d^{13/2} \left(b^2-4 a c\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-\frac{11 c d^3 (b d+2 c d x)^{7/2}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{11/2}}{2 \left(a+b x+c x^2\right)^2}+\frac{154}{3} c^2 d^5 (b d+2 c d x)^{3/2}",1,"(-4*d^5*(d*(b + 2*c*x))^(3/2)*(-27*b^4 + 180*b^3*c*x - 80*b*c^2*x*(11*a + 2*c*x^2) + 4*b^2*c*(99*a + 25*c*x^2) - 16*c^2*(77*a^2 + 55*a*c*x^2 + 5*c^2*x^4) + 1232*c^2*(a + x*(b + c*x))^2*Hypergeometric2F1[3/4, 3, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(15*(a + x*(b + c*x))^2)","C",1
1310,1,226,191,0.3329714,"\int \frac{(b d+2 c d x)^{11/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x]","-\frac{d^5 \sqrt{d (b+2 c x)} \left(\sqrt{b+2 c x} \left(-4 c^2 \left(45 a^2+81 a c x^2+32 c^2 x^4\right)+3 b^2 c \left(3 a-37 c x^2\right)-4 b c^2 x \left(81 a+64 c x^2\right)+b^4+17 b^3 c x\right)+90 c^2 \sqrt[4]{b^2-4 a c} (a+x (b+c x))^2 \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+90 c^2 \sqrt[4]{b^2-4 a c} (a+x (b+c x))^2 \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{2 \sqrt{b+2 c x} (a+x (b+c x))^2}","-45 c^2 d^{11/2} \sqrt[4]{b^2-4 a c} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-45 c^2 d^{11/2} \sqrt[4]{b^2-4 a c} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)-\frac{9 c d^3 (b d+2 c d x)^{5/2}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{9/2}}{2 \left(a+b x+c x^2\right)^2}+90 c^2 d^5 \sqrt{b d+2 c d x}",1,"-1/2*(d^5*Sqrt[d*(b + 2*c*x)]*(Sqrt[b + 2*c*x]*(b^4 + 17*b^3*c*x + 3*b^2*c*(3*a - 37*c*x^2) - 4*b*c^2*x*(81*a + 64*c*x^2) - 4*c^2*(45*a^2 + 81*a*c*x^2 + 32*c^2*x^4)) + 90*c^2*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))^2*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 90*c^2*(b^2 - 4*a*c)^(1/4)*(a + x*(b + c*x))^2*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(Sqrt[b + 2*c*x]*(a + x*(b + c*x))^2)","A",1
1311,1,119,170,0.143839,"\int \frac{(b d+2 c d x)^{9/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x]","\frac{4 (d (b+2 c x))^{9/2} \left(-112 c^2 (a+x (b+c x))^2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-5 \left(b^2-4 a c\right) (b+2 c x)^2+7 \left(b^2-4 a c\right)^2\right)}{5 \left(b^2-4 a c\right) (b+2 c x)^3 (a+x (b+c x))^2}","\frac{21 c^2 d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}-\frac{21 c^2 d^{9/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt[4]{b^2-4 a c}}-\frac{7 c d^3 (b d+2 c d x)^{3/2}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{7/2}}{2 \left(a+b x+c x^2\right)^2}",1,"(4*(d*(b + 2*c*x))^(9/2)*(7*(b^2 - 4*a*c)^2 - 5*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 112*c^2*(a + x*(b + c*x))^2*Hypergeometric2F1[3/4, 3, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(5*(b^2 - 4*a*c)*(b + 2*c*x)^3*(a + x*(b + c*x))^2)","C",1
1312,1,196,170,0.2778965,"\int \frac{(b d+2 c d x)^{7/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^3,x]","\frac{(d (b+2 c x))^{7/2} \left(-64 \left(b^2-4 a c\right)^{3/4} (b+2 c x)^{5/2}+40 \left(b^2-4 a c\right)^{7/4} \sqrt{b+2 c x}+20 c (a+x (b+c x)) \left(2 \left(b^2-4 a c\right)^{3/4} \sqrt{b+2 c x}-12 c (a+x (b+c x)) \left(\tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+\tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)\right)\right)}{48 \left(b^2-4 a c\right)^{3/4} (b+2 c x)^{7/2} (a+x (b+c x))^2}","-\frac{5 c^2 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4}}-\frac{5 c^2 d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/4}}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{5/2}}{2 \left(a+b x+c x^2\right)^2}",1,"((d*(b + 2*c*x))^(7/2)*(40*(b^2 - 4*a*c)^(7/4)*Sqrt[b + 2*c*x] - 64*(b^2 - 4*a*c)^(3/4)*(b + 2*c*x)^(5/2) + 20*c*(a + x*(b + c*x))*(2*(b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x] - 12*c*(a + x*(b + c*x))*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))))/(48*(b^2 - 4*a*c)^(3/4)*(b + 2*c*x)^(7/2)*(a + x*(b + c*x))^2)","A",1
1313,1,77,178,0.1093835,"\int \frac{(b d+2 c d x)^{5/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^3,x]","\frac{64}{5} c^2 d (d (b+2 c x))^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{\left(b^2-4 a c\right)^2}-\frac{1}{16 c^2 (a+x (b+c x))^2}\right)","-\frac{3 c^2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/4}}+\frac{3 c^2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/4}}-\frac{3 c d (b d+2 c d x)^{3/2}}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{d (b d+2 c d x)^{3/2}}{2 \left(a+b x+c x^2\right)^2}",1,"(64*c^2*d*(d*(b + 2*c*x))^(3/2)*(-1/16*1/(c^2*(a + x*(b + c*x))^2) + Hypergeometric2F1[3/4, 3, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]/(b^2 - 4*a*c)^2))/5","C",1
1314,1,171,178,0.3729493,"\int \frac{(b d+2 c d x)^{3/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^3,x]","\frac{(d (b+2 c x))^{3/2} \left(\frac{8 c (a+x (b+c x)) \left(-\left(b^2-4 a c\right)^{3/4} \sqrt{b+2 c x}+6 c (a+x (b+c x)) \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+6 c (a+x (b+c x)) \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{\left(b^2-4 a c\right)^{7/4}}-8 \sqrt{b+2 c x}\right)}{16 (b+2 c x)^{3/2} (a+x (b+c x))^2}","\frac{3 c^2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{d \sqrt{b d+2 c d x}}{2 \left(a+b x+c x^2\right)^2}",1,"((d*(b + 2*c*x))^(3/2)*(-8*Sqrt[b + 2*c*x] + (8*c*(a + x*(b + c*x))*(-((b^2 - 4*a*c)^(3/4)*Sqrt[b + 2*c*x]) + 6*c*(a + x*(b + c*x))*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 6*c*(a + x*(b + c*x))*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(b^2 - 4*a*c)^(7/4)))/(16*(b + 2*c*x)^(3/2)*(a + x*(b + c*x))^2)","A",1
1315,1,59,192,0.0428632,"\int \frac{\sqrt{b d+2 c d x}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^3,x]","-\frac{64 c^2 (d (b+2 c x))^{3/2} \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^3}","\frac{5 c^2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/4}}-\frac{5 c^2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/4}}+\frac{5 c (b d+2 c d x)^{3/2}}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{(b d+2 c d x)^{3/2}}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(-64*c^2*(d*(b + 2*c*x))^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^3*d)","C",1
1316,1,179,192,0.3679147,"\int \frac{1}{\sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^3),x]","\frac{c^2 \left(\frac{\left(b^2-4 a c\right) (b+2 c x) \left(-c \left(11 a+7 c x^2\right)+b^2-7 b c x\right)}{2 c^2 (a+x (b+c x))^2}+21 \sqrt[4]{b^2-4 a c} \sqrt{b+2 c x} \tan ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)+21 \sqrt[4]{b^2-4 a c} \sqrt{b+2 c x} \tanh ^{-1}\left(\frac{\sqrt{b+2 c x}}{\sqrt[4]{b^2-4 a c}}\right)\right)}{\left(4 a c-b^2\right)^3 \sqrt{d (b+2 c x)}}","-\frac{21 c^2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{11/4}}-\frac{21 c^2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{\sqrt{d} \left(b^2-4 a c\right)^{11/4}}+\frac{7 c \sqrt{b d+2 c d x}}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{\sqrt{b d+2 c d x}}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(c^2*(((b^2 - 4*a*c)*(b + 2*c*x)*(b^2 - 7*b*c*x - c*(11*a + 7*c*x^2)))/(2*c^2*(a + x*(b + c*x))^2) + 21*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x]*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + 21*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x]*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/((-b^2 + 4*a*c)^3*Sqrt[d*(b + 2*c*x)])","A",1
1317,1,57,223,0.0625021,"\int \frac{1}{(b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^3),x]","\frac{64 c^2 \, _2F_1\left(-\frac{1}{4},3;\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d \left(b^2-4 a c\right)^3 \sqrt{d (b+2 c x)}}","\frac{45 c^2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{13/4}}-\frac{45 c^2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{3/2} \left(b^2-4 a c\right)^{13/4}}+\frac{90 c^2}{d \left(b^2-4 a c\right)^3 \sqrt{b d+2 c d x}}+\frac{9 c}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right) \sqrt{b d+2 c d x}}-\frac{1}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2 \sqrt{b d+2 c d x}}",1,"(64*c^2*Hypergeometric2F1[-1/4, 3, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^3*d*Sqrt[d*(b + 2*c*x)])","C",1
1318,1,59,225,0.0729792,"\int \frac{1}{(b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x]","\frac{64 c^2 \, _2F_1\left(-\frac{3}{4},3;\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^3 (d (b+2 c x))^{3/2}}","-\frac{77 c^2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{15/4}}-\frac{77 c^2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{5/2} \left(b^2-4 a c\right)^{15/4}}+\frac{154 c^2}{3 d \left(b^2-4 a c\right)^3 (b d+2 c d x)^{3/2}}+\frac{11 c}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right) (b d+2 c d x)^{3/2}}-\frac{1}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2 (b d+2 c d x)^{3/2}}",1,"(64*c^2*Hypergeometric2F1[-3/4, 3, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^3*d*(d*(b + 2*c*x))^(3/2))","C",1
1319,1,59,256,0.0734722,"\int \frac{1}{(b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]","\frac{64 c^2 \, _2F_1\left(-\frac{5}{4},3;-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 d \left(b^2-4 a c\right)^3 (d (b+2 c x))^{5/2}}","\frac{117 c^2 \tan ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{17/4}}-\frac{117 c^2 \tanh ^{-1}\left(\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt[4]{b^2-4 a c}}\right)}{d^{7/2} \left(b^2-4 a c\right)^{17/4}}+\frac{234 c^2}{d^3 \left(b^2-4 a c\right)^4 \sqrt{b d+2 c d x}}+\frac{234 c^2}{5 d \left(b^2-4 a c\right)^3 (b d+2 c d x)^{5/2}}+\frac{13 c}{2 d \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right) (b d+2 c d x)^{5/2}}-\frac{1}{2 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2 (b d+2 c d x)^{5/2}}",1,"(64*c^2*Hypergeometric2F1[-5/4, 3, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)^3*d*(d*(b + 2*c*x))^(5/2))","C",1
1320,1,182,183,0.0966784,"\int \frac{(1+2 x)^{7/2}}{1+x+x^2} \, dx","Integrate[(1 + 2*x)^(7/2)/(1 + x + x^2),x]","\frac{16}{5} \sqrt{2 x+1} x^2+\frac{16}{5} \sqrt{2 x+1} x-\frac{56}{5} \sqrt{2 x+1}-\frac{3 \sqrt[4]{3} \log \left(2 x-\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)}{\sqrt{2}}+\frac{3 \sqrt[4]{3} \log \left(2 x+\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)}{\sqrt{2}}-3 \sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}\right)+3 \sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}+1\right)","\frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt[4]{3} \log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}+\frac{3 \sqrt[4]{3} \log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}-3 \sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)+3 \sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)",1,"(-56*Sqrt[1 + 2*x])/5 + (16*x*Sqrt[1 + 2*x])/5 + (16*x^2*Sqrt[1 + 2*x])/5 - 3*Sqrt[2]*3^(1/4)*ArcTan[1 - Sqrt[2 + 4*x]/3^(1/4)] + 3*Sqrt[2]*3^(1/4)*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] - (3*3^(1/4)*Log[1 + Sqrt[3] + 2*x - 3^(1/4)*Sqrt[2 + 4*x]])/Sqrt[2] + (3*3^(1/4)*Log[1 + Sqrt[3] + 2*x + 3^(1/4)*Sqrt[2 + 4*x]])/Sqrt[2]","A",1
1321,1,34,170,0.0105416,"\int \frac{(1+2 x)^{5/2}}{1+x+x^2} \, dx","Integrate[(1 + 2*x)^(5/2)/(1 + x + x^2),x]","-\frac{4}{3} (2 x+1)^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{1}{3} (2 x+1)^2\right)-1\right)","\frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}+\frac{3^{3/4} \log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)-\sqrt{2} 3^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)",1,"(-4*(1 + 2*x)^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -1/3*(1 + 2*x)^2]))/3","C",1
1322,1,148,168,0.0407143,"\int \frac{(1+2 x)^{3/2}}{1+x+x^2} \, dx","Integrate[(1 + 2*x)^(3/2)/(1 + x + x^2),x]","4 \sqrt{2 x+1}+\frac{\sqrt[4]{3} \log \left(2 x-\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)}{\sqrt{2}}-\frac{\sqrt[4]{3} \log \left(2 x+\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)}{\sqrt{2}}+\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}\right)-\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}+1\right)","4 \sqrt{2 x+1}+\frac{\sqrt[4]{3} \log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}-\frac{\sqrt[4]{3} \log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2}}+\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)-\sqrt{2} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)",1,"4*Sqrt[1 + 2*x] + Sqrt[2]*3^(1/4)*ArcTan[1 - Sqrt[2 + 4*x]/3^(1/4)] - Sqrt[2]*3^(1/4)*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] + (3^(1/4)*Log[1 + Sqrt[3] + 2*x - 3^(1/4)*Sqrt[2 + 4*x]])/Sqrt[2] - (3^(1/4)*Log[1 + Sqrt[3] + 2*x + 3^(1/4)*Sqrt[2 + 4*x]])/Sqrt[2]","A",1
1323,1,32,157,0.0058812,"\int \frac{\sqrt{1+2 x}}{1+x+x^2} \, dx","Integrate[Sqrt[1 + 2*x]/(1 + x + x^2),x]","\frac{4}{9} (2 x+1)^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\frac{1}{3} (2 x+1)^2\right)","\frac{\log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2} \sqrt[4]{3}}-\frac{\log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2} \sqrt[4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)}{\sqrt[4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)}{\sqrt[4]{3}}",1,"(4*(1 + 2*x)^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -1/3*(1 + 2*x)^2])/9","C",1
1324,1,108,157,0.0265601,"\int \frac{1}{\sqrt{1+2 x} \left(1+x+x^2\right)} \, dx","Integrate[1/(Sqrt[1 + 2*x]*(1 + x + x^2)),x]","\frac{-\log \left(2 x-\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)+\log \left(2 x+\sqrt[4]{3} \sqrt{4 x+2}+\sqrt{3}+1\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{4 x+2}}{\sqrt[4]{3}}+1\right)}{\sqrt{2} 3^{3/4}}","-\frac{\log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2} 3^{3/4}}+\frac{\log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{\sqrt{2} 3^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)}{3^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)}{3^{3/4}}",1,"(-2*ArcTan[1 - Sqrt[2 + 4*x]/3^(1/4)] + 2*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] - Log[1 + Sqrt[3] + 2*x - 3^(1/4)*Sqrt[2 + 4*x]] + Log[1 + Sqrt[3] + 2*x + 3^(1/4)*Sqrt[2 + 4*x]])/(Sqrt[2]*3^(3/4))","A",1
1325,1,32,180,0.0081255,"\int \frac{1}{(1+2 x)^{3/2} \left(1+x+x^2\right)} \, dx","Integrate[1/((1 + 2*x)^(3/2)*(1 + x + x^2)),x]","-\frac{4 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{1}{3} (2 x+1)^2\right)}{3 \sqrt{2 x+1}}","-\frac{4}{3 \sqrt{2 x+1}}-\frac{\log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{3 \sqrt{2} \sqrt[4]{3}}+\frac{\log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{3 \sqrt{2} \sqrt[4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)}{3 \sqrt[4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)}{3 \sqrt[4]{3}}",1,"(-4*Hypergeometric2F1[-1/4, 1, 3/4, -1/3*(1 + 2*x)^2])/(3*Sqrt[1 + 2*x])","C",1
1326,1,32,180,0.0081844,"\int \frac{1}{(1+2 x)^{5/2} \left(1+x+x^2\right)} \, dx","Integrate[1/((1 + 2*x)^(5/2)*(1 + x + x^2)),x]","-\frac{4 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{1}{3} (2 x+1)^2\right)}{9 (2 x+1)^{3/2}}","-\frac{4}{9 (2 x+1)^{3/2}}+\frac{\log \left(2 x-\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{3 \sqrt{2} 3^{3/4}}-\frac{\log \left(2 x+\sqrt{2} \sqrt[4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right)}{3 \sqrt{2} 3^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}\right)}{3\ 3^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt[4]{3}}+1\right)}{3\ 3^{3/4}}",1,"(-4*Hypergeometric2F1[-3/4, 1, 1/4, -1/3*(1 + 2*x)^2])/(9*(1 + 2*x)^(3/2))","C",1
1327,1,155,227,0.38473,"\int (b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2],x]","\frac{4 \sqrt{a+x (b+c x)} (d (b+2 c x))^{7/2} \left(7 (b+2 c x)^2 (a+x (b+c x))-10 c \left(a-\frac{b^2}{4 c}\right) \left(\frac{\left(b^2-4 a c\right) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{4 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))\right)\right)}{77 (b+2 c x)^3}","-\frac{5 d^{7/2} \left(b^2-4 a c\right)^{13/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{231 c^2 \sqrt{a+b x+c x^2}}-\frac{10 d^3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c}-\frac{2 d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}",1,"(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(7*(b + 2*c*x)^2*(a + x*(b + c*x)) - 10*(a - b^2/(4*c))*c*(2*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(77*(b + 2*c*x)^3)","C",1
1328,1,110,180,0.1349515,"\int (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]","\frac{1}{14} d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \left(\frac{\left(b^2-4 a c\right) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+8 (a+x (b+c x))\right)","-\frac{d^{3/2} \left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{21 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{21 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}",1,"(d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(8*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/14","C",1
1329,1,91,137,0.0561644,"\int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/Sqrt[b*d + 2*c*d*x],x]","\frac{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{2 c d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{3 c d}-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 c^2 \sqrt{d} \sqrt{a+b x+c x^2}}",1,"(Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2*c*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1330,1,91,137,0.0538875,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{5/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(5/2),x]","-\frac{\sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{6 c d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}}","\frac{\sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 c^2 d^{5/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{3 c d (b d+2 c d x)^{3/2}}",1,"-1/6*(Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/4, -1/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1331,1,99,184,0.0663159,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(9/2),x]","-\frac{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{14 c d^5 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{21 c^2 d^{9/2} \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c d^3 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}",1,"-1/14*(Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-7/4, -1/2, -3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d^5*(b + 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1332,1,99,231,0.0729388,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(13/2),x]","-\frac{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{11}{4},-\frac{1}{2};-\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{22 c d^7 (b+2 c x)^6 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{5 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{231 c^2 d^{13/2} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c d^5 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c d^3 \left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}",1,"-1/22*(Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-11/4, -1/2, -7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d^7*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1333,1,110,279,0.2063403,"\int (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2],x]","\frac{1}{18} d \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \left(\frac{\left(b^2-4 a c\right) \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+8 (a+x (b+c x))\right)","\frac{d^{5/2} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{45 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{9 c d}",1,"(d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*(8*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/18","C",1
1334,1,91,236,0.0480545,"\int \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{6 c d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{d} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c^2 \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{5 c d}",1,"((d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-1/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(6*c*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1335,1,91,229,0.0438008,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{3/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(3/2),x]","-\frac{\sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{2 c d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)}}","-\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c^2 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{c d \sqrt{b d+2 c d x}}",1,"-1/2*(Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-1/2, -1/4, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1336,1,91,283,0.0584361,"\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(7/2),x]","-\frac{\sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{10 c d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{5/2}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c^2 d^{7/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c^2 d^{7/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{5 c d^3 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}",1,"-1/10*(Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/4, -1/2, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*(d*(b + 2*c*x))^(5/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1337,1,161,274,0.3924564,"\int (b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x]","\frac{4 \sqrt{a+x (b+c x)} (d (b+2 c x))^{7/2} \left(11 (b+2 c x)^2 (a+x (b+c x))^2-10 c \left(a-\frac{b^2}{4 c}\right) \left(2 (a+x (b+c x))^2-\frac{\left(b^2-4 a c\right)^2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 c^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)\right)}{165 (b+2 c x)^3}","\frac{d^{7/2} \left(b^2-4 a c\right)^{17/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{462 c^3 \sqrt{a+b x+c x^2}}+\frac{d^3 \left(b^2-4 a c\right)^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c^2}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac{\left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{9/2}}{15 c d}",1,"(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(11*(b + 2*c*x)^2*(a + x*(b + c*x))^2 - 10*(a - b^2/(4*c))*c*(2*(a + x*(b + c*x))^2 - ((b^2 - 4*a*c)^2*Hypergeometric2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*c^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(165*(b + 2*c*x)^3)","C",1
1338,1,117,227,0.0871893,"\int (b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2),x]","\frac{2}{11} d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \left(2 (a+x (b+c x))^2-\frac{\left(b^2-4 a c\right)^2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 c^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)","\frac{d^{3/2} \left(b^2-4 a c\right)^{13/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{154 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{77 c^2}-\frac{3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{154 c^2 d}+\frac{\left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{5/2}}{11 c d}",1,"(2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^2 - ((b^2 - 4*a*c)^2*Hypergeometric2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*c^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/11","C",1
1339,1,99,182,0.0651831,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{\sqrt{b d+2 c d x}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/Sqrt[b*d + 2*c*d*x],x]","-\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{14 c^2 d}+\frac{\left(a+b x+c x^2\right)^{3/2} \sqrt{b d+2 c d x}}{7 c d}",1,"-1/8*((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^2*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1340,1,99,174,0.0549539,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(5/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{24 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}}","-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{6 c^3 d^{5/2} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{6 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 c d (b d+2 c d x)^{3/2}}",1,"((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -3/4, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(24*c^2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1341,1,107,174,0.0773955,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(9/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{56 c^2 d^5 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{14 c^3 d^{9/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{7 c d (b d+2 c d x)^{7/2}}",1,"((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-7/4, -3/2, -3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(56*c^2*d^5*(b + 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1342,1,107,221,0.0789804,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{13/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(13/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{88 c^2 d^7 (b+2 c x)^6 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{154 c^3 d^{13/2} \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{77 c^2 d^5 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}-\frac{3 \sqrt{a+b x+c x^2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{11 c d (b d+2 c d x)^{11/2}}",1,"((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-11/4, -3/2, -7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(88*c^2*d^7*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1343,1,107,268,0.085948,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{17/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(17/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{15}{4},-\frac{3}{2};-\frac{11}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{120 c^2 d^9 (b+2 c x)^8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{462 c^3 d^{17/2} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 d^7 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 d^5 \left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{15 c d (b d+2 c d x)^{15/2}}",1,"((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-15/4, -3/2, -11/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(120*c^2*d^9*(b + 2*c*x)^8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1344,1,117,326,0.1374011,"\int (b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2),x]","\frac{2}{13} d \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \left(2 (a+x (b+c x))^2-\frac{\left(b^2-4 a c\right)^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 c^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)","-\frac{d^{5/2} \left(b^2-4 a c\right)^{15/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d^{5/2} \left(b^2-4 a c\right)^{15/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{130 c^3 \sqrt{a+b x+c x^2}}+\frac{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac{\left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{7/2}}{13 c d}",1,"(2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^2 - ((b^2 - 4*a*c)^2*Hypergeometric2F1[-3/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*c^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/13","C",1
1345,1,99,281,0.0567337,"\int \sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2),x]","-\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{24 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{\sqrt{d} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{30 c^3 \sqrt{a+b x+c x^2}}+\frac{\sqrt{d} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{30 c^3 \sqrt{a+b x+c x^2}}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}+\frac{\left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{3/2}}{9 c d}",1,"-1/24*((b^2 - 4*a*c)*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^2*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1346,1,99,271,0.0527808,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(3/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)}}","\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{10 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{3/2}}{c d \sqrt{b d+2 c d x}}",1,"((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -1/4, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*c^2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1347,1,99,273,0.060069,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(7/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{40 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{5/2}}","-\frac{3 \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{a+b x+c x^2}}{10 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{5 c d (b d+2 c d x)^{5/2}}",1,"((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -5/4, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(40*c^2*d*(d*(b + 2*c*x))^(5/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1348,1,107,320,0.0805344,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(b d+2 c d x)^{11/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(11/2),x]","\frac{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{72 c^2 d^6 (b+2 c x)^5 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{30 c^3 d^{11/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{30 c^3 d^{11/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{15 c^2 d^5 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{30 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{9 c d (b d+2 c d x)^{9/2}}",1,"((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-9/4, -3/2, -5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(72*c^2*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1349,1,161,321,0.389392,"\int (b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]","\frac{4 \sqrt{a+x (b+c x)} (d (b+2 c x))^{7/2} \left(3 (b+2 c x)^2 (a+x (b+c x))^3-2 c \left(a-\frac{b^2}{4 c}\right) \left(\frac{\left(b^2-4 a c\right)^3 \, _2F_1\left(-\frac{5}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64 c^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))^3\right)\right)}{57 (b+2 c x)^3}","-\frac{5 d^{7/2} \left(b^2-4 a c\right)^{21/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{17556 c^4 \sqrt{a+b x+c x^2}}-\frac{5 d^3 \left(b^2-4 a c\right)^4 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{8778 c^3}-\frac{d \left(b^2-4 a c\right)^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{2926 c^3}+\frac{\left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{836 c^3 d}-\frac{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{9/2}}{114 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2} (b d+2 c d x)^{9/2}}{19 c d}",1,"(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(3*(b + 2*c*x)^2*(a + x*(b + c*x))^3 - 2*(a - b^2/(4*c))*c*(2*(a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3*Hypergeometric2F1[-5/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*c^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(57*(b + 2*c*x)^3)","C",1
1350,1,117,274,0.0958321,"\int (b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2),x]","\frac{2}{15} d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \left(\frac{\left(b^2-4 a c\right)^3 \, _2F_1\left(-\frac{5}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64 c^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))^3\right)","-\frac{d^{3/2} \left(b^2-4 a c\right)^{17/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{924 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left(b^2-4 a c\right)^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{462 c^3}+\frac{\left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{308 c^3 d}-\frac{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{5/2}}{66 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2} (b d+2 c d x)^{5/2}}{15 c d}",1,"(2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3*Hypergeometric2F1[-5/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*c^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/15","C",1
1351,1,101,229,0.0770103,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{\sqrt{b d+2 c d x}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/Sqrt[b*d + 2*c*d*x],x]","\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{5}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{32 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{5 \left(b^2-4 a c\right)^{13/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{308 c^4 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{5 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{308 c^3 d}-\frac{5 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} \sqrt{b d+2 c d x}}{154 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2} \sqrt{b d+2 c d x}}{11 c d}",1,"((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(32*c^3*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1352,1,101,219,0.0605732,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(5/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{3}{4};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{96 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}}","\frac{5 \left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}-\frac{5 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^3}+\frac{5 \left(a+b x+c x^2\right)^{3/2} \sqrt{b d+2 c d x}}{42 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{5/2}}{3 c d (b d+2 c d x)^{3/2}}",1,"-1/96*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -3/4, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1353,1,109,211,0.0821023,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(9/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{5}{2},-\frac{7}{4};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{224 c^3 d^5 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{5 \left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{84 c^4 d^{9/2} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^5}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{7 c d (b d+2 c d x)^{7/2}}",1,"-1/224*((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -7/4, -3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^5*(b + 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1354,1,109,211,0.0862654,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{13/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(13/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{11}{4},-\frac{5}{2};-\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{352 c^3 d^7 (b+2 c x)^6 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{5 \sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{308 c^4 d^{13/2} \sqrt{a+b x+c x^2}}-\frac{5 \sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{3/2}}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{11 c d (b d+2 c d x)^{11/2}}",1,"-1/352*((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-11/4, -5/2, -7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^7*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1355,1,109,258,0.0920317,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{17/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(17/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{15}{4},-\frac{5}{2};-\frac{11}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{480 c^3 d^9 (b+2 c x)^8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{924 c^4 d^{17/2} \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{462 c^3 d^7 \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{7/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{66 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{15 c d (b d+2 c d x)^{15/2}}",1,"-1/480*((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-15/4, -5/2, -11/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^9*(b + 2*c*x)^8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1356,1,109,305,0.0791289,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{21/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{19}{4},-\frac{5}{2};-\frac{15}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{608 c^3 d^9 (b+2 c x)^8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}}","\frac{5 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{17556 c^4 d^{21/2} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 d^9 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 d^7 \left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{19 c d (b d+2 c d x)^{19/2}}",1,"-1/608*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-19/4, -5/2, -15/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^9*(b + 2*c*x)^8*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1357,1,117,373,0.1755332,"\int (b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2),x]","\frac{2}{17} d \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \left(\frac{\left(b^2-4 a c\right)^3 \, _2F_1\left(-\frac{5}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64 c^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))^3\right)","\frac{d^{5/2} \left(b^2-4 a c\right)^{19/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left(b^2-4 a c\right)^{19/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left(b^2-4 a c\right)^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac{5 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac{5 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2} (b d+2 c d x)^{7/2}}{17 c d}",1,"(2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3*Hypergeometric2F1[-5/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*c^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/17","C",1
1358,1,101,328,0.0657068,"\int \sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(5/2),x]","\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2} \, _2F_1\left(-\frac{5}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{96 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{d} \left(b^2-4 a c\right)^{15/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{156 c^4 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left(b^2-4 a c\right)^{15/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{156 c^4 \sqrt{a+b x+c x^2}}+\frac{\left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac{5 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac{\left(a+b x+c x^2\right)^{5/2} (b d+2 c d x)^{3/2}}{13 c d}",1,"((b^2 - 4*a*c)^2*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(96*c^3*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1359,1,101,316,0.0540559,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{1}{4};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{32 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)}}","-\frac{\left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{12 c^3 d^3}+\frac{5 \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{3/2}}{18 c^2 d^3}-\frac{\left(a+b x+c x^2\right)^{5/2}}{c d \sqrt{b d+2 c d x}}",1,"-1/32*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -1/4, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1360,1,101,310,0.0629151,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(7/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \, _2F_1\left(-\frac{5}{2},-\frac{5}{4};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{160 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{5/2}}","\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{20 c^3 d^5}-\frac{\left(a+b x+c x^2\right)^{3/2}}{2 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{5 c d (b d+2 c d x)^{5/2}}",1,"-1/160*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -5/4, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d*(d*(b + 2*c*x))^(5/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1361,1,109,310,0.0870387,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{11/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(11/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{5}{2},-\frac{9}{4};-\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{288 c^3 d^6 (b+2 c x)^5 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}+\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{12 c^3 d^5 \sqrt{b d+2 c d x}}-\frac{\left(a+b x+c x^2\right)^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{9 c d (b d+2 c d x)^{9/2}}",1,"-1/288*((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -9/4, -5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1362,1,109,357,0.0886477,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(b d+2 c d x)^{15/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(15/2),x]","-\frac{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{13}{4},-\frac{5}{2};-\frac{9}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{416 c^3 d^8 (b+2 c x)^7 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{156 c^4 d^{15/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{156 c^4 d^{15/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{5 \left(a+b x+c x^2\right)^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left(a+b x+c x^2\right)^{5/2}}{13 c d (b d+2 c d x)^{13/2}}",1,"-1/416*((b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-13/4, -5/2, -9/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^8*(b + 2*c*x)^7*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","C",1
1363,1,166,174,0.1778339,"\int \frac{(b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 d^3 \sqrt{d (b+2 c x)} \left(8 c \left(-5 a^2 c+2 a \left(b^2-b c x-c^2 x^2\right)+x \left(2 b^3+5 b^2 c x+6 b c^2 x^2+3 c^3 x^3\right)\right)+5 \left(b^2-4 a c\right)^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{21 c \sqrt{a+x (b+c x)}}","\frac{10 d^{7/2} \left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{21 c \sqrt{a+b x+c x^2}}+\frac{20}{21} d^3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{4}{7} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}",1,"(2*d^3*Sqrt[d*(b + 2*c*x)]*(8*c*(-5*a^2*c + 2*a*(b^2 - b*c*x - c^2*x^2) + x*(2*b^3 + 5*b^2*c*x + 6*b*c^2*x^2 + 3*c^3*x^3)) + 5*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(21*c*Sqrt[a + x*(b + c*x)])","C",1
1364,1,111,132,0.1093663,"\int \frac{(b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 d \sqrt{d (b+2 c x)} \left(\left(b^2-4 a c\right) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+2 c (a+x (b+c x))\right)}{3 c \sqrt{a+x (b+c x)}}","\frac{2 d^{3/2} \left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 c \sqrt{a+b x+c x^2}}+\frac{4}{3} d \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}",1,"(2*d*Sqrt[d*(b + 2*c*x)]*(2*c*(a + x*(b + c*x)) + (b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*c*Sqrt[a + x*(b + c*x)])","C",1
1365,1,89,97,0.0581498,"\int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c d \sqrt{a+x (b+c x)}}","\frac{2 \sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c \sqrt{d} \sqrt{a+b x+c x^2}}",1,"(2*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*Sqrt[a + x*(b + c*x)])","C",1
1366,1,91,144,0.0510055,"\int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2]),x]","-\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 c d \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}}","\frac{2 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 c d^{5/2} \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{3 d \left(b^2-4 a c\right) (b d+2 c d x)^{3/2}}",1,"(-2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-3/4, 1/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)])","C",1
1367,1,99,188,0.0721245,"\int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2]),x]","-\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)} \, _2F_1\left(-\frac{7}{4},\frac{1}{2};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{7 c d^5 (b+2 c x)^4 \sqrt{a+x (b+c x)}}","\frac{10 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{21 c d^{9/2} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 d^3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}+\frac{4 \sqrt{a+b x+c x^2}}{7 d \left(b^2-4 a c\right) (b d+2 c d x)^{7/2}}",1,"(-2*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-7/4, 1/2, -3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(7*c*d^5*(b + 2*c*x)^4*Sqrt[a + x*(b + c*x)])","C",1
1368,1,167,273,0.1989684,"\int \frac{(b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 d^3 (d (b+2 c x))^{3/2} \left(8 c \left(-7 a^2 c+a \left(3 b^2-2 b c x-2 c^2 x^2\right)+x \left(3 b^3+8 b^2 c x+10 b c^2 x^2+5 c^3 x^3\right)\right)+7 \left(b^2-4 a c\right)^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{45 c \sqrt{a+x (b+c x)}}","-\frac{14 d^{9/2} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{15 c \sqrt{a+b x+c x^2}}+\frac{14 d^{9/2} \left(b^2-4 a c\right)^{11/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{15 c \sqrt{a+b x+c x^2}}+\frac{28}{45} d^3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}+\frac{4}{9} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}",1,"(2*d^3*(d*(b + 2*c*x))^(3/2)*(8*c*(-7*a^2*c + a*(3*b^2 - 2*b*c*x - 2*c^2*x^2) + x*(3*b^3 + 8*b^2*c*x + 10*b*c^2*x^2 + 5*c^3*x^3)) + 7*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(45*c*Sqrt[a + x*(b + c*x)])","C",1
1369,1,111,231,0.1246471,"\int \frac{(b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 d (d (b+2 c x))^{3/2} \left(\left(b^2-4 a c\right) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+2 c (a+x (b+c x))\right)}{5 c \sqrt{a+x (b+c x)}}","-\frac{6 d^{5/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c \sqrt{a+b x+c x^2}}+\frac{6 d^{5/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c \sqrt{a+b x+c x^2}}+\frac{4}{5} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}",1,"(2*d*(d*(b + 2*c*x))^(3/2)*(2*c*(a + x*(b + c*x)) + (b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(5*c*Sqrt[a + x*(b + c*x)])","C",1
1370,1,91,195,0.0461954,"\int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/Sqrt[a + b*x + c*x^2],x]","\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 c d \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{d} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c \sqrt{a+b x+c x^2}}",1,"(2*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c*d*Sqrt[a + x*(b + c*x)])","C",1
1371,1,89,237,0.0460342,"\int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]","-\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}}","\frac{2 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c d^{3/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{c d^{3/2} \sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{d \left(b^2-4 a c\right) \sqrt{b d+2 c d x}}",1,"(-2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-1/4, 1/2, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])","C",1
1372,1,91,287,0.0599007,"\int \frac{1}{(b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]","-\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 c d \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}}","\frac{6 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c d^{7/2} \left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 c d^{7/2} \left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{a+b x+c x^2}}{5 d^3 \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}+\frac{4 \sqrt{a+b x+c x^2}}{5 d \left(b^2-4 a c\right) (b d+2 c d x)^{5/2}}",1,"(-2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-5/4, 1/2, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*c*d*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])","C",1
1373,1,76,79,0.0308475,"\int \frac{(3-2 x)^{3/2}}{\sqrt{1-3 x+x^2}} \, dx","Integrate[(3 - 2*x)^(3/2)/Sqrt[1 - 3*x + x^2],x]","-\frac{2 \sqrt{3-2 x} \left(\sqrt{5} \sqrt{-x^2+3 x-1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{5} (3-2 x)^2\right)+2 x^2-6 x+2\right)}{3 \sqrt{x^2-3 x+1}}","-\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{3 \sqrt{x^2-3 x+1}}",1,"(-2*Sqrt[3 - 2*x]*(2 - 6*x + 2*x^2 + Sqrt[5]*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (3 - 2*x)^2/5]))/(3*Sqrt[1 - 3*x + x^2])","C",1
1374,1,63,51,0.0140484,"\int \frac{1}{\sqrt{3-2 x} \sqrt{1-3 x+x^2}} \, dx","Integrate[1/(Sqrt[3 - 2*x]*Sqrt[1 - 3*x + x^2]),x]","-\frac{2 \sqrt{3-2 x} \sqrt{-x^2+3 x-1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{5} (3-2 x)^2\right)}{\sqrt{5} \sqrt{x^2-3 x+1}}","-\frac{2 \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{\sqrt[4]{5} \sqrt{x^2-3 x+1}}",1,"(-2*Sqrt[3 - 2*x]*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (3 - 2*x)^2/5])/(Sqrt[5]*Sqrt[1 - 3*x + x^2])","C",1
1375,1,65,79,0.0144366,"\int \frac{1}{(3-2 x)^{5/2} \sqrt{1-3 x+x^2}} \, dx","Integrate[1/((3 - 2*x)^(5/2)*Sqrt[1 - 3*x + x^2]),x]","\frac{2 \sqrt{-x^2+3 x-1} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{1}{5} (3-2 x)^2\right)}{3 \sqrt{5} (3-2 x)^{3/2} \sqrt{x^2-3 x+1}}","-\frac{4 \sqrt{x^2-3 x+1}}{15 (3-2 x)^{3/2}}-\frac{2 \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{15 \sqrt[4]{5} \sqrt{x^2-3 x+1}}",1,"(2*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[-3/4, 1/2, 1/4, (3 - 2*x)^2/5])/(3*Sqrt[5]*(3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2])","C",1
1376,1,76,128,0.0245498,"\int \frac{(3-2 x)^{5/2}}{\sqrt{1-3 x+x^2}} \, dx","Integrate[(3 - 2*x)^(5/2)/Sqrt[1 - 3*x + x^2],x]","-\frac{2 (3-2 x)^{3/2} \left(\sqrt{5} \sqrt{-x^2+3 x-1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{1}{5} (3-2 x)^2\right)+2 x^2-6 x+2\right)}{5 \sqrt{x^2-3 x+1}}","-\frac{4}{5} \sqrt{x^2-3 x+1} (3-2 x)^{3/2}+\frac{6 \sqrt[4]{5} \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{\sqrt{x^2-3 x+1}}-\frac{6 \sqrt[4]{5} \sqrt{-x^2+3 x-1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{\sqrt{x^2-3 x+1}}",1,"(-2*(3 - 2*x)^(3/2)*(2 - 6*x + 2*x^2 + Sqrt[5]*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (3 - 2*x)^2/5]))/(5*Sqrt[1 - 3*x + x^2])","C",1
1377,1,65,103,0.0143742,"\int \frac{\sqrt{3-2 x}}{\sqrt{1-3 x+x^2}} \, dx","Integrate[Sqrt[3 - 2*x]/Sqrt[1 - 3*x + x^2],x]","-\frac{2 (3-2 x)^{3/2} \sqrt{-x^2+3 x-1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{1}{5} (3-2 x)^2\right)}{3 \sqrt{5} \sqrt{x^2-3 x+1}}","\frac{2 \sqrt[4]{5} \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{\sqrt{x^2-3 x+1}}-\frac{2 \sqrt[4]{5} \sqrt{-x^2+3 x-1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{\sqrt{x^2-3 x+1}}",1,"(-2*(3 - 2*x)^(3/2)*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (3 - 2*x)^2/5])/(3*Sqrt[5]*Sqrt[1 - 3*x + x^2])","C",1
1378,1,63,128,0.0131634,"\int \frac{1}{(3-2 x)^{3/2} \sqrt{1-3 x+x^2}} \, dx","Integrate[1/((3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2]),x]","\frac{2 \sqrt{-x^2+3 x-1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{1}{5} (3-2 x)^2\right)}{\sqrt{5} \sqrt{3-2 x} \sqrt{x^2-3 x+1}}","-\frac{4 \sqrt{x^2-3 x+1}}{5 \sqrt{3-2 x}}-\frac{2 \sqrt{-x^2+3 x-1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{5^{3/4} \sqrt{x^2-3 x+1}}+\frac{2 \sqrt{-x^2+3 x-1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-2 x}}{\sqrt[4]{5}}\right)\right|-1\right)}{5^{3/4} \sqrt{x^2-3 x+1}}",1,"(2*Sqrt[-1 + 3*x - x^2]*Hypergeometric2F1[-1/4, 1/2, 3/4, (3 - 2*x)^2/5])/(Sqrt[5]*Sqrt[3 - 2*x]*Sqrt[1 - 3*x + x^2])","C",1
1379,1,172,205,0.2042483,"\int \frac{(b d+2 c d x)^{11/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(3/2),x]","\frac{2 d^5 \sqrt{d (b+2 c x)} \left(16 c^2 \left(-15 a^2-6 a c x^2+2 c^2 x^4\right)+30 \left(b^2-4 a c\right)^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+24 b^2 c \left(4 a+3 c x^2\right)+32 b c^2 x \left(2 c x^2-3 a\right)-7 b^4+40 b^3 c x\right)}{7 \sqrt{a+x (b+c x)}}","\frac{60 d^{11/2} \left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{7 \sqrt{a+b x+c x^2}}+\frac{120}{7} c d^5 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{72}{7} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{2 d (b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}}",1,"(2*d^5*Sqrt[d*(b + 2*c*x)]*(-7*b^4 + 40*b^3*c*x + 32*b*c^2*x*(-3*a + 2*c*x^2) + 24*b^2*c*(4*a + 3*c*x^2) + 16*c^2*(-15*a^2 - 6*a*c*x^2 + 2*c^2*x^4) + 30*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(7*Sqrt[a + x*(b + c*x)])","C",1
1380,1,122,162,0.1238639,"\int \frac{(b d+2 c d x)^{7/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]","\frac{2 d^3 \sqrt{d (b+2 c x)} \left(10 \left(b^2-4 a c\right) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)}{3 \sqrt{a+x (b+c x)}}","\frac{20 d^{7/2} \left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 \sqrt{a+b x+c x^2}}+\frac{40}{3} c d^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}-\frac{2 d (b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}}",1,"(2*d^3*Sqrt[d*(b + 2*c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2) + 10*(b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*Sqrt[a + x*(b + c*x)])","C",1
1381,1,88,125,0.0628548,"\int \frac{(b d+2 c d x)^{3/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(3/2),x]","\frac{2 d \sqrt{d (b+2 c x)} \left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-1\right)}{\sqrt{a+x (b+c x)}}","\frac{4 d^{3/2} \sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}}",1,"(2*d*Sqrt[d*(b + 2*c*x)]*(-1 + 2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/Sqrt[a + x*(b + c*x)]","C",1
1382,1,100,137,0.0713526,"\int \frac{1}{\sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2)),x]","-\frac{2 \sqrt{d (b+2 c x)} \left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+1\right)}{d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{2 \sqrt{b d+2 c d x}}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{d} \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}",1,"(-2*Sqrt[d*(b + 2*c*x)]*(1 + 2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/((b^2 - 4*a*c)*d*Sqrt[a + x*(b + c*x)])","C",1
1383,1,98,184,0.0537112,"\int \frac{1}{(b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]","\frac{8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}}","-\frac{20 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 d^{5/2} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}-\frac{40 c \sqrt{a+b x+c x^2}}{3 d \left(b^2-4 a c\right)^2 (b d+2 c d x)^{3/2}}-\frac{2}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}",1,"(8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-3/4, 3/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)])","C",1
1384,1,122,258,0.1413362,"\int \frac{(b d+2 c d x)^{9/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(3/2),x]","-\frac{8 d^3 (d (b+2 c x))^{3/2} \left(7 \left(b^2-4 a c\right) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-2 \left(c \left(c x^2-7 a\right)+2 b^2+b c x\right)\right)}{5 \sqrt{a+x (b+c x)}}","-\frac{84 d^{9/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 \sqrt{a+b x+c x^2}}+\frac{84 d^{9/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 \sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}",1,"(-8*d^3*(d*(b + 2*c*x))^(3/2)*(-2*(2*b^2 + b*c*x + c*(-7*a + c*x^2)) + 7*(b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 3/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(5*Sqrt[a + x*(b + c*x)])","C",1
1385,1,88,219,0.0695818,"\int \frac{(b d+2 c d x)^{5/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]","-\frac{4 d (d (b+2 c x))^{3/2} \left(2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-1\right)}{\sqrt{a+x (b+c x)}}","-\frac{12 d^{5/2} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}+\frac{12 d^{5/2} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}}",1,"(-4*d*(d*(b + 2*c*x))^(3/2)*(-1 + 2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 3/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/Sqrt[a + x*(b + c*x)]","C",1
1386,1,98,231,0.0563607,"\int \frac{\sqrt{b d+2 c d x}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(3/2),x]","-\frac{8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{2 (b d+2 c d x)^{3/2}}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{d} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{d} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}",1,"(-8*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 3/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)*d*Sqrt[a + x*(b + c*x)])","C",1
1387,1,96,274,0.0547544,"\int \frac{1}{(b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]","\frac{8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}}","-\frac{12 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{d^{3/2} \left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{d^{3/2} \left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}-\frac{24 c \sqrt{a+b x+c x^2}}{d \left(b^2-4 a c\right)^2 \sqrt{b d+2 c d x}}-\frac{2}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}",1,"(8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-1/4, 3/2, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])","C",1
1388,1,98,325,0.0637871,"\int \frac{1}{(b d+2 c d x)^{7/2} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2)),x]","\frac{8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{5}{4},\frac{3}{2};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 d \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}}","-\frac{84 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 d^{7/2} \left(b^2-4 a c\right)^{9/4} \sqrt{a+b x+c x^2}}+\frac{84 \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 d^{7/2} \left(b^2-4 a c\right)^{9/4} \sqrt{a+b x+c x^2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 d^3 \left(b^2-4 a c\right)^3 \sqrt{b d+2 c d x}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 d \left(b^2-4 a c\right)^2 (b d+2 c d x)^{5/2}}-\frac{2}{d \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}",1,"(8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-5/4, 3/2, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])","C",1
1389,1,252,247,0.3215104,"\int \frac{(b d+2 c d x)^{15/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^(5/2),x]","\frac{2 d^7 \sqrt{d (b+2 c x)} \left(16 b^2 c^2 \left(156 a^2+117 a c x^2+116 c^2 x^4\right)+32 b c^3 x \left(-273 a^2-104 a c x^2+36 c^2 x^4\right)-32 c^3 \left(195 a^3+273 a^2 c x^2+52 a c^2 x^4-12 c^3 x^6\right)+2 b^4 c \left(219 c x^2-91 a\right)+16 b^3 c^2 x \left(221 a+112 c x^2\right)+780 c \left(b^2-4 a c\right)^2 (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-7 b^6-266 b^5 c x\right)}{21 (a+x (b+c x))^{3/2}}","\frac{1040}{7} c^2 d^7 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{520 c d^{15/2} \left(b^2-4 a c\right)^{9/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{7 \sqrt{a+b x+c x^2}}+\frac{624}{7} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}-\frac{52 c d^3 (b d+2 c d x)^{9/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{13/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(2*d^7*Sqrt[d*(b + 2*c*x)]*(-7*b^6 - 266*b^5*c*x + 16*b^3*c^2*x*(221*a + 112*c*x^2) + 2*b^4*c*(-91*a + 219*c*x^2) + 32*b*c^3*x*(-273*a^2 - 104*a*c*x^2 + 36*c^2*x^4) + 16*b^2*c^2*(156*a^2 + 117*a*c*x^2 + 116*c^2*x^4) - 32*c^3*(195*a^3 + 273*a^2*c*x^2 + 52*a*c^2*x^4 - 12*c^3*x^6) + 780*c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(21*(a + x*(b + c*x))^(3/2))","C",1
1390,1,201,196,0.21684,"\int \frac{(b d+2 c d x)^{11/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(5/2),x]","\frac{2 d^5 \sqrt{d (b+2 c x)} \left(-60 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 a^2 c+a \left(-b^2+4 b c x+4 c^2 x^2\right)-b^2 x (b+c x)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+8 c^2 \left(15 a^2+21 a c x^2+4 c^2 x^4\right)+6 b^2 c \left(c x^2-3 a\right)+8 b c^2 x \left(21 a+8 c x^2\right)-b^4-26 b^3 c x\right)}{3 (a+x (b+c x))^{3/2}}","\frac{40 c d^{11/2} \left(b^2-4 a c\right)^{5/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}+80 c^2 d^5 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}-\frac{12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{9/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(2*d^5*Sqrt[d*(b + 2*c*x)]*(-b^4 - 26*b^3*c*x + 6*b^2*c*(-3*a + c*x^2) + 8*b*c^2*x*(21*a + 8*c*x^2) + 8*c^2*(15*a^2 + 21*a*c*x^2 + 4*c^2*x^4) - 60*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(4*a^2*c - b^2*x*(b + c*x) + a*(-b^2 + 4*b*c*x + 4*c^2*x^2))*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b + c*x))^(3/2))","C",1
1391,1,122,165,0.12671,"\int \frac{(b d+2 c d x)^{7/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(7/2)/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 d^3 \sqrt{d (b+2 c x)} \left(-20 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+2 c \left(5 a+7 c x^2\right)+b^2+14 b c x\right)}{3 (a+x (b+c x))^{3/2}}","\frac{40 c d^{7/2} \sqrt[4]{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 \sqrt{a+b x+c x^2}}-\frac{20 c d^3 \sqrt{b d+2 c d x}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{5/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*d^3*Sqrt[d*(b + 2*c*x)]*(b^2 + 14*b*c*x + 2*c*(5*a + 7*c*x^2) - 20*c*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b + c*x))^(3/2))","C",1
1392,1,129,173,0.1526939,"\int \frac{(b d+2 c d x)^{3/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 d \sqrt{d (b+2 c x)} \left(4 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+2 c \left(c x^2-a\right)+b^2+2 b c x\right)}{3 \left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}","-\frac{8 c d^{3/2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 \left(b^2-4 a c\right)^{3/4} \sqrt{a+b x+c x^2}}-\frac{4 c d \sqrt{b d+2 c d x}}{3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*d*Sqrt[d*(b + 2*c*x)]*(b^2 + 2*b*c*x + 2*c*(-a + c*x^2) + 4*c*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2))","C",1
1393,1,132,187,0.1511426,"\int \frac{1}{\sqrt{b d+2 c d x} \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(5/2)),x]","-\frac{2 \sqrt{d (b+2 c x)} \left(-20 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-2 c \left(7 a+5 c x^2\right)+b^2-10 b c x\right)}{3 d \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{20 c \sqrt{b d+2 c d x}}{3 d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{b d+2 c d x}}{3 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{3 \sqrt{d} \left(b^2-4 a c\right)^{7/4} \sqrt{a+b x+c x^2}}",1,"(-2*Sqrt[d*(b + 2*c*x)]*(b^2 - 10*b*c*x - 2*c*(7*a + 5*c*x^2) - 20*c*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(b^2 - 4*a*c)^2*d*(a + x*(b + c*x))^(3/2))","C",1
1394,1,99,228,0.0680153,"\int \frac{1}{(b d+2 c d x)^{5/2} \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]","-\frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{3}{4},\frac{5}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}}","\frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left(b^2-4 a c\right)^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{d^{5/2} \left(b^2-4 a c\right)^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} (b d+2 c d x)^{3/2}}",1,"(-32*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-3/4, 5/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)])","C",1
1395,1,202,299,0.2171001,"\int \frac{(b d+2 c d x)^{13/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^(5/2),x]","\frac{4 d^5 (d (b+2 c x))^{3/2} \left(616 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(4 a^2 c+a \left(-b^2+4 b c x+4 c^2 x^2\right)-b^2 x (b+c x)\right) \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+16 c^2 \left(-77 a^2-33 a c x^2+3 c^2 x^4\right)+4 b^2 c \left(121 a+51 c x^2\right)+48 b c^2 x \left(2 c x^2-11 a\right)-41 b^4+156 b^3 c x\right)}{15 (a+x (b+c x))^{3/2}}","-\frac{616 c d^{13/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 \sqrt{a+b x+c x^2}}+\frac{616 c d^{13/2} \left(b^2-4 a c\right)^{7/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{5 \sqrt{a+b x+c x^2}}+\frac{1232}{15} c^2 d^5 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{11/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(4*d^5*(d*(b + 2*c*x))^(3/2)*(-41*b^4 + 156*b^3*c*x + 48*b*c^2*x*(-11*a + 2*c*x^2) + 4*b^2*c*(121*a + 51*c*x^2) + 16*c^2*(-77*a^2 - 33*a*c*x^2 + 3*c^2*x^4) + 616*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(4*a^2*c - b^2*x*(b + c*x) + a*(-b^2 + 4*b*c*x + 4*c^2*x^2))*Hypergeometric2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(15*(a + x*(b + c*x))^(3/2))","C",1
1396,1,122,258,0.1236433,"\int \frac{(b d+2 c d x)^{9/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]","-\frac{16 d^3 (d (b+2 c x))^{3/2} \left(14 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-c \left(7 a+3 c x^2\right)+b^2-3 b c x\right)}{3 (a+x (b+c x))^{3/2}}","-\frac{56 c d^{9/2} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}+\frac{56 c d^{9/2} \left(b^2-4 a c\right)^{3/4} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt{a+b x+c x^2}}-\frac{28 c d^3 (b d+2 c d x)^{3/2}}{3 \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{7/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-16*d^3*(d*(b + 2*c*x))^(3/2)*(b^2 - 3*b*c*x - c*(7*a + 3*c*x^2) + 14*c*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b + c*x))^(3/2))","C",1
1397,1,116,264,0.1116917,"\int \frac{(b d+2 c d x)^{5/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^(5/2)/(a + b*x + c*x^2)^(5/2),x]","-\frac{4 d (d (b+2 c x))^{3/2} \left(8 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-4 a c+b^2\right)}{3 \left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}","-\frac{8 c d^{5/2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{8 c d^{5/2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\sqrt[4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{4 c d (b d+2 c d x)^{3/2}}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{2 d (b d+2 c d x)^{3/2}}{3 \left(a+b x+c x^2\right)^{3/2}}",1,"(-4*d*(d*(b + 2*c*x))^(3/2)*(b^2 - 4*a*c + 8*c*(a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2))","C",1
1398,1,99,278,0.0558531,"\int \frac{\sqrt{b d+2 c d x}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(5/2),x]","\frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)}}","\frac{4 c (b d+2 c d x)^{3/2}}{d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (b d+2 c d x)^{3/2}}{3 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{8 c \sqrt{d} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{\left(b^2-4 a c\right)^{5/4} \sqrt{a+b x+c x^2}}",1,"(32*c*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^2*d*Sqrt[a + x*(b + c*x)])","C",1
1399,1,97,325,0.0589096,"\int \frac{1}{(b d+2 c d x)^{3/2} \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]","-\frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(-\frac{1}{4},\frac{5}{2};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}}","\frac{112 c^2 \sqrt{a+b x+c x^2}}{d \left(b^2-4 a c\right)^3 \sqrt{b d+2 c d x}}+\frac{56 c \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{d^{3/2} \left(b^2-4 a c\right)^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{b d+2 c x d}}{\sqrt[4]{b^2-4 a c} \sqrt{d}}\right)\right|-1\right)}{d^{3/2} \left(b^2-4 a c\right)^{9/4} \sqrt{a+b x+c x^2}}+\frac{28 c}{3 d \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}-\frac{2}{3 d \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} \sqrt{b d+2 c d x}}",1,"(-32*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-1/4, 5/2, 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])","C",1
1400,1,125,170,0.097672,"\int \frac{(c e+d e x)^{11/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","-\frac{2 e^5 \sqrt{e (c+d x)} \left(\sqrt{-c^2-2 c d x-d^2 x^2+1} \left(7 c^4+28 c^3 d x+c^2 \left(42 d^2 x^2+9\right)+2 c d x \left(14 d^2 x^2+9\right)+7 d^4 x^4+9 d^2 x^2+15\right)-15 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)\right)}{77 d}","-\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}+\frac{30 e^{11/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{77 d}",1,"(-2*e^5*Sqrt[e*(c + d*x)]*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(15 + 7*c^4 + 28*c^3*d*x + 9*d^2*x^2 + 7*d^4*x^4 + 2*c*d*x*(9 + 14*d^2*x^2) + c^2*(9 + 42*d^2*x^2)) - 15*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(77*d)","C",1
1401,1,86,124,0.0642921,"\int \frac{(c e+d e x)^{7/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[(c*e + d*e*x)^(7/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","-\frac{2 e^3 \sqrt{e (c+d x)} \left(\sqrt{-c^2-2 c d x-d^2 x^2+1} \left(3 c^2+6 c d x+3 d^2 x^2+5\right)-5 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)\right)}{21 d}","-\frac{10 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{21 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d}+\frac{10 e^{7/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{21 d}",1,"(-2*e^3*Sqrt[e*(c + d*x)]*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(5 + 3*c^2 + 6*c*d*x + 3*d^2*x^2) - 5*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(21*d)","C",1
1402,1,54,78,0.0204915,"\int \frac{(c e+d e x)^{3/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[(c*e + d*e*x)^(3/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","-\frac{2 e \sqrt{e (c+d x)} \left(\sqrt{1-(c+d x)^2}-\, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)\right)}{3 d}","\frac{2 e^{3/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{3 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{3 d}",1,"(-2*e*Sqrt[e*(c + d*x)]*(Sqrt[1 - (c + d*x)^2] - Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(3*d)","C",1
1403,1,38,31,0.0171681,"\int \frac{1}{\sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/(Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","\frac{2 (c+d x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)}{d \sqrt{e (c+d x)}}","\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{d \sqrt{e}}",1,"(2*(c + d*x)*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2])/(d*Sqrt[e*(c + d*x)])","C",1
1404,1,40,80,0.0208344,"\int \frac{1}{(c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/((c*e + d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","-\frac{2 (c+d x) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};(c+d x)^2\right)}{3 d (e (c+d x))^{5/2}}","\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{3 d e^{5/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}",1,"(-2*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, (c + d*x)^2])/(3*d*(e*(c + d*x))^(5/2))","C",1
1405,1,40,126,0.0261309,"\int \frac{1}{(c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/((c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","-\frac{2 (c+d x) \, _2F_1\left(-\frac{7}{4},\frac{1}{2};-\frac{3}{4};(c+d x)^2\right)}{7 d (e (c+d x))^{9/2}}","-\frac{10 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}+\frac{10 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{21 d e^{9/2}}",1,"(-2*(c + d*x)*Hypergeometric2F1[-7/4, 1/2, -3/4, (c + d*x)^2])/(7*d*(e*(c + d*x))^(9/2))","C",1
1406,1,40,172,0.0290039,"\int \frac{1}{(c e+d e x)^{13/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","-\frac{2 (c+d x) \, _2F_1\left(-\frac{11}{4},\frac{1}{2};-\frac{7}{4};(c+d x)^2\right)}{11 d (e (c+d x))^{13/2}}","-\frac{30 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac{18 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}+\frac{30 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{77 d e^{13/2}}",1,"(-2*(c + d*x)*Hypergeometric2F1[-11/4, 1/2, -7/4, (c + d*x)^2])/(11*d*(e*(c + d*x))^(13/2))","C",1
1407,1,86,157,0.0756243,"\int \frac{(c e+d e x)^{9/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[(c*e + d*e*x)^(9/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","-\frac{2 e^3 (e (c+d x))^{3/2} \left(\sqrt{-c^2-2 c d x-d^2 x^2+1} \left(5 c^2+10 c d x+5 d^2 x^2+7\right)-7 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)\right)}{45 d}","-\frac{14 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{45 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{7/2}}{9 d}-\frac{14 e^{9/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{15 d}+\frac{14 e^{9/2} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{15 d}",1,"(-2*e^3*(e*(c + d*x))^(3/2)*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(7 + 5*c^2 + 10*c*d*x + 5*d^2*x^2) - 7*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(45*d)","C",1
1408,1,54,111,0.0303026,"\int \frac{(c e+d e x)^{5/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[(c*e + d*e*x)^(5/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","-\frac{2 e (e (c+d x))^{3/2} \left(\sqrt{1-(c+d x)^2}-\, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)\right)}{5 d}","-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{5 d}-\frac{6 e^{5/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{5 d}+\frac{6 e^{5/2} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{5 d}",1,"(-2*e*(e*(c + d*x))^(3/2)*(Sqrt[1 - (c + d*x)^2] - Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(5*d)","C",1
1409,1,40,63,0.0172597,"\int \frac{\sqrt{c e+d e x}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[Sqrt[c*e + d*e*x]/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]","\frac{2 (c+d x) \sqrt{e (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)}{3 d}","\frac{2 \sqrt{e} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{d}-\frac{2 \sqrt{e} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{d}",1,"(2*(c + d*x)*Sqrt[e*(c + d*x)]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(3*d)","C",1
1410,1,38,107,0.016161,"\int \frac{1}{(c e+d e x)^{3/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/((c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","-\frac{2 (c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right)}{d (e (c+d x))^{3/2}}","-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt{c e+d e x}}+\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{d e^{3/2}}-\frac{2 E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{d e^{3/2}}",1,"(-2*(c + d*x)*Hypergeometric2F1[-1/4, 1/2, 3/4, (c + d*x)^2])/(d*(e*(c + d*x))^(3/2))","C",1
1411,1,40,159,0.0227277,"\int \frac{1}{(c e+d e x)^{7/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx","Integrate[1/((c*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]","-\frac{2 (c+d x) \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right)}{5 d (e (c+d x))^{7/2}}","-\frac{6 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt{c e+d e x}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}}+\frac{6 F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{5 d e^{7/2}}-\frac{6 E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right)\right|-1\right)}{5 d e^{7/2}}",1,"(-2*(c + d*x)*Hypergeometric2F1[-5/4, 1/2, -1/4, (c + d*x)^2])/(5*d*(e*(c + d*x))^(7/2))","C",1
1412,1,104,320,0.0914866,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{11/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(11/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \, _2F_1\left(-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64\ 2^{2/3} c^2 d \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{8/3}}","-\frac{3 \sqrt[3]{a+b x+c x^2} (d (b+2 c x))^{4/3}}{16 c^2 d^5 \left(b^2-4 a c\right)}+\frac{9 \left(a+b x+c x^2\right)^{4/3} (d (b+2 c x))^{4/3}}{16 c d^5 \left(b^2-4 a c\right)^2}-\frac{9 \left(a+b x+c x^2\right)^{7/3}}{4 d^3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{2/3}}+\frac{3 \left(a+b x+c x^2\right)^{7/3}}{4 d \left(b^2-4 a c\right) (b d+2 c d x)^{8/3}}-\frac{3 \log \left((d (b+2 c x))^{2/3}-2^{2/3} \sqrt[3]{c} d^{2/3} \sqrt[3]{a+b x+c x^2}\right)}{32\ 2^{2/3} c^{7/3} d^{11/3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (d (b+2 c x))^{2/3}}{\sqrt[3]{c} d^{2/3} \sqrt[3]{a+b x+c x^2}}+1}{\sqrt{3}}\right)}{16\ 2^{2/3} c^{7/3} d^{11/3}}",1,"(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -4/3, -1/3, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*2^(2/3)*c^2*d*(d*(b + 2*c*x))^(8/3)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
1413,1,50,44,0.0498507,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{17/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x]","\frac{3 (a+x (b+c x))^{7/3} \sqrt[3]{d (b+2 c x)}}{7 d^6 \left(b^2-4 a c\right) (b+2 c x)^5}","\frac{3 \left(a+b x+c x^2\right)^{7/3}}{7 d \left(b^2-4 a c\right) (b d+2 c d x)^{14/3}}",1,"(3*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(7/3))/(7*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)","A",1
1414,1,74,89,0.0759476,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{23/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(23/3),x]","\frac{3 (a+x (b+c x))^{7/3} \left(2 c \left(3 c x^2-7 a\right)+5 b^2+6 b c x\right) \sqrt[3]{d (b+2 c x)}}{35 d^8 \left(b^2-4 a c\right)^2 (b+2 c x)^7}","\frac{9 \left(a+b x+c x^2\right)^{7/3}}{70 d^3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{14/3}}+\frac{3 \left(a+b x+c x^2\right)^{7/3}}{10 d \left(b^2-4 a c\right) (b d+2 c d x)^{20/3}}",1,"(3*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(7/3)*(5*b^2 + 6*b*c*x + 2*c*(-7*a + 3*c*x^2)))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^7)","A",1
1415,1,122,133,0.0988953,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{29/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(29/3),x]","\frac{3 (a+x (b+c x))^{7/3} \left(16 c^2 \left(35 a^2-21 a c x^2+9 c^2 x^4\right)+4 b^2 c \left(75 c x^2-91 a\right)+48 b c^2 x \left(6 c x^2-7 a\right)+65 b^4+156 b^3 c x\right)}{455 d^9 \left(b^2-4 a c\right)^3 (b+2 c x)^8 (d (b+2 c x))^{2/3}}","\frac{27 \left(a+b x+c x^2\right)^{7/3}}{455 d^5 \left(b^2-4 a c\right)^3 (b d+2 c d x)^{14/3}}+\frac{9 \left(a+b x+c x^2\right)^{7/3}}{65 d^3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{20/3}}+\frac{3 \left(a+b x+c x^2\right)^{7/3}}{13 d \left(b^2-4 a c\right) (b d+2 c d x)^{26/3}}",1,"(3*(a + x*(b + c*x))^(7/3)*(65*b^4 + 156*b^3*c*x + 48*b*c^2*x*(-7*a + 6*c*x^2) + 4*b^2*c*(-91*a + 75*c*x^2) + 16*c^2*(35*a^2 - 21*a*c*x^2 + 9*c^2*x^4)))/(455*(b^2 - 4*a*c)^3*d^9*(b + 2*c*x)^8*(d*(b + 2*c*x))^(2/3))","A",1
1416,1,104,597,0.076055,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{2/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(2/3),x]","-\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \sqrt[3]{d (b+2 c x)} \, _2F_1\left(-\frac{4}{3},\frac{1}{6};\frac{7}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8\ 2^{2/3} c^2 d \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\left(b^2-4 a c\right) \left(-4 a c+b^2-(b+2 c x)^2\right) \sqrt[3]{d (b+2 c x)} \left(2 \sqrt[3]{c} d^{2/3}-\frac{\sqrt[3]{2} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right) \sqrt{\frac{\frac{2^{2/3} \sqrt[3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left(a+b x+c x^2\right)^{2/3}}+2 \sqrt[3]{2} c^{2/3} d^{4/3}}{\left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}} F\left(\cos ^{-1}\left(\frac{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1-\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{72 \sqrt[4]{3} c^{10/3} d^{5/3} \left(a+b x+c x^2\right)^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)}{\sqrt[3]{a+b x+c x^2} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}}}-\frac{\left(b^2-4 a c\right) \sqrt[3]{a+b x+c x^2} \sqrt[3]{d (b+2 c x)}}{9 c^2 d}+\frac{\left(a+b x+c x^2\right)^{4/3} \sqrt[3]{d (b+2 c x)}}{6 c d}",1,"(-3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, 1/6, 7/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*2^(2/3)*c^2*d*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
1417,1,104,581,0.0608006,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{8/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(8/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \, _2F_1\left(-\frac{4}{3},-\frac{5}{6};\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{40\ 2^{2/3} c^2 d \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{5/3}}","-\frac{\left(-4 a c+b^2-(b+2 c x)^2\right) \sqrt[3]{d (b+2 c x)} \left(2 \sqrt[3]{c} d^{2/3}-\frac{\sqrt[3]{2} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right) \sqrt{\frac{\frac{2^{2/3} \sqrt[3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left(a+b x+c x^2\right)^{2/3}}+2 \sqrt[3]{2} c^{2/3} d^{4/3}}{\left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}} F\left(\cos ^{-1}\left(\frac{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1-\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{40 \sqrt[4]{3} c^{10/3} d^{11/3} \left(a+b x+c x^2\right)^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)}{\sqrt[3]{a+b x+c x^2} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}}}+\frac{\sqrt[3]{a+b x+c x^2} \sqrt[3]{d (b+2 c x)}}{5 c^2 d^3}-\frac{3 \left(a+b x+c x^2\right)^{4/3}}{10 c d (d (b+2 c x))^{5/3}}",1,"(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -5/6, 1/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(40*2^(2/3)*c^2*d*(d*(b + 2*c*x))^(5/3)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
1418,1,112,591,0.087809,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{14/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(14/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \sqrt[3]{d (b+2 c x)} \, _2F_1\left(-\frac{11}{6},-\frac{4}{3};-\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{88\ 2^{2/3} c^2 d^5 (b+2 c x)^4 \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{3^{3/4} \left(-4 a c+b^2-(b+2 c x)^2\right) \sqrt[3]{d (b+2 c x)} \left(2 \sqrt[3]{c} d^{2/3}-\frac{\sqrt[3]{2} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right) \sqrt{\frac{\frac{2^{2/3} \sqrt[3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left(a+b x+c x^2\right)^{2/3}}+2 \sqrt[3]{2} c^{2/3} d^{4/3}}{\left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}} F\left(\cos ^{-1}\left(\frac{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1-\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{440 c^{10/3} d^{17/3} \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)}{\sqrt[3]{a+b x+c x^2} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}}}-\frac{3 \sqrt[3]{a+b x+c x^2}}{55 c^2 d^3 (d (b+2 c x))^{5/3}}-\frac{3 \left(a+b x+c x^2\right)^{4/3}}{22 c d (d (b+2 c x))^{11/3}}",1,"(3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-11/6, -4/3, -5/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(88*2^(2/3)*c^2*d^5*(b + 2*c*x)^4*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
1419,1,112,637,0.0982252,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{20/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(20/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \sqrt[3]{d (b+2 c x)} \, _2F_1\left(-\frac{17}{6},-\frac{4}{3};-\frac{11}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{136\ 2^{2/3} c^2 d^7 (b+2 c x)^6 \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{3\ 3^{3/4} \left(-4 a c+b^2-(b+2 c x)^2\right) \sqrt[3]{d (b+2 c x)} \left(2 \sqrt[3]{c} d^{2/3}-\frac{\sqrt[3]{2} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right) \sqrt{\frac{\frac{2^{2/3} \sqrt[3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}+\frac{(d (b+2 c x))^{4/3}}{\left(a+b x+c x^2\right)^{2/3}}+2 \sqrt[3]{2} c^{2/3} d^{4/3}}{\left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}} F\left(\cos ^{-1}\left(\frac{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1-\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}{2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{c x^2+b x+a}}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7480 c^{10/3} d^{23/3} \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{2/3} \sqrt{-\frac{(d (b+2 c x))^{2/3} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{(d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)}{\sqrt[3]{a+b x+c x^2} \left(2^{2/3} \sqrt[3]{c} d^{2/3}-\frac{\left(1+\sqrt{3}\right) (d (b+2 c x))^{2/3}}{\sqrt[3]{a+b x+c x^2}}\right)^2}}}+\frac{6 \sqrt[3]{a+b x+c x^2}}{935 c^2 d^5 \left(b^2-4 a c\right) (d (b+2 c x))^{5/3}}-\frac{3 \sqrt[3]{a+b x+c x^2}}{187 c^2 d^3 (d (b+2 c x))^{11/3}}-\frac{3 \left(a+b x+c x^2\right)^{4/3}}{34 c d (d (b+2 c x))^{17/3}}",1,"(3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-17/6, -4/3, -11/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(136*2^(2/3)*c^2*d^7*(b + 2*c*x)^6*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
1420,1,104,99,0.0587337,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{4/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(4/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \, _2F_1\left(-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8\ 2^{2/3} c^2 d \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt[3]{d (b+2 c x)}}","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+b x+c x^2} \, _2F_1\left(-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8 c^2 d \sqrt[3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[3]{d (b+2 c x)}}",1,"(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -1/6, 5/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*2^(2/3)*c^2*d*(d*(b + 2*c*x))^(1/3)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","A",1
1421,1,104,99,0.0669913,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{10/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(10/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} \, _2F_1\left(-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{56\ 2^{2/3} c^2 d \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{7/3}}","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+b x+c x^2} \, _2F_1\left(-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{56 c^2 d \sqrt[3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{7/3}}",1,"(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -7/6, -1/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(56*2^(2/3)*c^2*d*(d*(b + 2*c*x))^(7/3)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","A",1
1422,1,112,99,0.0882769,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(b d+2 c d x)^{16/3}} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(16/3),x]","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+x (b+c x)} (d (b+2 c x))^{2/3} \, _2F_1\left(-\frac{13}{6},-\frac{4}{3};-\frac{7}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{104\ 2^{2/3} c^2 d^6 (b+2 c x)^5 \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{3 \left(b^2-4 a c\right) \sqrt[3]{a+b x+c x^2} \, _2F_1\left(-\frac{13}{6},-\frac{4}{3};-\frac{7}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{104 c^2 d \sqrt[3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{13/3}}",1,"(3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(2/3)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-13/6, -4/3, -7/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(104*2^(2/3)*c^2*d^6*(b + 2*c*x)^5*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","A",1
1423,1,103,141,0.0943667,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right)^3 \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]","\frac{(b+2 c x) \left(-\frac{3 \left(b^2-4 a c\right) (b+2 c x)^4}{m+5}+\frac{3 \left(b^2-4 a c\right)^2 (b+2 c x)^2}{m+3}-\frac{\left(b^2-4 a c\right)^3}{m+1}+\frac{(b+2 c x)^6}{m+7}\right) (d (b+2 c x))^m}{128 c^4}","-\frac{3 \left(b^2-4 a c\right) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac{3 \left(b^2-4 a c\right)^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac{\left(b^2-4 a c\right)^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac{(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)}",1,"((b + 2*c*x)*(d*(b + 2*c*x))^m*(-((b^2 - 4*a*c)^3/(1 + m)) + (3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(3 + m) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^4)/(5 + m) + (b + 2*c*x)^6/(7 + m)))/(128*c^4)","A",1
1424,1,77,103,0.0491421,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right)^2 \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^2,x]","\frac{(b+2 c x) \left(-\frac{2 \left(b^2-4 a c\right) (b+2 c x)^2}{m+3}+\frac{\left(b^2-4 a c\right)^2}{m+1}+\frac{(b+2 c x)^4}{m+5}\right) (d (b+2 c x))^m}{32 c^3}","-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{m+3}}{16 c^3 d^3 (m+3)}+\frac{\left(b^2-4 a c\right)^2 (b d+2 c d x)^{m+1}}{32 c^3 d (m+1)}+\frac{(b d+2 c d x)^{m+5}}{32 c^3 d^5 (m+5)}",1,"((b + 2*c*x)*(d*(b + 2*c*x))^m*((b^2 - 4*a*c)^2/(1 + m) - (2*(b^2 - 4*a*c)*(b + 2*c*x)^2)/(3 + m) + (b + 2*c*x)^4/(5 + m)))/(32*c^3)","A",1
1425,1,64,65,0.0352453,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right) \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2),x]","\frac{(b+2 c x) \left(2 c \left(a (m+3)+c (m+1) x^2\right)-b^2+2 b c (m+1) x\right) (d (b+2 c x))^m}{4 c^2 (m+1) (m+3)}","\frac{(b d+2 c d x)^{m+3}}{8 c^2 d^3 (m+3)}-\frac{\left(b^2-4 a c\right) (b d+2 c d x)^{m+1}}{8 c^2 d (m+1)}",1,"((b + 2*c*x)*(d*(b + 2*c*x))^m*(-b^2 + 2*b*c*(1 + m)*x + 2*c*(a*(3 + m) + c*(1 + m)*x^2)))/(4*c^2*(1 + m)*(3 + m))","A",1
1426,1,68,67,0.0360593,"\int \frac{(b d+2 c d x)^m}{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2),x]","-\frac{2 (b+2 c x) (d (b+2 c x))^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{(m+1) \left(b^2-4 a c\right)}","-\frac{2 (d (b+2 c x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)}",1,"(-2*(b + 2*c*x)*(d*(b + 2*c*x))^m*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*(1 + m))","A",1
1427,1,69,68,0.0366108,"\int \frac{(b d+2 c d x)^m}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^2,x]","\frac{8 c (b+2 c x) (d (b+2 c x))^m \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{(m+1) \left(b^2-4 a c\right)^2}","\frac{8 c (d (b+2 c x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)^2}",1,"(8*c*(b + 2*c*x)*(d*(b + 2*c*x))^m*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^2*(1 + m))","A",1
1428,1,71,70,0.0401159,"\int \frac{(b d+2 c d x)^m}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^3,x]","-\frac{32 c^2 (b+2 c x) (d (b+2 c x))^m \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{(m+1) \left(b^2-4 a c\right)^3}","-\frac{32 c^2 (d (b+2 c x))^{m+1} \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)^3}",1,"(-32*c^2*(b + 2*c*x)*(d*(b + 2*c*x))^m*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^3*(1 + m))","A",1
1429,1,115,82,0.0771107,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^(5/2),x]","\frac{\left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{64 c^3 (m+1) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{2 \left(a+b x+c x^2\right)^{7/2} (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m+8}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)}",1,"((b^2 - 4*a*c)^2*(b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*c^3*(1 + m)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","A",1
1430,1,113,98,0.0615748,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^(3/2),x]","-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 c^2 (m+1) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{\left(4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}\right)^{5/2} (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{16 d (m+1) \left(b^2-4 a c\right)}",1,"-1/16*((b^2 - 4*a*c)*(b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^2*(1 + m)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","A",1
1431,1,105,98,0.0504318,"\int (b d+2 c d x)^m \sqrt{a+b x+c x^2} \, dx","Integrate[(b*d + 2*c*d*x)^m*Sqrt[a + b*x + c*x^2],x]","\frac{(b+2 c x) \sqrt{a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{4 c (m+1) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","-\frac{\left(4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}\right)^{3/2} (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{4 d (m+1) \left(b^2-4 a c\right)}",1,"((b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c*(1 + m)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])","A",1
1432,1,102,96,0.0447799,"\int \frac{(b d+2 c d x)^m}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(b*d + 2*c*d*x)^m/Sqrt[a + b*x + c*x^2],x]","\frac{(b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c (m+1) \sqrt{a+x (b+c x)}}","-\frac{\sqrt{4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}} (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)}",1,"((b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*(1 + m)*Sqrt[a + x*(b + c*x)])","A",1
1433,1,110,94,0.0521125,"\int \frac{(b d+2 c d x)^m}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^(3/2),x]","-\frac{4 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{(m+1) \left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{4 (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right) \sqrt{4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}}}",1,"(-4*(b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*(1 + m)*Sqrt[a + x*(b + c*x)])","A",1
1434,1,111,96,0.0531283,"\int \frac{(b d+2 c d x)^m}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^(5/2),x]","\frac{16 c (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{(m+1) \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)}}","-\frac{16 (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{m-2}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right) \left(4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}\right)^{3/2}}",1,"(16*c*(b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)^2*(1 + m)*Sqrt[a + x*(b + c*x)])","A",1
1435,1,107,107,0.054423,"\int (b d+2 c d x)^m \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^p,x]","\frac{2^{-2 p-1} (b+2 c x) (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} (d (b+2 c x))^m \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c (m+1)}","-\frac{2 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} (b d+2 c d x)^{m+1} \, _2F_1\left(1,\frac{1}{2} (m+2 p+3);\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (m+1) \left(b^2-4 a c\right)}",1,"(2^(-1 - 2*p)*(b + 2*c*x)*(d*(b + 2*c*x))^m*(a + x*(b + c*x))^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*(1 + m)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)","A",1
1436,1,145,121,0.0991025,"\int (b d+2 c d x)^5 \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^p,x]","\frac{d^5 (a+x (b+c x))^{p+1} \left(16 c^2 \left(2 a^2-2 a c (p+1) x^2+c^2 \left(p^2+3 p+2\right) x^4\right)-8 b^2 c \left(a (p+3)-c \left(3 p^2+10 p+7\right) x^2\right)-32 b c^2 (p+1) x \left(a-c (p+2) x^2\right)+b^4 \left(p^2+5 p+6\right)+8 b^3 c \left(p^2+4 p+3\right) x\right)}{(p+1) (p+2) (p+3)}","\frac{2 d^5 \left(b^2-4 a c\right) (b+2 c x)^2 \left(a+b x+c x^2\right)^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left(a+b x+c x^2\right)^{p+1}}{p+3}",1,"(d^5*(a + x*(b + c*x))^(1 + p)*(b^4*(6 + 5*p + p^2) + 8*b^3*c*(3 + 4*p + p^2)*x - 32*b*c^2*(1 + p)*x*(a - c*(2 + p)*x^2) - 8*b^2*c*(a*(3 + p) - c*(7 + 10*p + 3*p^2)*x^2) + 16*c^2*(2*a^2 - 2*a*c*(1 + p)*x^2 + c^2*(2 + 3*p + p^2)*x^4)))/((1 + p)*(2 + p)*(3 + p))","A",1
1437,1,92,90,0.0404545,"\int (b d+2 c d x)^4 \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^p,x]","\frac{d^4 2^{-2 p-1} (b+2 c x)^5 (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 c}","-\frac{2 d^4 (b+2 c x)^5 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{2};\frac{7}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 \left(b^2-4 a c\right)}",1,"(2^(-1 - 2*p)*d^4*(b + 2*c*x)^5*(a + x*(b + c*x))^p*Hypergeometric2F1[5/2, -p, 7/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)","A",1
1438,1,58,68,0.0491232,"\int (b d+2 c d x)^3 \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^p,x]","\frac{d^3 (a+x (b+c x))^{p+1} \left(4 c \left(c (p+1) x^2-a\right)+b^2 (p+2)+4 b c (p+1) x\right)}{(p+1) (p+2)}","\frac{d^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{p+1}}{(p+1) (p+2)}+\frac{d^3 (b+2 c x)^2 \left(a+b x+c x^2\right)^{p+1}}{p+2}",1,"(d^3*(a + x*(b + c*x))^(1 + p)*(b^2*(2 + p) + 4*b*c*(1 + p)*x + 4*c*(-a + c*(1 + p)*x^2)))/((1 + p)*(2 + p))","A",1
1439,1,92,90,0.0338536,"\int (b d+2 c d x)^2 \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^p,x]","\frac{d^2 2^{-2 p-1} (b+2 c x)^3 (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 c}","-\frac{2 d^2 (b+2 c x)^3 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{2};\frac{5}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 \left(b^2-4 a c\right)}",1,"(2^(-1 - 2*p)*d^2*(b + 2*c*x)^3*(a + x*(b + c*x))^p*Hypergeometric2F1[3/2, -p, 5/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)","A",1
1440,1,20,21,0.0073222,"\int (b d+2 c d x) \left(a+b x+c x^2\right)^p \, dx","Integrate[(b*d + 2*c*d*x)*(a + b*x + c*x^2)^p,x]","\frac{d (a+x (b+c x))^{p+1}}{p+1}","\frac{d \left(a+b x+c x^2\right)^{p+1}}{p+1}",1,"(d*(a + x*(b + c*x))^(1 + p))/(1 + p)","A",1
1441,1,64,63,0.0271776,"\int \frac{\left(a+b x+c x^2\right)^p}{b d+2 c d x} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x),x]","\frac{(a+x (b+c x))^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{d (p+1) \left(b^2-4 a c\right)}","\frac{\left(a+b x+c x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d (p+1) \left(b^2-4 a c\right)}",1,"((a + x*(b + c*x))^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)*d*(1 + p))","A",1
1442,1,90,88,0.0373059,"\int \frac{\left(a+b x+c x^2\right)^p}{(b d+2 c d x)^2} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^2,x]","-\frac{2^{-2 p-1} (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{c d^2 (b+2 c x)}","\frac{2 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} \, _2F_1\left(1,p+\frac{1}{2};\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d^2 \left(b^2-4 a c\right) (b+2 c x)}",1,"-((2^(-1 - 2*p)*(a + x*(b + c*x))^p*Hypergeometric2F1[-1/2, -p, 1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d^2*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p))","A",1
1443,1,64,63,0.037088,"\int \frac{\left(a+b x+c x^2\right)^p}{(b d+2 c d x)^3} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^3,x]","\frac{(a+x (b+c x))^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{d^3 (p+1) \left(b^2-4 a c\right)^2}","\frac{\left(a+b x+c x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d^3 (p+1) \left(b^2-4 a c\right)^2}",1,"((a + x*(b + c*x))^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)^2*d^3*(1 + p))","A",1
1444,1,92,90,0.042854,"\int \frac{\left(a+b x+c x^2\right)^p}{(b d+2 c d x)^4} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^4,x]","-\frac{2^{-2 p-1} (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 c d^4 (b+2 c x)^3}","\frac{2 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} \, _2F_1\left(1,p-\frac{1}{2};-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 d^4 \left(b^2-4 a c\right) (b+2 c x)^3}",1,"-1/3*(2^(-1 - 2*p)*(a + x*(b + c*x))^p*Hypergeometric2F1[-3/2, -p, -1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d^4*(b + 2*c*x)^3*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)","A",1
1445,1,64,63,0.0408181,"\int \frac{\left(a+b x+c x^2\right)^p}{(b d+2 c d x)^5} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^5,x]","\frac{(a+x (b+c x))^{p+1} \, _2F_1\left(3,p+1;p+2;\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right)}{d^5 (p+1) \left(b^2-4 a c\right)^3}","\frac{\left(a+b x+c x^2\right)^{p+1} \, _2F_1\left(3,p+1;p+2;1-\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{d^5 (p+1) \left(b^2-4 a c\right)^3}",1,"((a + x*(b + c*x))^(1 + p)*Hypergeometric2F1[3, 1 + p, 2 + p, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)^3*d^5*(1 + p))","A",1
1446,1,92,90,0.0458974,"\int \frac{\left(a+b x+c x^2\right)^p}{(b d+2 c d x)^6} \, dx","Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^6,x]","-\frac{2^{-2 p-1} (a+x (b+c x))^p \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{-p} \, _2F_1\left(-\frac{5}{2},-p;-\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 c d^6 (b+2 c x)^5}","\frac{2 \left(\frac{1}{4} \left(4 a-\frac{b^2}{c}\right)+\frac{(b+2 c x)^2}{4 c}\right)^{p+1} \, _2F_1\left(1,p-\frac{3}{2};-\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{5 d^6 \left(b^2-4 a c\right) (b+2 c x)^5}",1,"-1/5*(2^(-1 - 2*p)*(a + x*(b + c*x))^p*Hypergeometric2F1[-5/2, -p, -3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d^6*(b + 2*c*x)^5*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)","A",1
1447,1,16,16,0.0043079,"\int \frac{1+x}{\left(-3+2 x+x^2\right)^{2/3}} \, dx","Integrate[(1 + x)/(-3 + 2*x + x^2)^(2/3),x]","\frac{3}{2} \sqrt[3]{x^2+2 x-3}","\frac{3}{2} \sqrt[3]{x^2+2 x-3}",1,"(3*(-3 + 2*x + x^2)^(1/3))/2","A",1
1448,1,19,19,0.012029,"\int \frac{b+c x}{\left(a+2 b x+c x^2\right)^{3/7}} \, dx","Integrate[(b + c*x)/(a + 2*b*x + c*x^2)^(3/7),x]","\frac{7}{8} (a+x (2 b+c x))^{4/7}","\frac{7}{8} \left(a+2 b x+c x^2\right)^{4/7}",1,"(7*(a + x*(2*b + c*x))^(4/7))/8","A",1
1449,1,23,26,0.011915,"\int (1+x)^m \left(1+2 x+x^2\right)^n \, dx","Integrate[(1 + x)^m*(1 + 2*x + x^2)^n,x]","\frac{(x+1)^{m+1} \left((x+1)^2\right)^n}{m+2 n+1}","\frac{(x+1)^{m+1} \left(x^2+2 x+1\right)^n}{m+2 n+1}",1,"((1 + x)^(1 + m)*((1 + x)^2)^n)/(1 + m + 2*n)","A",1
1450,1,54,50,0.0287031,"\int \left(\frac{b e}{2 c}+e x\right)^m \left(\frac{b^2}{4 c}+b x+c x^2\right)^n \, dx","Integrate[((b*e)/(2*c) + e*x)^m*(b^2/(4*c) + b*x + c*x^2)^n,x]","\frac{2^{-2 n-1} (b+2 c x) \left(\frac{(b+2 c x)^2}{c}\right)^n \left(\frac{b e}{2 c}+e x\right)^m}{c (m+2 n+1)}","\frac{\left(\frac{b^2}{4 c}+b x+c x^2\right)^n \left(\frac{b e}{2 c}+e x\right)^{m+1}}{e (m+2 n+1)}",1,"(2^(-1 - 2*n)*(b + 2*c*x)*((b + 2*c*x)^2/c)^n*((b*e)/(2*c) + e*x)^m)/(c*(1 + m + 2*n))","A",1
1451,1,148,65,0.0265362,"\int (d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{1}{5} e^2 x^5 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)+d e x^4 \left(a^2 e^2+3 a b d e+b^2 d^2\right)+\frac{1}{3} d^2 x^3 \left(6 a^2 e^2+8 a b d e+b^2 d^2\right)+a^2 d^4 x+a d^3 x^2 (2 a e+b d)+\frac{1}{3} b e^3 x^6 (a e+2 b d)+\frac{1}{7} b^2 e^4 x^7","-\frac{b (d+e x)^6 (b d-a e)}{3 e^3}+\frac{(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac{b^2 (d+e x)^7}{7 e^3}",1,"a^2*d^4*x + a*d^3*(b*d + 2*a*e)*x^2 + (d^2*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2)*x^3)/3 + d*e*(b^2*d^2 + 3*a*b*d*e + a^2*e^2)*x^4 + (e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)*x^5)/5 + (b*e^3*(2*b*d + a*e)*x^6)/3 + (b^2*e^4*x^7)/7","B",1
1452,1,122,65,0.0170009,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{1}{4} e x^4 \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)+\frac{1}{3} d x^3 \left(3 a^2 e^2+6 a b d e+b^2 d^2\right)+a^2 d^3 x+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{5} b e^2 x^5 (2 a e+3 b d)+\frac{1}{6} b^2 e^3 x^6","-\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3}",1,"a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^2)/2 + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^3)/3 + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^4)/4 + (b*e^2*(3*b*d + 2*a*e)*x^5)/5 + (b^2*e^3*x^6)/6","A",1
1453,1,79,65,0.0136049,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{1}{3} x^3 \left(a^2 e^2+4 a b d e+b^2 d^2\right)+a^2 d^2 x+\frac{1}{2} b e x^4 (a e+b d)+a d x^2 (a e+b d)+\frac{1}{5} b^2 e^2 x^5","\frac{e (a+b x)^4 (b d-a e)}{2 b^3}+\frac{(a+b x)^3 (b d-a e)^2}{3 b^3}+\frac{e^2 (a+b x)^5}{5 b^3}",1,"a^2*d^2*x + a*d*(b*d + a*e)*x^2 + ((b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3)/3 + (b*e*(b*d + a*e)*x^4)/2 + (b^2*e^2*x^5)/5","A",1
1454,1,46,38,0.0098033,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{1}{12} x \left(6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right)","\frac{(a+b x)^3 (b d-a e)}{3 b^2}+\frac{e (a+b x)^4}{4 b^2}",1,"(x*(6*a^2*(2*d + e*x) + 4*a*b*x*(3*d + 2*e*x) + b^2*x^2*(4*d + 3*e*x)))/12","A",1
1455,1,22,22,0.0000916,"\int \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[a^2 + 2*a*b*x + b^2*x^2,x]","a^2 x+a b x^2+\frac{b^2 x^3}{3}","a^2 x+a b x^2+\frac{b^2 x^3}{3}",1,"a^2*x + a*b*x^2 + (b^2*x^3)/3","A",1
1456,1,43,50,0.0178289,"\int \frac{a^2+2 a b x+b^2 x^2}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x),x]","\frac{b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3}","\frac{(b d-a e)^2 \log (d+e x)}{e^3}-\frac{b x (b d-a e)}{e^2}+\frac{(a+b x)^2}{2 e}",1,"(b*e*x*(-2*b*d + 4*a*e + b*e*x) + 2*(b*d - a*e)^2*Log[d + e*x])/(2*e^3)","A",1
1457,1,47,51,0.0367978,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^2,x]","\frac{-\frac{(b d-a e)^2}{d+e x}+2 b (a e-b d) \log (d+e x)+b^2 e x}{e^3}","-\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}+\frac{b^2 x}{e^2}",1,"(b^2*e*x - (b*d - a*e)^2/(d + e*x) + 2*b*(-(b*d) + a*e)*Log[d + e*x])/e^3","A",1
1458,1,48,59,0.0237089,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^3,x]","\frac{\frac{(b d-a e) (a e+3 b d+4 b e x)}{(d+e x)^2}+2 b^2 \log (d+e x)}{2 e^3}","\frac{2 b (b d-a e)}{e^3 (d+e x)}-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{b^2 \log (d+e x)}{e^3}",1,"(((b*d - a*e)*(3*b*d + a*e + 4*b*e*x))/(d + e*x)^2 + 2*b^2*Log[d + e*x])/(2*e^3)","A",1
1459,1,53,28,0.0247443,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^4,x]","-\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left(d^2+3 d e x+3 e^2 x^2\right)}{3 e^3 (d+e x)^3}","\frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)}",1,"-1/3*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(e^3*(d + e*x)^3)","A",1
1460,1,55,65,0.0202342,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]","-\frac{3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left(d^2+4 d e x+6 e^2 x^2\right)}{12 e^3 (d+e x)^4}","\frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2}",1,"-1/12*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)","A",1
1461,1,55,65,0.0262369,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^6,x]","-\frac{6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left(d^2+5 d e x+10 e^2 x^2\right)}{30 e^3 (d+e x)^5}","\frac{b (b d-a e)}{2 e^3 (d+e x)^4}-\frac{(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac{b^2}{3 e^3 (d+e x)^3}",1,"-1/30*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2))/(e^3*(d + e*x)^5)","A",1
1462,1,55,65,0.0208298,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]","-\frac{10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left(d^2+6 d e x+15 e^2 x^2\right)}{60 e^3 (d+e x)^6}","\frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4}",1,"-1/60*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^2))/(e^3*(d + e*x)^6)","A",1
1463,1,398,119,0.0632547,"\int (d+e x)^6 \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","a^4 d^6 x+a^3 d^5 x^2 (3 a e+2 b d)+\frac{1}{3} b^2 e^4 x^9 \left(2 a^2 e^2+8 a b d e+5 b^2 d^2\right)+a^2 d^4 x^3 \left(5 a^2 e^2+8 a b d e+2 b^2 d^2\right)+\frac{1}{2} b e^3 x^8 \left(a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right)+a d^3 x^4 \left(5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right)+\frac{1}{7} e^2 x^7 \left(a^4 e^4+24 a^3 b d e^3+90 a^2 b^2 d^2 e^2+80 a b^3 d^3 e+15 b^4 d^4\right)+d e x^6 \left(a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right)+\frac{1}{5} d^2 x^5 \left(15 a^4 e^4+80 a^3 b d e^3+90 a^2 b^2 d^2 e^2+24 a b^3 d^3 e+b^4 d^4\right)+\frac{1}{5} b^3 e^5 x^{10} (2 a e+3 b d)+\frac{1}{11} b^4 e^6 x^{11}","-\frac{2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac{2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac{b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac{(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac{b^4 (d+e x)^{11}}{11 e^5}",1,"a^4*d^6*x + a^3*d^5*(2*b*d + 3*a*e)*x^2 + a^2*d^4*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^3 + a*d^3*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^4 + (d^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4)*x^5)/5 + d*e*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*x^6 + (e^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4)*x^7)/7 + (b*e^3*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^8)/2 + (b^2*e^4*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^9)/3 + (b^3*e^5*(3*b*d + 2*a*e)*x^10)/5 + (b^4*e^6*x^11)/11","B",1
1464,1,350,119,0.0495212,"\int (d+e x)^5 \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","a^4 d^5 x+\frac{1}{2} a^3 d^4 x^2 (5 a e+4 b d)+\frac{1}{4} b^2 e^3 x^8 \left(3 a^2 e^2+10 a b d e+5 b^2 d^2\right)+\frac{2}{3} a^2 d^3 x^3 \left(5 a^2 e^2+10 a b d e+3 b^2 d^2\right)+\frac{2}{7} b e^2 x^7 \left(2 a^3 e^3+15 a^2 b d e^2+20 a b^2 d^2 e+5 b^3 d^3\right)+\frac{1}{2} a d^2 x^4 \left(5 a^3 e^3+20 a^2 b d e^2+15 a b^2 d^2 e+2 b^3 d^3\right)+\frac{1}{6} e x^6 \left(a^4 e^4+20 a^3 b d e^3+60 a^2 b^2 d^2 e^2+40 a b^3 d^3 e+5 b^4 d^4\right)+\frac{1}{5} d x^5 \left(5 a^4 e^4+40 a^3 b d e^3+60 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+b^4 d^4\right)+\frac{1}{9} b^3 e^4 x^9 (4 a e+5 b d)+\frac{1}{10} b^4 e^5 x^{10}","-\frac{4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac{3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac{4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5}",1,"a^4*d^5*x + (a^3*d^4*(4*b*d + 5*a*e)*x^2)/2 + (2*a^2*d^3*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2)*x^3)/3 + (a*d^2*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3)*x^4)/2 + (d*(b^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4)*x^5)/5 + (e*(5*b^4*d^4 + 40*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4)*x^6)/6 + (2*b*e^2*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 2*a^3*e^3)*x^7)/7 + (b^2*e^3*(5*b^2*d^2 + 10*a*b*d*e + 3*a^2*e^2)*x^8)/4 + (b^3*e^4*(5*b*d + 4*a*e)*x^9)/9 + (b^4*e^5*x^10)/10","B",1
1465,1,273,119,0.0379861,"\int (d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","a^4 d^4 x+2 a^3 d^3 x^2 (a e+b d)+\frac{2}{7} b^2 e^2 x^7 \left(3 a^2 e^2+8 a b d e+3 b^2 d^2\right)+\frac{2}{3} a^2 d^2 x^3 \left(3 a^2 e^2+8 a b d e+3 b^2 d^2\right)+\frac{2}{3} b e x^6 \left(a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right)+a d x^4 \left(a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right)+\frac{1}{5} x^5 \left(a^4 e^4+16 a^3 b d e^3+36 a^2 b^2 d^2 e^2+16 a b^3 d^3 e+b^4 d^4\right)+\frac{1}{2} b^3 e^3 x^8 (a e+b d)+\frac{1}{9} b^4 e^4 x^9","\frac{e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac{6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac{2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac{(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac{e^4 (a+b x)^9}{9 b^5}",1,"a^4*d^4*x + 2*a^3*d^3*(b*d + a*e)*x^2 + (2*a^2*d^2*(3*b^2*d^2 + 8*a*b*d*e + 3*a^2*e^2)*x^3)/3 + a*d*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3)*x^4 + ((b^4*d^4 + 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e^3 + a^4*e^4)*x^5)/5 + (2*b*e*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3)*x^6)/3 + (2*b^2*e^2*(3*b^2*d^2 + 8*a*b*d*e + 3*a^2*e^2)*x^7)/7 + (b^3*e^3*(b*d + a*e)*x^8)/2 + (b^4*e^4*x^9)/9","B",1
1466,1,217,92,0.029763,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","a^4 d^3 x+\frac{1}{2} a^3 d^2 x^2 (3 a e+4 b d)+\frac{1}{2} b^2 e x^6 \left(2 a^2 e^2+4 a b d e+b^2 d^2\right)+a^2 d x^3 \left(a^2 e^2+4 a b d e+2 b^2 d^2\right)+\frac{1}{5} b x^5 \left(4 a^3 e^3+18 a^2 b d e^2+12 a b^2 d^2 e+b^3 d^3\right)+\frac{1}{4} a x^4 \left(a^3 e^3+12 a^2 b d e^2+18 a b^2 d^2 e+4 b^3 d^3\right)+\frac{1}{7} b^3 e^2 x^7 (4 a e+3 b d)+\frac{1}{8} b^4 e^3 x^8","\frac{3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac{e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac{(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac{e^3 (a+b x)^8}{8 b^4}",1,"a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + a^2*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3 + (a*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (b*(b^3*d^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3)*x^5)/5 + (b^2*e*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2)*x^6)/2 + (b^3*e^2*(3*b*d + 4*a*e)*x^7)/7 + (b^4*e^3*x^8)/8","B",1
1467,1,148,65,0.0242205,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","a^4 d^2 x+a^3 d x^2 (a e+2 b d)+\frac{1}{5} b^2 x^5 \left(6 a^2 e^2+8 a b d e+b^2 d^2\right)+a b x^4 \left(a^2 e^2+3 a b d e+b^2 d^2\right)+\frac{1}{3} a^2 x^3 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)+\frac{1}{3} b^3 e x^6 (2 a e+b d)+\frac{1}{7} b^4 e^2 x^7","\frac{e (a+b x)^6 (b d-a e)}{3 b^3}+\frac{(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac{e^2 (a+b x)^7}{7 b^3}",1,"a^4*d^2*x + a^3*d*(2*b*d + a*e)*x^2 + (a^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)*x^3)/3 + a*b*(b^2*d^2 + 3*a*b*d*e + a^2*e^2)*x^4 + (b^2*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2)*x^5)/5 + (b^3*e*(b*d + 2*a*e)*x^6)/3 + (b^4*e^2*x^7)/7","B",1
1468,1,84,38,0.0165279,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{1}{30} x \left(15 a^4 (2 d+e x)+20 a^3 b x (3 d+2 e x)+15 a^2 b^2 x^2 (4 d+3 e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right)","\frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2}",1,"(x*(15*a^4*(2*d + e*x) + 20*a^3*b*x*(3*d + 2*e*x) + 15*a^2*b^2*x^2*(4*d + 3*e*x) + 6*a*b^3*x^3*(5*d + 4*e*x) + b^4*x^4*(6*d + 5*e*x)))/30","B",1
1469,1,14,14,0.0015736,"\int \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{(a+b x)^5}{5 b}","\frac{(a+b x)^5}{5 b}",1,"(a + b*x)^5/(5*b)","A",1
1470,1,115,98,0.0443627,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x),x]","\frac{b e x \left(48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left(6 d^2-3 d e x+2 e^2 x^2\right)+b^3 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+12 (b d-a e)^4 \log (d+e x)}{12 e^5}","\frac{(b d-a e)^4 \log (d+e x)}{e^5}-\frac{b x (b d-a e)^3}{e^4}+\frac{(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac{(a+b x)^3 (b d-a e)}{3 e^2}+\frac{(a+b x)^4}{4 e}",1,"(b*e*x*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 12*(b*d - a*e)^4*Log[d + e*x])/(12*e^5)","A",1
1471,1,165,104,0.0581441,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^2,x]","\frac{-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left(-d^2+d e x+e^2 x^2\right)+6 a b^3 e \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)-12 b (d+e x) (b d-a e)^3 \log (d+e x)+b^4 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)}{3 e^5 (d+e x)}","-\frac{2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac{6 b^2 x (b d-a e)^2}{e^4}-\frac{(b d-a e)^4}{e^5 (d+e x)}-\frac{4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac{b^4 (d+e x)^3}{3 e^5}",1,"(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) - 12*b*(b*d - a*e)^3*(d + e*x)*Log[d + e*x])/(3*e^5*(d + e*x))","A",1
1472,1,167,103,0.0566491,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^3,x]","\frac{-a^4 e^4-4 a^3 b e^3 (d+2 e x)+6 a^2 b^2 d e^2 (3 d+4 e x)+4 a b^3 e \left(-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)+12 b^2 (d+e x)^2 (b d-a e)^2 \log (d+e x)+b^4 \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)}{2 e^5 (d+e x)^2}","-\frac{b^3 x (3 b d-4 a e)}{e^4}+\frac{6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}+\frac{4 b (b d-a e)^3}{e^5 (d+e x)}-\frac{(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac{b^4 x^2}{2 e^3}",1,"(-(a^4*e^4) - 4*a^3*b*e^3*(d + 2*e*x) + 6*a^2*b^2*d*e^2*(3*d + 4*e*x) + 4*a*b^3*e*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + b^4*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 12*b^2*(b*d - a*e)^2*(d + e*x)^2*Log[d + e*x])/(2*e^5*(d + e*x)^2)","A",1
1473,1,163,103,0.0741532,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]","-\frac{a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left(d^2+3 d e x+3 e^2 x^2\right)-2 a b^3 d e \left(11 d^2+27 d e x+18 e^2 x^2\right)+12 b^3 (d+e x)^3 (b d-a e) \log (d+e x)+b^4 \left(13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right)}{3 e^5 (d+e x)^3}","-\frac{4 b^3 (b d-a e) \log (d+e x)}{e^5}-\frac{6 b^2 (b d-a e)^2}{e^5 (d+e x)}+\frac{2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac{(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac{b^4 x}{e^4}",1,"-1/3*(a^4*e^4 + 2*a^3*b*e^3*(d + 3*e*x) + 6*a^2*b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) - 2*a*b^3*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + b^4*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4) + 12*b^3*(b*d - a*e)*(d + e*x)^3*Log[d + e*x])/(e^5*(d + e*x)^3)","A",1
1474,1,119,111,0.0621958,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^5,x]","\frac{\frac{(b d-a e) \left(3 a^3 e^3+a^2 b e^2 (7 d+16 e x)+a b^2 e \left(13 d^2+40 d e x+36 e^2 x^2\right)+b^3 \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)\right)}{(d+e x)^4}+12 b^4 \log (d+e x)}{12 e^5}","\frac{4 b^3 (b d-a e)}{e^5 (d+e x)}-\frac{3 b^2 (b d-a e)^2}{e^5 (d+e x)^2}+\frac{4 b (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac{(b d-a e)^4}{4 e^5 (d+e x)^4}+\frac{b^4 \log (d+e x)}{e^5}",1,"(((b*d - a*e)*(3*a^3*e^3 + a^2*b*e^2*(7*d + 16*e*x) + a*b^2*e*(13*d^2 + 40*d*e*x + 36*e^2*x^2) + b^3*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)))/(d + e*x)^4 + 12*b^4*Log[d + e*x])/(12*e^5)","A",1
1475,1,140,28,0.052439,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^6,x]","-\frac{a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+a b^3 e \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+b^4 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)}{5 e^5 (d+e x)^5}","\frac{(a+b x)^5}{5 (d+e x)^5 (b d-a e)}",1,"-1/5*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a*b^3*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(e^5*(d + e*x)^5)","B",1
1476,1,144,58,0.045264,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^7,x]","-\frac{5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+2 a b^3 e \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+b^4 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)}{30 e^5 (d+e x)^6}","\frac{b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac{(a+b x)^5}{6 (d+e x)^6 (b d-a e)}",1,"-1/30*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))/(e^5*(d + e*x)^6)","B",1
1477,1,144,89,0.0521238,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^8,x]","-\frac{15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+3 a b^3 e \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+b^4 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)}{105 e^5 (d+e x)^7}","\frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)}",1,"-1/105*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(e^5*(d + e*x)^7)","A",1
1478,1,144,117,0.0467199,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^9,x]","-\frac{35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left(d^2+8 d e x+28 e^2 x^2\right)+4 a b^3 e \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+b^4 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)}{280 e^5 (d+e x)^8}","\frac{4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac{b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac{4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac{(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac{b^4}{4 e^5 (d+e x)^4}",1,"-1/280*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4))/(e^5*(d + e*x)^8)","A",1
1479,1,144,119,0.051071,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^{10}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^10,x]","-\frac{70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left(d^2+9 d e x+36 e^2 x^2\right)+5 a b^3 e \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+b^4 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)}{630 e^5 (d+e x)^9}","\frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac{b^4}{5 e^5 (d+e x)^5}",1,"-1/630*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4))/(e^5*(d + e*x)^9)","A",1
1480,1,144,119,0.0459145,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^{11}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]","-\frac{126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left(d^2+10 d e x+45 e^2 x^2\right)+6 a b^3 e \left(d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right)+b^4 \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)}{1260 e^5 (d+e x)^{10}}","\frac{4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac{3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac{4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac{(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac{b^4}{6 e^5 (d+e x)^6}",1,"-1/1260*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4))/(e^5*(d + e*x)^10)","A",1
1481,1,771,173,0.1137885,"\int (d+e x)^8 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","a^6 d^8 x+a^5 d^7 x^2 (4 a e+3 b d)+\frac{1}{13} b^4 e^6 x^{13} \left(15 a^2 e^2+48 a b d e+28 b^2 d^2\right)+\frac{1}{3} a^4 d^6 x^3 \left(28 a^2 e^2+48 a b d e+15 b^2 d^2\right)+\frac{1}{3} b^3 e^5 x^{12} \left(5 a^3 e^3+30 a^2 b d e^2+42 a b^2 d^2 e+14 b^3 d^3\right)+a^3 d^5 x^4 \left(14 a^3 e^3+42 a^2 b d e^2+30 a b^2 d^2 e+5 b^3 d^3\right)+\frac{1}{11} b^2 e^4 x^{11} \left(15 a^4 e^4+160 a^3 b d e^3+420 a^2 b^2 d^2 e^2+336 a b^3 d^3 e+70 b^4 d^4\right)+\frac{1}{5} a^2 d^4 x^5 \left(70 a^4 e^4+336 a^3 b d e^3+420 a^2 b^2 d^2 e^2+160 a b^3 d^3 e+15 b^4 d^4\right)+\frac{1}{5} b e^3 x^{10} \left(3 a^5 e^5+60 a^4 b d e^4+280 a^3 b^2 d^2 e^3+420 a^2 b^3 d^3 e^2+210 a b^4 d^4 e+28 b^5 d^5\right)+\frac{1}{3} a d^3 x^6 \left(28 a^5 e^5+210 a^4 b d e^4+420 a^3 b^2 d^2 e^3+280 a^2 b^3 d^3 e^2+60 a b^4 d^4 e+3 b^5 d^5\right)+\frac{1}{9} e^2 x^9 \left(a^6 e^6+48 a^5 b d e^5+420 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+1050 a^2 b^4 d^4 e^2+336 a b^5 d^5 e+28 b^6 d^6\right)+d e x^8 \left(a^6 e^6+21 a^5 b d e^5+105 a^4 b^2 d^2 e^4+175 a^3 b^3 d^3 e^3+105 a^2 b^4 d^4 e^2+21 a b^5 d^5 e+b^6 d^6\right)+\frac{1}{7} d^2 x^7 \left(28 a^6 e^6+336 a^5 b d e^5+1050 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+420 a^2 b^4 d^4 e^2+48 a b^5 d^5 e+b^6 d^6\right)+\frac{1}{7} b^5 e^7 x^{14} (3 a e+4 b d)+\frac{1}{15} b^6 e^8 x^{15}","-\frac{3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac{15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac{5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac{15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac{3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac{(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac{b^6 (d+e x)^{15}}{15 e^7}",1,"a^6*d^8*x + a^5*d^7*(3*b*d + 4*a*e)*x^2 + (a^4*d^6*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2)*x^3)/3 + a^3*d^5*(5*b^3*d^3 + 30*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 14*a^3*e^3)*x^4 + (a^2*d^4*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4)*x^5)/5 + (a*d^3*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5)*x^6)/3 + (d^2*(b^6*d^6 + 48*a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^2*e^4 + 336*a^5*b*d*e^5 + 28*a^6*e^6)*x^7)/7 + d*e*(b^6*d^6 + 21*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 + 175*a^3*b^3*d^3*e^3 + 105*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + a^6*e^6)*x^8 + (e^2*(28*b^6*d^6 + 336*a*b^5*d^5*e + 1050*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 420*a^4*b^2*d^2*e^4 + 48*a^5*b*d*e^5 + a^6*e^6)*x^9)/9 + (b*e^3*(28*b^5*d^5 + 210*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 280*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 + 3*a^5*e^5)*x^10)/5 + (b^2*e^4*(70*b^4*d^4 + 336*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 160*a^3*b*d*e^3 + 15*a^4*e^4)*x^11)/11 + (b^3*e^5*(14*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 5*a^3*e^3)*x^12)/3 + (b^4*e^6*(28*b^2*d^2 + 48*a*b*d*e + 15*a^2*e^2)*x^13)/13 + (b^5*e^7*(4*b*d + 3*a*e)*x^14)/7 + (b^6*e^8*x^15)/15","B",1
1482,1,684,173,0.0973672,"\int (d+e x)^7 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","a^6 d^7 x+\frac{1}{2} a^5 d^6 x^2 (7 a e+6 b d)+\frac{1}{4} b^4 e^5 x^{12} \left(5 a^2 e^2+14 a b d e+7 b^2 d^2\right)+a^4 d^5 x^3 \left(7 a^2 e^2+14 a b d e+5 b^2 d^2\right)+\frac{1}{11} b^3 e^4 x^{11} \left(20 a^3 e^3+105 a^2 b d e^2+126 a b^2 d^2 e+35 b^3 d^3\right)+\frac{1}{4} a^3 d^4 x^4 \left(35 a^3 e^3+126 a^2 b d e^2+105 a b^2 d^2 e+20 b^3 d^3\right)+\frac{1}{2} b^2 e^3 x^{10} \left(3 a^4 e^4+28 a^3 b d e^3+63 a^2 b^2 d^2 e^2+42 a b^3 d^3 e+7 b^4 d^4\right)+a^2 d^3 x^5 \left(7 a^4 e^4+42 a^3 b d e^3+63 a^2 b^2 d^2 e^2+28 a b^3 d^3 e+3 b^4 d^4\right)+\frac{1}{3} b e^2 x^9 \left(2 a^5 e^5+35 a^4 b d e^4+140 a^3 b^2 d^2 e^3+175 a^2 b^3 d^3 e^2+70 a b^4 d^4 e+7 b^5 d^5\right)+\frac{1}{2} a d^2 x^6 \left(7 a^5 e^5+70 a^4 b d e^4+175 a^3 b^2 d^2 e^3+140 a^2 b^3 d^3 e^2+35 a b^4 d^4 e+2 b^5 d^5\right)+\frac{1}{8} e x^8 \left(a^6 e^6+42 a^5 b d e^5+315 a^4 b^2 d^2 e^4+700 a^3 b^3 d^3 e^3+525 a^2 b^4 d^4 e^2+126 a b^5 d^5 e+7 b^6 d^6\right)+\frac{1}{7} d x^7 \left(7 a^6 e^6+126 a^5 b d e^5+525 a^4 b^2 d^2 e^4+700 a^3 b^3 d^3 e^3+315 a^2 b^4 d^4 e^2+42 a b^5 d^5 e+b^6 d^6\right)+\frac{1}{13} b^5 e^6 x^{13} (6 a e+7 b d)+\frac{1}{14} b^6 e^7 x^{14}","-\frac{6 b^5 (d+e x)^{13} (b d-a e)}{13 e^7}+\frac{5 b^4 (d+e x)^{12} (b d-a e)^2}{4 e^7}-\frac{20 b^3 (d+e x)^{11} (b d-a e)^3}{11 e^7}+\frac{3 b^2 (d+e x)^{10} (b d-a e)^4}{2 e^7}-\frac{2 b (d+e x)^9 (b d-a e)^5}{3 e^7}+\frac{(d+e x)^8 (b d-a e)^6}{8 e^7}+\frac{b^6 (d+e x)^{14}}{14 e^7}",1,"a^6*d^7*x + (a^5*d^6*(6*b*d + 7*a*e)*x^2)/2 + a^4*d^5*(5*b^2*d^2 + 14*a*b*d*e + 7*a^2*e^2)*x^3 + (a^3*d^4*(20*b^3*d^3 + 105*a*b^2*d^2*e + 126*a^2*b*d*e^2 + 35*a^3*e^3)*x^4)/4 + a^2*d^3*(3*b^4*d^4 + 28*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 42*a^3*b*d*e^3 + 7*a^4*e^4)*x^5 + (a*d^2*(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 175*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5)*x^6)/2 + (d*(b^6*d^6 + 42*a*b^5*d^5*e + 315*a^2*b^4*d^4*e^2 + 700*a^3*b^3*d^3*e^3 + 525*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 7*a^6*e^6)*x^7)/7 + (e*(7*b^6*d^6 + 126*a*b^5*d^5*e + 525*a^2*b^4*d^4*e^2 + 700*a^3*b^3*d^3*e^3 + 315*a^4*b^2*d^2*e^4 + 42*a^5*b*d*e^5 + a^6*e^6)*x^8)/8 + (b*e^2*(7*b^5*d^5 + 70*a*b^4*d^4*e + 175*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 + 2*a^5*e^5)*x^9)/3 + (b^2*e^3*(7*b^4*d^4 + 42*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 28*a^3*b*d*e^3 + 3*a^4*e^4)*x^10)/2 + (b^3*e^4*(35*b^3*d^3 + 126*a*b^2*d^2*e + 105*a^2*b*d*e^2 + 20*a^3*e^3)*x^11)/11 + (b^4*e^5*(7*b^2*d^2 + 14*a*b*d*e + 5*a^2*e^2)*x^12)/4 + (b^5*e^6*(7*b*d + 6*a*e)*x^13)/13 + (b^6*e^7*x^14)/14","B",1
1483,1,573,171,0.0802121,"\int (d+e x)^6 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","a^6 d^6 x+3 a^5 d^5 x^2 (a e+b d)+\frac{3}{11} b^4 e^4 x^{11} \left(5 a^2 e^2+12 a b d e+5 b^2 d^2\right)+a^4 d^4 x^3 \left(5 a^2 e^2+12 a b d e+5 b^2 d^2\right)+b^3 e^3 x^{10} \left(2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right)+\frac{5}{2} a^3 d^3 x^4 \left(2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right)+\frac{5}{3} b^2 e^2 x^9 \left(a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right)+3 a^2 d^2 x^5 \left(a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right)+\frac{3}{4} b e x^8 \left(a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right)+a d x^6 \left(a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right)+\frac{1}{7} x^7 \left(a^6 e^6+36 a^5 b d e^5+225 a^4 b^2 d^2 e^4+400 a^3 b^3 d^3 e^3+225 a^2 b^4 d^4 e^2+36 a b^5 d^5 e+b^6 d^6\right)+\frac{1}{2} b^5 e^5 x^{12} (a e+b d)+\frac{1}{13} b^6 e^6 x^{13}","\frac{e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac{15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac{2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac{5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac{3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac{(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7}",1,"a^6*d^6*x + 3*a^5*d^5*(b*d + a*e)*x^2 + a^4*d^4*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2)*x^3 + (5*a^3*d^3*(2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^4)/2 + 3*a^2*d^2*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4)*x^5 + a*d*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5)*x^6 + ((b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7)/7 + (3*b*e*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5)*x^8)/4 + (5*b^2*e^2*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4)*x^9)/3 + b^3*e^3*(2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3)*x^10 + (3*b^4*e^4*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2)*x^11)/11 + (b^5*e^5*(b*d + a*e)*x^12)/2 + (b^6*e^6*x^13)/13","B",1
1484,1,501,143,0.0701841,"\int (d+e x)^5 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","a^6 d^5 x+\frac{1}{2} a^5 d^4 x^2 (5 a e+6 b d)+\frac{1}{2} b^4 e^3 x^{10} \left(3 a^2 e^2+6 a b d e+2 b^2 d^2\right)+\frac{5}{3} a^4 d^3 x^3 \left(2 a^2 e^2+6 a b d e+3 b^2 d^2\right)+\frac{5}{9} b^3 e^2 x^9 \left(4 a^3 e^3+15 a^2 b d e^2+12 a b^2 d^2 e+2 b^3 d^3\right)+\frac{5}{4} a^3 d^2 x^4 \left(2 a^3 e^3+12 a^2 b d e^2+15 a b^2 d^2 e+4 b^3 d^3\right)+\frac{5}{8} b^2 e x^8 \left(3 a^4 e^4+20 a^3 b d e^3+30 a^2 b^2 d^2 e^2+12 a b^3 d^3 e+b^4 d^4\right)+a^2 d x^5 \left(a^4 e^4+12 a^3 b d e^3+30 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+3 b^4 d^4\right)+\frac{1}{7} b x^7 \left(6 a^5 e^5+75 a^4 b d e^4+200 a^3 b^2 d^2 e^3+150 a^2 b^3 d^3 e^2+30 a b^4 d^4 e+b^5 d^5\right)+\frac{1}{6} a x^6 \left(a^5 e^5+30 a^4 b d e^4+150 a^3 b^2 d^2 e^3+200 a^2 b^3 d^3 e^2+75 a b^4 d^4 e+6 b^5 d^5\right)+\frac{1}{11} b^5 e^4 x^{11} (6 a e+5 b d)+\frac{1}{12} b^6 e^5 x^{12}","\frac{5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac{e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac{10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac{5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac{(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6}",1,"a^6*d^5*x + (a^5*d^4*(6*b*d + 5*a*e)*x^2)/2 + (5*a^4*d^3*(3*b^2*d^2 + 6*a*b*d*e + 2*a^2*e^2)*x^3)/3 + (5*a^3*d^2*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 + 2*a^3*e^3)*x^4)/4 + a^2*d*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*(6*b^5*d^5 + 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5)*x^6)/6 + (b*(b^5*d^5 + 30*a*b^4*d^4*e + 150*a^2*b^3*d^3*e^2 + 200*a^3*b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5)*x^7)/7 + (5*b^2*e*(b^4*d^4 + 12*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 3*a^4*e^4)*x^8)/8 + (5*b^3*e^2*(2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 4*a^3*e^3)*x^9)/9 + (b^4*e^3*(2*b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^10)/2 + (b^5*e^4*(5*b*d + 6*a*e)*x^11)/11 + (b^6*e^5*x^12)/12","B",1
1485,1,398,119,0.0603592,"\int (d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","a^6 d^4 x+a^5 d^3 x^2 (2 a e+3 b d)+\frac{1}{3} b^4 e^2 x^9 \left(5 a^2 e^2+8 a b d e+2 b^2 d^2\right)+a^4 d^2 x^3 \left(2 a^2 e^2+8 a b d e+5 b^2 d^2\right)+\frac{1}{2} b^3 e x^8 \left(5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right)+a^3 d x^4 \left(a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right)+\frac{1}{7} b^2 x^7 \left(15 a^4 e^4+80 a^3 b d e^3+90 a^2 b^2 d^2 e^2+24 a b^3 d^3 e+b^4 d^4\right)+a b x^6 \left(a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right)+\frac{1}{5} a^2 x^5 \left(a^4 e^4+24 a^3 b d e^3+90 a^2 b^2 d^2 e^2+80 a b^3 d^3 e+15 b^4 d^4\right)+\frac{1}{5} b^5 e^3 x^{10} (3 a e+2 b d)+\frac{1}{11} b^6 e^4 x^{11}","\frac{2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac{2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac{e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac{(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5}",1,"a^6*d^4*x + a^5*d^3*(3*b*d + 2*a*e)*x^2 + a^4*d^2*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^3 + a^3*d*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^4 + (a^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4)*x^5)/5 + a*b*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*x^6 + (b^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4)*x^7)/7 + (b^3*e*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^8)/2 + (b^4*e^2*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^9)/3 + (b^5*e^3*(2*b*d + 3*a*e)*x^10)/5 + (b^6*e^4*x^11)/11","B",1
1486,1,276,92,0.0790734,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{1}{840} x \left(210 a^6 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+252 a^5 b x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)+210 a^4 b^2 x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+120 a^3 b^3 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)+45 a^2 b^4 x^4 \left(56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right)+10 a b^5 x^5 \left(84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right)+b^6 x^6 \left(120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right)\right)","\frac{e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac{3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac{(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4}",1,"(x*(210*a^6*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 252*a^5*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 210*a^4*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 120*a^3*b^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 45*a^2*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 10*a*b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + b^6*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3)))/840","B",1
1487,1,199,65,0.0659964,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{1}{252} x \left(84 a^6 \left(3 d^2+3 d e x+e^2 x^2\right)+126 a^5 b x \left(6 d^2+8 d e x+3 e^2 x^2\right)+126 a^4 b^2 x^2 \left(10 d^2+15 d e x+6 e^2 x^2\right)+84 a^3 b^3 x^3 \left(15 d^2+24 d e x+10 e^2 x^2\right)+36 a^2 b^4 x^4 \left(21 d^2+35 d e x+15 e^2 x^2\right)+9 a b^5 x^5 \left(28 d^2+48 d e x+21 e^2 x^2\right)+b^6 x^6 \left(36 d^2+63 d e x+28 e^2 x^2\right)\right)","\frac{e (a+b x)^8 (b d-a e)}{4 b^3}+\frac{(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac{e^2 (a+b x)^9}{9 b^3}",1,"(x*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2)))/252","B",1
1488,1,122,38,0.0369597,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{1}{56} x \left(28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right)","\frac{(a+b x)^7 (b d-a e)}{7 b^2}+\frac{e (a+b x)^8}{8 b^2}",1,"(x*(28*a^6*(2*d + e*x) + 56*a^5*b*x*(3*d + 2*e*x) + 70*a^4*b^2*x^2*(4*d + 3*e*x) + 56*a^3*b^3*x^3*(5*d + 4*e*x) + 28*a^2*b^4*x^4*(6*d + 5*e*x) + 8*a*b^5*x^5*(7*d + 6*e*x) + b^6*x^6*(8*d + 7*e*x)))/56","B",1
1489,1,14,14,0.0013224,"\int \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{(a+b x)^7}{7 b}","\frac{(a+b x)^7}{7 b}",1,"(a + b*x)^7/(7*b)","A",1
1490,1,230,146,0.0872042,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x),x]","\frac{b e x \left(360 a^5 e^5+450 a^4 b e^4 (e x-2 d)+200 a^3 b^2 e^3 \left(6 d^2-3 d e x+2 e^2 x^2\right)+75 a^2 b^3 e^2 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+6 a b^4 e \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)+b^5 \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)+60 (b d-a e)^6 \log (d+e x)}{60 e^7}","\frac{(b d-a e)^6 \log (d+e x)}{e^7}-\frac{b x (b d-a e)^5}{e^6}+\frac{(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac{(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac{(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac{(a+b x)^5 (b d-a e)}{5 e^2}+\frac{(a+b x)^6}{6 e}",1,"(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200*a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 75*a^2*b^3*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 60*(b*d - a*e)^6*Log[d + e*x])/(60*e^7)","A",1
1491,1,302,156,0.0981834,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^2,x]","\frac{-10 a^6 e^6+60 a^5 b d e^5+150 a^4 b^2 e^4 \left(-d^2+d e x+e^2 x^2\right)+100 a^3 b^3 e^3 \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)+50 a^2 b^4 e^2 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+5 a b^5 e \left(12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right)-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left(-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right)}{10 e^7 (d+e x)}","-\frac{3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac{5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac{10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac{15 b^2 x (b d-a e)^4}{e^6}-\frac{(b d-a e)^6}{e^7 (d+e x)}-\frac{6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac{b^6 (d+e x)^5}{5 e^7}",1,"(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x + e^2*x^2) + 100*a^3*b^3*e^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 50*a^2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b^5*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*d - a*e)^5*(d + e*x)*Log[d + e*x])/(10*e^7*(d + e*x))","A",1
1492,1,303,158,0.1052709,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^3,x]","\frac{-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left(-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)+30 a^2 b^4 e^2 \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)+4 a b^5 e \left(-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right)+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left(22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right)}{4 e^7 (d+e x)^2}","-\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7}",1,"(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3*b^3*e^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*a*b^5*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + b^6*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + 60*b^2*(b*d - a*e)^4*(d + e*x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)","A",1
1493,1,302,156,0.1106804,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]","\frac{-a^6 e^6-3 a^5 b e^5 (d+3 e x)-15 a^4 b^2 e^4 \left(d^2+3 d e x+3 e^2 x^2\right)+10 a^3 b^3 d e^3 \left(11 d^2+27 d e x+18 e^2 x^2\right)+15 a^2 b^4 e^2 \left(-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right)+3 a b^5 e \left(47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right)-60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left(-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right)}{3 e^7 (d+e x)^3}","-\frac{3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac{15 b^4 x (b d-a e)^2}{e^6}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{b^6 (d+e x)^3}{3 e^7}",1,"(-(a^6*e^6) - 3*a^5*b*e^5*(d + 3*e*x) - 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^3*b^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + 3*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-37*d^6 - 51*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 60*b^3*(b*d - a*e)^3*(d + e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)","A",1
1494,1,301,155,0.1177378,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5,x]","-\frac{a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left(d^2+4 d e x+6 e^2 x^2\right)+20 a^3 b^3 e^3 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)-5 a^2 b^4 d e^2 \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+2 a b^5 e \left(77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right)-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)-\left(b^6 \left(57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right)\right)}{4 e^7 (d+e x)^4}","-\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5}",1,"-1/4*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 2*a*b^5*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - b^6*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - 60*b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e*x])/(e^7*(d + e*x)^4)","A",1
1495,1,297,155,0.118527,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]","-\frac{2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left(d^2+5 d e x+10 e^2 x^2\right)+10 a^3 b^3 e^3 \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+30 a^2 b^4 e^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)-a b^5 d e \left(137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right)+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left(87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right)}{10 e^7 (d+e x)^5}","-\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{b^6 x}{e^6}",1,"-1/10*(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + b^6*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*b^5*(b*d - a*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)","A",1
1496,1,233,167,0.1187797,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]","\frac{\frac{(b d-a e) \left(10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left(37 d^2+162 d e x+225 e^2 x^2\right)+a^2 b^3 e^2 \left(57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right)+a b^4 e \left(87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right)+b^5 \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)\right)}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7}","\frac{6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac{15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac{6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac{(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac{b^6 \log (d+e x)}{e^7}",1,"(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)","A",1
1497,1,271,28,0.0955849,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^8,x]","-\frac{a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left(d^2+7 d e x+21 e^2 x^2\right)+a^3 b^3 e^3 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+a^2 b^4 e^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+a b^5 e \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)+b^6 \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)}{7 e^7 (d+e x)^7}","\frac{(a+b x)^7}{7 (d+e x)^7 (b d-a e)}",1,"-1/7*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a^3*b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + a*b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b^6*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)","B",1
1498,1,277,58,0.0892268,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]","-\frac{7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left(d^2+8 d e x+28 e^2 x^2\right)+4 a^3 b^3 e^3 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+3 a^2 b^4 e^2 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+2 a b^5 e \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)+b^6 \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)}{56 e^7 (d+e x)^8}","\frac{b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac{(a+b x)^7}{8 (d+e x)^8 (b d-a e)}",1,"-1/56*(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a^3*b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a^2*b^4*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 2*a*b^5*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + b^6*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)","B",1
1499,1,277,89,0.0941755,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{10}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^10,x]","-\frac{28 a^6 e^6+21 a^5 b e^5 (d+9 e x)+15 a^4 b^2 e^4 \left(d^2+9 d e x+36 e^2 x^2\right)+10 a^3 b^3 e^3 \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+6 a^2 b^4 e^2 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+3 a b^5 e \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)+b^6 \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)}{252 e^7 (d+e x)^9}","\frac{b^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac{b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac{(a+b x)^7}{9 (d+e x)^9 (b d-a e)}",1,"-1/252*(28*a^6*e^6 + 21*a^5*b*e^5*(d + 9*e*x) + 15*a^4*b^2*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 6*a^2*b^4*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 3*a*b^5*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + b^6*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)","B",1
1500,1,277,120,0.0867052,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{11}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]","-\frac{84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left(d^2+10 d e x+45 e^2 x^2\right)+20 a^3 b^3 e^3 \left(d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right)+10 a^2 b^4 e^2 \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)+4 a b^5 e \left(d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right)+b^6 \left(d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right)}{840 e^7 (d+e x)^{10}}","\frac{b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac{b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac{(a+b x)^7}{10 (d+e x)^{10} (b d-a e)}",1,"-1/840*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6))/(e^7*(d + e*x)^10)","B",1
1501,1,277,170,0.0910848,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{12}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12,x]","-\frac{210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left(d^2+11 d e x+55 e^2 x^2\right)+35 a^3 b^3 e^3 \left(d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right)+15 a^2 b^4 e^2 \left(d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right)+5 a b^5 e \left(d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right)+b^6 \left(d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right)}{2310 e^7 (d+e x)^{11}}","\frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac{b^6}{5 e^7 (d+e x)^5}",1,"-1/2310*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6))/(e^7*(d + e*x)^11)","A",1
1502,1,277,173,0.0881466,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{13}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]","-\frac{462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left(d^2+12 d e x+66 e^2 x^2\right)+56 a^3 b^3 e^3 \left(d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right)+21 a^2 b^4 e^2 \left(d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right)+6 a b^5 e \left(d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right)+b^6 \left(d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right)}{5544 e^7 (d+e x)^{12}}","\frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac{b^6}{6 e^7 (d+e x)^6}",1,"-1/5544*(462*a^6*e^6 + 252*a^5*b*e^5*(d + 12*e*x) + 126*a^4*b^2*e^4*(d^2 + 12*d*e*x + 66*e^2*x^2) + 56*a^3*b^3*e^3*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + 21*a^2*b^4*e^2*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4) + 6*a*b^5*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + b^6*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6))/(e^7*(d + e*x)^12)","A",1
1503,1,277,171,0.0951925,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{14}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]","-\frac{924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left(d^2+13 d e x+78 e^2 x^2\right)+84 a^3 b^3 e^3 \left(d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right)+28 a^2 b^4 e^2 \left(d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right)+7 a b^5 e \left(d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right)+b^6 \left(d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right)}{12012 e^7 (d+e x)^{13}}","\frac{3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac{5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac{2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac{15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac{b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac{(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac{b^6}{7 e^7 (d+e x)^7}",1,"-1/12012*(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 210*a^4*b^2*e^4*(d^2 + 13*d*e*x + 78*e^2*x^2) + 84*a^3*b^3*e^3*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^5) + b^6*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^6))/(e^7*(d + e*x)^13)","A",1
1504,1,277,173,0.0947844,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{15}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^15,x]","-\frac{1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left(d^2+14 d e x+91 e^2 x^2\right)+120 a^3 b^3 e^3 \left(d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right)+36 a^2 b^4 e^2 \left(d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right)+8 a b^5 e \left(d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right)+b^6 \left(d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right)}{24024 e^7 (d+e x)^{14}}","\frac{2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac{3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac{20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac{5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac{6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac{(b d-a e)^6}{14 e^7 (d+e x)^{14}}-\frac{b^6}{8 e^7 (d+e x)^8}",1,"-1/24024*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 330*a^4*b^2*e^4*(d^2 + 14*d*e*x + 91*e^2*x^2) + 120*a^3*b^3*e^3*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5) + b^6*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6))/(e^7*(d + e*x)^14)","A",1
1505,1,230,131,0.0752783,"\int \frac{(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{12 a^5 e^5-12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left(4 d^2+6 d e x-e^2 x^2\right)+10 a^2 b^3 e^2 \left(-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right)-5 a b^4 e \left(-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right)+60 e (a+b x) (b d-a e)^4 \log (a+b x)+b^5 \left(-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right)}{12 b^6 (a+b x)}","\frac{5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac{5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e^5 (a+b x)^4}{4 b^6}+\frac{10 e^2 x (b d-a e)^3}{b^5}",1,"(12*a^5*e^5 - 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(4*d^2 + 6*d*e*x - e^2*x^2) + 10*a^2*b^3*e^2*(-12*d^3 - 24*d^2*e*x + 12*d*e^2*x^2 + e^3*x^3) - 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^4*x^4) + b^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e*(b*d - a*e)^4*(a + b*x)*Log[a + b*x])/(12*b^6*(a + b*x))","A",1
1506,1,166,104,0.0643149,"\int \frac{(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{-3 a^4 e^4+3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left(-3 d^2-4 d e x+e^2 x^2\right)-2 a b^3 e \left(-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right)-12 e (a+b x) (a e-b d)^3 \log (a+b x)+b^4 \left(-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right)}{3 b^5 (a+b x)}","\frac{2 e^3 (a+b x)^2 (b d-a e)}{b^5}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5}+\frac{6 e^2 x (b d-a e)^2}{b^4}",1,"(-3*a^4*e^4 + 3*a^3*b*e^3*(4*d + 3*e*x) + 6*a^2*b^2*e^2*(-3*d^2 - 4*d*e*x + e^2*x^2) - 2*a*b^3*e*(-6*d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(-(b*d) + a*e)^3*(a + b*x)*Log[a + b*x])/(3*b^5*(a + b*x))","A",1
1507,1,72,75,0.0515789,"\int \frac{(d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 b e^2 x (3 b d-2 a e)-\frac{2 (b d-a e)^3}{a+b x}+6 e (b d-a e)^2 \log (a+b x)+b^2 e^3 x^2}{2 b^4}","-\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e^2 x (3 b d-2 a e)}{b^3}+\frac{e^3 x^2}{2 b^2}",1,"(2*b*e^2*(3*b*d - 2*a*e)*x + b^2*e^3*x^2 - (2*(b*d - a*e)^3)/(a + b*x) + 6*e*(b*d - a*e)^2*Log[a + b*x])/(2*b^4)","A",1
1508,1,47,51,0.0387042,"\int \frac{(d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{-\frac{(b d-a e)^2}{a+b x}+2 e (b d-a e) \log (a+b x)+b e^2 x}{b^3}","-\frac{(b d-a e)^2}{b^3 (a+b x)}+\frac{2 e (b d-a e) \log (a+b x)}{b^3}+\frac{e^2 x}{b^2}",1,"(b*e^2*x - (b*d - a*e)^2/(a + b*x) + 2*e*(b*d - a*e)*Log[a + b*x])/b^3","A",1
1509,1,31,32,0.0106001,"\int \frac{d+e x}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{a e-b d}{b^2 (a+b x)}+\frac{e \log (a+b x)}{b^2}","\frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)}",1,"(-(b*d) + a*e)/(b^2*(a + b*x)) + (e*Log[a + b*x])/b^2","A",1
1510,1,12,12,0.0028816,"\int \frac{1}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-1),x]","-\frac{1}{b (a+b x)}","-\frac{1}{b (a+b x)}",1,"-(1/(b*(a + b*x)))","A",1
1511,1,53,57,0.0254758,"\int \frac{1}{(d+e x) \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)),x]","\frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{(a+b x) (b d-a e)^2}","-\frac{1}{(a+b x) (b d-a e)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2}",1,"(-(b*d) + a*e - e*(a + b*x)*Log[a + b*x] + e*(a + b*x)*Log[d + e*x])/((b*d - a*e)^2*(a + b*x))","A",1
1512,1,66,81,0.0673633,"\int \frac{1}{(d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)),x]","\frac{\frac{b (a e-b d)}{a+b x}+\frac{e (a e-b d)}{d+e x}-2 b e \log (a+b x)+2 b e \log (d+e x)}{(b d-a e)^3}","-\frac{b}{(a+b x) (b d-a e)^2}-\frac{e}{(d+e x) (b d-a e)^2}-\frac{2 b e \log (a+b x)}{(b d-a e)^3}+\frac{2 b e \log (d+e x)}{(b d-a e)^3}",1,"((b*(-(b*d) + a*e))/(a + b*x) + (e*(-(b*d) + a*e))/(d + e*x) - 2*b*e*Log[a + b*x] + 2*b*e*Log[d + e*x])/(b*d - a*e)^3","A",1
1513,1,97,110,0.0957767,"\int \frac{1}{(d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)),x]","-\frac{\frac{2 b^2 (b d-a e)}{a+b x}+6 b^2 e \log (a+b x)+\frac{4 b e (b d-a e)}{d+e x}+\frac{e (b d-a e)^2}{(d+e x)^2}-6 b^2 e \log (d+e x)}{2 (b d-a e)^4}","-\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}-\frac{2 b e}{(d+e x) (b d-a e)^3}-\frac{e}{2 (d+e x)^2 (b d-a e)^2}",1,"-1/2*((2*b^2*(b*d - a*e))/(a + b*x) + (e*(b*d - a*e)^2)/(d + e*x)^2 + (4*b*e*(b*d - a*e))/(d + e*x) + 6*b^2*e*Log[a + b*x] - 6*b^2*e*Log[d + e*x])/(b*d - a*e)^4","A",1
1514,1,120,133,0.1220578,"\int \frac{1}{(d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]","\frac{-\frac{3 b^3 (b d-a e)}{a+b x}-12 b^3 e \log (a+b x)-\frac{9 b^2 e (b d-a e)}{d+e x}-\frac{3 b e (b d-a e)^2}{(d+e x)^2}+\frac{e (a e-b d)^3}{(d+e x)^3}+12 b^3 e \log (d+e x)}{3 (b d-a e)^5}","-\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2}",1,"((-3*b^3*(b*d - a*e))/(a + b*x) + (e*(-(b*d) + a*e)^3)/(d + e*x)^3 - (3*b*e*(b*d - a*e)^2)/(d + e*x)^2 - (9*b^2*e*(b*d - a*e))/(d + e*x) - 12*b^3*e*Log[a + b*x] + 12*b^3*e*Log[d + e*x])/(3*(b*d - a*e)^5)","A",1
1515,1,301,156,0.114252,"\int \frac{(d+e x)^6}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{-37 a^6 e^6+3 a^5 b e^5 (47 d-17 e x)+3 a^4 b^2 e^4 \left(-65 d^2+81 d e x+13 e^2 x^2\right)+a^3 b^3 e^3 \left(110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right)+3 a^2 b^4 e^2 \left(-5 d^4+90 d^3 e x-45 d^2 e^2 x^2-63 d e^3 x^3+5 e^4 x^4\right)-3 a b^5 e \left(d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right)-60 e^3 (a+b x)^3 (a e-b d)^3 \log (a+b x)+b^6 \left(-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right)}{3 b^7 (a+b x)^3}","\frac{3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac{20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac{15 e^2 (b d-a e)^4}{b^7 (a+b x)}-\frac{3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac{(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac{e^6 (a+b x)^3}{3 b^7}+\frac{15 e^4 x (b d-a e)^2}{b^6}",1,"(-37*a^6*e^6 + 3*a^5*b*e^5*(47*d - 17*e*x) + 3*a^4*b^2*e^4*(-65*d^2 + 81*d*e*x + 13*e^2*x^2) + a^3*b^3*e^3*(110*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 73*e^3*x^3) + 3*a^2*b^4*e^2*(-5*d^4 + 90*d^3*e*x - 45*d^2*e^2*x^2 - 63*d*e^3*x^3 + 5*e^4*x^4) - 3*a*b^5*e*(d^5 + 15*d^4*e*x - 60*d^3*e^2*x^2 - 45*d^2*e^3*x^3 + 15*d*e^4*x^4 + e^5*x^5) + b^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 45*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) - 60*e^3*(-(b*d) + a*e)^3*(a + b*x)^3*Log[a + b*x])/(3*b^7*(a + b*x)^3)","A",1
1516,1,228,129,0.0814677,"\int \frac{(d+e x)^5}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{47 a^5 e^5+a^4 b e^4 (81 e x-130 d)+a^3 b^2 e^3 \left(110 d^2-270 d e x-9 e^2 x^2\right)-a^2 b^3 e^2 \left(20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right)-5 a b^4 e \left(d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right)+60 e^3 (a+b x)^3 (b d-a e)^2 \log (a+b x)+b^5 \left(-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right)}{6 b^6 (a+b x)^3}","\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac{e^4 x (5 b d-4 a e)}{b^5}+\frac{e^5 x^2}{2 b^4}",1,"(47*a^5*e^5 + a^4*b*e^4*(-130*d + 81*e*x) + a^3*b^2*e^3*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*b^3*e^2*(20*d^3 - 270*d^2*e*x + 90*d*e^2*x^2 + 63*e^3*x^3) - 5*a*b^4*e*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 - 18*d*e^3*x^3 + 3*e^4*x^4) + b^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^3*(b*d - a*e)^2*(a + b*x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)","A",1
1517,1,166,103,0.0591297,"\int \frac{(d+e x)^4}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{-13 a^4 e^4+a^3 b e^3 (22 d-27 e x)-3 a^2 b^2 e^2 \left(2 d^2-18 d e x+3 e^2 x^2\right)+a b^3 e \left(-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right)-12 e^3 (a+b x)^3 (a e-b d) \log (a+b x)-\left(b^4 \left(d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right)\right)}{3 b^5 (a+b x)^3}","\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac{e^4 x}{b^4}",1,"(-13*a^4*e^4 + a^3*b*e^3*(22*d - 27*e*x) - 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4) - 12*e^3*(-(b*d) + a*e)*(a + b*x)^3*Log[a + b*x])/(3*b^5*(a + b*x)^3)","A",1
1518,1,80,86,0.0406358,"\int \frac{(d+e x)^3}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{6 e^3 \log (a+b x)-\frac{(b d-a e) \left(11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left(2 d^2+9 d e x+18 e^2 x^2\right)\right)}{(a+b x)^3}}{6 b^4}","-\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4}",1,"(-(((b*d - a*e)*(11*a^2*e^2 + a*b*e*(5*d + 27*e*x) + b^2*(2*d^2 + 9*d*e*x + 18*e^2*x^2)))/(a + b*x)^3) + 6*e^3*Log[a + b*x])/(6*b^4)","A",1
1519,1,53,28,0.0216538,"\int \frac{(d+e x)^2}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","-\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left(d^2+3 d e x+3 e^2 x^2\right)}{3 b^3 (a+b x)^3}","-\frac{(d+e x)^3}{3 (a+b x)^3 (b d-a e)}",1,"-1/3*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(b^3*(a + b*x)^3)","A",1
1520,1,27,38,0.0083038,"\int \frac{d+e x}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","-\frac{a e+2 b d+3 b e x}{6 b^2 (a+b x)^3}","-\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2}",1,"-1/6*(2*b*d + a*e + 3*b*e*x)/(b^2*(a + b*x)^3)","A",1
1521,1,14,14,0.0029062,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-2),x]","-\frac{1}{3 b (a+b x)^3}","-\frac{1}{3 b (a+b x)^3}",1,"-1/3*1/(b*(a + b*x)^3)","A",1
1522,1,107,107,0.0437639,"\int \frac{1}{(d+e x) \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}-\frac{e^2}{(a+b x) (b d-a e)^3}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}+\frac{1}{3 (a+b x)^3 (a e-b d)}","-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}-\frac{e^2}{(a+b x) (b d-a e)^3}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)}",1,"1/(3*(-(b*d) + a*e)*(a + b*x)^3) + e/(2*(b*d - a*e)^2*(a + b*x)^2) - e^2/((b*d - a*e)^3*(a + b*x)) - (e^3*Log[a + b*x])/(b*d - a*e)^4 + (e^3*Log[d + e*x])/(b*d - a*e)^4","A",1
1523,1,121,132,0.1187638,"\int \frac{1}{(d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","\frac{\frac{3 e^3 (a e-b d)}{d+e x}-\frac{9 b e^2 (b d-a e)}{a+b x}+\frac{3 b e (b d-a e)^2}{(a+b x)^2}-\frac{b (b d-a e)^3}{(a+b x)^3}-12 b e^3 \log (a+b x)+12 b e^3 \log (d+e x)}{3 (b d-a e)^5}","-\frac{e^3}{(d+e x) (b d-a e)^4}-\frac{4 b e^3 \log (a+b x)}{(b d-a e)^5}+\frac{4 b e^3 \log (d+e x)}{(b d-a e)^5}-\frac{3 b e^2}{(a+b x) (b d-a e)^4}+\frac{b e}{(a+b x)^2 (b d-a e)^3}-\frac{b}{3 (a+b x)^3 (b d-a e)^2}",1,"(-((b*(b*d - a*e)^3)/(a + b*x)^3) + (3*b*e*(b*d - a*e)^2)/(a + b*x)^2 - (9*b*e^2*(b*d - a*e))/(a + b*x) + (3*e^3*(-(b*d) + a*e))/(d + e*x) - 12*b*e^3*Log[a + b*x] + 12*b*e^3*Log[d + e*x])/(3*(b*d - a*e)^5)","A",1
1524,1,154,170,0.2097029,"\int \frac{1}{(d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","-\frac{\frac{36 b^2 e^2 (b d-a e)}{a+b x}-\frac{9 b^2 e (b d-a e)^2}{(a+b x)^2}+\frac{2 b^2 (b d-a e)^3}{(a+b x)^3}+60 b^2 e^3 \log (a+b x)+\frac{24 b e^3 (b d-a e)}{d+e x}+\frac{3 e^3 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^3 \log (d+e x)}{6 (b d-a e)^6}","-\frac{10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac{10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}-\frac{6 b^2 e^2}{(a+b x) (b d-a e)^5}+\frac{3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac{b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac{4 b e^3}{(d+e x) (b d-a e)^5}-\frac{e^3}{2 (d+e x)^2 (b d-a e)^4}",1,"-1/6*((2*b^2*(b*d - a*e)^3)/(a + b*x)^3 - (9*b^2*e*(b*d - a*e)^2)/(a + b*x)^2 + (36*b^2*e^2*(b*d - a*e))/(a + b*x) + (3*e^3*(b*d - a*e)^2)/(d + e*x)^2 + (24*b*e^3*(b*d - a*e))/(d + e*x) + 60*b^2*e^3*Log[a + b*x] - 60*b^2*e^3*Log[d + e*x])/(b*d - a*e)^6","A",1
1525,1,195,208,0.161665,"\int \frac{(d+e x)^8}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{15 b e^6 x \left(21 a^2 e^2-48 a b d e+28 b^2 d^2\right)+15 b^2 e^7 x^2 (4 b d-3 a e)+840 e^5 (b d-a e)^3 \log (a+b x)-\frac{1050 e^4 (b d-a e)^4}{a+b x}+\frac{420 e^3 (a e-b d)^5}{(a+b x)^2}-\frac{140 e^2 (b d-a e)^6}{(a+b x)^3}+\frac{30 e (a e-b d)^7}{(a+b x)^4}-\frac{3 (b d-a e)^8}{(a+b x)^5}+5 b^3 e^8 x^3}{15 b^9}","\frac{4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac{e^8 (a+b x)^3}{3 b^9}+\frac{28 e^6 x (b d-a e)^2}{b^8}",1,"(15*b*e^6*(28*b^2*d^2 - 48*a*b*d*e + 21*a^2*e^2)*x + 15*b^2*e^7*(4*b*d - 3*a*e)*x^2 + 5*b^3*e^8*x^3 - (3*(b*d - a*e)^8)/(a + b*x)^5 + (30*e*(-(b*d) + a*e)^7)/(a + b*x)^4 - (140*e^2*(b*d - a*e)^6)/(a + b*x)^3 + (420*e^3*(-(b*d) + a*e)^5)/(a + b*x)^2 - (1050*e^4*(b*d - a*e)^4)/(a + b*x) + 840*e^5*(b*d - a*e)^3*Log[a + b*x])/(15*b^9)","A",1
1526,1,389,181,0.1457529,"\int \frac{(d+e x)^7}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^7/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{459 a^7 e^7+3 a^6 b e^6 (625 e x-406 d)+a^5 b^2 e^5 \left(959 d^2-5250 d e x+2700 e^2 x^2\right)+5 a^4 b^3 e^4 \left(-28 d^3+875 d^2 e x-1680 d e^2 x^2+260 e^3 x^3\right)-5 a^3 b^4 e^3 \left(7 d^4+140 d^3 e x-1540 d^2 e^2 x^2+1120 d e^3 x^3+80 e^4 x^4\right)-a^2 b^5 e^2 \left(14 d^5+175 d^4 e x+1400 d^3 e^2 x^2-6300 d^2 e^3 x^3+700 d e^4 x^4+500 e^5 x^5\right)-7 a b^6 e \left(d^6+10 d^5 e x+50 d^4 e^2 x^2+200 d^3 e^3 x^3-300 d^2 e^4 x^4-100 d e^5 x^5+10 e^6 x^6\right)+420 e^5 (a+b x)^5 (b d-a e)^2 \log (a+b x)-\left(b^7 \left(4 d^7+35 d^6 e x+140 d^5 e^2 x^2+350 d^4 e^3 x^3+700 d^3 e^4 x^4-140 d e^6 x^6-10 e^7 x^7\right)\right)}{20 b^8 (a+b x)^5}","\frac{21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}-\frac{35 e^4 (b d-a e)^3}{b^8 (a+b x)}-\frac{35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac{7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}-\frac{7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac{(b d-a e)^7}{5 b^8 (a+b x)^5}+\frac{e^6 x (7 b d-6 a e)}{b^7}+\frac{e^7 x^2}{2 b^6}",1,"(459*a^7*e^7 + 3*a^6*b*e^6*(-406*d + 625*e*x) + a^5*b^2*e^5*(959*d^2 - 5250*d*e*x + 2700*e^2*x^2) + 5*a^4*b^3*e^4*(-28*d^3 + 875*d^2*e*x - 1680*d*e^2*x^2 + 260*e^3*x^3) - 5*a^3*b^4*e^3*(7*d^4 + 140*d^3*e*x - 1540*d^2*e^2*x^2 + 1120*d*e^3*x^3 + 80*e^4*x^4) - a^2*b^5*e^2*(14*d^5 + 175*d^4*e*x + 1400*d^3*e^2*x^2 - 6300*d^2*e^3*x^3 + 700*d*e^4*x^4 + 500*e^5*x^5) - 7*a*b^6*e*(d^6 + 10*d^5*e*x + 50*d^4*e^2*x^2 + 200*d^3*e^3*x^3 - 300*d^2*e^4*x^4 - 100*d*e^5*x^5 + 10*e^6*x^6) - b^7*(4*d^7 + 35*d^6*e*x + 140*d^5*e^2*x^2 + 350*d^4*e^3*x^3 + 700*d^3*e^4*x^4 - 140*d*e^6*x^6 - 10*e^7*x^7) + 420*e^5*(b*d - a*e)^2*(a + b*x)^5*Log[a + b*x])/(20*b^8*(a + b*x)^5)","B",1
1527,1,300,155,0.1538275,"\int \frac{(d+e x)^6}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","-\frac{87 a^6 e^6+a^5 b e^5 (375 e x-137 d)+5 a^4 b^2 e^4 \left(6 d^2-125 d e x+120 e^2 x^2\right)+10 a^3 b^3 e^3 \left(d^3+15 d^2 e x-110 d e^2 x^2+40 e^3 x^3\right)+5 a^2 b^4 e^2 \left(d^4+10 d^3 e x+60 d^2 e^2 x^2-180 d e^3 x^3+10 e^4 x^4\right)+a b^5 e \left(3 d^5+25 d^4 e x+100 d^3 e^2 x^2+300 d^2 e^3 x^3-300 d e^4 x^4-50 e^5 x^5\right)+60 e^5 (a+b x)^5 (a e-b d) \log (a+b x)+b^6 \left(2 d^6+15 d^5 e x+50 d^4 e^2 x^2+100 d^3 e^3 x^3+150 d^2 e^4 x^4-10 e^6 x^6\right)}{10 b^7 (a+b x)^5}","\frac{6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac{15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac{10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac{5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac{3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac{(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac{e^6 x}{b^6}",1,"-1/10*(87*a^6*e^6 + a^5*b*e^5*(-137*d + 375*e*x) + 5*a^4*b^2*e^4*(6*d^2 - 125*d*e*x + 120*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 15*d^2*e*x - 110*d*e^2*x^2 + 40*e^3*x^3) + 5*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 60*d^2*e^2*x^2 - 180*d*e^3*x^3 + 10*e^4*x^4) + a*b^5*e*(3*d^5 + 25*d^4*e*x + 100*d^3*e^2*x^2 + 300*d^2*e^3*x^3 - 300*d*e^4*x^4 - 50*e^5*x^5) + b^6*(2*d^6 + 15*d^5*e*x + 50*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 150*d^2*e^4*x^4 - 10*e^6*x^6) + 60*e^5*(-(b*d) + a*e)*(a + b*x)^5*Log[a + b*x])/(b^7*(a + b*x)^5)","A",1
1528,1,171,138,0.0860086,"\int \frac{(d+e x)^5}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{e^5 \log (a+b x)}{b^6}-\frac{(b d-a e) \left(137 a^4 e^4+a^3 b e^3 (77 d+625 e x)+a^2 b^2 e^2 \left(47 d^2+325 d e x+1100 e^2 x^2\right)+a b^3 e \left(27 d^3+175 d^2 e x+500 d e^2 x^2+900 e^3 x^3\right)+b^4 \left(12 d^4+75 d^3 e x+200 d^2 e^2 x^2+300 d e^3 x^3+300 e^4 x^4\right)\right)}{60 b^6 (a+b x)^5}","-\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6}",1,"-1/60*((b*d - a*e)*(137*a^4*e^4 + a^3*b*e^3*(77*d + 625*e*x) + a^2*b^2*e^2*(47*d^2 + 325*d*e*x + 1100*e^2*x^2) + a*b^3*e*(27*d^3 + 175*d^2*e*x + 500*d*e^2*x^2 + 900*e^3*x^3) + b^4*(12*d^4 + 75*d^3*e*x + 200*d^2*e^2*x^2 + 300*d*e^3*x^3 + 300*e^4*x^4)))/(b^6*(a + b*x)^5) + (e^5*Log[a + b*x])/b^6","A",1
1529,1,140,28,0.0457105,"\int \frac{(d+e x)^4}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","-\frac{a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+a b^3 e \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+b^4 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)}{5 b^5 (a+b x)^5}","-\frac{(d+e x)^5}{5 (a+b x)^5 (b d-a e)}",1,"-1/5*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a*b^3*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(b^5*(a + b*x)^5)","B",1
1530,1,97,58,0.034515,"\int \frac{(d+e x)^3}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","-\frac{a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left(3 d^2+10 d e x+10 e^2 x^2\right)+b^3 \left(4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right)}{20 b^4 (a+b x)^5}","\frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)}",1,"-1/20*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(3*d^2 + 10*d*e*x + 10*e^2*x^2) + b^3*(4*d^3 + 15*d^2*e*x + 20*d*e^2*x^2 + 10*e^3*x^3))/(b^4*(a + b*x)^5)","A",1
1531,1,57,65,0.0237082,"\int \frac{(d+e x)^2}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","-\frac{a^2 e^2+a b e (3 d+5 e x)+b^2 \left(6 d^2+15 d e x+10 e^2 x^2\right)}{30 b^3 (a+b x)^5}","-\frac{e (b d-a e)}{2 b^3 (a+b x)^4}-\frac{(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac{e^2}{3 b^3 (a+b x)^3}",1,"-1/30*(a^2*e^2 + a*b*e*(3*d + 5*e*x) + b^2*(6*d^2 + 15*d*e*x + 10*e^2*x^2))/(b^3*(a + b*x)^5)","A",1
1532,1,27,38,0.0093641,"\int \frac{d+e x}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","-\frac{a e+4 b d+5 b e x}{20 b^2 (a+b x)^5}","-\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4}",1,"-1/20*(4*b*d + a*e + 5*b*e*x)/(b^2*(a + b*x)^5)","A",1
1533,1,14,14,0.0033143,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3),x]","-\frac{1}{5 b (a+b x)^5}","-\frac{1}{5 b (a+b x)^5}",1,"-1/5*1/(b*(a + b*x)^5)","A",1
1534,1,152,155,0.0726217,"\int \frac{1}{(d+e x) \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","\frac{60 e^5 (a+b x)^5 \log (d+e x)+60 e^4 (a+b x)^4 (a e-b d)+30 e^3 (a+b x)^3 (b d-a e)^2+20 e^2 (a+b x)^2 (a e-b d)^3+15 e (a+b x) (b d-a e)^4-12 (b d-a e)^5-60 e^5 (a+b x)^5 \log (a+b x)}{60 (a+b x)^5 (b d-a e)^6}","-\frac{e^5 \log (a+b x)}{(b d-a e)^6}+\frac{e^5 \log (d+e x)}{(b d-a e)^6}-\frac{e^4}{(a+b x) (b d-a e)^5}+\frac{e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac{e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac{e}{4 (a+b x)^4 (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (b d-a e)}",1,"(-12*(b*d - a*e)^5 + 15*e*(b*d - a*e)^4*(a + b*x) + 20*e^2*(-(b*d) + a*e)^3*(a + b*x)^2 + 30*e^3*(b*d - a*e)^2*(a + b*x)^3 + 60*e^4*(-(b*d) + a*e)*(a + b*x)^4 - 60*e^5*(a + b*x)^5*Log[a + b*x] + 60*e^5*(a + b*x)^5*Log[d + e*x])/(60*(b*d - a*e)^6*(a + b*x)^5)","A",1
1535,1,167,181,0.1271322,"\int \frac{1}{(d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","\frac{\frac{10 e^5 (a e-b d)}{d+e x}-\frac{50 b e^4 (b d-a e)}{a+b x}+\frac{20 b e^3 (b d-a e)^2}{(a+b x)^2}-\frac{10 b e^2 (b d-a e)^3}{(a+b x)^3}+\frac{5 b e (b d-a e)^4}{(a+b x)^4}-\frac{2 b (b d-a e)^5}{(a+b x)^5}-60 b e^5 \log (a+b x)+60 b e^5 \log (d+e x)}{10 (b d-a e)^7}","-\frac{e^5}{(d+e x) (b d-a e)^6}-\frac{6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac{6 b e^5 \log (d+e x)}{(b d-a e)^7}-\frac{5 b e^4}{(a+b x) (b d-a e)^6}+\frac{2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac{b e^2}{(a+b x)^3 (b d-a e)^4}+\frac{b e}{2 (a+b x)^4 (b d-a e)^3}-\frac{b}{5 (a+b x)^5 (b d-a e)^2}",1,"((-2*b*(b*d - a*e)^5)/(a + b*x)^5 + (5*b*e*(b*d - a*e)^4)/(a + b*x)^4 - (10*b*e^2*(b*d - a*e)^3)/(a + b*x)^3 + (20*b*e^3*(b*d - a*e)^2)/(a + b*x)^2 - (50*b*e^4*(b*d - a*e))/(a + b*x) + (10*e^5*(-(b*d) + a*e))/(d + e*x) - 60*b*e^5*Log[a + b*x] + 60*b*e^5*Log[d + e*x])/(10*(b*d - a*e)^7)","A",1
1536,1,204,220,0.1606259,"\int \frac{1}{(d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","-\frac{\frac{300 b^2 e^4 (b d-a e)}{a+b x}-\frac{100 b^2 e^3 (b d-a e)^2}{(a+b x)^2}+\frac{40 b^2 e^2 (b d-a e)^3}{(a+b x)^3}-\frac{15 b^2 e (b d-a e)^4}{(a+b x)^4}+\frac{4 b^2 (b d-a e)^5}{(a+b x)^5}+420 b^2 e^5 \log (a+b x)+\frac{120 b e^5 (b d-a e)}{d+e x}+\frac{10 e^5 (b d-a e)^2}{(d+e x)^2}-420 b^2 e^5 \log (d+e x)}{20 (b d-a e)^8}","-\frac{21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac{21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}-\frac{15 b^2 e^4}{(a+b x) (b d-a e)^7}+\frac{5 b^2 e^3}{(a+b x)^2 (b d-a e)^6}-\frac{2 b^2 e^2}{(a+b x)^3 (b d-a e)^5}+\frac{3 b^2 e}{4 (a+b x)^4 (b d-a e)^4}-\frac{b^2}{5 (a+b x)^5 (b d-a e)^3}-\frac{6 b e^5}{(d+e x) (b d-a e)^7}-\frac{e^5}{2 (d+e x)^2 (b d-a e)^6}",1,"-1/20*((4*b^2*(b*d - a*e)^5)/(a + b*x)^5 - (15*b^2*e*(b*d - a*e)^4)/(a + b*x)^4 + (40*b^2*e^2*(b*d - a*e)^3)/(a + b*x)^3 - (100*b^2*e^3*(b*d - a*e)^2)/(a + b*x)^2 + (300*b^2*e^4*(b*d - a*e))/(a + b*x) + (10*e^5*(b*d - a*e)^2)/(d + e*x)^2 + (120*b*e^5*(b*d - a*e))/(d + e*x) + 420*b^2*e^5*Log[a + b*x] - 420*b^2*e^5*Log[d + e*x])/(b*d - a*e)^8","A",1
1537,1,81,31,0.0159163,"\int (d+e x) \left(9+12 x+4 x^2\right)^3 \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^3,x]","\frac{64}{7} x^7 (d+9 e)+24 x^6 (4 d+15 e)+432 x^5 (d+2 e)+135 x^4 (8 d+9 e)+324 x^3 (5 d+3 e)+\frac{729}{2} x^2 (4 d+e)+729 d x+8 e x^8","\frac{1}{28} (2 x+3)^7 (2 d-3 e)+\frac{1}{32} e (2 x+3)^8",1,"729*d*x + (729*(4*d + e)*x^2)/2 + 324*(5*d + 3*e)*x^3 + 135*(8*d + 9*e)*x^4 + 432*(d + 2*e)*x^5 + 24*(4*d + 15*e)*x^6 + (64*(d + 9*e)*x^7)/7 + 8*e*x^8","B",1
1538,1,59,31,0.0093727,"\int (d+e x) \left(9+12 x+4 x^2\right)^2 \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]","\frac{16}{5} x^5 (d+6 e)+6 x^4 (4 d+9 e)+72 x^3 (d+e)+\frac{27}{2} x^2 (8 d+3 e)+81 d x+\frac{8 e x^6}{3}","\frac{1}{20} (2 x+3)^5 (2 d-3 e)+\frac{1}{24} e (2 x+3)^6",1,"81*d*x + (27*(8*d + 3*e)*x^2)/2 + 72*(d + e)*x^3 + 6*(4*d + 9*e)*x^4 + (16*(d + 6*e)*x^5)/5 + (8*e*x^6)/3","A",1
1539,1,36,31,0.0066216,"\int (d+e x) \left(9+12 x+4 x^2\right) \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2),x]","\frac{4}{3} x^3 (d+3 e)+\frac{3}{2} x^2 (4 d+3 e)+9 d x+e x^4","\frac{1}{12} (2 x+3)^3 (2 d-3 e)+\frac{1}{16} e (2 x+3)^4",1,"9*d*x + (3*(4*d + 3*e)*x^2)/2 + (4*(d + 3*e)*x^3)/3 + e*x^4","A",1
1540,1,30,30,0.0080133,"\int \frac{d+e x}{9+12 x+4 x^2} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2),x]","\frac{3 e-2 d}{4 (2 x+3)}+\frac{1}{4} e \log (2 x+3)","\frac{1}{4} e \log (2 x+3)-\frac{2 d-3 e}{4 (2 x+3)}",1,"(-2*d + 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4","A",1
1541,1,22,31,0.0077151,"\int \frac{d+e x}{\left(9+12 x+4 x^2\right)^2} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^2,x]","-\frac{4 d+6 e x+3 e}{24 (2 x+3)^3}","-\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2}",1,"-1/24*(4*d + 3*e + 6*e*x)/(3 + 2*x)^3","A",1
1542,1,22,31,0.0074364,"\int \frac{d+e x}{\left(9+12 x+4 x^2\right)^3} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^3,x]","-\frac{8 d+e (10 x+3)}{80 (2 x+3)^5}","-\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4}",1,"-1/80*(8*d + e*(3 + 10*x))/(3 + 2*x)^5","A",1
1543,1,111,92,0.0386867,"\int (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} \left(6 a \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+b x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)\right)}{30 (a+b x)}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)}{5 e^2 (a+b x)}",1,"(x*Sqrt[(a + b*x)^2]*(6*a*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)))/(30*(a + b*x))","A",1
1544,1,89,92,0.0287697,"\int (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} \left(5 a \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+b x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)\right)}{20 (a+b x)}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)}",1,"(x*Sqrt[(a + b*x)^2]*(5*a*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)))/(20*(a + b*x))","A",1
1545,1,67,92,0.0213849,"\int (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} \left(4 a \left(3 d^2+3 d e x+e^2 x^2\right)+b x \left(6 d^2+8 d e x+3 e^2 x^2\right)\right)}{12 (a+b x)}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)}",1,"(x*Sqrt[(a + b*x)^2]*(4*a*(3*d^2 + 3*d*e*x + e^2*x^2) + b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/(12*(a + b*x))","A",1
1546,1,45,69,0.0157537,"\int (d+e x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} (3 a (2 d+e x)+b x (3 d+2 e x))}{6 (a+b x)}","\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 b^2}+\frac{e \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{3 b^2}",1,"(x*Sqrt[(a + b*x)^2]*(3*a*(2*d + e*x) + b*x*(3*d + 2*e*x)))/(6*(a + b*x))","A",1
1547,1,30,32,0.0085752,"\int \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{x \sqrt{(a+b x)^2} (2 a+b x)}{2 (a+b x)}","\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b}",1,"(x*Sqrt[(a + b*x)^2]*(2*a + b*x))/(2*(a + b*x))","A",1
1548,1,42,80,0.0156717,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x),x]","\frac{\sqrt{(a+b x)^2} ((a e-b d) \log (d+e x)+b e x)}{e^2 (a+b x)}","\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(b*e*x + (-(b*d) + a*e)*Log[d + e*x]))/(e^2*(a + b*x))","A",1
1549,1,50,85,0.0192191,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^2,x]","\frac{\sqrt{(a+b x)^2} (-a e+b (d+e x) \log (d+e x)+b d)}{e^2 (a+b x) (d+e x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(b*d - a*e + b*(d + e*x)*Log[d + e*x]))/(e^2*(a + b*x)*(d + e*x))","A",1
1550,1,44,46,0.0163612,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^3,x]","-\frac{\sqrt{(a+b x)^2} (a e+b (d+2 e x))}{2 e^2 (a+b x) (d+e x)^2}","\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)}",1,"-1/2*(Sqrt[(a + b*x)^2]*(a*e + b*(d + 2*e*x)))/(e^2*(a + b*x)*(d + e*x)^2)","A",1
1551,1,45,92,0.0175499,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^4,x]","-\frac{\sqrt{(a+b x)^2} (2 a e+b (d+3 e x))}{6 e^2 (a+b x) (d+e x)^3}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^3}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x) (d+e x)^2}",1,"-1/6*(Sqrt[(a + b*x)^2]*(2*a*e + b*(d + 3*e*x)))/(e^2*(a + b*x)*(d + e*x)^3)","A",1
1552,1,45,92,0.0171446,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^5,x]","-\frac{\sqrt{(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3}",1,"-1/12*(Sqrt[(a + b*x)^2]*(3*a*e + b*(d + 4*e*x)))/(e^2*(a + b*x)*(d + e*x)^4)","A",1
1553,1,45,92,0.0195278,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^6} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^6,x]","-\frac{\sqrt{(a+b x)^2} (4 a e+b (d+5 e x))}{20 e^2 (a+b x) (d+e x)^5}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}",1,"-1/20*(Sqrt[(a + b*x)^2]*(4*a*e + b*(d + 5*e*x)))/(e^2*(a + b*x)*(d + e*x)^5)","A",1
1554,1,259,200,0.0877253,"\int (d+e x)^5 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x \sqrt{(a+b x)^2} \left(84 a^3 \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+36 a^2 b x \left(21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right)+9 a b^2 x^2 \left(56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right)+b^3 x^3 \left(126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right)\right)}{504 (a+b x)}","-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)}{8 e^4 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2}{7 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3}{6 e^4 (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^4 (a+b x)}",1,"(x*Sqrt[(a + b*x)^2]*(84*a^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 36*a^2*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*b^2*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/(504*(a + b*x))","A",1
1555,1,215,200,0.0708737,"\int (d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x \sqrt{(a+b x)^2} \left(56 a^3 \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+28 a^2 b x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)+8 a b^2 x^2 \left(35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right)+b^3 x^3 \left(70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right)\right)}{280 (a+b x)}","-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)}",1,"(x*Sqrt[(a + b*x)^2]*(56*a^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 28*a^2*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 8*a*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)))/(280*(a + b*x))","A",1
1556,1,171,172,0.0617206,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x \sqrt{(a+b x)^2} \left(35 a^3 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+21 a^2 b x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)+7 a b^2 x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+b^3 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)\right)}{140 (a+b x)}","\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{2 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^3}{4 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}",1,"(x*Sqrt[(a + b*x)^2]*(35*a^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 21*a^2*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 7*a*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + b^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)))/(140*(a + b*x))","A",1
1557,1,127,114,0.0449258,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x \sqrt{(a+b x)^2} \left(20 a^3 \left(3 d^2+3 d e x+e^2 x^2\right)+15 a^2 b x \left(6 d^2+8 d e x+3 e^2 x^2\right)+6 a b^2 x^2 \left(10 d^2+15 d e x+6 e^2 x^2\right)+b^3 x^3 \left(15 d^2+24 d e x+10 e^2 x^2\right)\right)}{60 (a+b x)}","\frac{2 e \left(a^2+2 a b x+b^2 x^2\right)^{5/2} (b d-a e)}{5 b^3}+\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2} (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{6 b^3}",1,"(x*Sqrt[(a + b*x)^2]*(20*a^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 6*a*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2)))/(60*(a + b*x))","A",1
1558,1,83,69,0.034011,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{x \sqrt{(a+b x)^2} \left(10 a^3 (2 d+e x)+10 a^2 b x (3 d+2 e x)+5 a b^2 x^2 (4 d+3 e x)+b^3 x^3 (5 d+4 e x)\right)}{20 (a+b x)}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2} (b d-a e)}{4 b^2}+\frac{e \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{5 b^2}",1,"(x*Sqrt[(a + b*x)^2]*(10*a^3*(2*d + e*x) + 10*a^2*b*x*(3*d + 2*e*x) + 5*a*b^2*x^2*(4*d + 3*e*x) + b^3*x^3*(5*d + 4*e*x)))/(20*(a + b*x))","A",1
1559,1,23,32,0.008887,"\int \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{(a+b x) \left((a+b x)^2\right)^{3/2}}{4 b}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{4 b}",1,"((a + b*x)*((a + b*x)^2)^(3/2))/(4*b)","A",1
1560,1,92,166,0.0496819,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x),x]","\frac{\sqrt{(a+b x)^2} \left(b e x \left(18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-6 (b d-a e)^3 \log (d+e x)\right)}{6 e^4 (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}+\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e}",1,"(Sqrt[(a + b*x)^2]*(b*e*x*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(b*d - a*e)^3*Log[d + e*x]))/(6*e^4*(a + b*x))","A",1
1561,1,132,183,0.0854581,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^2,x]","\frac{\sqrt{(a+b x)^2} \left(-2 a^3 e^3+6 a^2 b d e^2+6 a b^2 e \left(-d^2+d e x+e^2 x^2\right)+6 b (d+e x) (b d-a e)^2 \log (d+e x)+b^3 \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)\right)}{2 e^4 (a+b x) (d+e x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}-\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(6*a^2*b*d*e^2 - 2*a^3*e^3 + 6*a*b^2*e*(-d^2 + d*e*x + e^2*x^2) + b^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 6*b*(b*d - a*e)^2*(d + e*x)*Log[d + e*x]))/(2*e^4*(a + b*x)*(d + e*x))","A",1
1562,1,131,186,0.0644466,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^3,x]","-\frac{\sqrt{(a+b x)^2} \left(a^3 e^3+3 a^2 b e^2 (d+2 e x)-3 a b^2 d e (3 d+4 e x)+6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left(5 d^3+4 d^2 e x-4 d e^2 x^2-2 e^3 x^3\right)\right)}{2 e^4 (a+b x) (d+e x)^2}","-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}",1,"-1/2*(Sqrt[(a + b*x)^2]*(a^3*e^3 + 3*a^2*b*e^2*(d + 2*e*x) - 3*a*b^2*d*e*(3*d + 4*e*x) + b^3*(5*d^3 + 4*d^2*e*x - 4*d*e^2*x^2 - 2*e^3*x^3) + 6*b^2*(b*d - a*e)*(d + e*x)^2*Log[d + e*x]))/(e^4*(a + b*x)*(d + e*x)^2)","A",1
1563,1,104,194,0.058687,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^4,x]","\frac{\sqrt{(a+b x)^2} \left((b d-a e) \left(2 a^2 e^2+a b e (5 d+9 e x)+b^2 \left(11 d^2+27 d e x+18 e^2 x^2\right)\right)+6 b^3 (d+e x)^3 \log (d+e x)\right)}{6 e^4 (a+b x) (d+e x)^3}","\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) (d+e x)}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^3}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*((b*d - a*e)*(2*a^2*e^2 + a*b*e*(5*d + 9*e*x) + b^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + 6*b^3*(d + e*x)^3*Log[d + e*x]))/(6*e^4*(a + b*x)*(d + e*x)^3)","A",1
1564,1,109,48,0.0411329,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^5,x]","-\frac{\sqrt{(a+b x)^2} \left(a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left(d^2+4 d e x+6 e^2 x^2\right)+b^3 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)\right)}{4 e^4 (a+b x) (d+e x)^4}","\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)}",1,"-1/4*(Sqrt[(a + b*x)^2]*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)))/(e^4*(a + b*x)*(d + e*x)^4)","B",1
1565,1,112,98,0.0463578,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^6,x]","-\frac{\sqrt{(a+b x)^2} \left(4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left(d^2+5 d e x+10 e^2 x^2\right)+b^3 \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)\right)}{20 e^4 (a+b x) (d+e x)^5}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)}",1,"-1/20*(Sqrt[(a + b*x)^2]*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))/(e^4*(a + b*x)*(d + e*x)^5)","A",1
1566,1,112,143,0.0451477,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^7,x]","-\frac{\sqrt{(a+b x)^2} \left(10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left(d^2+6 d e x+15 e^2 x^2\right)+b^3 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)\right)}{60 e^4 (a+b x) (d+e x)^6}","\frac{b^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{60 (d+e x)^4 (b d-a e)^3}+\frac{b (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{15 (d+e x)^5 (b d-a e)^2}+\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{6 (d+e x)^6 (b d-a e)}",1,"-1/60*(Sqrt[(a + b*x)^2]*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)))/(e^4*(a + b*x)*(d + e*x)^6)","A",1
1567,1,112,200,0.0413055,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^8,x]","-\frac{\sqrt{(a+b x)^2} \left(20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left(d^2+7 d e x+21 e^2 x^2\right)+b^3 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)\right)}{140 e^4 (a+b x) (d+e x)^7}","\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}",1,"-1/140*(Sqrt[(a + b*x)^2]*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4*a*b^2*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + b^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)))/(e^4*(a + b*x)*(d + e*x)^7)","A",1
1568,1,112,200,0.0461137,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^9,x]","-\frac{\sqrt{(a+b x)^2} \left(35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left(d^2+8 d e x+28 e^2 x^2\right)+b^3 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)\right)}{280 e^4 (a+b x) (d+e x)^8}","\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^4 (a+b x) (d+e x)^6}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{8 e^4 (a+b x) (d+e x)^8}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5}",1,"-1/280*(Sqrt[(a + b*x)^2]*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)))/(e^4*(a + b*x)*(d + e*x)^8)","A",1
1569,1,385,266,0.1210433,"\int (d+e x)^5 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x \sqrt{(a+b x)^2} \left(462 a^5 \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+330 a^4 b x \left(21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right)+165 a^3 b^2 x^2 \left(56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right)+55 a^2 b^3 x^3 \left(126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right)+11 a b^4 x^4 \left(252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right)+b^5 x^5 \left(462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right)\right)}{2772 (a+b x)}","\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6}",1,"(x*Sqrt[(a + b*x)^2]*(462*a^5*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 330*a^4*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 165*a^3*b^2*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 55*a^2*b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 11*a*b^4*x^4*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + b^5*x^5*(462*d^5 + 1980*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)))/(2772*(a + b*x))","A",1
1570,1,319,219,0.1055226,"\int (d+e x)^4 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x \sqrt{(a+b x)^2} \left(252 a^5 \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+210 a^4 b x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)+120 a^3 b^2 x^2 \left(35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right)+45 a^2 b^3 x^3 \left(70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right)+10 a b^4 x^4 \left(126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right)+b^5 x^5 \left(210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right)\right)}{1260 (a+b x)}","\frac{4 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac{4 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}",1,"(x*Sqrt[(a + b*x)^2]*(252*a^5*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 210*a^4*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 120*a^3*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 45*a^2*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 10*a*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + b^5*x^5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)))/(1260*(a + b*x))","A",1
1571,1,253,172,0.084753,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x \sqrt{(a+b x)^2} \left(126 a^5 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+126 a^4 b x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)+84 a^3 b^2 x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+36 a^2 b^3 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)+9 a b^4 x^4 \left(56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right)+b^5 x^5 \left(84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right)\right)}{504 (a+b x)}","\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}",1,"(x*Sqrt[(a + b*x)^2]*(126*a^5*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 126*a^4*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 84*a^3*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 36*a^2*b^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 9*a*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + b^5*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)))/(504*(a + b*x))","A",1
1572,1,187,125,0.068163,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x \sqrt{(a+b x)^2} \left(56 a^5 \left(3 d^2+3 d e x+e^2 x^2\right)+70 a^4 b x \left(6 d^2+8 d e x+3 e^2 x^2\right)+56 a^3 b^2 x^2 \left(10 d^2+15 d e x+6 e^2 x^2\right)+28 a^2 b^3 x^3 \left(15 d^2+24 d e x+10 e^2 x^2\right)+8 a b^4 x^4 \left(21 d^2+35 d e x+15 e^2 x^2\right)+b^5 x^5 \left(28 d^2+48 d e x+21 e^2 x^2\right)\right)}{168 (a+b x)}","\frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{6 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}",1,"(x*Sqrt[(a + b*x)^2]*(56*a^5*(3*d^2 + 3*d*e*x + e^2*x^2) + 70*a^4*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 56*a^3*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 28*a^2*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 8*a*b^4*x^4*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2)))/(168*(a + b*x))","A",1
1573,1,121,69,0.0441445,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{x \sqrt{(a+b x)^2} \left(21 a^5 (2 d+e x)+35 a^4 b x (3 d+2 e x)+35 a^3 b^2 x^2 (4 d+3 e x)+21 a^2 b^3 x^3 (5 d+4 e x)+7 a b^4 x^4 (6 d+5 e x)+b^5 x^5 (7 d+6 e x)\right)}{42 (a+b x)}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2} (b d-a e)}{6 b^2}+\frac{e \left(a^2+2 a b x+b^2 x^2\right)^{7/2}}{7 b^2}",1,"(x*Sqrt[(a + b*x)^2]*(21*a^5*(2*d + e*x) + 35*a^4*b*x*(3*d + 2*e*x) + 35*a^3*b^2*x^2*(4*d + 3*e*x) + 21*a^2*b^3*x^3*(5*d + 4*e*x) + 7*a*b^4*x^4*(6*d + 5*e*x) + b^5*x^5*(7*d + 6*e*x)))/(42*(a + b*x))","A",1
1574,1,23,32,0.0133841,"\int \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{(a+b x) \left((a+b x)^2\right)^{5/2}}{6 b}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{6 b}",1,"((a + b*x)*((a + b*x)^2)^(5/2))/(6*b)","A",1
1575,1,185,254,0.0947954,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x),x]","\frac{\sqrt{(a+b x)^2} \left(b e x \left(300 a^4 e^4+300 a^3 b e^3 (e x-2 d)+100 a^2 b^2 e^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)+25 a b^3 e \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+b^4 \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)\right)-60 (b d-a e)^5 \log (d+e x)\right)}{60 e^6 (a+b x)}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^6 (a+b x)}+\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^5 (a+b x)}-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4}+\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^3}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2}+\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e}",1,"(Sqrt[(a + b*x)^2]*(b*e*x*(300*a^4*e^4 + 300*a^3*b*e^3*(-2*d + e*x) + 100*a^2*b^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b^3*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) - 60*(b*d - a*e)^5*Log[d + e*x]))/(60*e^6*(a + b*x))","A",1
1576,1,246,292,0.1550448,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^2,x]","\frac{\sqrt{(a+b x)^2} \left(-12 a^5 e^5+60 a^4 b d e^4+120 a^3 b^2 e^3 \left(-d^2+d e x+e^2 x^2\right)+60 a^2 b^3 e^2 \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)+20 a b^4 e \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \left(12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right)\right)}{12 e^6 (a+b x) (d+e x)}","\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) (d+e x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x)}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^6 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{e^6 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(60*a^4*b*d*e^4 - 12*a^5*e^5 + 120*a^3*b^2*e^3*(-d^2 + d*e*x + e^2*x^2) + 60*a^2*b^3*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 20*a*b^4*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^5*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + 60*b*(b*d - a*e)^4*(d + e*x)*Log[d + e*x]))/(12*e^6*(a + b*x)*(d + e*x))","A",1
1577,1,248,295,0.107414,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^3,x]","\frac{\sqrt{(a+b x)^2} \left(-3 a^5 e^5-15 a^4 b e^4 (d+2 e x)+30 a^3 b^2 d e^3 (3 d+4 e x)+30 a^2 b^3 e^2 \left(-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)+15 a b^4 e \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)-60 b^2 (d+e x)^2 (b d-a e)^3 \log (d+e x)+b^5 \left(-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right)\right)}{6 e^6 (a+b x) (d+e x)^2}","-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^6 (a+b x) (d+e x)^2}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{2 e^6 (a+b x)}+\frac{10 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(-3*a^5*e^5 - 15*a^4*b*e^4*(d + 2*e*x) + 30*a^3*b^2*d*e^3*(3*d + 4*e*x) + 30*a^2*b^3*e^2*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 15*a*b^4*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + b^5*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) - 60*b^2*(b*d - a*e)^3*(d + e*x)^2*Log[d + e*x]))/(6*e^6*(a + b*x)*(d + e*x)^2)","A",1
1578,1,247,292,0.1118608,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^4,x]","\frac{\sqrt{(a+b x)^2} \left(-2 a^5 e^5-5 a^4 b e^4 (d+3 e x)-20 a^3 b^2 e^3 \left(d^2+3 d e x+3 e^2 x^2\right)+10 a^2 b^3 d e^2 \left(11 d^2+27 d e x+18 e^2 x^2\right)+10 a b^4 e \left(-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right)+60 b^3 (d+e x)^3 (b d-a e)^2 \log (d+e x)+b^5 \left(47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right)\right)}{6 e^6 (a+b x) (d+e x)^3}","\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^3}+\frac{b^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)}-\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (4 b d-5 a e)}{e^5 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^6 (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(-2*a^5*e^5 - 5*a^4*b*e^4*(d + 3*e*x) - 20*a^3*b^2*e^3*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^2*b^3*d*e^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 10*a*b^4*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + b^5*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + 60*b^3*(b*d - a*e)^2*(d + e*x)^3*Log[d + e*x]))/(6*e^6*(a + b*x)*(d + e*x)^3)","A",1
1579,1,243,292,0.1191827,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^5} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^5,x]","-\frac{\sqrt{(a+b x)^2} \left(3 a^5 e^5+5 a^4 b e^4 (d+4 e x)+10 a^3 b^2 e^3 \left(d^2+4 d e x+6 e^2 x^2\right)+30 a^2 b^3 e^2 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)-5 a b^4 d e \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+60 b^4 (d+e x)^4 (b d-a e) \log (d+e x)+b^5 \left(77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right)\right)}{12 e^6 (a+b x) (d+e x)^4}","\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)}",1,"-1/12*(Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 5*a^4*b*e^4*(d + 4*e*x) + 10*a^3*b^2*e^3*(d^2 + 4*d*e*x + 6*e^2*x^2) + 30*a^2*b^3*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a*b^4*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + b^5*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + 60*b^4*(b*d - a*e)*(d + e*x)^4*Log[d + e*x]))/(e^6*(a + b*x)*(d + e*x)^4)","A",1
1580,1,196,300,0.1249543,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^6} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^6,x]","\frac{\sqrt{(a+b x)^2} \left((b d-a e) \left(12 a^4 e^4+3 a^3 b e^3 (9 d+25 e x)+a^2 b^2 e^2 \left(47 d^2+175 d e x+200 e^2 x^2\right)+a b^3 e \left(77 d^3+325 d^2 e x+500 d e^2 x^2+300 e^3 x^3\right)+b^4 \left(137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right)\right)+60 b^5 (d+e x)^5 \log (d+e x)\right)}{60 e^6 (a+b x) (d+e x)^5}","\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2}",1,"(Sqrt[(a + b*x)^2]*((b*d - a*e)*(12*a^4*e^4 + 3*a^3*b*e^3*(9*d + 25*e*x) + a^2*b^2*e^2*(47*d^2 + 175*d*e*x + 200*e^2*x^2) + a*b^3*e*(77*d^3 + 325*d^2*e*x + 500*d*e^2*x^2 + 300*e^3*x^3) + b^4*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*b^5*(d + e*x)^5*Log[d + e*x]))/(60*e^6*(a + b*x)*(d + e*x)^5)","A",1
1581,1,218,48,0.0762026,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^7} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]","-\frac{\sqrt{(a+b x)^2} \left(a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left(d^2+6 d e x+15 e^2 x^2\right)+a^2 b^3 e^2 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+a b^4 e \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)+b^5 \left(d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right)\right)}{6 e^6 (a+b x) (d+e x)^6}","\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)}",1,"-1/6*(Sqrt[(a + b*x)^2]*(a^5*e^5 + a^4*b*e^4*(d + 6*e*x) + a^3*b^2*e^3*(d^2 + 6*d*e*x + 15*e^2*x^2) + a^2*b^3*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + a*b^4*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + b^5*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^6)","B",1
1582,1,223,98,0.0750349,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^8} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^8,x]","-\frac{\sqrt{(a+b x)^2} \left(6 a^5 e^5+5 a^4 b e^4 (d+7 e x)+4 a^3 b^2 e^3 \left(d^2+7 d e x+21 e^2 x^2\right)+3 a^2 b^3 e^2 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+2 a b^4 e \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+b^5 \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)\right)}{42 e^6 (a+b x) (d+e x)^7}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{42 (d+e x)^6 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{7 (d+e x)^7 (b d-a e)}",1,"-1/42*(Sqrt[(a + b*x)^2]*(6*a^5*e^5 + 5*a^4*b*e^4*(d + 7*e*x) + 4*a^3*b^2*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a^2*b^3*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*a*b^4*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + b^5*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^7)","B",1
1583,1,223,149,0.0767079,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^9} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^9,x]","-\frac{\sqrt{(a+b x)^2} \left(21 a^5 e^5+15 a^4 b e^4 (d+8 e x)+10 a^3 b^2 e^3 \left(d^2+8 d e x+28 e^2 x^2\right)+6 a^2 b^3 e^2 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+3 a b^4 e \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+b^5 \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)\right)}{168 e^6 (a+b x) (d+e x)^8}","\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)}",1,"-1/168*(Sqrt[(a + b*x)^2]*(21*a^5*e^5 + 15*a^4*b*e^4*(d + 8*e*x) + 10*a^3*b^2*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a^2*b^3*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a*b^4*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + b^5*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^8)","A",1
1584,1,223,200,0.0839529,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{10}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]","-\frac{\sqrt{(a+b x)^2} \left(56 a^5 e^5+35 a^4 b e^4 (d+9 e x)+20 a^3 b^2 e^3 \left(d^2+9 d e x+36 e^2 x^2\right)+10 a^2 b^3 e^2 \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+4 a b^4 e \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+b^5 \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)\right)}{504 e^6 (a+b x) (d+e x)^9}","\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{84 (d+e x)^7 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{24 (d+e x)^8 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{9 (d+e x)^9 (b d-a e)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{504 (d+e x)^6 (b d-a e)^4}",1,"-1/504*(Sqrt[(a + b*x)^2]*(56*a^5*e^5 + 35*a^4*b*e^4*(d + 9*e*x) + 20*a^3*b^2*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2) + 10*a^2*b^3*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*a*b^4*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + b^5*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^9)","A",1
1585,1,223,308,0.0790986,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{11}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^11,x]","-\frac{\sqrt{(a+b x)^2} \left(126 a^5 e^5+70 a^4 b e^4 (d+10 e x)+35 a^3 b^2 e^3 \left(d^2+10 d e x+45 e^2 x^2\right)+15 a^2 b^3 e^2 \left(d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right)+5 a b^4 e \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)+b^5 \left(d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right)\right)}{1260 e^6 (a+b x) (d+e x)^{10}}","\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{4 e^6 (a+b x) (d+e x)^8}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{10 e^6 (a+b x) (d+e x)^{10}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{6 e^6 (a+b x) (d+e x)^6}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^7}",1,"-1/1260*(Sqrt[(a + b*x)^2]*(126*a^5*e^5 + 70*a^4*b*e^4*(d + 10*e*x) + 35*a^3*b^2*e^3*(d^2 + 10*d*e*x + 45*e^2*x^2) + 15*a^2*b^3*e^2*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 5*a*b^4*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + b^5*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^10)","A",1
1586,1,223,308,0.0780236,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{12}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^12,x]","-\frac{\sqrt{(a+b x)^2} \left(252 a^5 e^5+126 a^4 b e^4 (d+11 e x)+56 a^3 b^2 e^3 \left(d^2+11 d e x+55 e^2 x^2\right)+21 a^2 b^3 e^2 \left(d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right)+6 a b^4 e \left(d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right)+b^5 \left(d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right)\right)}{2772 e^6 (a+b x) (d+e x)^{11}}","\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8}",1,"-1/2772*(Sqrt[(a + b*x)^2]*(252*a^5*e^5 + 126*a^4*b*e^4*(d + 11*e*x) + 56*a^3*b^2*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) + 21*a^2*b^3*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 6*a*b^4*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + b^5*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^11)","A",1
1587,1,130,222,0.0770349,"\int \frac{(d+e x)^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^4/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(b e x \left(-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left(18 d^2+6 d e x+e^2 x^2\right)+b^3 \left(48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right)\right)+12 (b d-a e)^4 \log (a+b x)\right)}{12 b^5 \sqrt{(a+b x)^2}}","\frac{(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*e*x*(-12*a^3*e^3 + 6*a^2*b*e^2*(8*d + e*x) - 4*a*b^2*e*(18*d^2 + 6*d*e*x + e^2*x^2) + b^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 12*(b*d - a*e)^4*Log[a + b*x]))/(12*b^5*Sqrt[(a + b*x)^2])","A",1
1588,1,90,173,0.05067,"\int \frac{(d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(b e x \left(6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)\right)+6 (b d-a e)^3 \log (a+b x)\right)}{6 b^4 \sqrt{(a+b x)^2}}","\frac{(a+b x) (d+e x)^2 (b d-a e)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^3 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*e*x*(6*a^2*e^2 - 3*a*b*e*(6*d + e*x) + b^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 6*(b*d - a*e)^3*Log[a + b*x]))/(6*b^4*Sqrt[(a + b*x)^2])","A",1
1589,1,59,124,0.0308874,"\int \frac{(d+e x)^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)\right)}{2 b^3 \sqrt{(a+b x)^2}}","\frac{e x (a+b x) (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^2 \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(b*e*x*(4*b*d - 2*a*e + b*e*x) + 2*(b*d - a*e)^2*Log[a + b*x]))/(2*b^3*Sqrt[(a + b*x)^2])","A",1
1590,1,40,69,0.0168265,"\int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) ((b d-a e) \log (a+b x)+b e x)}{b^2 \sqrt{(a+b x)^2}}","\frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}",1,"((a + b*x)*(b*e*x + (b*d - a*e)*Log[a + b*x]))/(b^2*Sqrt[(a + b*x)^2])","A",1
1591,1,26,35,0.0075237,"\int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \log (a+b x)}{b \sqrt{(a+b x)^2}}","\frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*Log[a + b*x])/(b*Sqrt[(a + b*x)^2])","A",1
1592,1,42,86,0.0185723,"\int \frac{1}{(d+e x) \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) (\log (a+b x)-\log (d+e x))}{\sqrt{(a+b x)^2} (b d-a e)}","\frac{(a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"((a + b*x)*(Log[a + b*x] - Log[d + e*x]))/((b*d - a*e)*Sqrt[(a + b*x)^2])","A",1
1593,1,69,131,0.0447949,"\int \frac{1}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) (b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d)}{\sqrt{(a+b x)^2} (d+e x) (b d-a e)^2}","\frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac{b (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{b (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}",1,"((a + b*x)*(b*d - a*e + b*(d + e*x)*Log[a + b*x] - b*(d + e*x)*Log[d + e*x]))/((b*d - a*e)^2*Sqrt[(a + b*x)^2]*(d + e*x))","A",1
1594,1,97,182,0.0664287,"\int \frac{1}{(d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) \left(2 b^2 (d+e x)^2 \log (a+b x)+(b d-a e) (-a e+3 b d+2 b e x)-2 b^2 (d+e x)^2 \log (d+e x)\right)}{2 \sqrt{(a+b x)^2} (d+e x)^2 (b d-a e)^3}","\frac{b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}+\frac{a+b x}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac{b^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}",1,"((a + b*x)*((b*d - a*e)*(3*b*d - a*e + 2*b*e*x) + 2*b^2*(d + e*x)^2*Log[a + b*x] - 2*b^2*(d + e*x)^2*Log[d + e*x]))/(2*(b*d - a*e)^3*Sqrt[(a + b*x)^2]*(d + e*x)^2)","A",1
1595,1,124,231,0.0768114,"\int \frac{1}{(d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) \left(6 b^3 (d+e x)^3 \log (a+b x)+6 b^2 (d+e x)^2 (b d-a e)+3 b (d+e x) (b d-a e)^2+2 (b d-a e)^3-6 b^3 (d+e x)^3 \log (d+e x)\right)}{6 \sqrt{(a+b x)^2} (d+e x)^3 (b d-a e)^4}","\frac{b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{b (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}+\frac{a+b x}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}",1,"((a + b*x)*(2*(b*d - a*e)^3 + 3*b*(b*d - a*e)^2*(d + e*x) + 6*b^2*(b*d - a*e)*(d + e*x)^2 + 6*b^3*(d + e*x)^3*Log[a + b*x] - 6*b^3*(d + e*x)^3*Log[d + e*x]))/(6*(b*d - a*e)^4*Sqrt[(a + b*x)^2]*(d + e*x)^3)","A",1
1596,1,174,210,0.08021,"\int \frac{(d+e x)^4}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{7 a^4 e^4+2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left(18 d^2-16 d e x-11 e^2 x^2\right)-4 a b^3 e \left(d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right)+12 e^2 (a+b x)^2 (b d-a e)^2 \log (a+b x)+b^4 \left(-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right)}{2 b^5 (a+b x) \sqrt{(a+b x)^2}}","\frac{6 e^2 (a+b x) (b d-a e)^2 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x) (4 b d-3 a e)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(7*a^4*e^4 + 2*a^3*b*e^3*(-10*d + e*x) + a^2*b^2*e^2*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*b^3*e*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + b^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(b*d - a*e)^2*(a + b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1597,1,125,161,0.0573656,"\int \frac{(d+e x)^3}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{-5 a^3 e^3+a^2 b e^2 (9 d-4 e x)+a b^2 e \left(-3 d^2+12 d e x+4 e^2 x^2\right)-6 e^2 (a+b x)^2 (a e-b d) \log (a+b x)-\left(b^3 \left(d^3+6 d^2 e x-2 e^3 x^3\right)\right)}{2 b^4 (a+b x) \sqrt{(a+b x)^2}}","\frac{3 e^2 (a+b x) (b d-a e) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-5*a^3*e^3 + a^2*b*e^2*(9*d - 4*e*x) + a*b^2*e*(-3*d^2 + 12*d*e*x + 4*e^2*x^2) - b^3*(d^3 + 6*d^2*e*x - 2*e^3*x^3) - 6*e^2*(-(b*d) + a*e)*(a + b*x)^2*Log[a + b*x])/(2*b^4*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1598,1,67,117,0.0329025,"\int \frac{(d+e x)^2}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{2 e^2 (a+b x)^2 \log (a+b x)-(b d-a e) (3 a e+b (d+4 e x))}{2 b^3 (a+b x) \sqrt{(a+b x)^2}}","-\frac{2 e (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-((b*d - a*e)*(3*a*e + b*(d + 4*e*x))) + 2*e^2*(a + b*x)^2*Log[a + b*x])/(2*b^3*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1599,1,39,69,0.0150267,"\int \frac{d+e x}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{-a e-b (d+2 e x)}{2 b^2 (a+b x) \sqrt{(a+b x)^2}}","-\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-(a*e) - b*(d + 2*e*x))/(2*b^2*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1600,1,23,34,0.0099897,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]","-\frac{a+b x}{2 b \left((a+b x)^2\right)^{3/2}}","-\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"-1/2*(a + b*x)/(b*((a + b*x)^2)^(3/2))","A",1
1601,1,92,165,0.0525708,"\int \frac{1}{(d+e x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{-2 e^2 (a+b x)^2 \log (d+e x)-(b d-a e) (b (d-2 e x)-3 a e)+2 e^2 (a+b x)^2 \log (a+b x)}{2 (a+b x) \sqrt{(a+b x)^2} (b d-a e)^3}","\frac{e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"(-((b*d - a*e)*(-3*a*e + b*(d - 2*e*x))) + 2*e^2*(a + b*x)^2*Log[a + b*x] - 2*e^2*(a + b*x)^2*Log[d + e*x])/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1602,1,141,217,0.0974622,"\int \frac{1}{(d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{-(b d-a e) \left(-2 a^2 e^2-a b e (5 d+9 e x)+b^2 \left(d^2-3 d e x-6 e^2 x^2\right)\right)+6 b e^2 (a+b x)^2 (d+e x) \log (a+b x)-6 b e^2 (a+b x)^2 (d+e x) \log (d+e x)}{2 (a+b x) \sqrt{(a+b x)^2} (d+e x) (b d-a e)^4}","\frac{e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{3 b e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{3 b e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}",1,"(-((b*d - a*e)*(-2*a^2*e^2 - a*b*e*(5*d + 9*e*x) + b^2*(d^2 - 3*d*e*x - 6*e^2*x^2))) + 6*b*e^2*(a + b*x)^2*(d + e*x)*Log[a + b*x] - 6*b*e^2*(a + b*x)^2*(d + e*x)*Log[d + e*x])/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[(a + b*x)^2]*(d + e*x))","A",1
1603,1,163,276,0.0990753,"\int \frac{1}{(d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{(a+b x) \left(-12 b^2 e^2 (a+b x)^2 \log (d+e x)+6 b^2 e (a+b x) (b d-a e)+b^2 \left(-(b d-a e)^2\right)+12 b^2 e^2 (a+b x)^2 \log (a+b x)+\frac{6 b e^2 (a+b x)^2 (b d-a e)}{d+e x}+\frac{e^2 (a+b x)^2 (b d-a e)^2}{(d+e x)^2}\right)}{2 \left((a+b x)^2\right)^{3/2} (b d-a e)^5}","\frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}",1,"((a + b*x)*(-(b^2*(b*d - a*e)^2) + 6*b^2*e*(b*d - a*e)*(a + b*x) + (e^2*(b*d - a*e)^2*(a + b*x)^2)/(d + e*x)^2 + (6*b*e^2*(b*d - a*e)*(a + b*x)^2)/(d + e*x) + 12*b^2*e^2*(a + b*x)^2*Log[a + b*x] - 12*b^2*e^2*(a + b*x)^2*Log[d + e*x]))/(2*(b*d - a*e)^5*((a + b*x)^2)^(3/2))","A",1
1604,1,313,302,0.1463747,"\int \frac{(d+e x)^6}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{57 a^6 e^6+14 a^5 b e^5 (12 e x-11 d)+a^4 b^2 e^4 \left(125 d^2-496 d e x+132 e^2 x^2\right)-4 a^3 b^3 e^3 \left(5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right)-a^2 b^4 e^2 \left(5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right)-2 a b^5 e \left(d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right)+60 e^4 (a+b x)^4 (b d-a e)^2 \log (a+b x)-\left(b^6 \left(d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right)\right)}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(57*a^6*e^6 + 14*a^5*b*e^5*(-11*d + 12*e*x) + a^4*b^2*e^4*(125*d^2 - 496*d*e*x + 132*e^2*x^2) - 4*a^3*b^3*e^3*(5*d^3 - 110*d^2*e*x + 126*d*e^2*x^2 + 8*e^3*x^3) - a^2*b^4*e^2*(5*d^4 + 80*d^3*e*x - 540*d^2*e^2*x^2 + 96*d*e^3*x^3 + 68*e^4*x^4) - 2*a*b^5*e*(d^5 + 10*d^4*e*x + 60*d^3*e^2*x^2 - 120*d^2*e^3*x^3 - 48*d*e^4*x^4 + 6*e^5*x^5) - b^6*(d^6 + 8*d^5*e*x + 30*d^4*e^2*x^2 + 80*d^3*e^3*x^3 - 24*d*e^5*x^5 - 2*e^6*x^6) + 60*e^4*(b*d - a*e)^2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1605,1,242,253,0.1238485,"\int \frac{(d+e x)^5}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-77 a^5 e^5+a^4 b e^4 (125 d-248 e x)-2 a^3 b^2 e^3 \left(15 d^2-220 d e x+126 e^2 x^2\right)-2 a^2 b^3 e^2 \left(5 d^3+60 d^2 e x-270 d e^2 x^2+24 e^3 x^3\right)+a b^4 e \left(-5 d^4-40 d^3 e x-180 d^2 e^2 x^2+240 d e^3 x^3+48 e^4 x^4\right)-60 e^4 (a+b x)^4 (a e-b d) \log (a+b x)-\left(b^5 \left(3 d^5+20 d^4 e x+60 d^3 e^2 x^2+120 d^2 e^3 x^3-12 e^5 x^5\right)\right)}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}}","\frac{5 e^4 (a+b x) (b d-a e) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 e^3 (b d-a e)^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-77*a^5*e^5 + a^4*b*e^4*(125*d - 248*e*x) - 2*a^3*b^2*e^3*(15*d^2 - 220*d*e*x + 126*e^2*x^2) - 2*a^2*b^3*e^2*(5*d^3 + 60*d^2*e*x - 270*d*e^2*x^2 + 24*e^3*x^3) + a*b^4*e*(-5*d^4 - 40*d^3*e*x - 180*d^2*e^2*x^2 + 240*d*e^3*x^3 + 48*e^4*x^4) - b^5*(3*d^5 + 20*d^4*e*x + 60*d^3*e^2*x^2 + 120*d^2*e^3*x^3 - 12*e^5*x^5) - 60*e^4*(-(b*d) + a*e)*(a + b*x)^4*Log[a + b*x])/(12*b^6*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1606,1,138,209,0.0731865,"\int \frac{(d+e x)^4}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{12 e^4 (a+b x)^4 \log (a+b x)-(b d-a e) \left(25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left(7 d^2+40 d e x+108 e^2 x^2\right)+b^3 \left(3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right)\right)}{12 b^5 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{4 e^3 (b d-a e)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (b d-a e)^2}{b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{4 b^5 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-((b*d - a*e)*(25*a^3*e^3 + a^2*b*e^2*(13*d + 88*e*x) + a*b^2*e*(7*d^2 + 40*d*e*x + 108*e^2*x^2) + b^3*(3*d^3 + 16*d^2*e*x + 36*d*e^2*x^2 + 48*e^3*x^3))) + 12*e^4*(a + b*x)^4*Log[a + b*x])/(12*b^5*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1607,1,106,48,0.0452289,"\int \frac{(d+e x)^3}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a^3 e^3-a^2 b e^2 (d+4 e x)-a b^2 e \left(d^2+4 d e x+6 e^2 x^2\right)-\left(b^3 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)\right)}{4 b^4 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{(d+e x)^4}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"(-(a^3*e^3) - a^2*b*e^2*(d + 4*e*x) - a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) - b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3))/(4*b^4*(a + b*x)^3*Sqrt[(a + b*x)^2])","B",1
1608,1,69,125,0.0285295,"\int \frac{(d+e x)^2}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a^2 e^2-2 a b e (d+2 e x)-\left(b^2 \left(3 d^2+8 d e x+6 e^2 x^2\right)\right)}{12 b^3 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-(a^2*e^2) - 2*a*b*e*(d + 2*e*x) - b^2*(3*d^2 + 8*d*e*x + 6*e^2*x^2))/(12*b^3*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1609,1,39,71,0.015806,"\int \frac{d+e x}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{-a e-3 b d-4 b e x}{12 b^2 (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{b d-a e}{4 b^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}-\frac{e}{3 b^2 \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}",1,"(-3*b*d - a*e - 4*b*e*x)/(12*b^2*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1610,1,23,34,0.0112707,"\int \frac{1}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-5/2),x]","-\frac{a+b x}{4 b \left((a+b x)^2\right)^{5/2}}","-\frac{1}{4 b (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^{3/2}}",1,"-1/4*(a + b*x)/(b*((a + b*x)^2)^(5/2))","A",1
1611,1,163,253,0.1185666,"\int \frac{1}{(d+e x) \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{-(b d-a e) \left(-25 a^3 e^3+a^2 b e^2 (23 d-52 e x)+a b^2 e \left(-13 d^2+20 d e x-42 e^2 x^2\right)+b^3 \left(3 d^3-4 d^2 e x+6 d e^2 x^2-12 e^3 x^3\right)\right)-12 e^4 (a+b x)^4 \log (d+e x)+12 e^4 (a+b x)^4 \log (a+b x)}{12 (a+b x)^3 \sqrt{(a+b x)^2} (b d-a e)^5}","\frac{e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"(-((b*d - a*e)*(-25*a^3*e^3 + a^2*b*e^2*(23*d - 52*e*x) + a*b^2*e*(-13*d^2 + 20*d*e*x - 42*e^2*x^2) + b^3*(3*d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 12*e^3*x^3))) + 12*e^4*(a + b*x)^4*Log[a + b*x] - 12*e^4*(a + b*x)^4*Log[d + e*x])/(12*(b*d - a*e)^5*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1612,1,167,307,0.1167129,"\int \frac{1}{(d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{\frac{12 e^4 (a+b x)^3 (b d-a e)}{d+e x}-60 b e^4 (a+b x)^3 \log (d+e x)+48 b e^3 (a+b x)^2 (b d-a e)-18 b e^2 (a+b x) (b d-a e)^2-\frac{3 b (b d-a e)^4}{a+b x}+8 b e (b d-a e)^3+60 b e^4 (a+b x)^3 \log (a+b x)}{12 \left((a+b x)^2\right)^{3/2} (b d-a e)^6}","\frac{e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac{5 b e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{5 b e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{4 b e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{3 b e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}",1,"(8*b*e*(b*d - a*e)^3 - (3*b*(b*d - a*e)^4)/(a + b*x) - 18*b*e^2*(b*d - a*e)^2*(a + b*x) + 48*b*e^3*(b*d - a*e)*(a + b*x)^2 + (12*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e*x) + 60*b*e^4*(a + b*x)^3*Log[a + b*x] - 60*b*e^4*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^6*((a + b*x)^2)^(3/2))","A",1
1613,1,209,365,0.1535454,"\int \frac{1}{(d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{-60 b^2 e^4 (a+b x)^3 \log (d+e x)+40 b^2 e^3 (a+b x)^2 (b d-a e)-12 b^2 e^2 (a+b x) (b d-a e)^2-\frac{b^2 (b d-a e)^4}{a+b x}+4 b^2 e (b d-a e)^3+60 b^2 e^4 (a+b x)^3 \log (a+b x)+\frac{20 b e^4 (a+b x)^3 (b d-a e)}{d+e x}+\frac{2 e^4 (a+b x)^3 (b d-a e)^2}{(d+e x)^2}}{4 \left((a+b x)^2\right)^{3/2} (b d-a e)^7}","\frac{5 b e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac{e^4 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac{15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac{15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{10 b^2 e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{b^2 e}{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}",1,"(4*b^2*e*(b*d - a*e)^3 - (b^2*(b*d - a*e)^4)/(a + b*x) - 12*b^2*e^2*(b*d - a*e)^2*(a + b*x) + 40*b^2*e^3*(b*d - a*e)*(a + b*x)^2 + (2*e^4*(b*d - a*e)^2*(a + b*x)^3)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e*x) + 60*b^2*e^4*(a + b*x)^3*Log[a + b*x] - 60*b^2*e^4*(a + b*x)^3*Log[d + e*x])/(4*(b*d - a*e)^7*((a + b*x)^2)^(3/2))","A",1
1614,1,81,50,0.0337429,"\int (d+e x) \left(9+12 x+4 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^(5/2),x]","\frac{x \sqrt{(2 x+3)^2} \left(14 d \left(16 x^5+144 x^4+540 x^3+1080 x^2+1215 x+729\right)+3 e x \left(64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right)\right)}{42 (2 x+3)}","\frac{1}{24} (2 x+3) \left(4 x^2+12 x+9\right)^{5/2} (2 d-3 e)+\frac{1}{28} e \left(4 x^2+12 x+9\right)^{7/2}",1,"(x*Sqrt[(3 + 2*x)^2]*(14*d*(729 + 1215*x + 1080*x^2 + 540*x^3 + 144*x^4 + 16*x^5) + 3*e*x*(1701 + 3780*x + 3780*x^2 + 2016*x^3 + 560*x^4 + 64*x^5)))/(42*(3 + 2*x))","A",1
1615,1,57,50,0.0222826,"\int (d+e x) \left(9+12 x+4 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^(3/2),x]","\frac{x \sqrt{(2 x+3)^2} \left(10 d \left(2 x^3+12 x^2+27 x+27\right)+e x \left(16 x^3+90 x^2+180 x+135\right)\right)}{20 x+30}","\frac{1}{16} (2 x+3) \left(4 x^2+12 x+9\right)^{3/2} (2 d-3 e)+\frac{1}{20} e \left(4 x^2+12 x+9\right)^{5/2}",1,"(x*Sqrt[(3 + 2*x)^2]*(10*d*(27 + 27*x + 12*x^2 + 2*x^3) + e*x*(135 + 180*x + 90*x^2 + 16*x^3)))/(30 + 20*x)","A",1
1616,1,38,50,0.0114758,"\int (d+e x) \sqrt{9+12 x+4 x^2} \, dx","Integrate[(d + e*x)*Sqrt[9 + 12*x + 4*x^2],x]","\frac{x \sqrt{(2 x+3)^2} (6 d (x+3)+e x (4 x+9))}{6 (2 x+3)}","\frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left(4 x^2+12 x+9\right)^{3/2}",1,"(x*Sqrt[(3 + 2*x)^2]*(6*d*(3 + x) + e*x*(9 + 4*x)))/(6*(3 + 2*x))","A",1
1617,1,42,56,0.0157724,"\int \frac{d+e x}{\sqrt{9+12 x+4 x^2}} \, dx","Integrate[(d + e*x)/Sqrt[9 + 12*x + 4*x^2],x]","\frac{(2 x+3) ((2 d-3 e) \log (2 x+3)+e (2 x+3))}{4 \sqrt{(2 x+3)^2}}","\frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9}",1,"((3 + 2*x)*(e*(3 + 2*x) + (2*d - 3*e)*Log[3 + 2*x]))/(4*Sqrt[(3 + 2*x)^2])","A",1
1618,1,34,52,0.0119189,"\int \frac{d+e x}{\left(9+12 x+4 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]","\frac{-2 d-e (4 x+3)}{8 (2 x+3) \sqrt{(2 x+3)^2}}","-\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}}",1,"(-2*d - e*(3 + 4*x))/(8*(3 + 2*x)*Sqrt[(3 + 2*x)^2])","A",1
1619,1,34,52,0.0145928,"\int \frac{d+e x}{\left(9+12 x+4 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(5/2),x]","\frac{-6 d-e (8 x+3)}{48 (2 x+3)^3 \sqrt{(2 x+3)^2}}","-\frac{2 d-3 e}{16 (2 x+3) \left(4 x^2+12 x+9\right)^{3/2}}-\frac{e}{12 \left(4 x^2+12 x+9\right)^{3/2}}",1,"(-6*d - e*(3 + 8*x))/(48*(3 + 2*x)^3*Sqrt[(3 + 2*x)^2])","A",1
1620,1,34,52,0.0160615,"\int \frac{d+e x}{\left(9+12 x+4 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]","\frac{-10 d-3 (4 e x+e)}{120 (2 x+3)^5 \sqrt{(2 x+3)^2}}","-\frac{2 d-3 e}{24 (2 x+3) \left(4 x^2+12 x+9\right)^{5/2}}-\frac{e}{20 \left(4 x^2+12 x+9\right)^{5/2}}",1,"(-10*d - 3*(e + 4*e*x))/(120*(3 + 2*x)^5*Sqrt[(3 + 2*x)^2])","A",1
1621,1,61,71,0.0538297,"\int (d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 (d+e x)^{9/2} \left(143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left(8 d^2-36 d e x+99 e^2 x^2\right)\right)}{1287 e^3}","-\frac{4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac{2 b^2 (d+e x)^{13/2}}{13 e^3}",1,"(2*(d + e*x)^(9/2)*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)","A",1
1622,1,61,71,0.0414079,"\int (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 (d+e x)^{7/2} \left(99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)}{693 e^3}","-\frac{4 b (d+e x)^{9/2} (b d-a e)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e)^2}{7 e^3}+\frac{2 b^2 (d+e x)^{11/2}}{11 e^3}",1,"(2*(d + e*x)^(7/2)*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)","A",1
1623,1,61,71,0.0380972,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 (d+e x)^{5/2} \left(63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)}{315 e^3}","-\frac{4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3}",1,"(2*(d + e*x)^(5/2)*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)","A",1
1624,1,61,71,0.0338861,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 (d+e x)^{3/2} \left(35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)}{105 e^3}","-\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3}",1,"(2*(d + e*x)^(3/2)*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)","A",1
1625,1,60,69,0.0331707,"\int \frac{a^2+2 a b x+b^2 x^2}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)}{15 e^3}","-\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3}",1,"(2*Sqrt[d + e*x]*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)","A",1
1626,1,59,67,0.0309359,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(3/2),x]","\frac{2 \left(-3 a^2 e^2+6 a b e (2 d+e x)+b^2 \left(-8 d^2-4 d e x+e^2 x^2\right)\right)}{3 e^3 \sqrt{d+e x}}","-\frac{4 b \sqrt{d+e x} (b d-a e)}{e^3}-\frac{2 (b d-a e)^2}{e^3 \sqrt{d+e x}}+\frac{2 b^2 (d+e x)^{3/2}}{3 e^3}",1,"(2*(-3*a^2*e^2 + 6*a*b*e*(2*d + e*x) + b^2*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])","A",1
1627,1,62,67,0.0354299,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(5/2),x]","\frac{-2 a^2 e^2-4 a b e (2 d+3 e x)+2 b^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)}{3 e^3 (d+e x)^{3/2}}","\frac{4 b (b d-a e)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 b^2 \sqrt{d+e x}}{e^3}",1,"(-2*a^2*e^2 - 4*a*b*e*(2*d + 3*e*x) + 2*b^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2))/(3*e^3*(d + e*x)^(3/2))","A",1
1628,1,61,69,0.0338685,"\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]","-\frac{2 \left(3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)\right)}{15 e^3 (d+e x)^{5/2}}","\frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}}",1,"(-2*(3*a^2*e^2 + 2*a*b*e*(2*d + 5*e*x) + b^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))","A",1
1629,1,101,129,0.1172965,"\int (d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 (d+e x)^{9/2} \left(-29172 b^3 (d+e x)^3 (b d-a e)+50490 b^2 (d+e x)^2 (b d-a e)^2-39780 b (d+e x) (b d-a e)^3+12155 (b d-a e)^4+6435 b^4 (d+e x)^4\right)}{109395 e^5}","-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac{12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac{8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5}",1,"(2*(d + e*x)^(9/2)*(12155*(b*d - a*e)^4 - 39780*b*(b*d - a*e)^3*(d + e*x) + 50490*b^2*(b*d - a*e)^2*(d + e*x)^2 - 29172*b^3*(b*d - a*e)*(d + e*x)^3 + 6435*b^4*(d + e*x)^4))/(109395*e^5)","A",1
1630,1,101,129,0.0967028,"\int (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 (d+e x)^{7/2} \left(-13860 b^3 (d+e x)^3 (b d-a e)+24570 b^2 (d+e x)^2 (b d-a e)^2-20020 b (d+e x) (b d-a e)^3+6435 (b d-a e)^4+3003 b^4 (d+e x)^4\right)}{45045 e^5}","-\frac{8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac{12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac{8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac{2 b^4 (d+e x)^{15/2}}{15 e^5}",1,"(2*(d + e*x)^(7/2)*(6435*(b*d - a*e)^4 - 20020*b*(b*d - a*e)^3*(d + e*x) + 24570*b^2*(b*d - a*e)^2*(d + e*x)^2 - 13860*b^3*(b*d - a*e)*(d + e*x)^3 + 3003*b^4*(d + e*x)^4))/(45045*e^5)","A",1
1631,1,101,129,0.0835203,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 (d+e x)^{5/2} \left(-5460 b^3 (d+e x)^3 (b d-a e)+10010 b^2 (d+e x)^2 (b d-a e)^2-8580 b (d+e x) (b d-a e)^3+3003 (b d-a e)^4+1155 b^4 (d+e x)^4\right)}{15015 e^5}","-\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5}",1,"(2*(d + e*x)^(5/2)*(3003*(b*d - a*e)^4 - 8580*b*(b*d - a*e)^3*(d + e*x) + 10010*b^2*(b*d - a*e)^2*(d + e*x)^2 - 5460*b^3*(b*d - a*e)*(d + e*x)^3 + 1155*b^4*(d + e*x)^4))/(15015*e^5)","A",1
1632,1,101,129,0.0783278,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 (d+e x)^{3/2} \left(-1540 b^3 (d+e x)^3 (b d-a e)+2970 b^2 (d+e x)^2 (b d-a e)^2-2772 b (d+e x) (b d-a e)^3+1155 (b d-a e)^4+315 b^4 (d+e x)^4\right)}{3465 e^5}","-\frac{8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac{12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac{8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac{2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac{2 b^4 (d+e x)^{11/2}}{11 e^5}",1,"(2*(d + e*x)^(3/2)*(1155*(b*d - a*e)^4 - 2772*b*(b*d - a*e)^3*(d + e*x) + 2970*b^2*(b*d - a*e)^2*(d + e*x)^2 - 1540*b^3*(b*d - a*e)*(d + e*x)^3 + 315*b^4*(d + e*x)^4))/(3465*e^5)","A",1
1633,1,101,127,0.0724918,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(-180 b^3 (d+e x)^3 (b d-a e)+378 b^2 (d+e x)^2 (b d-a e)^2-420 b (d+e x) (b d-a e)^3+315 (b d-a e)^4+35 b^4 (d+e x)^4\right)}{315 e^5}","-\frac{8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac{12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac{8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^4}{e^5}+\frac{2 b^4 (d+e x)^{9/2}}{9 e^5}",1,"(2*Sqrt[d + e*x]*(315*(b*d - a*e)^4 - 420*b*(b*d - a*e)^3*(d + e*x) + 378*b^2*(b*d - a*e)^2*(d + e*x)^2 - 180*b^3*(b*d - a*e)*(d + e*x)^3 + 35*b^4*(d + e*x)^4))/(315*e^5)","A",1
1634,1,101,123,0.0711052,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(3/2),x]","\frac{2 \left(-28 b^3 (d+e x)^3 (b d-a e)+70 b^2 (d+e x)^2 (b d-a e)^2-140 b (d+e x) (b d-a e)^3-35 (b d-a e)^4+5 b^4 (d+e x)^4\right)}{35 e^5 \sqrt{d+e x}}","-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5}",1,"(2*(-35*(b*d - a*e)^4 - 140*b*(b*d - a*e)^3*(d + e*x) + 70*b^2*(b*d - a*e)^2*(d + e*x)^2 - 28*b^3*(b*d - a*e)*(d + e*x)^3 + 5*b^4*(d + e*x)^4))/(35*e^5*Sqrt[d + e*x])","A",1
1635,1,101,125,0.0733587,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(5/2),x]","\frac{2 \left(-20 b^3 (d+e x)^3 (b d-a e)+90 b^2 (d+e x)^2 (b d-a e)^2+60 b (d+e x) (b d-a e)^3-5 (b d-a e)^4+3 b^4 (d+e x)^4\right)}{15 e^5 (d+e x)^{3/2}}","-\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5}",1,"(2*(-5*(b*d - a*e)^4 + 60*b*(b*d - a*e)^3*(d + e*x) + 90*b^2*(b*d - a*e)^2*(d + e*x)^2 - 20*b^3*(b*d - a*e)*(d + e*x)^3 + 3*b^4*(d + e*x)^4))/(15*e^5*(d + e*x)^(3/2))","A",1
1636,1,101,125,0.0725432,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^2}{(d+e x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(7/2),x]","\frac{2 \left(-60 b^3 (d+e x)^3 (b d-a e)-90 b^2 (d+e x)^2 (b d-a e)^2+20 b (d+e x) (b d-a e)^3-3 (b d-a e)^4+5 b^4 (d+e x)^4\right)}{15 e^5 (d+e x)^{5/2}}","-\frac{8 b^3 \sqrt{d+e x} (b d-a e)}{e^5}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5}",1,"(2*(-3*(b*d - a*e)^4 + 20*b*(b*d - a*e)^3*(d + e*x) - 90*b^2*(b*d - a*e)^2*(d + e*x)^2 - 60*b^3*(b*d - a*e)*(d + e*x)^3 + 5*b^4*(d + e*x)^4))/(15*e^5*(d + e*x)^(5/2))","A",1
1637,1,145,187,0.1488635,"\int (d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 (d+e x)^{9/2} \left(-918918 b^5 (d+e x)^5 (b d-a e)+2567565 b^4 (d+e x)^4 (b d-a e)^2-3879876 b^3 (d+e x)^3 (b d-a e)^3+3357585 b^2 (d+e x)^2 (b d-a e)^4-1587222 b (d+e x) (b d-a e)^5+323323 (b d-a e)^6+138567 b^6 (d+e x)^6\right)}{2909907 e^7}","-\frac{12 b^5 (d+e x)^{19/2} (b d-a e)}{19 e^7}+\frac{30 b^4 (d+e x)^{17/2} (b d-a e)^2}{17 e^7}-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 (d+e x)^{13/2} (b d-a e)^4}{13 e^7}-\frac{12 b (d+e x)^{11/2} (b d-a e)^5}{11 e^7}+\frac{2 (d+e x)^{9/2} (b d-a e)^6}{9 e^7}+\frac{2 b^6 (d+e x)^{21/2}}{21 e^7}",1,"(2*(d + e*x)^(9/2)*(323323*(b*d - a*e)^6 - 1587222*b*(b*d - a*e)^5*(d + e*x) + 3357585*b^2*(b*d - a*e)^4*(d + e*x)^2 - 3879876*b^3*(b*d - a*e)^3*(d + e*x)^3 + 2567565*b^4*(b*d - a*e)^2*(d + e*x)^4 - 918918*b^5*(b*d - a*e)*(d + e*x)^5 + 138567*b^6*(d + e*x)^6))/(2909907*e^7)","A",1
1638,1,145,185,0.1052822,"\int (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 (d+e x)^{7/2} \left(-342342 b^5 (d+e x)^5 (b d-a e)+969969 b^4 (d+e x)^4 (b d-a e)^2-1492260 b^3 (d+e x)^3 (b d-a e)^3+1322685 b^2 (d+e x)^2 (b d-a e)^4-646646 b (d+e x) (b d-a e)^5+138567 (b d-a e)^6+51051 b^6 (d+e x)^6\right)}{969969 e^7}","-\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7}",1,"(2*(d + e*x)^(7/2)*(138567*(b*d - a*e)^6 - 646646*b*(b*d - a*e)^5*(d + e*x) + 1322685*b^2*(b*d - a*e)^4*(d + e*x)^2 - 1492260*b^3*(b*d - a*e)^3*(d + e*x)^3 + 969969*b^4*(b*d - a*e)^2*(d + e*x)^4 - 342342*b^5*(b*d - a*e)*(d + e*x)^5 + 51051*b^6*(d + e*x)^6))/(969969*e^7)","A",1
1639,1,145,187,0.096623,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 (d+e x)^{5/2} \left(-102102 b^5 (d+e x)^5 (b d-a e)+294525 b^4 (d+e x)^4 (b d-a e)^2-464100 b^3 (d+e x)^3 (b d-a e)^3+425425 b^2 (d+e x)^2 (b d-a e)^4-218790 b (d+e x) (b d-a e)^5+51051 (b d-a e)^6+15015 b^6 (d+e x)^6\right)}{255255 e^7}","-\frac{4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac{30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac{40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac{12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac{2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac{2 b^6 (d+e x)^{17/2}}{17 e^7}",1,"(2*(d + e*x)^(5/2)*(51051*(b*d - a*e)^6 - 218790*b*(b*d - a*e)^5*(d + e*x) + 425425*b^2*(b*d - a*e)^4*(d + e*x)^2 - 464100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 294525*b^4*(b*d - a*e)^2*(d + e*x)^4 - 102102*b^5*(b*d - a*e)*(d + e*x)^5 + 15015*b^6*(d + e*x)^6))/(255255*e^7)","A",1
1640,1,145,187,0.0899253,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 (d+e x)^{3/2} \left(-20790 b^5 (d+e x)^5 (b d-a e)+61425 b^4 (d+e x)^4 (b d-a e)^2-100100 b^3 (d+e x)^3 (b d-a e)^3+96525 b^2 (d+e x)^2 (b d-a e)^4-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right)}{45045 e^7}","-\frac{12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac{30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac{40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac{30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac{12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac{2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac{2 b^6 (d+e x)^{15/2}}{15 e^7}",1,"(2*(d + e*x)^(3/2)*(15015*(b*d - a*e)^6 - 54054*b*(b*d - a*e)^5*(d + e*x) + 96525*b^2*(b*d - a*e)^4*(d + e*x)^2 - 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 61425*b^4*(b*d - a*e)^2*(d + e*x)^4 - 20790*b^5*(b*d - a*e)*(d + e*x)^5 + 3003*b^6*(d + e*x)^6))/(45045*e^7)","A",1
1641,1,145,181,0.080638,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(-1638 b^5 (d+e x)^5 (b d-a e)+5005 b^4 (d+e x)^4 (b d-a e)^2-8580 b^3 (d+e x)^3 (b d-a e)^3+9009 b^2 (d+e x)^2 (b d-a e)^4-6006 b (d+e x) (b d-a e)^5+3003 (b d-a e)^6+231 b^6 (d+e x)^6\right)}{3003 e^7}","-\frac{12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac{10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac{40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac{6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac{4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac{2 \sqrt{d+e x} (b d-a e)^6}{e^7}+\frac{2 b^6 (d+e x)^{13/2}}{13 e^7}",1,"(2*Sqrt[d + e*x]*(3003*(b*d - a*e)^6 - 6006*b*(b*d - a*e)^5*(d + e*x) + 9009*b^2*(b*d - a*e)^4*(d + e*x)^2 - 8580*b^3*(b*d - a*e)^3*(d + e*x)^3 + 5005*b^4*(b*d - a*e)^2*(d + e*x)^4 - 1638*b^5*(b*d - a*e)*(d + e*x)^5 + 231*b^6*(d + e*x)^6))/(3003*e^7)","A",1
1642,1,145,179,0.0793741,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(3/2),x]","\frac{2 \left(-154 b^5 (d+e x)^5 (b d-a e)+495 b^4 (d+e x)^4 (b d-a e)^2-924 b^3 (d+e x)^3 (b d-a e)^3+1155 b^2 (d+e x)^2 (b d-a e)^4-1386 b (d+e x) (b d-a e)^5-231 (b d-a e)^6+21 b^6 (d+e x)^6\right)}{231 e^7 \sqrt{d+e x}}","-\frac{4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac{30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac{12 b \sqrt{d+e x} (b d-a e)^5}{e^7}-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7}",1,"(2*(-231*(b*d - a*e)^6 - 1386*b*(b*d - a*e)^5*(d + e*x) + 1155*b^2*(b*d - a*e)^4*(d + e*x)^2 - 924*b^3*(b*d - a*e)^3*(d + e*x)^3 + 495*b^4*(b*d - a*e)^2*(d + e*x)^4 - 154*b^5*(b*d - a*e)*(d + e*x)^5 + 21*b^6*(d + e*x)^6))/(231*e^7*Sqrt[d + e*x])","A",1
1643,1,145,181,0.0808148,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(5/2),x]","\frac{2 \left(-54 b^5 (d+e x)^5 (b d-a e)+189 b^4 (d+e x)^4 (b d-a e)^2-420 b^3 (d+e x)^3 (b d-a e)^3+945 b^2 (d+e x)^2 (b d-a e)^4+378 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+7 b^6 (d+e x)^6\right)}{63 e^7 (d+e x)^{3/2}}","-\frac{12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac{30 b^2 \sqrt{d+e x} (b d-a e)^4}{e^7}+\frac{12 b (b d-a e)^5}{e^7 \sqrt{d+e x}}-\frac{2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac{2 b^6 (d+e x)^{9/2}}{9 e^7}",1,"(2*(-21*(b*d - a*e)^6 + 378*b*(b*d - a*e)^5*(d + e*x) + 945*b^2*(b*d - a*e)^4*(d + e*x)^2 - 420*b^3*(b*d - a*e)^3*(d + e*x)^3 + 189*b^4*(b*d - a*e)^2*(d + e*x)^4 - 54*b^5*(b*d - a*e)*(d + e*x)^5 + 7*b^6*(d + e*x)^6))/(63*e^7*(d + e*x)^(3/2))","A",1
1644,1,145,179,0.0826012,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^3}{(d+e x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^(7/2),x]","\frac{2 \left(-42 b^5 (d+e x)^5 (b d-a e)+175 b^4 (d+e x)^4 (b d-a e)^2-700 b^3 (d+e x)^3 (b d-a e)^3-525 b^2 (d+e x)^2 (b d-a e)^4+70 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+5 b^6 (d+e x)^6\right)}{35 e^7 (d+e x)^{5/2}}","-\frac{12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac{10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac{40 b^3 \sqrt{d+e x} (b d-a e)^3}{e^7}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7}",1,"(2*(-7*(b*d - a*e)^6 + 70*b*(b*d - a*e)^5*(d + e*x) - 525*b^2*(b*d - a*e)^4*(d + e*x)^2 - 700*b^3*(b*d - a*e)^3*(d + e*x)^3 + 175*b^4*(b*d - a*e)^2*(d + e*x)^4 - 42*b^5*(b*d - a*e)*(d + e*x)^5 + 5*b^6*(d + e*x)^6))/(35*e^7*(d + e*x)^(5/2))","A",1
1645,1,50,162,0.025741,"\int \frac{(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 e (d+e x)^{11/2} \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right)}{11 (a e-b d)^2}","-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{11/2}}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2}",1,"(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(11*(-(b*d) + a*e)^2)","C",1
1646,1,50,137,0.0156631,"\int \frac{(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 e (d+e x)^{9/2} \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right)}{9 (a e-b d)^2}","-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{9/2}}+\frac{7 e \sqrt{d+e x} (b d-a e)^2}{b^4}+\frac{7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{7 e (d+e x)^{5/2}}{5 b^2}",1,"(2*e*(d + e*x)^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(9*(-(b*d) + a*e)^2)","C",1
1647,1,50,110,0.0153918,"\int \frac{(d+e x)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 e (d+e x)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{b (d+e x)}{a e-b d}\right)}{7 (a e-b d)^2}","-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{7/2}}+\frac{5 e \sqrt{d+e x} (b d-a e)}{b^3}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{5 e (d+e x)^{3/2}}{3 b^2}",1,"(2*e*(d + e*x)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(7*(-(b*d) + a*e)^2)","C",1
1648,1,50,85,0.013084,"\int \frac{(d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{2 e (d+e x)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (a e-b d)^2}","-\frac{3 e \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{5/2}}-\frac{(d+e x)^{3/2}}{b (a+b x)}+\frac{3 e \sqrt{d+e x}}{b^2}",1,"(2*e*(d + e*x)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^2)","C",1
1649,1,69,70,0.0779328,"\int \frac{\sqrt{d+e x}}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{e \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{b^{3/2} \sqrt{a e-b d}}-\frac{\sqrt{d+e x}}{b (a+b x)}","-\frac{e \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{3/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x}}{b (a+b x)}",1,"-(Sqrt[d + e*x]/(b*(a + b*x))) + (e*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(b^(3/2)*Sqrt[-(b*d) + a*e])","A",1
1650,1,76,76,0.0551753,"\int \frac{1}{\sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)),x]","\frac{e \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{\sqrt{b} (a e-b d)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{b} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x}}{(a+b x) (b d-a e)}",1,"-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(3/2))","A",1
1651,1,48,99,0.0135934,"\int \frac{1}{(d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]","-\frac{2 e \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{\sqrt{d+e x} (a e-b d)^2}","-\frac{3 e}{\sqrt{d+e x} (b d-a e)^2}-\frac{1}{(a+b x) \sqrt{d+e x} (b d-a e)}+\frac{3 \sqrt{b} e \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{(b d-a e)^{5/2}}",1,"(-2*e*Hypergeometric2F1[-1/2, 2, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^2*Sqrt[d + e*x])","C",1
1652,1,50,124,0.0146176,"\int \frac{1}{(d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]","-\frac{2 e \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{3 (d+e x)^{3/2} (a e-b d)^2}","\frac{5 b^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2}",1,"(-2*e*Hypergeometric2F1[-3/2, 2, -1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^2*(d + e*x)^(3/2))","C",1
1653,1,50,151,0.0175951,"\int \frac{1}{(d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)} \, dx","Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]","-\frac{2 e \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (d+e x)^{5/2} (a e-b d)^2}","\frac{7 b^{5/2} e \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{(b d-a e)^{9/2}}-\frac{7 b^2 e}{\sqrt{d+e x} (b d-a e)^4}-\frac{7 b e}{3 (d+e x)^{3/2} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{5/2} (b d-a e)}-\frac{7 e}{5 (d+e x)^{5/2} (b d-a e)^2}",1,"(-2*e*Hypergeometric2F1[-5/2, 2, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^2*(d + e*x)^(5/2))","C",1
1654,1,52,201,0.0268341,"\int \frac{(d+e x)^{11/2}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 e^3 (d+e x)^{13/2} \, _2F_1\left(4,\frac{13}{2};\frac{15}{2};-\frac{b (d+e x)}{a e-b d}\right)}{13 (a e-b d)^4}","-\frac{231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{13/2}}+\frac{231 e^3 \sqrt{d+e x} (b d-a e)^2}{8 b^6}+\frac{77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}-\frac{33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac{11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac{231 e^3 (d+e x)^{5/2}}{40 b^4}",1,"(2*e^3*(d + e*x)^(13/2)*Hypergeometric2F1[4, 13/2, 15/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(13*(-(b*d) + a*e)^4)","C",1
1655,1,52,172,0.0192331,"\int \frac{(d+e x)^{9/2}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 e^3 (d+e x)^{11/2} \, _2F_1\left(4,\frac{11}{2};\frac{13}{2};-\frac{b (d+e x)}{a e-b d}\right)}{11 (a e-b d)^4}","-\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{11/2}}+\frac{105 e^3 \sqrt{d+e x} (b d-a e)}{8 b^5}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4}",1,"(2*e^3*(d + e*x)^(11/2)*Hypergeometric2F1[4, 11/2, 13/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(11*(-(b*d) + a*e)^4)","C",1
1656,1,52,145,0.017339,"\int \frac{(d+e x)^{7/2}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 e^3 (d+e x)^{9/2} \, _2F_1\left(4,\frac{9}{2};\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right)}{9 (a e-b d)^4}","-\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{9/2}}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{35 e^3 \sqrt{d+e x}}{8 b^4}",1,"(2*e^3*(d + e*x)^(9/2)*Hypergeometric2F1[4, 9/2, 11/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(9*(-(b*d) + a*e)^4)","C",1
1657,1,119,126,0.1426346,"\int \frac{(d+e x)^{5/2}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{5 e^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{8 b^{7/2} \sqrt{a e-b d}}-\frac{\sqrt{d+e x} \left(15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left(8 d^2+26 d e x+33 e^2 x^2\right)\right)}{24 b^3 (a+b x)^3}","-\frac{5 e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{7/2} \sqrt{b d-a e}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 (a+b x)}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{5/2}}{3 b (a+b x)^3}",1,"-1/24*(Sqrt[d + e*x]*(15*a^2*e^2 + 10*a*b*e*(d + 4*e*x) + b^2*(8*d^2 + 26*d*e*x + 33*e^2*x^2)))/(b^3*(a + b*x)^3) + (5*e^3*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*b^(7/2)*Sqrt[-(b*d) + a*e])","A",1
1658,1,52,136,0.0150774,"\int \frac{(d+e x)^{3/2}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 e^3 (d+e x)^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (a e-b d)^4}","\frac{e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{5/2} (b d-a e)^{3/2}}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (a+b x) (b d-a e)}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}",1,"(2*e^3*(d + e*x)^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^4)","C",1
1659,1,52,146,0.0133481,"\int \frac{\sqrt{d+e x}}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{2 e^3 (d+e x)^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};-\frac{b (d+e x)}{a e-b d}\right)}{3 (a e-b d)^4}","-\frac{e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 b^{3/2} (b d-a e)^{5/2}}+\frac{e^2 \sqrt{d+e x}}{8 b (a+b x) (b d-a e)^2}-\frac{e \sqrt{d+e x}}{12 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x}}{3 b (a+b x)^3}",1,"(2*e^3*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^4)","C",1
1660,1,50,147,0.0121112,"\int \frac{1}{\sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","\frac{2 e^3 \sqrt{d+e x} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right)}{(a e-b d)^4}","\frac{5 e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 \sqrt{b} (b d-a e)^{7/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 (a+b x) (b d-a e)^3}+\frac{5 e \sqrt{d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x}}{3 (a+b x)^3 (b d-a e)}",1,"(2*e^3*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 4, 3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(-(b*d) + a*e)^4","C",1
1661,1,50,173,0.0129613,"\int \frac{1}{(d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","-\frac{2 e^3 \, _2F_1\left(-\frac{1}{2},4;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{\sqrt{d+e x} (a e-b d)^4}","-\frac{35 e^3}{8 \sqrt{d+e x} (b d-a e)^4}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 (b d-a e)^{9/2}}-\frac{35 e^2}{24 (a+b x) \sqrt{d+e x} (b d-a e)^3}+\frac{7 e}{12 (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 \sqrt{d+e x} (b d-a e)}",1,"(-2*e^3*Hypergeometric2F1[-1/2, 4, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^4*Sqrt[d + e*x])","C",1
1662,1,52,200,0.0182526,"\int \frac{1}{(d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","-\frac{2 e^3 \, _2F_1\left(-\frac{3}{2},4;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{3 (d+e x)^{3/2} (a e-b d)^4}","\frac{105 b^{3/2} e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 (b d-a e)^{11/2}}-\frac{105 b e^3}{8 \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}",1,"(-2*e^3*Hypergeometric2F1[-3/2, 4, -1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^4*(d + e*x)^(3/2))","C",1
1663,1,52,229,0.020192,"\int \frac{1}{(d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]","-\frac{2 e^3 \, _2F_1\left(-\frac{5}{2},4;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (d+e x)^{5/2} (a e-b d)^4}","\frac{231 b^{5/2} e^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{8 (b d-a e)^{13/2}}-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}",1,"(-2*e^3*Hypergeometric2F1[-5/2, 4, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^4*(d + e*x)^(5/2))","C",1
1664,1,52,253,0.028456,"\int \frac{(d+e x)^{15/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{17/2} \, _2F_1\left(6,\frac{17}{2};\frac{19}{2};-\frac{b (d+e x)}{a e-b d}\right)}{17 (a e-b d)^6}","-\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{17/2}}+\frac{9009 e^5 \sqrt{d+e x} (b d-a e)^2}{128 b^8}+\frac{3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}",1,"(2*e^5*(d + e*x)^(17/2)*Hypergeometric2F1[6, 17/2, 19/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(17*(-(b*d) + a*e)^6)","C",1
1665,1,52,224,0.0220757,"\int \frac{(d+e x)^{13/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{15/2} \, _2F_1\left(6,\frac{15}{2};\frac{17}{2};-\frac{b (d+e x)}{a e-b d}\right)}{15 (a e-b d)^6}","-\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{15/2}}+\frac{3003 e^5 \sqrt{d+e x} (b d-a e)}{128 b^7}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6}",1,"(2*e^5*(d + e*x)^(15/2)*Hypergeometric2F1[6, 15/2, 17/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(15*(-(b*d) + a*e)^6)","C",1
1666,1,52,197,0.0229365,"\int \frac{(d+e x)^{11/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{13/2} \, _2F_1\left(6,\frac{13}{2};\frac{15}{2};-\frac{b (d+e x)}{a e-b d}\right)}{13 (a e-b d)^6}","-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{13/2}}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6}",1,"(2*e^5*(d + e*x)^(13/2)*Hypergeometric2F1[6, 13/2, 15/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(13*(-(b*d) + a*e)^6)","C",1
1667,1,178,178,0.3048694,"\int \frac{(d+e x)^{9/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{63 e^5 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{128 b^{11/2} \sqrt{a e-b d}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}","-\frac{63 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{11/2} \sqrt{b d-a e}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5}",1,"(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5) + (63*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*Sqrt[-(b*d) + a*e])","A",1
1668,1,52,188,0.0183893,"\int \frac{(d+e x)^{7/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{9/2} \, _2F_1\left(\frac{9}{2},6;\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right)}{9 (a e-b d)^6}","\frac{7 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{9/2} (b d-a e)^{3/2}}-\frac{7 e^4 \sqrt{d+e x}}{128 b^4 (a+b x) (b d-a e)}-\frac{7 e^3 \sqrt{d+e x}}{64 b^4 (a+b x)^2}-\frac{7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac{7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{7/2}}{5 b (a+b x)^5}",1,"(2*e^5*(d + e*x)^(9/2)*Hypergeometric2F1[9/2, 6, 11/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(9*(-(b*d) + a*e)^6)","C",1
1669,1,52,198,0.0181477,"\int \frac{(d+e x)^{5/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{7/2} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};-\frac{b (d+e x)}{a e-b d}\right)}{7 (a e-b d)^6}","-\frac{3 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{7/2} (b d-a e)^{5/2}}+\frac{3 e^4 \sqrt{d+e x}}{128 b^3 (a+b x) (b d-a e)^2}-\frac{e^3 \sqrt{d+e x}}{64 b^3 (a+b x)^2 (b d-a e)}-\frac{e^2 \sqrt{d+e x}}{16 b^3 (a+b x)^3}-\frac{e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{5/2}}{5 b (a+b x)^5}",1,"(2*e^5*(d + e*x)^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(7*(-(b*d) + a*e)^6)","C",1
1670,1,52,208,0.0168532,"\int \frac{(d+e x)^{3/2}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{5/2} \, _2F_1\left(\frac{5}{2},6;\frac{7}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (a e-b d)^6}","\frac{3 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (a+b x) (b d-a e)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (a+b x)^2 (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (a+b x)^3 (b d-a e)}-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}",1,"(2*e^5*(d + e*x)^(5/2)*Hypergeometric2F1[5/2, 6, 7/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^6)","C",1
1671,1,52,218,0.0152211,"\int \frac{\sqrt{d+e x}}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{2 e^5 (d+e x)^{3/2} \, _2F_1\left(\frac{3}{2},6;\frac{5}{2};-\frac{b (d+e x)}{a e-b d}\right)}{3 (a e-b d)^6}","-\frac{7 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 b^{3/2} (b d-a e)^{9/2}}+\frac{7 e^4 \sqrt{d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac{7 e^3 \sqrt{d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac{7 e^2 \sqrt{d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac{e \sqrt{d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac{\sqrt{d+e x}}{5 b (a+b x)^5}",1,"(2*e^5*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 6, 5/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^6)","C",1
1672,1,50,213,0.0113825,"\int \frac{1}{\sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","\frac{2 e^5 \sqrt{d+e x} \, _2F_1\left(\frac{1}{2},6;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right)}{(a e-b d)^6}","\frac{63 e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 \sqrt{b} (b d-a e)^{11/2}}-\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)}",1,"(2*e^5*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 6, 3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(-(b*d) + a*e)^6","C",1
1673,1,50,239,0.0168952,"\int \frac{1}{(d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","-\frac{2 e^5 \, _2F_1\left(-\frac{1}{2},6;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{\sqrt{d+e x} (a e-b d)^6}","-\frac{693 e^5}{128 \sqrt{d+e x} (b d-a e)^6}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 (b d-a e)^{13/2}}-\frac{231 e^4}{128 (a+b x) \sqrt{d+e x} (b d-a e)^5}+\frac{231 e^3}{320 (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{33 e^2}{80 (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}+\frac{11 e}{40 (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 \sqrt{d+e x} (b d-a e)}",1,"(-2*e^5*Hypergeometric2F1[-1/2, 6, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^6*Sqrt[d + e*x])","C",1
1674,1,52,266,0.0208272,"\int \frac{1}{(d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","-\frac{2 e^5 \, _2F_1\left(-\frac{3}{2},6;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right)}{3 (d+e x)^{3/2} (a e-b d)^6}","\frac{3003 b^{3/2} e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 (b d-a e)^{15/2}}-\frac{3003 b e^5}{128 \sqrt{d+e x} (b d-a e)^7}-\frac{1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac{3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac{429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac{13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}",1,"(-2*e^5*Hypergeometric2F1[-3/2, 6, -1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^6*(d + e*x)^(3/2))","C",1
1675,1,52,295,0.0252273,"\int \frac{1}{(d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]","-\frac{2 e^5 \, _2F_1\left(-\frac{5}{2},6;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right)}{5 (d+e x)^{5/2} (a e-b d)^6}","\frac{9009 b^{5/2} e^5 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{128 (b d-a e)^{17/2}}-\frac{9009 b^2 e^5}{128 \sqrt{d+e x} (b d-a e)^8}-\frac{3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac{9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac{1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac{143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac{13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)}",1,"(-2*e^5*Hypergeometric2F1[-5/2, 6, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^6*(d + e*x)^(5/2))","C",1
1676,1,48,96,0.0334048,"\int (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} (9 a e-2 b d+7 b e x)}{63 e^2 (a+b x)}","\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^2 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(-2*b*d + 9*a*e + 7*b*e*x))/(63*e^2*(a + b*x))","A",1
1677,1,48,96,0.0255003,"\int (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} (7 a e-2 b d+5 b e x)}{35 e^2 (a+b x)}","\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^2 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(-2*b*d + 7*a*e + 5*b*e*x))/(35*e^2*(a + b*x))","A",1
1678,1,48,96,0.0223206,"\int \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} (5 a e-2 b d+3 b e x)}{15 e^2 (a+b x)}","\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^2 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(-2*b*d + 5*a*e + 3*b*e*x))/(15*e^2*(a + b*x))","A",1
1679,1,47,94,0.0268997,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/Sqrt[d + e*x],x]","\frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} (3 a e-2 b d+b e x)}{3 e^2 (a+b x)}","\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^2 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(-2*b*d + 3*a*e + b*e*x))/(3*e^2*(a + b*x))","A",1
1680,1,45,92,0.0241708,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(3/2),x]","\frac{2 \sqrt{(a+b x)^2} (-a e+2 b d+b e x)}{e^2 (a+b x) \sqrt{d+e x}}","\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^2 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt{d+e x}}",1,"(2*Sqrt[(a + b*x)^2]*(2*b*d - a*e + b*e*x))/(e^2*(a + b*x)*Sqrt[d + e*x])","A",1
1681,1,47,94,0.023703,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(5/2),x]","-\frac{2 \sqrt{(a+b x)^2} (a e+2 b d+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}}","\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}}",1,"(-2*Sqrt[(a + b*x)^2]*(2*b*d + a*e + 3*b*e*x))/(3*e^2*(a + b*x)*(d + e*x)^(3/2))","A",1
1682,1,48,96,0.0299314,"\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(7/2),x]","-\frac{2 \sqrt{(a+b x)^2} (3 a e+2 b d+5 b e x)}{15 e^2 (a+b x) (d+e x)^{5/2}}","\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^{5/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^{3/2}}",1,"(-2*Sqrt[(a + b*x)^2]*(2*b*d + 3*a*e + 5*b*e*x))/(15*e^2*(a + b*x)*(d + e*x)^(5/2))","A",1
1683,1,120,208,0.0888031,"\int (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left(429 a^3 e^3+143 a^2 b e^2 (7 e x-2 d)+13 a b^2 e \left(8 d^2-28 d e x+63 e^2 x^2\right)+b^3 \left(-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right)\right)}{3003 e^4 (a+b x)}","-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^4 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^4 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3)))/(3003*e^4*(a + b*x))","A",1
1684,1,120,208,0.0673134,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} \left(231 a^3 e^3+99 a^2 b e^2 (5 e x-2 d)+11 a b^2 e \left(8 d^2-20 d e x+35 e^2 x^2\right)+b^3 \left(-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right)\right)}{1155 e^4 (a+b x)}","-\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^4 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/(1155*e^4*(a + b*x))","A",1
1685,1,120,208,0.0629985,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left(105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left(8 d^2-12 d e x+15 e^2 x^2\right)+b^3 \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)\right)}{315 e^4 (a+b x)}","-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) + 9*a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)))/(315*e^4*(a + b*x))","A",1
1686,1,119,204,0.0687433,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left(35 a^3 e^3+35 a^2 b e^2 (e x-2 d)+7 a b^2 e \left(8 d^2-4 d e x+3 e^2 x^2\right)+b^3 \left(-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right)\right)}{35 e^4 (a+b x)}","-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^4 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^4 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(35*a^3*e^3 + 35*a^2*b*e^2*(-2*d + e*x) + 7*a*b^2*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + b^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)))/(35*e^4*(a + b*x))","A",1
1687,1,119,202,0.0589641,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(3/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(5 a^3 e^3-15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left(8 d^2+4 d e x-e^2 x^2\right)-\left(b^3 \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)\right)\right)}{5 e^4 (a+b x) \sqrt{d+e x}}","-\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^4 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^4 (a+b x)}",1,"(-2*Sqrt[(a + b*x)^2]*(5*a^3*e^3 - 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(8*d^2 + 4*d*e*x - e^2*x^2) - b^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(5*e^4*(a + b*x)*Sqrt[d + e*x])","A",1
1688,1,119,204,0.0644769,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(5/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left(8 d^2+12 d e x+3 e^2 x^2\right)+b^3 \left(16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right)\right)}{3 e^4 (a+b x) (d+e x)^{3/2}}","-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^4 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)}",1,"(-2*Sqrt[(a + b*x)^2]*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)))/(3*e^4*(a + b*x)*(d + e*x)^(3/2))","A",1
1689,1,118,202,0.0635642,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(7/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left(8 d^2+20 d e x+15 e^2 x^2\right)-\left(b^3 \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)\right)\right)}{5 e^4 (a+b x) (d+e x)^{5/2}}","\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) \sqrt{d+e x}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^4 (a+b x)}",1,"(-2*Sqrt[(a + b*x)^2]*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)))/(5*e^4*(a + b*x)*(d + e*x)^(5/2))","A",1
1690,1,119,204,0.0625701,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(9/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(5 a^3 e^3+3 a^2 b e^2 (2 d+7 e x)+a b^2 e \left(8 d^2+28 d e x+35 e^2 x^2\right)+b^3 \left(16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right)\right)}{35 e^4 (a+b x) (d+e x)^{7/2}}","\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^4 (a+b x) (d+e x)^{3/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^4 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^{7/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt{d+e x}}",1,"(-2*Sqrt[(a + b*x)^2]*(5*a^3*e^3 + 3*a^2*b*e^2*(2*d + 7*e*x) + a*b^2*e*(8*d^2 + 28*d*e*x + 35*e^2*x^2) + b^3*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3)))/(35*e^4*(a + b*x)*(d + e*x)^(7/2))","A",1
1691,1,120,208,0.0654223,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(11/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(35 a^3 e^3+15 a^2 b e^2 (2 d+9 e x)+3 a b^2 e \left(8 d^2+36 d e x+63 e^2 x^2\right)+b^3 \left(16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right)\right)}{315 e^4 (a+b x) (d+e x)^{9/2}}","\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}",1,"(-2*Sqrt[(a + b*x)^2]*(35*a^3*e^3 + 15*a^2*b*e^2*(2*d + 9*e*x) + 3*a*b^2*e*(8*d^2 + 36*d*e*x + 63*e^2*x^2) + b^3*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3)))/(315*e^4*(a + b*x)*(d + e*x)^(9/2))","A",1
1692,1,141,320,0.1740358,"\int (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{2 \left((a+b x)^2\right)^{5/2} (d+e x)^{7/2} \left(-51051 b^4 (d+e x)^4 (b d-a e)+117810 b^3 (d+e x)^3 (b d-a e)^2-139230 b^2 (d+e x)^2 (b d-a e)^3+85085 b (d+e x) (b d-a e)^4-21879 (b d-a e)^5+9009 b^5 (d+e x)^5\right)}{153153 e^6 (a+b x)^5}","-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)}",1,"(2*((a + b*x)^2)^(5/2)*(d + e*x)^(7/2)*(-21879*(b*d - a*e)^5 + 85085*b*(b*d - a*e)^4*(d + e*x) - 139230*b^2*(b*d - a*e)^3*(d + e*x)^2 + 117810*b^3*(b*d - a*e)^2*(d + e*x)^3 - 51051*b^4*(b*d - a*e)*(d + e*x)^4 + 9009*b^5*(d + e*x)^5))/(153153*e^6*(a + b*x)^5)","A",1
1693,1,141,320,0.1349997,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{2 \left((a+b x)^2\right)^{5/2} (d+e x)^{5/2} \left(-17325 b^4 (d+e x)^4 (b d-a e)+40950 b^3 (d+e x)^3 (b d-a e)^2-50050 b^2 (d+e x)^2 (b d-a e)^3+32175 b (d+e x) (b d-a e)^4-9009 (b d-a e)^5+3003 b^5 (d+e x)^5\right)}{45045 e^6 (a+b x)^5}","-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^6 (a+b x)}",1,"(2*((a + b*x)^2)^(5/2)*(d + e*x)^(5/2)*(-9009*(b*d - a*e)^5 + 32175*b*(b*d - a*e)^4*(d + e*x) - 50050*b^2*(b*d - a*e)^3*(d + e*x)^2 + 40950*b^3*(b*d - a*e)^2*(d + e*x)^3 - 17325*b^4*(b*d - a*e)*(d + e*x)^4 + 3003*b^5*(d + e*x)^5))/(45045*e^6*(a + b*x)^5)","A",1
1694,1,235,318,0.1140911,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left(3003 a^5 e^5+3003 a^4 b e^4 (3 e x-2 d)+858 a^3 b^2 e^3 \left(8 d^2-12 d e x+15 e^2 x^2\right)+286 a^2 b^3 e^2 \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)+13 a b^4 e \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)+b^5 \left(-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right)\right)}{9009 e^6 (a+b x)}","-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5 + 3003*a^4*b*e^4*(-2*d + 3*e*x) + 858*a^3*b^2*e^3*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 286*a^2*b^3*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 13*a*b^4*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + b^5*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5)))/(9009*e^6*(a + b*x))","A",1
1695,1,234,316,0.119393,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left(693 a^5 e^5+1155 a^4 b e^4 (e x-2 d)+462 a^3 b^2 e^3 \left(8 d^2-4 d e x+3 e^2 x^2\right)+198 a^2 b^3 e^2 \left(-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right)+11 a b^4 e \left(128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right)+b^5 \left(-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right)\right)}{693 e^6 (a+b x)}","-\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(693*a^5*e^5 + 1155*a^4*b*e^4*(-2*d + e*x) + 462*a^3*b^2*e^3*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 198*a^2*b^3*e^2*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 11*a*b^4*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + b^5*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5)))/(693*e^6*(a + b*x))","A",1
1696,1,232,314,0.1298419,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(3/2),x]","\frac{2 \sqrt{(a+b x)^2} \left(-63 a^5 e^5+315 a^4 b e^4 (2 d+e x)+210 a^3 b^2 e^3 \left(-8 d^2-4 d e x+e^2 x^2\right)+126 a^2 b^3 e^2 \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)+9 a b^4 e \left(-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right)+b^5 \left(256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right)\right)}{63 e^6 (a+b x) \sqrt{d+e x}}","-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^6 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^6 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^6 (a+b x)}",1,"(2*Sqrt[(a + b*x)^2]*(-63*a^5*e^5 + 315*a^4*b*e^4*(2*d + e*x) + 210*a^3*b^2*e^3*(-8*d^2 - 4*d*e*x + e^2*x^2) + 126*a^2*b^3*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 9*a*b^4*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + b^5*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(63*e^6*(a + b*x)*Sqrt[d + e*x])","A",1
1697,1,235,314,0.1126205,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(5/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(7 a^5 e^5+35 a^4 b e^4 (2 d+3 e x)-70 a^3 b^2 e^3 \left(8 d^2+12 d e x+3 e^2 x^2\right)+70 a^2 b^3 e^2 \left(16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right)-7 a b^4 e \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)+b^5 \left(256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right)\right)}{21 e^6 (a+b x) (d+e x)^{3/2}}","-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)}",1,"(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 35*a^4*b*e^4*(2*d + 3*e*x) - 70*a^3*b^2*e^3*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 70*a^2*b^3*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3) - 7*a*b^4*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + b^5*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(21*e^6*(a + b*x)*(d + e*x)^(3/2))","A",1
1698,1,236,316,0.1117828,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(7/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(3 a^5 e^5+5 a^4 b e^4 (2 d+5 e x)+10 a^3 b^2 e^3 \left(8 d^2+20 d e x+15 e^2 x^2\right)-30 a^2 b^3 e^2 \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)+5 a b^4 e \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)-\left(b^5 \left(256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right)\right)\right)}{15 e^6 (a+b x) (d+e x)^{5/2}}","\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) \sqrt{d+e x}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^6 (a+b x)}",1,"(-2*Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 5*a^4*b*e^4*(2*d + 5*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 30*a^2*b^3*e^2*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 5*a*b^4*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) - b^5*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5)))/(15*e^6*(a + b*x)*(d + e*x)^(5/2))","A",1
1699,1,235,314,0.1094898,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(9/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(3 a^5 e^5+3 a^4 b e^4 (2 d+7 e x)+2 a^3 b^2 e^3 \left(8 d^2+28 d e x+35 e^2 x^2\right)+6 a^2 b^3 e^2 \left(16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right)-3 a b^4 e \left(128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right)+b^5 \left(256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right)\right)}{21 e^6 (a+b x) (d+e x)^{7/2}}","\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^6 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) \sqrt{d+e x}}",1,"(-2*Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 3*a^4*b*e^4*(2*d + 7*e*x) + 2*a^3*b^2*e^3*(8*d^2 + 28*d*e*x + 35*e^2*x^2) + 6*a^2*b^3*e^2*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) - 3*a*b^4*e*(128*d^4 + 448*d^3*e*x + 560*d^2*e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4) + b^5*(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d*e^4*x^4 - 7*e^5*x^5)))/(21*e^6*(a + b*x)*(d + e*x)^(7/2))","A",1
1700,1,235,314,0.112656,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(7 a^5 e^5+5 a^4 b e^4 (2 d+9 e x)+2 a^3 b^2 e^3 \left(8 d^2+36 d e x+63 e^2 x^2\right)+2 a^2 b^3 e^2 \left(16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right)+a b^4 e \left(128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right)-\left(b^5 \left(256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right)\right)\right)}{63 e^6 (a+b x) (d+e x)^{9/2}}","\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^6 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}}",1,"(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 5*a^4*b*e^4*(2*d + 9*e*x) + 2*a^3*b^2*e^3*(8*d^2 + 36*d*e*x + 63*e^2*x^2) + 2*a^2*b^3*e^2*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + a*b^4*e*(128*d^4 + 576*d^3*e*x + 1008*d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4) - b^5*(256*d^5 + 1152*d^4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 + 630*d*e^4*x^4 + 63*e^5*x^5)))/(63*e^6*(a + b*x)*(d + e*x)^(9/2))","A",1
1701,1,234,316,0.1107513,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{13/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(63 a^5 e^5+35 a^4 b e^4 (2 d+11 e x)+10 a^3 b^2 e^3 \left(8 d^2+44 d e x+99 e^2 x^2\right)+6 a^2 b^3 e^2 \left(16 d^3+88 d^2 e x+198 d e^2 x^2+231 e^3 x^3\right)+a b^4 e \left(128 d^4+704 d^3 e x+1584 d^2 e^2 x^2+1848 d e^3 x^3+1155 e^4 x^4\right)+b^5 \left(256 d^5+1408 d^4 e x+3168 d^3 e^2 x^2+3696 d^2 e^3 x^3+2310 d e^4 x^4+693 e^5 x^5\right)\right)}{693 e^6 (a+b x) (d+e x)^{11/2}}","\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}}",1,"(-2*Sqrt[(a + b*x)^2]*(63*a^5*e^5 + 35*a^4*b*e^4*(2*d + 11*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 44*d*e*x + 99*e^2*x^2) + 6*a^2*b^3*e^2*(16*d^3 + 88*d^2*e*x + 198*d*e^2*x^2 + 231*e^3*x^3) + a*b^4*e*(128*d^4 + 704*d^3*e*x + 1584*d^2*e^2*x^2 + 1848*d*e^3*x^3 + 1155*e^4*x^4) + b^5*(256*d^5 + 1408*d^4*e*x + 3168*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 + 2310*d*e^4*x^4 + 693*e^5*x^5)))/(693*e^6*(a + b*x)*(d + e*x)^(11/2))","A",1
1702,1,235,318,0.1157798,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}}{(d+e x)^{15/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(15/2),x]","-\frac{2 \sqrt{(a+b x)^2} \left(693 a^5 e^5+315 a^4 b e^4 (2 d+13 e x)+70 a^3 b^2 e^3 \left(8 d^2+52 d e x+143 e^2 x^2\right)+30 a^2 b^3 e^2 \left(16 d^3+104 d^2 e x+286 d e^2 x^2+429 e^3 x^3\right)+3 a b^4 e \left(128 d^4+832 d^3 e x+2288 d^2 e^2 x^2+3432 d e^3 x^3+3003 e^4 x^4\right)+b^5 \left(256 d^5+1664 d^4 e x+4576 d^3 e^2 x^2+6864 d^2 e^3 x^3+6006 d e^4 x^4+3003 e^5 x^5\right)\right)}{9009 e^6 (a+b x) (d+e x)^{13/2}}","\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^6 (a+b x) (d+e x)^{11/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{13 e^6 (a+b x) (d+e x)^{13/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^{7/2}}",1,"(-2*Sqrt[(a + b*x)^2]*(693*a^5*e^5 + 315*a^4*b*e^4*(2*d + 13*e*x) + 70*a^3*b^2*e^3*(8*d^2 + 52*d*e*x + 143*e^2*x^2) + 30*a^2*b^3*e^2*(16*d^3 + 104*d^2*e*x + 286*d*e^2*x^2 + 429*e^3*x^3) + 3*a*b^4*e*(128*d^4 + 832*d^3*e*x + 2288*d^2*e^2*x^2 + 3432*d*e^3*x^3 + 3003*e^4*x^4) + b^5*(256*d^5 + 1664*d^4*e*x + 4576*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 + 6006*d*e^4*x^4 + 3003*e^5*x^5)))/(9009*e^6*(a + b*x)*(d + e*x)^(13/2))","A",1
1703,1,150,263,0.2490986,"\int \frac{(d+e x)^{7/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{(a+b x) \left(\frac{14 (b d-a e) \left(5 (b d-a e) \left(\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)\right)+3 b^{5/2} (d+e x)^{5/2}\right)}{15 b^{7/2}}+2 (d+e x)^{7/2}\right)}{7 b \sqrt{(a+b x)^2}}","\frac{2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(2*(d + e*x)^(7/2) + (14*(b*d - a*e)*(3*b^(5/2)*(d + e*x)^(5/2) + 5*(b*d - a*e)*(Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) - 3*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])))/(15*b^(7/2))))/(7*b*Sqrt[(a + b*x)^2])","A",1
1704,1,127,212,0.1037935,"\int \frac{(d+e x)^{5/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 (a+b x) \left(\sqrt{b} \sqrt{d+e x} \left(15 a^2 e^2-5 a b e (7 d+e x)+b^2 \left(23 d^2+11 d e x+3 e^2 x^2\right)\right)-15 (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)\right)}{15 b^{7/2} \sqrt{(a+b x)^2}}","\frac{2 (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x]*(15*a^2*e^2 - 5*a*b*e*(7*d + e*x) + b^2*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) - 15*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))/(15*b^(7/2)*Sqrt[(a + b*x)^2])","A",1
1705,1,96,161,0.054912,"\int \frac{(d+e x)^{3/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 (a+b x) \left(\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)\right)}{3 b^{5/2} \sqrt{(a+b x)^2}}","\frac{2 (a+b x) \sqrt{d+e x} (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) - 3*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))/(3*b^(5/2)*Sqrt[(a + b*x)^2])","A",1
1706,1,81,112,0.0210955,"\int \frac{\sqrt{d+e x}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{2 (a+b x) \left(\sqrt{b} \sqrt{d+e x}-\sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)\right)}{b^{3/2} \sqrt{(a+b x)^2}}","\frac{2 (a+b x) \sqrt{d+e x}}{b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x] - Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))/(b^(3/2)*Sqrt[(a + b*x)^2])","A",1
1707,1,63,72,0.0193102,"\int \frac{1}{\sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{b} \sqrt{(a+b x)^2} \sqrt{b d-a e}}","-\frac{2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}",1,"(-2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*Sqrt[b*d - a*e]*Sqrt[(a + b*x)^2])","A",1
1708,1,62,119,0.0162548,"\int \frac{1}{(d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","-\frac{2 (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{\sqrt{(a+b x)^2} \sqrt{d+e x} (a e-b d)}","\frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}",1,"(-2*(a + b*x)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(d + e*x))/(b*d - a*e)])/((-(b*d) + a*e)*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])","C",1
1709,1,64,168,0.016668,"\int \frac{1}{(d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{2 (a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (b d-a e)}","\frac{2 b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}+\frac{2 (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}",1,"(2*(a + b*x)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2))","C",1
1710,1,64,219,0.0185444,"\int \frac{1}{(d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/((d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{2 (a+b x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 \sqrt{(a+b x)^2} (d+e x)^{5/2} (b d-a e)}","\frac{2 b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 b (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac{2 (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{5/2} (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}",1,"(2*(a + b*x)*Hypergeometric2F1[-5/2, 1, -3/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2))","C",1
1711,1,67,308,0.045652,"\int \frac{(d+e x)^{9/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","-\frac{2 e^2 (a+b x) (d+e x)^{11/2} \, _2F_1\left(3,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right)}{11 \sqrt{(a+b x)^2} (b d-a e)^3}","-\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^2*(a + b*x)*(d + e*x)^(11/2)*Hypergeometric2F1[3, 11/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(11*(b*d - a*e)^3*Sqrt[(a + b*x)^2])","C",1
1712,1,67,254,0.0372901,"\int \frac{(d+e x)^{7/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","-\frac{2 e^2 (a+b x) (d+e x)^{9/2} \, _2F_1\left(3,\frac{9}{2};\frac{11}{2};\frac{b (d+e x)}{b d-a e}\right)}{9 \sqrt{(a+b x)^2} (b d-a e)^3}","-\frac{(d+e x)^{7/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) \sqrt{d+e x} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^2*(a + b*x)*(d + e*x)^(9/2)*Hypergeometric2F1[3, 9/2, 11/2, (b*(d + e*x))/(b*d - a*e)])/(9*(b*d - a*e)^3*Sqrt[(a + b*x)^2])","C",1
1713,1,67,202,0.0373257,"\int \frac{(d+e x)^{5/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","-\frac{2 e^2 (a+b x) (d+e x)^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{b (d+e x)}{b d-a e}\right)}{7 \sqrt{(a+b x)^2} (b d-a e)^3}","-\frac{(d+e x)^{5/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 e^2 (a+b x) \sqrt{d+e x}}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^2*(a + b*x)*(d + e*x)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, (b*(d + e*x))/(b*d - a*e)])/(7*(b*d - a*e)^3*Sqrt[(a + b*x)^2])","C",1
1714,1,110,158,0.1053761,"\int \frac{(d+e x)^{3/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{\frac{3 e^2 (a+b x)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{\sqrt{a e-b d}}-\sqrt{b} \sqrt{d+e x} (3 a e+2 b d+5 b e x)}{4 b^{5/2} (a+b x) \sqrt{(a+b x)^2}}","-\frac{3 e \sqrt{d+e x}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}",1,"(-(Sqrt[b]*Sqrt[d + e*x]*(2*b*d + 3*a*e + 5*b*e*x)) + (3*e^2*(a + b*x)^2*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/Sqrt[-(b*d) + a*e])/(4*b^(5/2)*(a + b*x)*Sqrt[(a + b*x)^2])","A",1
1715,1,67,168,0.0273142,"\int \frac{\sqrt{d+e x}}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","-\frac{2 e^2 (a+b x) (d+e x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 \sqrt{(a+b x)^2} (b d-a e)^3}","-\frac{e \sqrt{d+e x}}{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}",1,"(-2*e^2*(a + b*x)*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)^3*Sqrt[(a + b*x)^2])","C",1
1716,1,65,172,0.0205015,"\int \frac{1}{\sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","-\frac{2 e^2 (a+b x) \sqrt{d+e x} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{\sqrt{(a+b x)^2} (b d-a e)^3}","-\frac{3 e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac{3 e \sqrt{d+e x}}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"(-2*e^2*(a + b*x)*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 3, 3/2, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)^3*Sqrt[(a + b*x)^2])","C",1
1717,1,65,223,0.0217119,"\int \frac{1}{(d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{2 e^2 (a+b x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{\sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)^3}","\frac{15 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}+\frac{5 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}",1,"(2*e^2*(a + b*x)*Hypergeometric2F1[-1/2, 3, 1/2, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)^3*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])","C",1
1718,1,67,275,0.0244295,"\int \frac{1}{(d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{2 e^2 (a+b x) \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (b d-a e)^3}","\frac{35 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2 (a+b x)}{12 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}",1,"(2*e^2*(a + b*x)*Hypergeometric2F1[-3/2, 3, -1/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)^3*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2))","C",1
1719,1,67,329,0.0284722,"\int \frac{1}{(d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]","\frac{2 e^2 (a+b x) \, _2F_1\left(-\frac{5}{2},3;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 \sqrt{(a+b x)^2} (d+e x)^{5/2} (b d-a e)^3}","\frac{63 b^2 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2 (a+b x)}{20 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{9 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}",1,"(2*e^2*(a + b*x)*Hypergeometric2F1[-5/2, 3, -3/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)^3*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2))","C",1
1720,1,67,400,0.0567673,"\int \frac{(d+e x)^{13/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{15/2} \, _2F_1\left(5,\frac{15}{2};\frac{17}{2};\frac{b (d+e x)}{b d-a e}\right)}{15 \sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) \sqrt{d+e x} (b d-a e)^2}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(15/2)*Hypergeometric2F1[5, 15/2, 17/2, (b*(d + e*x))/(b*d - a*e)])/(15*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1721,1,67,346,0.0482765,"\int \frac{(d+e x)^{11/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{13/2} \, _2F_1\left(5,\frac{13}{2};\frac{15}{2};\frac{b (d+e x)}{b d-a e}\right)}{13 \sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1155 e^4 (a+b x) \sqrt{d+e x} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(13/2)*Hypergeometric2F1[5, 13/2, 15/2, (b*(d + e*x))/(b*d - a*e)])/(13*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1722,1,67,294,0.0431048,"\int \frac{(d+e x)^{9/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{11/2} \, _2F_1\left(5,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right)}{11 \sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 e^4 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x) \sqrt{d+e x}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(11/2)*Hypergeometric2F1[5, 11/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(11*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1723,1,182,250,0.2193697,"\int \frac{(d+e x)^{7/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{\frac{105 e^4 (a+b x)^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right)}{\sqrt{a e-b d}}-\sqrt{b} \sqrt{d+e x} \left(105 a^3 e^3+35 a^2 b e^2 (2 d+11 e x)+7 a b^2 e \left(8 d^2+36 d e x+73 e^2 x^2\right)+b^3 \left(48 d^3+200 d^2 e x+326 d e^2 x^2+279 e^3 x^3\right)\right)}{192 b^{9/2} (a+b x)^3 \sqrt{(a+b x)^2}}","-\frac{7 e (d+e x)^{5/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{7/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{35 e^3 \sqrt{d+e x}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (d+e x)^{3/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-(Sqrt[b]*Sqrt[d + e*x]*(105*a^3*e^3 + 35*a^2*b*e^2*(2*d + 11*e*x) + 7*a*b^2*e*(8*d^2 + 36*d*e*x + 73*e^2*x^2) + b^3*(48*d^3 + 200*d^2*e*x + 326*d*e^2*x^2 + 279*e^3*x^3))) + (105*e^4*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/Sqrt[-(b*d) + a*e])/(192*b^(9/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])","A",1
1724,1,67,260,0.0367509,"\int \frac{(d+e x)^{5/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{b (d+e x)}{b d-a e}\right)}{7 \sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac{5 e^3 \sqrt{d+e x}}{64 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e^2 \sqrt{d+e x}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (b*(d + e*x))/(b*d - a*e)])/(7*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1725,1,67,270,0.0307984,"\int \frac{(d+e x)^{3/2}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 \sqrt{(a+b x)^2} (b d-a e)^5}","\frac{3 e^3 \sqrt{d+e x}}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{32 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e \sqrt{d+e x}}{8 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1726,1,67,280,0.0273442,"\int \frac{\sqrt{d+e x}}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[Sqrt[d + e*x]/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","-\frac{2 e^4 (a+b x) (d+e x)^{3/2} \, _2F_1\left(\frac{3}{2},5;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 \sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{5 e^3 \sqrt{d+e x}}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{5 e^2 \sqrt{d+e x}}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e \sqrt{d+e x}}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}",1,"(-2*e^4*(a + b*x)*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 5, 5/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1727,1,65,278,0.0230036,"\int \frac{1}{\sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","-\frac{2 e^4 (a+b x) \sqrt{d+e x} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{\sqrt{(a+b x)^2} (b d-a e)^5}","-\frac{35 e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{35 e^3 \sqrt{d+e x}}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{35 e^2 \sqrt{d+e x}}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{7 e \sqrt{d+e x}}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"(-2*e^4*(a + b*x)*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 5, 3/2, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)^5*Sqrt[(a + b*x)^2])","C",1
1728,1,65,329,0.0233129,"\int \frac{1}{(d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{2 e^4 (a+b x) \, _2F_1\left(-\frac{1}{2},5;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{\sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)^5}","\frac{315 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{105 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}",1,"(2*e^4*(a + b*x)*Hypergeometric2F1[-1/2, 5, 1/2, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)^5*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])","C",1
1729,1,67,381,0.0285947,"\int \frac{1}{(d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{2 e^4 (a+b x) \, _2F_1\left(-\frac{3}{2},5;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (b d-a e)^5}","\frac{1155 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}+\frac{385 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac{231 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{33 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{11 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}",1,"(2*e^4*(a + b*x)*Hypergeometric2F1[-3/2, 5, -1/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)^5*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2))","C",1
1730,1,67,435,0.0319083,"\int \frac{1}{(d+e x)^{7/2} \left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]","\frac{2 e^4 (a+b x) \, _2F_1\left(-\frac{5}{2},5;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 \sqrt{(a+b x)^2} (d+e x)^{5/2} (b d-a e)^5}","\frac{3003 b^2 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^7}+\frac{1001 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac{3003 e^4 (a+b x)}{320 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac{429 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac{143 e^2}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac{13 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}",1,"(2*e^4*(a + b*x)*Hypergeometric2F1[-5/2, 5, -3/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)^5*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2))","C",1
1731,1,175,206,0.1787417,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{(d+e x)^{m+1} \left(-\frac{6 b^5 (d+e x)^5 (b d-a e)}{m+6}+\frac{15 b^4 (d+e x)^4 (b d-a e)^2}{m+5}-\frac{20 b^3 (d+e x)^3 (b d-a e)^3}{m+4}+\frac{15 b^2 (d+e x)^2 (b d-a e)^4}{m+3}-\frac{6 b (d+e x) (b d-a e)^5}{m+2}+\frac{(b d-a e)^6}{m+1}+\frac{b^6 (d+e x)^6}{m+7}\right)}{e^7}","-\frac{6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}+\frac{(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{b^6 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*((b*d - a*e)^6/(1 + m) - (6*b*(b*d - a*e)^5*(d + e*x))/(2 + m) + (15*b^2*(b*d - a*e)^4*(d + e*x)^2)/(3 + m) - (20*b^3*(b*d - a*e)^3*(d + e*x)^3)/(4 + m) + (15*b^4*(b*d - a*e)^2*(d + e*x)^4)/(5 + m) - (6*b^5*(b*d - a*e)*(d + e*x)^5)/(6 + m) + (b^6*(d + e*x)^6)/(7 + m)))/e^7","A",1
1732,1,121,142,0.1079366,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(-\frac{4 b^3 (d+e x)^3 (b d-a e)}{m+4}+\frac{6 b^2 (d+e x)^2 (b d-a e)^2}{m+3}-\frac{4 b (d+e x) (b d-a e)^3}{m+2}+\frac{(b d-a e)^4}{m+1}+\frac{b^4 (d+e x)^4}{m+5}\right)}{e^5}","-\frac{4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac{(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac{b^4 (d+e x)^{m+5}}{e^5 (m+5)}",1,"((d + e*x)^(1 + m)*((b*d - a*e)^4/(1 + m) - (4*b*(b*d - a*e)^3*(d + e*x))/(2 + m) + (6*b^2*(b*d - a*e)^2*(d + e*x)^2)/(3 + m) - (4*b^3*(b*d - a*e)*(d + e*x)^3)/(4 + m) + (b^4*(d + e*x)^4)/(5 + m)))/e^5","A",1
1733,1,67,78,0.0703847,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right) \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{(d+e x)^{m+1} \left(-\frac{2 b (d+e x) (b d-a e)}{m+2}+\frac{(b d-a e)^2}{m+1}+\frac{b^2 (d+e x)^2}{m+3}\right)}{e^3}","\frac{(b d-a e)^2 (d+e x)^{m+1}}{e^3 (m+1)}-\frac{2 b (b d-a e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{b^2 (d+e x)^{m+3}}{e^3 (m+3)}",1,"((d + e*x)^(1 + m)*((b*d - a*e)^2/(1 + m) - (2*b*(b*d - a*e)*(d + e*x))/(2 + m) + (b^2*(d + e*x)^2)/(3 + m)))/e^3","A",1
1734,1,52,51,0.0135895,"\int \frac{(d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2),x]","\frac{e (d+e x)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right)}{(m+1) (a e-b d)^2}","\frac{e (d+e x)^{m+1} \, _2F_1\left(2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) (b d-a e)^2}",1,"(e*(d + e*x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^2*(1 + m))","A",1
1735,1,54,53,0.0153769,"\int \frac{(d+e x)^m}{\left(a^2+2 a b x+b^2 x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^2,x]","\frac{e^3 (d+e x)^{m+1} \, _2F_1\left(4,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right)}{(m+1) (a e-b d)^4}","\frac{e^3 (d+e x)^{m+1} \, _2F_1\left(4,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) (b d-a e)^4}",1,"(e^3*(d + e*x)^(1 + m)*Hypergeometric2F1[4, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^4*(1 + m))","A",1
1736,1,54,53,0.0158826,"\int \frac{(d+e x)^m}{\left(a^2+2 a b x+b^2 x^2\right)^3} \, dx","Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^3,x]","\frac{e^5 (d+e x)^{m+1} \, _2F_1\left(6,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right)}{(m+1) (a e-b d)^6}","\frac{e^5 (d+e x)^{m+1} \, _2F_1\left(6,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) (b d-a e)^6}",1,"(e^5*(d + e*x)^(1 + m)*Hypergeometric2F1[6, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^6*(1 + m))","A",1
1737,1,167,337,0.2569182,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{\left((a+b x)^2\right)^{5/2} (d+e x)^{m+1} \left(-\frac{5 b^4 (d+e x)^4 (b d-a e)}{m+5}+\frac{10 b^3 (d+e x)^3 (b d-a e)^2}{m+4}-\frac{10 b^2 (d+e x)^2 (b d-a e)^3}{m+3}+\frac{5 b (d+e x) (b d-a e)^4}{m+2}-\frac{(b d-a e)^5}{m+1}+\frac{b^5 (d+e x)^5}{m+6}\right)}{e^6 (a+b x)^5}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1) (a+b x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2) (a+b x)}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3) (a+b x)}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+6}}{e^6 (m+6) (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+5}}{e^6 (m+5) (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4) (a+b x)}",1,"(((a + b*x)^2)^(5/2)*(d + e*x)^(1 + m)*(-((b*d - a*e)^5/(1 + m)) + (5*b*(b*d - a*e)^4*(d + e*x))/(2 + m) - (10*b^2*(b*d - a*e)^3*(d + e*x)^2)/(3 + m) + (10*b^3*(b*d - a*e)^2*(d + e*x)^3)/(4 + m) - (5*b^4*(b*d - a*e)*(d + e*x)^4)/(5 + m) + (b^5*(d + e*x)^5)/(6 + m)))/(e^6*(a + b*x)^5)","A",1
1738,1,113,219,0.1209173,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{\left((a+b x)^2\right)^{3/2} (d+e x)^{m+1} \left(-\frac{3 b^2 (d+e x)^2 (b d-a e)}{m+3}+\frac{3 b (d+e x) (b d-a e)^2}{m+2}-\frac{(b d-a e)^3}{m+1}+\frac{b^3 (d+e x)^3}{m+4}\right)}{e^4 (a+b x)^3}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)}",1,"(((a + b*x)^2)^(3/2)*(d + e*x)^(1 + m)*(-((b*d - a*e)^3/(1 + m)) + (3*b*(b*d - a*e)^2*(d + e*x))/(2 + m) - (3*b^2*(b*d - a*e)*(d + e*x)^2)/(3 + m) + (b^3*(d + e*x)^3)/(4 + m)))/(e^4*(a + b*x)^3)","A",1
1739,1,59,101,0.0465857,"\int (d+e x)^m \sqrt{a^2+2 a b x+b^2 x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","\frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} (a e (m+2)-b d+b e (m+1) x)}{e^2 (m+1) (m+2) (a+b x)}","\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+2}}{e^2 (m+2) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+1}}{e^2 (m+1) (a+b x)}",1,"(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*(-(b*d) + a*e*(2 + m) + b*e*(1 + m)*x))/(e^2*(1 + m)*(2 + m)*(a + b*x))","A",1
1740,1,67,76,0.0228845,"\int \frac{(d+e x)^m}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]","-\frac{(a+b x) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) \sqrt{(a+b x)^2} (b d-a e)}","-\frac{(a+b x) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}",1,"-(((a + b*x)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)*(1 + m)*Sqrt[(a + b*x)^2]))","A",1
1741,1,72,79,0.032059,"\int \frac{(d+e x)^m}{\left(a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{e^2 (a+b x)^3 (d+e x)^{m+1} \, _2F_1\left(3,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right)}{(m+1) \left((a+b x)^2\right)^{3/2} (a e-b d)^3}","-\frac{e^2 (a+b x) (d+e x)^{m+1} \, _2F_1\left(3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}",1,"(e^2*(a + b*x)^3*(d + e*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^3*(1 + m)*((a + b*x)^2)^(3/2))","A",1
1742,1,72,79,0.031506,"\int \frac{(d+e x)^m}{\left(a^2+2 a b x+b^2 x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]","\frac{e^4 (a+b x)^5 (d+e x)^{m+1} \, _2F_1\left(5,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right)}{(m+1) \left((a+b x)^2\right)^{5/2} (a e-b d)^5}","-\frac{e^4 (a+b x) (d+e x)^{m+1} \, _2F_1\left(5,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right)}{(m+1) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}",1,"(e^4*(a + b*x)^5*(d + e*x)^(1 + m)*Hypergeometric2F1[5, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^5*(1 + m)*((a + b*x)^2)^(5/2))","A",1
1743,1,75,85,0.0353144,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{\left((a+b x)^2\right)^p (d+e x)^{m+1} \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right)}{e (m+1)}","\frac{\left(a^2+2 a b x+b^2 x^2\right)^p (d+e x)^{m+1} \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right)}{e (m+1)}",1,"(((a + b*x)^2)^p*(d + e*x)^(1 + m)*Hypergeometric2F1[1 + m, -2*p, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(e*(1 + m)*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))","A",1
1744,1,107,181,0.0887576,"\int (d+e x)^3 \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{(a+b x) \left((a+b x)^2\right)^p \left(\frac{6 e^2 (a+b x)^2 (b d-a e)}{2 p+3}+\frac{3 e (a+b x) (b d-a e)^2}{p+1}+\frac{2 (b d-a e)^3}{2 p+1}+\frac{e^3 (a+b x)^3}{p+2}\right)}{2 b^4}","\frac{3 e^2 (a+b x)^3 (b d-a e) \left(a^2+2 a b x+b^2 x^2\right)^p}{b^4 (2 p+3)}+\frac{3 e (a+b x)^2 (b d-a e)^2 \left(a^2+2 a b x+b^2 x^2\right)^p}{2 b^4 (p+1)}+\frac{(a+b x) (b d-a e)^3 \left(a^2+2 a b x+b^2 x^2\right)^p}{b^4 (2 p+1)}+\frac{e^3 (a+b x)^4 \left(a^2+2 a b x+b^2 x^2\right)^p}{2 b^4 (p+2)}",1,"((a + b*x)*((a + b*x)^2)^p*((2*(b*d - a*e)^3)/(1 + 2*p) + (3*e*(b*d - a*e)^2*(a + b*x))/(1 + p) + (6*e^2*(b*d - a*e)*(a + b*x)^2)/(3 + 2*p) + (e^3*(a + b*x)^3)/(2 + p)))/(2*b^4)","A",1
1745,1,75,127,0.0836482,"\int (d+e x)^2 \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{(a+b x) \left((a+b x)^2\right)^p \left(\frac{e (a+b x) (b d-a e)}{p+1}+\frac{(b d-a e)^2}{2 p+1}+\frac{e^2 (a+b x)^2}{2 p+3}\right)}{b^3}","\frac{e (a+b x)^2 (b d-a e) \left(a^2+2 a b x+b^2 x^2\right)^p}{b^3 (p+1)}+\frac{(a+b x) (b d-a e)^2 \left(a^2+2 a b x+b^2 x^2\right)^p}{b^3 (2 p+1)}+\frac{e^2 (a+b x)^3 \left(a^2+2 a b x+b^2 x^2\right)^p}{b^3 (2 p+3)}",1,"((a + b*x)*((a + b*x)^2)^p*((b*d - a*e)^2/(1 + 2*p) + (e*(b*d - a*e)*(a + b*x))/(1 + p) + (e^2*(a + b*x)^2)/(3 + 2*p)))/b^3","A",1
1746,1,54,76,0.0367985,"\int (d+e x) \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{(a+b x) \left((a+b x)^2\right)^p (-a e+2 b d (p+1)+b e (2 p+1) x)}{2 b^2 (p+1) (2 p+1)}","\frac{(a+b x) (b d-a e) \left(a^2+2 a b x+b^2 x^2\right)^p}{b^2 (2 p+1)}+\frac{e \left(a^2+2 a b x+b^2 x^2\right)^{p+1}}{2 b^2 (p+1)}",1,"((a + b*x)*((a + b*x)^2)^p*(-(a*e) + 2*b*d*(1 + p) + b*e*(1 + 2*p)*x))/(2*b^2*(1 + p)*(1 + 2*p))","A",1
1747,1,23,34,0.0116327,"\int \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{(a+b x) \left((a+b x)^2\right)^p}{2 b p+b}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p}{b (2 p+1)}",1,"((a + b*x)*((a + b*x)^2)^p)/(b + 2*b*p)","A",1
1748,1,62,71,0.0172862,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{d+e x} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x),x]","-\frac{(a+b x) \left((a+b x)^2\right)^p \, _2F_1\left(1,2 p+1;2 p+2;\frac{e (a+b x)}{a e-b d}\right)}{(2 p+1) (a e-b d)}","\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p \, _2F_1\left(1,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right)}{(2 p+1) (b d-a e)}",1,"-(((a + b*x)*((a + b*x)^2)^p*Hypergeometric2F1[1, 1 + 2*p, 2 + 2*p, (e*(a + b*x))/(-(b*d) + a*e)])/((-(b*d) + a*e)*(1 + 2*p)))","A",1
1749,1,63,72,0.0176615,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^2,x]","\frac{b (a+b x) \left((a+b x)^2\right)^p \, _2F_1\left(2,2 p+1;2 p+2;-\frac{e (a+b x)}{b d-a e}\right)}{(2 p+1) (b d-a e)^2}","\frac{b (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p \, _2F_1\left(2,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right)}{(2 p+1) (b d-a e)^2}",1,"(b*(a + b*x)*((a + b*x)^2)^p*Hypergeometric2F1[2, 1 + 2*p, 2 + 2*p, -((e*(a + b*x))/(b*d - a*e))])/((b*d - a*e)^2*(1 + 2*p))","A",1
1750,1,65,74,0.0146203,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^3,x]","\frac{b^2 (a+b x) \left((a+b x)^2\right)^p \, _2F_1\left(3,2 p+1;2 p+2;-\frac{e (a+b x)}{b d-a e}\right)}{(2 p+1) (b d-a e)^3}","\frac{b^2 (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p \, _2F_1\left(3,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right)}{(2 p+1) (b d-a e)^3}",1,"(b^2*(a + b*x)*((a + b*x)^2)^p*Hypergeometric2F1[3, 1 + 2*p, 2 + 2*p, -((e*(a + b*x))/(b*d - a*e))])/((b*d - a*e)^3*(1 + 2*p))","A",1
1751,1,73,83,0.0278479,"\int (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{2 (d+e x)^{5/2} \left((a+b x)^2\right)^p \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(\frac{5}{2},-2 p;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 e}","\frac{2 (d+e x)^{5/2} \left(a^2+2 a b x+b^2 x^2\right)^p \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(\frac{5}{2},-2 p;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right)}{5 e}",1,"(2*((a + b*x)^2)^p*(d + e*x)^(5/2)*Hypergeometric2F1[5/2, -2*p, 7/2, (b*(d + e*x))/(b*d - a*e)])/(5*e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))","A",1
1752,1,73,83,0.0226259,"\int \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{2 (d+e x)^{3/2} \left((a+b x)^2\right)^p \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 e}","\frac{2 (d+e x)^{3/2} \left(a^2+2 a b x+b^2 x^2\right)^p \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 e}",1,"(2*((a + b*x)^2)^p*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, -2*p, 5/2, (b*(d + e*x))/(b*d - a*e)])/(3*e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))","A",1
1753,1,71,81,0.0163961,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{\sqrt{d+e x}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left((a+b x)^2\right)^p \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(\frac{1}{2},-2 p;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{e}","\frac{2 \sqrt{d+e x} \left(a^2+2 a b x+b^2 x^2\right)^p \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(\frac{1}{2},-2 p;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right)}{e}",1,"(2*((a + b*x)^2)^p*Sqrt[d + e*x]*Hypergeometric2F1[1/2, -2*p, 3/2, (b*(d + e*x))/(b*d - a*e)])/(e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))","A",1
1754,1,71,81,0.0176383,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{(d+e x)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^(3/2),x]","-\frac{2 \left((a+b x)^2\right)^p \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(-\frac{1}{2},-2 p;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{e \sqrt{d+e x}}","-\frac{2 \left(a^2+2 a b x+b^2 x^2\right)^p \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(-\frac{1}{2},-2 p;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{e \sqrt{d+e x}}",1,"(-2*((a + b*x)^2)^p*Hypergeometric2F1[-1/2, -2*p, 1/2, (b*(d + e*x))/(b*d - a*e)])/(e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p)*Sqrt[d + e*x])","A",1
1755,1,73,83,0.0211836,"\int \frac{\left(a^2+2 a b x+b^2 x^2\right)^p}{(d+e x)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^(5/2),x]","-\frac{2 \left((a+b x)^2\right)^p \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(-\frac{3}{2},-2 p;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 e (d+e x)^{3/2}}","-\frac{2 \left(a^2+2 a b x+b^2 x^2\right)^p \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(-\frac{3}{2},-2 p;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right)}{3 e (d+e x)^{3/2}}",1,"(-2*((a + b*x)^2)^p*Hypergeometric2F1[-3/2, -2*p, -1/2, (b*(d + e*x))/(b*d - a*e)])/(3*e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p)*(d + e*x)^(3/2))","A",1
1756,1,87,97,0.0567557,"\int (d+e x)^m \left(a^2+2 a b x+b^2 x^2\right)^{5+p} \, dx","Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5 + p),x]","\frac{(b d-a e)^{10} \left((a+b x)^2\right)^p (d+e x)^{m+1} \left(\frac{e (a+b x)}{a e-b d}\right)^{-2 p} \, _2F_1\left(m+1,-2 (p+5);m+2;\frac{b (d+e x)}{b d-a e}\right)}{e^{11} (m+1)}","\frac{(b d-a e)^{10} \left(a^2+2 a b x+b^2 x^2\right)^p (d+e x)^{m+1} \left(-\frac{e (a+b x)}{b d-a e}\right)^{-2 p} \, _2F_1\left(m+1,-2 (p+5);m+2;\frac{b (d+e x)}{b d-a e}\right)}{e^{11} (m+1)}",1,"((b*d - a*e)^10*((a + b*x)^2)^p*(d + e*x)^(1 + m)*Hypergeometric2F1[1 + m, -2*(5 + p), 2 + m, (b*(d + e*x))/(b*d - a*e)])/(e^11*(1 + m)*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))","A",1
1757,1,72,115,0.0402298,"\int (d+e x)^{-3-2 p} \left(a^2+2 a b x+b^2 x^2\right)^p \, dx","Integrate[(d + e*x)^(-3 - 2*p)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]","\frac{(a+b x) \left((a+b x)^2\right)^p (d+e x)^{-2 (p+1)} (-a e (2 p+1)+2 b d (p+1)+b e x)}{2 (p+1) (2 p+1) (b d-a e)^2}","\frac{b (a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac{(a+b x) \left(a^2+2 a b x+b^2 x^2\right)^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)}",1,"((a + b*x)*((a + b*x)^2)^p*(2*b*d*(1 + p) - a*e*(1 + 2*p) + b*e*x))/(2*(b*d - a*e)^2*(1 + p)*(1 + 2*p)*(d + e*x)^(2*(1 + p)))","A",1
1758,1,48,60,0.0267216,"\int (d+e x) \left(9+12 x+4 x^2\right)^p \, dx","Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^p,x]","\frac{(2 x+3) \left((2 x+3)^2\right)^p (4 d (p+1)+e ((4 p+2) x-3))}{8 (p+1) (2 p+1)}","\frac{(2 x+3) (2 d-3 e) \left(4 x^2+12 x+9\right)^p}{4 (2 p+1)}+\frac{e \left(4 x^2+12 x+9\right)^{p+1}}{8 (p+1)}",1,"((3 + 2*x)*((3 + 2*x)^2)^p*(4*d*(1 + p) + e*(-3 + (2 + 4*p)*x)))/(8*(1 + p)*(1 + 2*p))","A",1
1759,1,84,38,0.0191858,"\int (a+b x)^3 \left(a c+(b c+a d) x+b d x^2\right) \, dx","Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{1}{30} x \left(15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right)","\frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2}",1,"(x*(15*a^4*(2*c + d*x) + 20*a^3*b*x*(3*c + 2*d*x) + 15*a^2*b^2*x^2*(4*c + 3*d*x) + 6*a*b^3*x^3*(5*c + 4*d*x) + b^4*x^4*(6*c + 5*d*x)))/30","B",1
1760,1,67,38,0.0103732,"\int (a+b x)^2 \left(a c+(b c+a d) x+b d x^2\right) \, dx","Integrate[(a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2),x]","a^3 c x+\frac{1}{2} a^2 x^2 (a d+3 b c)+\frac{1}{4} b^2 x^4 (3 a d+b c)+a b x^3 (a d+b c)+\frac{1}{5} b^3 d x^5","\frac{(a+b x)^4 (b c-a d)}{4 b^2}+\frac{d (a+b x)^5}{5 b^2}",1,"a^3*c*x + (a^2*(3*b*c + a*d)*x^2)/2 + a*b*(b*c + a*d)*x^3 + (b^2*(b*c + 3*a*d)*x^4)/4 + (b^3*d*x^5)/5","A",1
1761,1,46,38,0.008006,"\int (a+b x) \left(a c+(b c+a d) x+b d x^2\right) \, dx","Integrate[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{1}{12} x \left(6 a^2 (2 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right)","\frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2}",1,"(x*(6*a^2*(2*c + d*x) + 4*a*b*x*(3*c + 2*d*x) + b^2*x^2*(4*c + 3*d*x)))/12","A",1
1762,1,32,28,0.0000851,"\int \left(a c+(b c+a d) x+b d x^2\right) \, dx","Integrate[a*c + (b*c + a*d)*x + b*d*x^2,x]","a c x+\frac{1}{2} a d x^2+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3","\frac{1}{2} x^2 (a d+b c)+a c x+\frac{1}{3} b d x^3",1,"a*c*x + (b*c*x^2)/2 + (a*d*x^2)/2 + (b*d*x^3)/3","A",1
1763,1,12,12,0.0007042,"\int \frac{a c+(b c+a d) x+b d x^2}{a+b x} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x),x]","c x+\frac{d x^2}{2}","c x+\frac{d x^2}{2}",1,"c*x + (d*x^2)/2","A",1
1764,1,25,25,0.008636,"\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^2,x]","\frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b}","\frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b}",1,"(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2","A",1
1765,1,31,32,0.0110883,"\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^3} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^3,x]","\frac{a d-b c}{b^2 (a+b x)}+\frac{d \log (a+b x)}{b^2}","\frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)}",1,"(-(b*c) + a*d)/(b^2*(a + b*x)) + (d*Log[a + b*x])/b^2","A",1
1766,1,26,28,0.0092901,"\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^4,x]","-\frac{a d+b (c+2 d x)}{2 b^2 (a+b x)^2}","-\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)}",1,"-1/2*(a*d + b*(c + 2*d*x))/(b^2*(a + b*x)^2)","A",1
1767,1,27,38,0.0089866,"\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^5,x]","-\frac{a d+2 b c+3 b d x}{6 b^2 (a+b x)^3}","-\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2}",1,"-1/6*(2*b*c + a*d + 3*b*d*x)/(b^2*(a + b*x)^3)","A",1
1768,1,27,38,0.0095298,"\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^6} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,x]","-\frac{a d+3 b c+4 b d x}{12 b^2 (a+b x)^4}","-\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3}",1,"-1/12*(3*b*c + a*d + 4*b*d*x)/(b^2*(a + b*x)^4)","A",1
1769,1,189,65,0.0333295,"\int (a+b x)^3 \left(a c+(b c+a d) x+b d x^2\right)^2 \, dx","Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","a^5 c^2 x+\frac{1}{2} a^4 c x^2 (2 a d+5 b c)+a b^2 x^5 \left(2 a^2 d^2+4 a b c d+b^2 c^2\right)+\frac{5}{4} a^2 b x^4 \left(a^2 d^2+4 a b c d+2 b^2 c^2\right)+\frac{1}{6} b^3 x^6 \left(10 a^2 d^2+10 a b c d+b^2 c^2\right)+\frac{1}{3} a^3 x^3 \left(a^2 d^2+10 a b c d+10 b^2 c^2\right)+\frac{1}{7} b^4 d x^7 (5 a d+2 b c)+\frac{1}{8} b^5 d^2 x^8","\frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3}",1,"a^5*c^2*x + (a^4*c*(5*b*c + 2*a*d)*x^2)/2 + (a^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^3)/3 + (5*a^2*b*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + a*b^2*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^5 + (b^3*(b^2*c^2 + 10*a*b*c*d + 10*a^2*d^2)*x^6)/6 + (b^4*d*(2*b*c + 5*a*d)*x^7)/7 + (b^5*d^2*x^8)/8","B",1
1770,1,148,65,0.0207476,"\int (a+b x)^2 \left(a c+(b c+a d) x+b d x^2\right)^2 \, dx","Integrate[(a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","a^4 c^2 x+a^3 c x^2 (a d+2 b c)+\frac{1}{5} b^2 x^5 \left(6 a^2 d^2+8 a b c d+b^2 c^2\right)+a b x^4 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{1}{3} a^2 x^3 \left(a^2 d^2+8 a b c d+6 b^2 c^2\right)+\frac{1}{3} b^3 d x^6 (2 a d+b c)+\frac{1}{7} b^4 d^2 x^7","\frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}",1,"a^4*c^2*x + a^3*c*(2*b*c + a*d)*x^2 + (a^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^3)/3 + a*b*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^4 + (b^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^2)*x^5)/5 + (b^3*d*(b*c + 2*a*d)*x^6)/3 + (b^4*d^2*x^7)/7","B",1
1771,1,122,65,0.0152334,"\int (a+b x) \left(a c+(b c+a d) x+b d x^2\right)^2 \, dx","Integrate[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","a^3 c^2 x+\frac{1}{4} b x^4 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+\frac{1}{3} a x^3 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{2} a^2 c x^2 (2 a d+3 b c)+\frac{1}{5} b^2 d x^5 (3 a d+2 b c)+\frac{1}{6} b^3 d^2 x^6","\frac{2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac{(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac{d^2 (a+b x)^6}{6 b^3}",1,"a^3*c^2*x + (a^2*c*(3*b*c + 2*a*d)*x^2)/2 + (a*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^3)/3 + (b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4)/4 + (b^2*d*(2*b*c + 3*a*d)*x^5)/5 + (b^3*d^2*x^6)/6","A",1
1772,1,79,65,0.0103071,"\int \left(a c+(b c+a d) x+b d x^2\right)^2 \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{1}{3} x^3 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{1}{2} b d x^4 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{5} b^2 d^2 x^5","-\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3}",1,"a^2*c^2*x + a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3)/3 + (b*d*(b*c + a*d)*x^4)/2 + (b^2*d^2*x^5)/5","A",1
1773,1,47,38,0.0109266,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{a+b x} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x),x]","\frac{1}{12} x \left(4 d x^2 (a d+2 b c)+6 c x (2 a d+b c)+12 a c^2+3 b d^2 x^3\right)","\frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2}",1,"(x*(12*a*c^2 + 6*c*(b*c + 2*a*d)*x + 4*d*(2*b*c + a*d)*x^2 + 3*b*d^2*x^3))/12","A",1
1774,1,14,14,0.0015057,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^2} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^2,x]","\frac{(c+d x)^3}{3 d}","\frac{(c+d x)^3}{3 d}",1,"(c + d*x)^3/(3*d)","A",1
1775,1,43,49,0.0179091,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^3} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^3,x]","\frac{b d x (-2 a d+4 b c+b d x)+2 (b c-a d)^2 \log (a+b x)}{2 b^3}","\frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{d x (b c-a d)}{b^2}+\frac{(c+d x)^2}{2 b}",1,"(b*d*x*(4*b*c - 2*a*d + b*d*x) + 2*(b*c - a*d)^2*Log[a + b*x])/(2*b^3)","A",1
1776,1,47,51,0.0374405,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^4} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^4,x]","\frac{-\frac{(b c-a d)^2}{a+b x}+2 d (b c-a d) \log (a+b x)+b d^2 x}{b^3}","-\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2}",1,"(b*d^2*x - (b*c - a*d)^2/(a + b*x) + 2*d*(b*c - a*d)*Log[a + b*x])/b^3","A",1
1777,1,49,59,0.0238162,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^5} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^5,x]","\frac{2 d^2 \log (a+b x)-\frac{(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3}","-\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3}",1,"(-(((b*c - a*d)*(3*a*d + b*(c + 4*d*x)))/(a + b*x)^2) + 2*d^2*Log[a + b*x])/(2*b^3)","A",1
1778,1,53,28,0.0224274,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^6} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^6,x]","-\frac{a^2 d^2+a b d (c+3 d x)+b^2 \left(c^2+3 c d x+3 d^2 x^2\right)}{3 b^3 (a+b x)^3}","-\frac{(c+d x)^3}{3 (a+b x)^3 (b c-a d)}",1,"-1/3*(a^2*d^2 + a*b*d*(c + 3*d*x) + b^2*(c^2 + 3*c*d*x + 3*d^2*x^2))/(b^3*(a + b*x)^3)","A",1
1779,1,56,65,0.0191465,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^7} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^7,x]","-\frac{a^2 d^2+2 a b d (c+2 d x)+b^2 \left(3 c^2+8 c d x+6 d^2 x^2\right)}{12 b^3 (a+b x)^4}","-\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2}",1,"-1/12*(a^2*d^2 + 2*a*b*d*(c + 2*d*x) + b^2*(3*c^2 + 8*c*d*x + 6*d^2*x^2))/(b^3*(a + b*x)^4)","A",1
1780,1,57,65,0.0239108,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^8} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^8,x]","-\frac{a^2 d^2+a b d (3 c+5 d x)+b^2 \left(6 c^2+15 c d x+10 d^2 x^2\right)}{30 b^3 (a+b x)^5}","-\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3}",1,"-1/30*(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2))/(b^3*(a + b*x)^5)","A",1
1781,1,58,65,0.0204428,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^9} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]","-\frac{a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left(10 c^2+24 c d x+15 d^2 x^2\right)}{60 b^3 (a+b x)^6}","-\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4}",1,"-1/60*(a^2*d^2 + 2*a*b*d*(2*c + 3*d*x) + b^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2))/(b^3*(a + b*x)^6)","A",1
1782,1,57,65,0.0234171,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^2}{(a+b x)^{10}} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^10,x]","-\frac{a^2 d^2+a b d (5 c+7 d x)+b^2 \left(15 c^2+35 c d x+21 d^2 x^2\right)}{105 b^3 (a+b x)^7}","-\frac{d (b c-a d)}{3 b^3 (a+b x)^6}-\frac{(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac{d^2}{5 b^3 (a+b x)^5}",1,"-1/105*(a^2*d^2 + a*b*d*(5*c + 7*d*x) + b^2*(15*c^2 + 35*c*d*x + 21*d^2*x^2))/(b^3*(a + b*x)^7)","A",1
1783,1,276,92,0.0845587,"\int (a+b x)^3 \left(a c+(b c+a d) x+b d x^2\right)^3 \, dx","Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{1}{840} x \left(210 a^6 \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)+252 a^5 b x \left(10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right)+210 a^4 b^2 x^2 \left(20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right)+120 a^3 b^3 x^3 \left(35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right)+45 a^2 b^4 x^4 \left(56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right)+10 a b^5 x^5 \left(84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right)+b^6 x^6 \left(120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right)\right)","\frac{d^2 (a+b x)^9 (b c-a d)}{3 b^4}+\frac{3 d (a+b x)^8 (b c-a d)^2}{8 b^4}+\frac{(a+b x)^7 (b c-a d)^3}{7 b^4}+\frac{d^3 (a+b x)^{10}}{10 b^4}",1,"(x*(210*a^6*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 252*a^5*b*x*(10*c^3 + 20*c^2*d*x + 15*c*d^2*x^2 + 4*d^3*x^3) + 210*a^4*b^2*x^2*(20*c^3 + 45*c^2*d*x + 36*c*d^2*x^2 + 10*d^3*x^3) + 120*a^3*b^3*x^3*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + 45*a^2*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + 10*a*b^5*x^5*(84*c^3 + 216*c^2*d*x + 189*c*d^2*x^2 + 56*d^3*x^3) + b^6*x^6*(120*c^3 + 315*c^2*d*x + 280*c*d^2*x^2 + 84*d^3*x^3)))/840","B",1
1784,1,235,92,0.0571915,"\int (a+b x)^2 \left(a c+(b c+a d) x+b d x^2\right)^3 \, dx","Integrate[(a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{1}{504} x \left(126 a^5 \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)+126 a^4 b x \left(10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right)+84 a^3 b^2 x^2 \left(20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right)+36 a^2 b^3 x^3 \left(35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right)+9 a b^4 x^4 \left(56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right)+b^5 x^5 \left(84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right)\right)","\frac{3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac{3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac{(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac{d^3 (a+b x)^9}{9 b^4}",1,"(x*(126*a^5*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 126*a^4*b*x*(10*c^3 + 20*c^2*d*x + 15*c*d^2*x^2 + 4*d^3*x^3) + 84*a^3*b^2*x^2*(20*c^3 + 45*c^2*d*x + 36*c*d^2*x^2 + 10*d^3*x^3) + 36*a^2*b^3*x^3*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + 9*a*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + b^5*x^5*(84*c^3 + 216*c^2*d*x + 189*c*d^2*x^2 + 56*d^3*x^3)))/504","B",1
1785,1,217,92,0.0255905,"\int (a+b x) \left(a c+(b c+a d) x+b d x^2\right)^3 \, dx","Integrate[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","a^4 c^3 x+\frac{1}{2} a^3 c^2 x^2 (3 a d+4 b c)+\frac{1}{2} b^2 d x^6 \left(2 a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c x^3 \left(a^2 d^2+4 a b c d+2 b^2 c^2\right)+\frac{1}{5} b x^5 \left(4 a^3 d^3+18 a^2 b c d^2+12 a b^2 c^2 d+b^3 c^3\right)+\frac{1}{4} a x^4 \left(a^3 d^3+12 a^2 b c d^2+18 a b^2 c^2 d+4 b^3 c^3\right)+\frac{1}{7} b^3 d^2 x^7 (4 a d+3 b c)+\frac{1}{8} b^4 d^3 x^8","\frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4}",1,"a^4*c^3*x + (a^3*c^2*(4*b*c + 3*a*d)*x^2)/2 + a^2*c*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3 + (a*(4*b^3*c^3 + 18*a*b^2*c^2*d + 12*a^2*b*c*d^2 + a^3*d^3)*x^4)/4 + (b*(b^3*c^3 + 12*a*b^2*c^2*d + 18*a^2*b*c*d^2 + 4*a^3*d^3)*x^5)/5 + (b^2*d*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^6)/2 + (b^3*d^2*(3*b*c + 4*a*d)*x^7)/7 + (b^4*d^3*x^8)/8","B",1
1786,1,161,92,0.0189608,"\int \left(a c+(b c+a d) x+b d x^2\right)^3 \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","a^3 c^3 x+\frac{3}{5} b d x^5 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a c x^3 \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{3}{2} a^2 c^2 x^2 (a d+b c)+\frac{1}{4} x^4 \left(a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right)+\frac{1}{2} b^2 d^2 x^6 (a d+b c)+\frac{1}{7} b^3 d^3 x^7","-\frac{b^2 (c+d x)^6 (b c-a d)}{2 d^4}+\frac{3 b (c+d x)^5 (b c-a d)^2}{5 d^4}-\frac{(c+d x)^4 (b c-a d)^3}{4 d^4}+\frac{b^3 (c+d x)^7}{7 d^4}",1,"a^3*c^3*x + (3*a^2*c^2*(b*c + a*d)*x^2)/2 + a*c*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^3 + ((b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x^4)/4 + (3*b*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^5)/5 + (b^2*d^2*(b*c + a*d)*x^6)/2 + (b^3*d^3*x^7)/7","A",1
1787,1,122,65,0.0131324,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{a+b x} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x),x]","\frac{1}{4} d x^4 \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{3} c x^3 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{6} b^2 d^3 x^6","-\frac{2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac{(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac{b^2 (c+d x)^6}{6 d^3}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^2)/2 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^3)/3 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4)/4 + (b*d^2*(3*b*c + 2*a*d)*x^5)/5 + (b^2*d^3*x^6)/6","A",1
1788,1,67,38,0.0092541,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^2} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^2,x]","\frac{1}{2} c^2 x^2 (3 a d+b c)+\frac{1}{4} d^2 x^4 (a d+3 b c)+c d x^3 (a d+b c)+a c^3 x+\frac{1}{5} b d^3 x^5","\frac{b (c+d x)^5}{5 d^2}-\frac{(c+d x)^4 (b c-a d)}{4 d^2}",1,"a*c^3*x + (c^2*(b*c + 3*a*d)*x^2)/2 + c*d*(b*c + a*d)*x^3 + (d^2*(3*b*c + a*d)*x^4)/4 + (b*d^3*x^5)/5","A",1
1789,1,14,14,0.0021306,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^3} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^3,x]","\frac{(c+d x)^4}{4 d}","\frac{(c+d x)^4}{4 d}",1,"(c + d*x)^4/(4*d)","A",1
1790,1,74,73,0.0308025,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^4} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^4,x]","\frac{b d x \left(6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left(18 c^2+9 c d x+2 d^2 x^2\right)\right)+6 (b c-a d)^3 \log (a+b x)}{6 b^4}","\frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(c+d x)^3}{3 b}",1,"(b*d*x*(6*a^2*d^2 - 3*a*b*d*(6*c + d*x) + b^2*(18*c^2 + 9*c*d*x + 2*d^2*x^2)) + 6*(b*c - a*d)^3*Log[a + b*x])/(6*b^4)","A",1
1791,1,72,75,0.0498499,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^5} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^5,x]","\frac{2 b d^2 x (3 b c-2 a d)-\frac{2 (b c-a d)^3}{a+b x}+6 d (b c-a d)^2 \log (a+b x)+b^2 d^3 x^2}{2 b^4}","-\frac{(b c-a d)^3}{b^4 (a+b x)}+\frac{3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{d^3 x^2}{2 b^2}",1,"(2*b*d^2*(3*b*c - 2*a*d)*x + b^2*d^3*x^2 - (2*(b*c - a*d)^3)/(a + b*x) + 6*d*(b*c - a*d)^2*Log[a + b*x])/(2*b^4)","A",1
1792,1,114,78,0.0421235,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^6} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^6,x]","\frac{-5 a^3 d^3+a^2 b d^2 (9 c-4 d x)+a b^2 d \left(-3 c^2+12 c d x+4 d^2 x^2\right)-6 d^2 (a+b x)^2 (a d-b c) \log (a+b x)-\left(b^3 \left(c^3+6 c^2 d x-2 d^3 x^3\right)\right)}{2 b^4 (a+b x)^2}","\frac{3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac{3 d (b c-a d)^2}{b^4 (a+b x)}-\frac{(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac{d^3 x}{b^3}",1,"(-5*a^3*d^3 + a^2*b*d^2*(9*c - 4*d*x) + a*b^2*d*(-3*c^2 + 12*c*d*x + 4*d^2*x^2) - b^3*(c^3 + 6*c^2*d*x - 2*d^3*x^3) - 6*d^2*(-(b*c) + a*d)*(a + b*x)^2*Log[a + b*x])/(2*b^4*(a + b*x)^2)","A",1
1793,1,80,86,0.0404663,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^7} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^7,x]","\frac{6 d^3 \log (a+b x)-\frac{(b c-a d) \left(11 a^2 d^2+a b d (5 c+27 d x)+b^2 \left(2 c^2+9 c d x+18 d^2 x^2\right)\right)}{(a+b x)^3}}{6 b^4}","-\frac{3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac{3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac{(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac{d^3 \log (a+b x)}{b^4}",1,"(-(((b*c - a*d)*(11*a^2*d^2 + a*b*d*(5*c + 27*d*x) + b^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2)))/(a + b*x)^3) + 6*d^3*Log[a + b*x])/(6*b^4)","A",1
1794,1,91,28,0.0304886,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^8} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]","-\frac{a^3 d^3+a^2 b d^2 (c+4 d x)+a b^2 d \left(c^2+4 c d x+6 d^2 x^2\right)+b^3 \left(c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right)}{4 b^4 (a+b x)^4}","-\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)}",1,"-1/4*(a^3*d^3 + a^2*b*d^2*(c + 4*d*x) + a*b^2*d*(c^2 + 4*c*d*x + 6*d^2*x^2) + b^3*(c^3 + 4*c^2*d*x + 6*c*d^2*x^2 + 4*d^3*x^3))/(b^4*(a + b*x)^4)","B",1
1795,1,97,58,0.0357427,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^9} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^9,x]","-\frac{a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left(3 c^2+10 c d x+10 d^2 x^2\right)+b^3 \left(4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right)}{20 b^4 (a+b x)^5}","\frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)}",1,"-1/20*(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3))/(b^4*(a + b*x)^5)","A",1
1796,1,97,92,0.0308247,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^{10}} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]","-\frac{a^3 d^3+3 a^2 b d^2 (c+2 d x)+3 a b^2 d \left(2 c^2+6 c d x+5 d^2 x^2\right)+b^3 \left(10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right)}{60 b^4 (a+b x)^6}","-\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3}",1,"-1/60*(a^3*d^3 + 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*d*(2*c^2 + 6*c*d*x + 5*d^2*x^2) + b^3*(10*c^3 + 36*c^2*d*x + 45*c*d^2*x^2 + 20*d^3*x^3))/(b^4*(a + b*x)^6)","A",1
1797,1,97,92,0.0317795,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^{11}} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^11,x]","-\frac{a^3 d^3+a^2 b d^2 (4 c+7 d x)+a b^2 d \left(10 c^2+28 c d x+21 d^2 x^2\right)+b^3 \left(20 c^3+70 c^2 d x+84 c d^2 x^2+35 d^3 x^3\right)}{140 b^4 (a+b x)^7}","-\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4}",1,"-1/140*(a^3*d^3 + a^2*b*d^2*(4*c + 7*d*x) + a*b^2*d*(10*c^2 + 28*c*d*x + 21*d^2*x^2) + b^3*(20*c^3 + 70*c^2*d*x + 84*c*d^2*x^2 + 35*d^3*x^3))/(b^4*(a + b*x)^7)","A",1
1798,1,97,92,0.0362611,"\int \frac{\left(a c+(b c+a d) x+b d x^2\right)^3}{(a+b x)^{12}} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^12,x]","-\frac{a^3 d^3+a^2 b d^2 (5 c+8 d x)+a b^2 d \left(15 c^2+40 c d x+28 d^2 x^2\right)+b^3 \left(35 c^3+120 c^2 d x+140 c d^2 x^2+56 d^3 x^3\right)}{280 b^4 (a+b x)^8}","-\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{d^3}{5 b^4 (a+b x)^5}",1,"-1/280*(a^3*d^3 + a^2*b*d^2*(5*c + 8*d*x) + a*b^2*d*(15*c^2 + 40*c*d*x + 28*d^2*x^2) + b^3*(35*c^3 + 120*c^2*d*x + 140*c*d^2*x^2 + 56*d^3*x^3))/(b^4*(a + b*x)^8)","A",1
1799,1,167,122,0.065537,"\int \frac{(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{b d x \left(300 a^4 d^4+300 a^3 b d^3 (d x-2 c)+100 a^2 b^2 d^2 \left(6 c^2-3 c d x+2 d^2 x^2\right)+25 a b^3 d \left(-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right)+b^4 \left(60 c^4-30 c^3 d x+20 c^2 d^2 x^2-15 c d^3 x^3+12 d^4 x^4\right)\right)-60 (b c-a d)^5 \log (c+d x)}{60 d^6}","-\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}+\frac{(a+b x)^5}{5 d}",1,"(b*d*x*(300*a^4*d^4 + 300*a^3*b*d^3*(-2*c + d*x) + 100*a^2*b^2*d^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 25*a*b^3*d*(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3) + b^4*(60*c^4 - 30*c^3*d*x + 20*c^2*d^2*x^2 - 15*c*d^3*x^3 + 12*d^4*x^4)) - 60*(b*c - a*d)^5*Log[c + d*x])/(60*d^6)","A",1
1800,1,115,98,0.0408565,"\int \frac{(a+b x)^5}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{b d x \left(48 a^3 d^3+36 a^2 b d^2 (d x-2 c)+8 a b^2 d \left(6 c^2-3 c d x+2 d^2 x^2\right)+b^3 \left(-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right)\right)+12 (b c-a d)^4 \log (c+d x)}{12 d^5}","\frac{(b c-a d)^4 \log (c+d x)}{d^5}-\frac{b x (b c-a d)^3}{d^4}+\frac{(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac{(a+b x)^3 (b c-a d)}{3 d^2}+\frac{(a+b x)^4}{4 d}",1,"(b*d*x*(48*a^3*d^3 + 36*a^2*b*d^2*(-2*c + d*x) + 8*a*b^2*d*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + b^3*(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3)) + 12*(b*c - a*d)^4*Log[c + d*x])/(12*d^5)","A",1
1801,1,74,74,0.0269445,"\int \frac{(a+b x)^4}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{b d x \left(18 a^2 d^2+9 a b d (d x-2 c)+b^2 \left(6 c^2-3 c d x+2 d^2 x^2\right)\right)-6 (b c-a d)^3 \log (c+d x)}{6 d^4}","-\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}+\frac{(a+b x)^3}{3 d}",1,"(b*d*x*(18*a^2*d^2 + 9*a*b*d*(-2*c + d*x) + b^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2)) - 6*(b*c - a*d)^3*Log[c + d*x])/(6*d^4)","A",1
1802,1,43,50,0.0161411,"\int \frac{(a+b x)^3}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3}","\frac{(b c-a d)^2 \log (c+d x)}{d^3}-\frac{b x (b c-a d)}{d^2}+\frac{(a+b x)^2}{2 d}",1,"(b*d*x*(-2*b*c + 4*a*d + b*d*x) + 2*(b*c - a*d)^2*Log[c + d*x])/(2*d^3)","A",1
1803,1,25,26,0.0071577,"\int \frac{(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{(a d-b c) \log (c+d x)}{d^2}+\frac{b x}{d}","\frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2}",1,"(b*x)/d + ((-(b*c) + a*d)*Log[c + d*x])/d^2","A",1
1804,1,10,10,0.0011853,"\int \frac{a+b x}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2),x]","\frac{\log (c+d x)}{d}","\frac{\log (c+d x)}{d}",1,"Log[c + d*x]/d","A",1
1805,1,26,36,0.0136953,"\int \frac{1}{a c+(b c+a d) x+b d x^2} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]","\frac{\log (a+b x)-\log (c+d x)}{b c-a d}","\frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d}",1,"(Log[a + b*x] - Log[c + d*x])/(b*c - a*d)","A",1
1806,1,53,57,0.0239823,"\int \frac{1}{(a+b x) \left(a c+(b c+a d) x+b d x^2\right)} \, dx","Integrate[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)),x]","\frac{d (a+b x) \log (c+d x)-d (a+b x) \log (a+b x)+a d-b c}{(a+b x) (b c-a d)^2}","-\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2}",1,"(-(b*c) + a*d - d*(a + b*x)*Log[a + b*x] + d*(a + b*x)*Log[c + d*x])/((b*c - a*d)^2*(a + b*x))","A",1
1807,1,67,82,0.0619065,"\int \frac{1}{(a+b x)^2 \left(a c+(b c+a d) x+b d x^2\right)} \, dx","Integrate[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)),x]","\frac{\frac{(b c-a d) (3 a d-b c+2 b d x)}{(a+b x)^2}+2 d^2 \log (a+b x)-2 d^2 \log (c+d x)}{2 (b c-a d)^3}","\frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)}",1,"(((b*c - a*d)*(-(b*c) + 3*a*d + 2*b*d*x))/(a + b*x)^2 + 2*d^2*Log[a + b*x] - 2*d^2*Log[c + d*x])/(2*(b*c - a*d)^3)","A",1
1808,1,107,107,0.0415082,"\int \frac{1}{(a+b x)^3 \left(a c+(b c+a d) x+b d x^2\right)} \, dx","Integrate[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)),x]","-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}-\frac{d^2}{(a+b x) (b c-a d)^3}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}+\frac{1}{3 (a+b x)^3 (a d-b c)}","-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}-\frac{d^2}{(a+b x) (b c-a d)^3}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)}",1,"1/(3*(-(b*c) + a*d)*(a + b*x)^3) + d/(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x])/(b*c - a*d)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4","A",1
1809,1,130,130,0.0481474,"\int \frac{1}{(a+b x)^4 \left(a c+(b c+a d) x+b d x^2\right)} \, dx","Integrate[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]","\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}+\frac{1}{4 (a+b x)^4 (a d-b c)}","\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}-\frac{1}{4 (a+b x)^4 (b c-a d)}",1,"1/(4*(-(b*c) + a*d)*(a + b*x)^4) + d/(3*(b*c - a*d)^2*(a + b*x)^3) - d^2/(2*(b*c - a*d)^3*(a + b*x)^2) + d^3/((b*c - a*d)^4*(a + b*x)) + (d^4*Log[a + b*x])/(b*c - a*d)^5 - (d^4*Log[c + d*x])/(b*c - a*d)^5","A",1
1810,1,165,104,0.0614825,"\int \frac{(a+b x)^6}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{-3 a^4 d^4+12 a^3 b c d^3+18 a^2 b^2 d^2 \left(-c^2+c d x+d^2 x^2\right)+6 a b^3 d \left(2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right)-12 b (c+d x) (b c-a d)^3 \log (c+d x)+b^4 \left(-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right)}{3 d^5 (c+d x)}","-\frac{2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac{6 b^2 x (b c-a d)^2}{d^4}-\frac{(b c-a d)^4}{d^5 (c+d x)}-\frac{4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac{b^4 (c+d x)^3}{3 d^5}",1,"(12*a^3*b*c*d^3 - 3*a^4*d^4 + 18*a^2*b^2*d^2*(-c^2 + c*d*x + d^2*x^2) + 6*a*b^3*d*(2*c^3 - 4*c^2*d*x - 3*c*d^2*x^2 + d^3*x^3) + b^4*(-3*c^4 + 9*c^3*d*x + 6*c^2*d^2*x^2 - 2*c*d^3*x^3 + d^4*x^4) - 12*b*(b*c - a*d)^3*(c + d*x)*Log[c + d*x])/(3*d^5*(c + d*x))","A",1
1811,1,114,75,0.0349212,"\int \frac{(a+b x)^5}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{3 \left(a^2 b d^2-2 a b^2 c d+b^3 c^2\right) \log (c+d x)}{d^4}+\frac{-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3}{d^4 (c+d x)}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{b^3 x^2}{2 d^2}","-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2}",1,"-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (b^3*x^2)/(2*d^2) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(d^4*(c + d*x)) + (3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*Log[c + d*x])/d^4","A",1
1812,1,47,51,0.0357224,"\int \frac{(a+b x)^4}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{-\frac{(b c-a d)^2}{c+d x}+2 b (a d-b c) \log (c+d x)+b^2 d x}{d^3}","-\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2}",1,"(b^2*d*x - (b*c - a*d)^2/(c + d*x) + 2*b*(-(b*c) + a*d)*Log[c + d*x])/d^3","A",1
1813,1,31,31,0.0101429,"\int \frac{(a+b x)^3}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2}","\frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2}",1,"(b*c - a*d)/(d^2*(c + d*x)) + (b*Log[c + d*x])/d^2","A",1
1814,1,12,12,0.0029112,"\int \frac{(a+b x)^2}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","-\frac{1}{d (c+d x)}","-\frac{1}{d (c+d x)}",1,"-(1/(d*(c + d*x)))","A",1
1815,1,53,56,0.0282284,"\int \frac{a+b x}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]","\frac{b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c}{(c+d x) (b c-a d)^2}","\frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2}",1,"(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x])/((b*c - a*d)^2*(c + d*x))","A",1
1816,1,66,86,0.0660724,"\int \frac{1}{\left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-2),x]","\frac{\frac{b (a d-b c)}{a+b x}+\frac{d (a d-b c)}{c+d x}-2 b d \log (a+b x)+2 b d \log (c+d x)}{(b c-a d)^3}","-\frac{a d+b c+2 b d x}{(b c-a d)^2 \left(x (a d+b c)+a c+b d x^2\right)}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3}",1,"((b*(-(b*c) + a*d))/(a + b*x) + (d*(-(b*c) + a*d))/(c + d*x) - 2*b*d*Log[a + b*x] + 2*b*d*Log[c + d*x])/(b*c - a*d)^3","A",1
1817,1,98,109,0.0740934,"\int \frac{1}{(a+b x) \left(a c+(b c+a d) x+b d x^2\right)^2} \, dx","Integrate[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^2),x]","\frac{\frac{2 d^2 (b c-a d)}{c+d x}+\frac{4 b d (b c-a d)}{a+b x}-\frac{b (b c-a d)^2}{(a+b x)^2}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4}","\frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2}",1,"(-((b*(b*c - a*d)^2)/(a + b*x)^2) + (4*b*d*(b*c - a*d))/(a + b*x) + (2*d^2*(b*c - a*d))/(c + d*x) + 6*b*d^2*Log[a + b*x] - 6*b*d^2*Log[c + d*x])/(2*(b*c - a*d)^4)","A",1
1818,1,230,133,0.0801802,"\int \frac{(a+b x)^8}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^8/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left(-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right)+15 a b^4 d \left(7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right)-60 b^2 (c+d x)^2 (b c-a d)^3 \log (c+d x)+b^5 \left(-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right)}{6 d^6 (c+d x)^2}","-\frac{5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac{10 b^3 x (b c-a d)^2}{d^5}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac{b^5 (c+d x)^3}{3 d^6}",1,"(-3*a^5*d^5 - 15*a^4*b*d^4*(c + 2*d*x) + 30*a^3*b^2*c*d^3*(3*c + 4*d*x) + 30*a^2*b^3*d^2*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) + 15*a*b^4*d*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + b^5*(-27*c^5 + 6*c^4*d*x + 63*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5) - 60*b^2*(b*c - a*d)^3*(c + d*x)^2*Log[c + d*x])/(6*d^6*(c + d*x)^2)","A",1
1819,1,167,103,0.0539605,"\int \frac{(a+b x)^7}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^7/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{-a^4 d^4-4 a^3 b d^3 (c+2 d x)+6 a^2 b^2 c d^2 (3 c+4 d x)+4 a b^3 d \left(-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right)+12 b^2 (c+d x)^2 (b c-a d)^2 \log (c+d x)+b^4 \left(7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right)}{2 d^5 (c+d x)^2}","-\frac{b^3 x (3 b c-4 a d)}{d^4}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{b^4 x^2}{2 d^3}",1,"(-(a^4*d^4) - 4*a^3*b*d^3*(c + 2*d*x) + 6*a^2*b^2*c*d^2*(3*c + 4*d*x) + 4*a*b^3*d*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) + b^4*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*Log[c + d*x])/(2*d^5*(c + d*x)^2)","A",1
1820,1,114,78,0.0395115,"\int \frac{(a+b x)^6}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{-a^3 d^3-3 a^2 b d^2 (c+2 d x)+3 a b^2 c d (3 c+4 d x)-6 b^2 (c+d x)^2 (b c-a d) \log (c+d x)+b^3 \left(-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right)}{2 d^4 (c+d x)^2}","-\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac{b^3 x}{d^3}",1,"(-(a^3*d^3) - 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*c*d*(3*c + 4*d*x) + b^3*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) - 6*b^2*(b*c - a*d)*(c + d*x)^2*Log[c + d*x])/(2*d^4*(c + d*x)^2)","A",1
1821,1,48,59,0.023112,"\int \frac{(a+b x)^5}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{\frac{(b c-a d) (a d+3 b c+4 b d x)}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d^3}","\frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3}",1,"(((b*c - a*d)*(3*b*c + a*d + 4*b*d*x))/(c + d*x)^2 + 2*b^2*Log[c + d*x])/(2*d^3)","A",1
1822,1,26,28,0.0087128,"\int \frac{(a+b x)^4}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","-\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2}","\frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)}",1,"-1/2*(a*d + b*(c + 2*d*x))/(d^2*(c + d*x)^2)","A",1
1823,1,14,14,0.0029313,"\int \frac{(a+b x)^3}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","-\frac{1}{2 d (c+d x)^2}","-\frac{1}{2 d (c+d x)^2}",1,"-1/2*1/(d*(c + d*x)^2)","A",1
1824,1,67,82,0.0505897,"\int \frac{(a+b x)^2}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","\frac{2 b^2 \log (a+b x)+\frac{(b c-a d) (-a d+3 b c+2 b d x)}{(c+d x)^2}-2 b^2 \log (c+d x)}{2 (b c-a d)^3}","\frac{b^2 \log (a+b x)}{(b c-a d)^3}-\frac{b^2 \log (c+d x)}{(b c-a d)^3}+\frac{b}{(c+d x) (b c-a d)^2}+\frac{1}{2 (c+d x)^2 (b c-a d)}",1,"(((b*c - a*d)*(3*b*c - a*d + 2*b*d*x))/(c + d*x)^2 + 2*b^2*Log[a + b*x] - 2*b^2*Log[c + d*x])/(2*(b*c - a*d)^3)","A",1
1825,1,97,110,0.0961088,"\int \frac{a+b x}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]","-\frac{\frac{2 b^2 (b c-a d)}{a+b x}+6 b^2 d \log (a+b x)+\frac{4 b d (b c-a d)}{c+d x}+\frac{d (b c-a d)^2}{(c+d x)^2}-6 b^2 d \log (c+d x)}{2 (b c-a d)^4}","-\frac{b^2}{(a+b x) (b c-a d)^3}-\frac{3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac{3 b^2 d \log (c+d x)}{(b c-a d)^4}-\frac{2 b d}{(c+d x) (b c-a d)^3}-\frac{d}{2 (c+d x)^2 (b c-a d)^2}",1,"-1/2*((2*b^2*(b*c - a*d))/(a + b*x) + (d*(b*c - a*d)^2)/(c + d*x)^2 + (4*b*d*(b*c - a*d))/(c + d*x) + 6*b^2*d*Log[a + b*x] - 6*b^2*d*Log[c + d*x])/(b*c - a*d)^4","A",1
1826,1,128,143,0.105616,"\int \frac{1}{\left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-3),x]","\frac{\frac{6 b^2 d (b c-a d)}{a+b x}-\frac{b^2 (b c-a d)^2}{(a+b x)^2}+12 b^2 d^2 \log (a+b x)+\frac{6 b d^2 (b c-a d)}{c+d x}+\frac{d^2 (b c-a d)^2}{(c+d x)^2}-12 b^2 d^2 \log (c+d x)}{2 (b c-a d)^5}","\frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac{3 b d (a d+b c+2 b d x)}{(b c-a d)^4 \left(x (a d+b c)+a c+b d x^2\right)}-\frac{a d+b c+2 b d x}{2 (b c-a d)^2 \left(x (a d+b c)+a c+b d x^2\right)^2}",1,"(-((b^2*(b*c - a*d)^2)/(a + b*x)^2) + (6*b^2*d*(b*c - a*d))/(a + b*x) + (d^2*(b*c - a*d)^2)/(c + d*x)^2 + (6*b*d^2*(b*c - a*d))/(c + d*x) + 12*b^2*d^2*Log[a + b*x] - 12*b^2*d^2*Log[c + d*x])/(2*(b*c - a*d)^5)","A",1
1827,1,154,170,0.2050935,"\int \frac{1}{(a+b x) \left(a c+(b c+a d) x+b d x^2\right)^3} \, dx","Integrate[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3),x]","-\frac{\frac{36 b^2 d^2 (b c-a d)}{a+b x}-\frac{9 b^2 d (b c-a d)^2}{(a+b x)^2}+\frac{2 b^2 (b c-a d)^3}{(a+b x)^3}+60 b^2 d^3 \log (a+b x)+\frac{24 b d^3 (b c-a d)}{c+d x}+\frac{3 d^3 (b c-a d)^2}{(c+d x)^2}-60 b^2 d^3 \log (c+d x)}{6 (b c-a d)^6}","-\frac{10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac{10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}-\frac{6 b^2 d^2}{(a+b x) (b c-a d)^5}+\frac{3 b^2 d}{2 (a+b x)^2 (b c-a d)^4}-\frac{b^2}{3 (a+b x)^3 (b c-a d)^3}-\frac{4 b d^3}{(c+d x) (b c-a d)^5}-\frac{d^3}{2 (c+d x)^2 (b c-a d)^4}",1,"-1/6*((2*b^2*(b*c - a*d)^3)/(a + b*x)^3 - (9*b^2*d*(b*c - a*d)^2)/(a + b*x)^2 + (36*b^2*d^2*(b*c - a*d))/(a + b*x) + (3*d^3*(b*c - a*d)^2)/(c + d*x)^2 + (24*b*d^3*(b*c - a*d))/(c + d*x) + 60*b^2*d^3*Log[a + b*x] - 60*b^2*d^3*Log[c + d*x])/(b*c - a*d)^6","A",1
1828,1,117,39,0.0304925,"\int (d+e x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{1}{42} x \left(7 a e \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+c d x \left(21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right)\right)","\frac{1}{6} (d+e x)^6 \left(a-\frac{c d^2}{e^2}\right)+\frac{c d (d+e x)^7}{7 e^2}",1,"(x*(7*a*e*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + c*d*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)))/42","B",1
1829,1,95,39,0.0178928,"\int (d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{1}{30} x \left(6 a e \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+c d x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)\right)","\frac{1}{5} (d+e x)^5 \left(a-\frac{c d^2}{e^2}\right)+\frac{c d (d+e x)^6}{6 e^2}",1,"(x*(6*a*e*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + c*d*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)))/30","B",1
1830,1,73,39,0.0198006,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{1}{20} x \left(5 a e \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+c d x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)\right)","\frac{1}{4} (d+e x)^4 \left(a-\frac{c d^2}{e^2}\right)+\frac{c d (d+e x)^5}{5 e^2}",1,"(x*(5*a*e*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + c*d*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)))/20","A",1
1831,1,51,39,0.0121153,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{1}{12} x \left(4 a e \left(3 d^2+3 d e x+e^2 x^2\right)+c d x \left(6 d^2+8 d e x+3 e^2 x^2\right)\right)","\frac{1}{3} (d+e x)^3 \left(a-\frac{c d^2}{e^2}\right)+\frac{c d (d+e x)^4}{4 e^2}",1,"(x*(4*a*e*(3*d^2 + 3*d*e*x + e^2*x^2) + c*d*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/12","A",1
1832,1,38,34,0.0000934,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2,x]","a d e x+\frac{1}{2} a e^2 x^2+\frac{1}{2} c d^2 x^2+\frac{1}{3} c d e x^3","\frac{1}{2} x^2 \left(a e^2+c d^2\right)+a d e x+\frac{1}{3} c d e x^3",1,"a*d*e*x + (c*d^2*x^2)/2 + (a*e^2*x^2)/2 + (c*d*e*x^3)/3","A",1
1833,1,14,14,0.0009774,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x),x]","a e x+\frac{1}{2} c d x^2","a e x+\frac{1}{2} c d x^2",1,"a*e*x + (c*d*x^2)/2","A",1
1834,1,30,26,0.0091746,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^2,x]","\frac{\left(a e^2-c d^2\right) \log (d+e x)}{e^2}+\frac{c d x}{e}","\left(a-\frac{c d^2}{e^2}\right) \log (d+e x)+\frac{c d x}{e}",1,"(c*d*x)/e + ((-(c*d^2) + a*e^2)*Log[d + e*x])/e^2","A",1
1835,1,36,33,0.012567,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^3,x]","\frac{c d^2-a e^2}{e^2 (d+e x)}+\frac{c d \log (d+e x)}{e^2}","\frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x}",1,"(c*d^2 - a*e^2)/(e^2*(d + e*x)) + (c*d*Log[d + e*x])/e^2","A",1
1836,1,29,35,0.0103179,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]","-\frac{a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2}","\frac{(a e+c d x)^2}{2 (d+e x)^2 \left(c d^2-a e^2\right)}",1,"-1/2*(a*e^2 + c*d*(d + 2*e*x))/(e^2*(d + e*x)^2)","A",1
1837,1,30,39,0.0121431,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^5,x]","-\frac{2 a e^2+c d (d+3 e x)}{6 e^2 (d+e x)^3}","-\frac{a-\frac{c d^2}{e^2}}{3 (d+e x)^3}-\frac{c d}{2 e^2 (d+e x)^2}",1,"-1/6*(2*a*e^2 + c*d*(d + 3*e*x))/(e^2*(d + e*x)^3)","A",1
1838,1,30,39,0.0114129,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]","-\frac{3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4}","-\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3}",1,"-1/12*(3*a*e^2 + c*d*(d + 4*e*x))/(e^2*(d + e*x)^4)","A",1
1839,1,160,77,0.0313581,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","a^2 \left(d^4 e^2 x+2 d^3 e^3 x^2+2 d^2 e^4 x^3+d e^5 x^4+\frac{e^6 x^5}{5}\right)+\frac{1}{15} a c d e x^2 \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)+\frac{1}{105} c^2 d^2 x^3 \left(35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right)","-\frac{c d (d+e x)^6 \left(c d^2-a e^2\right)}{3 e^3}+\frac{(d+e x)^5 \left(c d^2-a e^2\right)^2}{5 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3}",1,"(a*c*d*e*x^2*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4))/15 + (c^2*d^2*x^3*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))/105 + a^2*(d^4*e^2*x + 2*d^3*e^3*x^2 + 2*d^2*e^4*x^3 + d*e^5*x^4 + (e^6*x^5)/5)","B",1
1840,1,120,77,0.0326943,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2 \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{1}{60} x \left(15 a^2 e^2 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+6 a c d e x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)+c^2 d^2 x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)\right)","-\frac{2 c d (d+e x)^5 \left(c d^2-a e^2\right)}{5 e^3}+\frac{(d+e x)^4 \left(c d^2-a e^2\right)^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3}",1,"(x*(15*a^2*e^2*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*c*d*e*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + c^2*d^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)))/60","A",1
1841,1,87,77,0.0254456,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2 \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{1}{30} x \left(10 a^2 e^2 \left(3 d^2+3 d e x+e^2 x^2\right)+5 a c d e x \left(6 d^2+8 d e x+3 e^2 x^2\right)+c^2 d^2 x^2 \left(10 d^2+15 d e x+6 e^2 x^2\right)\right)","-\frac{c d (d+e x)^4 \left(c d^2-a e^2\right)}{2 e^3}+\frac{(d+e x)^3 \left(c d^2-a e^2\right)^2}{3 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3}",1,"(x*(10*a^2*e^2*(3*d^2 + 3*d*e*x + e^2*x^2) + 5*a*c*d*e*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + c^2*d^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2)))/30","A",1
1842,1,54,54,0.0124618,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x),x]","\frac{1}{12} x \left(6 a^2 e^2 (2 d+e x)+4 a c d e x (3 d+2 e x)+c^2 d^2 x^2 (4 d+3 e x)\right)","\frac{\left(c d^2-a e^2\right) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2}",1,"(x*(6*a^2*e^2*(2*d + e*x) + 4*a*c*d*e*x*(3*d + 2*e*x) + c^2*d^2*x^2*(4*d + 3*e*x)))/12","A",1
1843,1,20,20,0.0019333,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^2,x]","\frac{(a e+c d x)^3}{3 c d}","\frac{(a e+c d x)^3}{3 c d}",1,"(a*e + c*d*x)^3/(3*c*d)","A",1
1844,1,52,62,0.0212206,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^3,x]","\frac{2 \left(c d^2-a e^2\right)^2 \log (d+e x)+c d e x \left(4 a e^2+c d (e x-2 d)\right)}{2 e^3}","-\frac{c d x \left(c d^2-a e^2\right)}{e^2}+\frac{\left(c d^2-a e^2\right)^2 \log (d+e x)}{e^3}+\frac{(a e+c d x)^2}{2 e}",1,"(c*d*e*x*(4*a*e^2 + c*d*(-2*d + e*x)) + 2*(c*d^2 - a*e^2)^2*Log[d + e*x])/(2*e^3)","A",1
1845,1,59,63,0.0429368,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^4,x]","\frac{-\frac{\left(c d^2-a e^2\right)^2}{d+e x}+2 c d \left(a e^2-c d^2\right) \log (d+e x)+c^2 d^2 e x}{e^3}","-\frac{\left(c d^2-a e^2\right)^2}{e^3 (d+e x)}-\frac{2 c d \left(c d^2-a e^2\right) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2}",1,"(c^2*d^2*e*x - (c*d^2 - a*e^2)^2/(d + e*x) + 2*c*d*(-(c*d^2) + a*e^2)*Log[d + e*x])/e^3","A",1
1846,1,59,71,0.0308799,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^5,x]","\frac{\frac{\left(c d^2-a e^2\right) \left(a e^2+c d (3 d+4 e x)\right)}{(d+e x)^2}+2 c^2 d^2 \log (d+e x)}{2 e^3}","\frac{2 c d \left(c d^2-a e^2\right)}{e^3 (d+e x)}-\frac{\left(c d^2-a e^2\right)^2}{2 e^3 (d+e x)^2}+\frac{c^2 d^2 \log (d+e x)}{e^3}",1,"(((c*d^2 - a*e^2)*(a*e^2 + c*d*(3*d + 4*e*x)))/(d + e*x)^2 + 2*c^2*d^2*Log[d + e*x])/(2*e^3)","A",1
1847,1,59,35,0.0268678,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^6,x]","-\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left(d^2+3 d e x+3 e^2 x^2\right)}{3 e^3 (d+e x)^3}","\frac{(a e+c d x)^3}{3 (d+e x)^3 \left(c d^2-a e^2\right)}",1,"-1/3*(a^2*e^4 + a*c*d*e^2*(d + 3*e*x) + c^2*d^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(e^3*(d + e*x)^3)","A",1
1848,1,61,77,0.0221021,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^7,x]","-\frac{3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left(d^2+4 d e x+6 e^2 x^2\right)}{12 e^3 (d+e x)^4}","\frac{2 c d \left(c d^2-a e^2\right)}{3 e^3 (d+e x)^3}-\frac{\left(c d^2-a e^2\right)^2}{4 e^3 (d+e x)^4}-\frac{c^2 d^2}{2 e^3 (d+e x)^2}",1,"-1/12*(3*a^2*e^4 + 2*a*c*d*e^2*(d + 4*e*x) + c^2*d^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(e^3*(d + e*x)^4)","A",1
1849,1,61,77,0.0304699,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^8,x]","-\frac{6 a^2 e^4+3 a c d e^2 (d+5 e x)+c^2 d^2 \left(d^2+5 d e x+10 e^2 x^2\right)}{30 e^3 (d+e x)^5}","\frac{c d \left(c d^2-a e^2\right)}{2 e^3 (d+e x)^4}-\frac{\left(c d^2-a e^2\right)^2}{5 e^3 (d+e x)^5}-\frac{c^2 d^2}{3 e^3 (d+e x)^3}",1,"-1/30*(6*a^2*e^4 + 3*a*c*d*e^2*(d + 5*e*x) + c^2*d^2*(d^2 + 5*d*e*x + 10*e^2*x^2))/(e^3*(d + e*x)^5)","A",1
1850,1,61,77,0.025021,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^9} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^9,x]","-\frac{10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left(d^2+6 d e x+15 e^2 x^2\right)}{60 e^3 (d+e x)^6}","\frac{2 c d \left(c d^2-a e^2\right)}{5 e^3 (d+e x)^5}-\frac{\left(c d^2-a e^2\right)^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4}",1,"-1/60*(10*a^2*e^4 + 4*a*c*d*e^2*(d + 6*e*x) + c^2*d^2*(d^2 + 6*d*e*x + 15*e^2*x^2))/(e^3*(d + e*x)^6)","A",1
1851,1,255,111,0.0709781,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3 \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{1}{504} x \left(84 a^3 e^3 \left(6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right)+36 a^2 c d e^2 x \left(21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right)+9 a c^2 d^2 e x^2 \left(56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right)+c^3 d^3 x^3 \left(126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right)\right)","-\frac{3 c^2 d^2 (d+e x)^8 \left(c d^2-a e^2\right)}{8 e^4}+\frac{3 c d (d+e x)^7 \left(c d^2-a e^2\right)^2}{7 e^4}-\frac{(d+e x)^6 \left(c d^2-a e^2\right)^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4}",1,"(x*(84*a^3*e^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 36*a^2*c*d*e^2*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*c^2*d^2*e*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + c^3*d^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/504","B",1
1852,1,211,111,0.0641906,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3 \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{1}{280} x \left(56 a^3 e^3 \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+28 a^2 c d e^2 x \left(15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right)+8 a c^2 d^2 e x^2 \left(35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right)+c^3 d^3 x^3 \left(70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right)\right)","-\frac{3 c^2 d^2 (d+e x)^7 \left(c d^2-a e^2\right)}{7 e^4}+\frac{c d (d+e x)^6 \left(c d^2-a e^2\right)^2}{2 e^4}-\frac{(d+e x)^5 \left(c d^2-a e^2\right)^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4}",1,"(x*(56*a^3*e^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 28*a^2*c*d*e^2*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 8*a*c^2*d^2*e*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + c^3*d^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)))/280","A",1
1853,1,167,111,0.0503685,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3 \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{1}{140} x \left(35 a^3 e^3 \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+21 a^2 c d e^2 x \left(10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right)+7 a c^2 d^2 e x^2 \left(20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right)+c^3 d^3 x^3 \left(35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right)\right)","-\frac{c^2 d^2 (d+e x)^6 \left(c d^2-a e^2\right)}{2 e^4}+\frac{3 c d (d+e x)^5 \left(c d^2-a e^2\right)^2}{5 e^4}-\frac{(d+e x)^4 \left(c d^2-a e^2\right)^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4}",1,"(x*(35*a^3*e^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 21*a^2*c*d*e^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 7*a*c^2*d^2*e*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + c^3*d^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)))/140","A",1
1854,1,123,91,0.037953,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x),x]","\frac{1}{60} x \left(20 a^3 e^3 \left(3 d^2+3 d e x+e^2 x^2\right)+15 a^2 c d e^2 x \left(6 d^2+8 d e x+3 e^2 x^2\right)+6 a c^2 d^2 e x^2 \left(10 d^2+15 d e x+6 e^2 x^2\right)+c^3 d^3 x^3 \left(15 d^2+24 d e x+10 e^2 x^2\right)\right)","\frac{e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac{2 e \left(c d^2-a e^2\right) (a e+c d x)^5}{5 c^3 d^3}+\frac{\left(c d^2-a e^2\right)^2 (a e+c d x)^4}{4 c^3 d^3}",1,"(x*(20*a^3*e^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*c*d*e^2*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 6*a*c^2*d^2*e*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + c^3*d^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2)))/60","A",1
1855,1,79,54,0.0260364,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^2,x]","\frac{1}{20} x \left(10 a^3 e^3 (2 d+e x)+10 a^2 c d e^2 x (3 d+2 e x)+5 a c^2 d^2 e x^2 (4 d+3 e x)+c^3 d^3 x^3 (5 d+4 e x)\right)","\frac{\left(c d^2-a e^2\right) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2}",1,"(x*(10*a^3*e^3*(2*d + e*x) + 10*a^2*c*d*e^2*x*(3*d + 2*e*x) + 5*a*c^2*d^2*e*x^2*(4*d + 3*e*x) + c^3*d^3*x^3*(5*d + 4*e*x)))/20","A",1
1856,1,20,20,0.0024208,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^3,x]","\frac{(a e+c d x)^4}{4 c d}","\frac{(a e+c d x)^4}{4 c d}",1,"(a*e + c*d*x)^4/(4*c*d)","A",1
1857,1,85,89,0.0354508,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^4,x]","\frac{c d e x \left(18 a^2 e^4+9 a c d e^2 (e x-2 d)+c^2 d^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-6 \left(c d^2-a e^2\right)^3 \log (d+e x)}{6 e^4}","\frac{1}{2} \left(a-\frac{c d^2}{e^2}\right) (a e+c d x)^2-\frac{\left(c d^2-a e^2\right)^3 \log (d+e x)}{e^4}+\frac{c d x \left(c d^2-a e^2\right)^2}{e^3}+\frac{(a e+c d x)^3}{3 e}",1,"(c*d*e*x*(18*a^2*e^4 + 9*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(c*d^2 - a*e^2)^3*Log[d + e*x])/(6*e^4)","A",1
1858,1,129,94,0.043187,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^5,x]","\frac{-2 a^3 e^6+6 a^2 c d^2 e^4+6 a c^2 d^2 e^2 \left(-d^2+d e x+e^2 x^2\right)+6 c d (d+e x) \left(c d^2-a e^2\right)^2 \log (d+e x)+c^3 d^3 \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)}{2 e^4 (d+e x)}","-\frac{c^2 d^2 x \left(2 c d^2-3 a e^2\right)}{e^3}+\frac{\left(c d^2-a e^2\right)^3}{e^4 (d+e x)}+\frac{3 c d \left(c d^2-a e^2\right)^2 \log (d+e x)}{e^4}+\frac{c^3 d^3 x^2}{2 e^2}",1,"(6*a^2*c*d^2*e^4 - 2*a^3*e^6 + 6*a*c^2*d^2*e^2*(-d^2 + d*e*x + e^2*x^2) + c^3*d^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 6*c*d*(c*d^2 - a*e^2)^2*(d + e*x)*Log[d + e*x])/(2*e^4*(d + e*x))","A",1
1859,1,129,97,0.0457604,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^6,x]","\frac{-a^3 e^6-3 a^2 c d e^4 (d+2 e x)+3 a c^2 d^3 e^2 (3 d+4 e x)-6 c^2 d^2 (d+e x)^2 \left(c d^2-a e^2\right) \log (d+e x)+c^3 d^3 \left(-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)}{2 e^4 (d+e x)^2}","-\frac{3 c^2 d^2 \left(c d^2-a e^2\right) \log (d+e x)}{e^4}-\frac{3 c d \left(c d^2-a e^2\right)^2}{e^4 (d+e x)}+\frac{\left(c d^2-a e^2\right)^3}{2 e^4 (d+e x)^2}+\frac{c^3 d^3 x}{e^3}",1,"(-(a^3*e^6) - 3*a^2*c*d*e^4*(d + 2*e*x) + 3*a*c^2*d^3*e^2*(3*d + 4*e*x) + c^3*d^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) - 6*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2*Log[d + e*x])/(2*e^4*(d + e*x)^2)","A",1
1860,1,92,105,0.0470615,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^7} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^7,x]","\frac{\frac{\left(c d^2-a e^2\right) \left(2 a^2 e^4+a c d e^2 (5 d+9 e x)+c^2 d^2 \left(11 d^2+27 d e x+18 e^2 x^2\right)\right)}{(d+e x)^3}+6 c^3 d^3 \log (d+e x)}{6 e^4}","\frac{3 c^2 d^2 \left(c d^2-a e^2\right)}{e^4 (d+e x)}-\frac{3 c d \left(c d^2-a e^2\right)^2}{2 e^4 (d+e x)^2}+\frac{\left(c d^2-a e^2\right)^3}{3 e^4 (d+e x)^3}+\frac{c^3 d^3 \log (d+e x)}{e^4}",1,"(((c*d^2 - a*e^2)*(2*a^2*e^4 + a*c*d*e^2*(5*d + 9*e*x) + c^2*d^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2)))/(d + e*x)^3 + 6*c^3*d^3*Log[d + e*x])/(6*e^4)","A",1
1861,1,100,35,0.039286,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^8} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^8,x]","-\frac{a^3 e^6+a^2 c d e^4 (d+4 e x)+a c^2 d^2 e^2 \left(d^2+4 d e x+6 e^2 x^2\right)+c^3 d^3 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)}{4 e^4 (d+e x)^4}","\frac{(a e+c d x)^4}{4 (d+e x)^4 \left(c d^2-a e^2\right)}",1,"-1/4*(a^3*e^6 + a^2*c*d*e^4*(d + 4*e*x) + a*c^2*d^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^3*d^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3))/(e^4*(d + e*x)^4)","B",1
1862,1,103,73,0.0402898,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^9} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]","-\frac{4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+c^3 d^3 \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)}{20 e^4 (d+e x)^5}","\frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left(c d^2-a e^2\right)^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left(c d^2-a e^2\right)}",1,"-1/20*(4*a^3*e^6 + 3*a^2*c*d*e^4*(d + 5*e*x) + 2*a*c^2*d^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + c^3*d^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3))/(e^4*(d + e*x)^5)","A",1
1863,1,103,111,0.0354652,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{10}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^10,x]","-\frac{10 a^3 e^6+6 a^2 c d e^4 (d+6 e x)+3 a c^2 d^2 e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+c^3 d^3 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)}{60 e^4 (d+e x)^6}","\frac{3 c^2 d^2 \left(c d^2-a e^2\right)}{4 e^4 (d+e x)^4}-\frac{3 c d \left(c d^2-a e^2\right)^2}{5 e^4 (d+e x)^5}+\frac{\left(c d^2-a e^2\right)^3}{6 e^4 (d+e x)^6}-\frac{c^3 d^3}{3 e^4 (d+e x)^3}",1,"-1/60*(10*a^3*e^6 + 6*a^2*c*d*e^4*(d + 6*e*x) + 3*a*c^2*d^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^3*d^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3))/(e^4*(d + e*x)^6)","A",1
1864,1,103,111,0.034216,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{11}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]","-\frac{20 a^3 e^6+10 a^2 c d e^4 (d+7 e x)+4 a c^2 d^2 e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+c^3 d^3 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)}{140 e^4 (d+e x)^7}","\frac{3 c^2 d^2 \left(c d^2-a e^2\right)}{5 e^4 (d+e x)^5}-\frac{c d \left(c d^2-a e^2\right)^2}{2 e^4 (d+e x)^6}+\frac{\left(c d^2-a e^2\right)^3}{7 e^4 (d+e x)^7}-\frac{c^3 d^3}{4 e^4 (d+e x)^4}",1,"-1/140*(20*a^3*e^6 + 10*a^2*c*d*e^4*(d + 7*e*x) + 4*a*c^2*d^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^3*d^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3))/(e^4*(d + e*x)^7)","A",1
1865,1,134,131,0.0525393,"\int \frac{(d+e x)^5}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{c d e x \left(-12 a^3 e^6+6 a^2 c d e^4 (8 d+e x)-4 a c^2 d^2 e^2 \left(18 d^2+6 d e x+e^2 x^2\right)+c^3 d^3 \left(48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right)\right)+12 \left(c d^2-a e^2\right)^4 \log (a e+c d x)}{12 c^5 d^5}","\frac{\left(c d^2-a e^2\right)^4 \log (a e+c d x)}{c^5 d^5}+\frac{e x \left(c d^2-a e^2\right)^3}{c^4 d^4}+\frac{(d+e x)^2 \left(c d^2-a e^2\right)^2}{2 c^3 d^3}+\frac{(d+e x)^3 \left(c d^2-a e^2\right)}{3 c^2 d^2}+\frac{(d+e x)^4}{4 c d}",1,"(c*d*e*x*(-12*a^3*e^6 + 6*a^2*c*d*e^4*(8*d + e*x) - 4*a*c^2*d^2*e^2*(18*d^2 + 6*d*e*x + e^2*x^2) + c^3*d^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 12*(c*d^2 - a*e^2)^4*Log[a*e + c*d*x])/(12*c^5*d^5)","A",1
1866,1,91,100,0.0349085,"\int \frac{(d+e x)^4}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{c d e x \left(6 a^2 e^4-3 a c d e^2 (6 d+e x)+c^2 d^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)\right)+6 \left(c d^2-a e^2\right)^3 \log (a e+c d x)}{6 c^4 d^4}","\frac{\left(c d^2-a e^2\right)^3 \log (a e+c d x)}{c^4 d^4}+\frac{e x \left(c d^2-a e^2\right)^2}{c^3 d^3}+\frac{(d+e x)^2 \left(c d^2-a e^2\right)}{2 c^2 d^2}+\frac{(d+e x)^3}{3 c d}",1,"(c*d*e*x*(6*a^2*e^4 - 3*a*c*d*e^2*(6*d + e*x) + c^2*d^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 6*(c*d^2 - a*e^2)^3*Log[a*e + c*d*x])/(6*c^4*d^4)","A",1
1867,1,58,69,0.021255,"\int \frac{(d+e x)^3}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 \left(c d^2-a e^2\right)^2 \log (a e+c d x)+c d e x \left(c d (4 d+e x)-2 a e^2\right)}{2 c^3 d^3}","\frac{\left(c d^2-a e^2\right)^2 \log (a e+c d x)}{c^3 d^3}+\frac{e x \left(c d^2-a e^2\right)}{c^2 d^2}+\frac{(d+e x)^2}{2 c d}",1,"(c*d*e*x*(-2*a*e^2 + c*d*(4*d + e*x)) + 2*(c*d^2 - a*e^2)^2*Log[a*e + c*d*x])/(2*c^3*d^3)","A",1
1868,1,35,38,0.0091326,"\int \frac{(d+e x)^2}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{\left(c d^2-a e^2\right) \log (a e+c d x)+c d e x}{c^2 d^2}","\frac{\left(c d^2-a e^2\right) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d}",1,"(c*d*e*x + (c*d^2 - a*e^2)*Log[a*e + c*d*x])/(c^2*d^2)","A",1
1869,1,16,16,0.0019011,"\int \frac{d+e x}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{\log (a e+c d x)}{c d}","\frac{\log (a e+c d x)}{c d}",1,"Log[a*e + c*d*x]/(c*d)","A",1
1870,1,33,47,0.0148651,"\int \frac{1}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1),x]","\frac{\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2}","\frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2}",1,"(Log[a*e + c*d*x] - Log[d + e*x])/(c*d^2 - a*e^2)","A",1
1871,1,66,73,0.0340632,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left(c d^2-a e^2\right)^2}","\frac{1}{(d+e x) \left(c d^2-a e^2\right)}+\frac{c d \log (a e+c d x)}{\left(c d^2-a e^2\right)^2}-\frac{c d \log (d+e x)}{\left(c d^2-a e^2\right)^2}",1,"(c*d^2 - a*e^2 + c*d*(d + e*x)*Log[a*e + c*d*x] - c*d*(d + e*x)*Log[d + e*x])/((c*d^2 - a*e^2)^2*(d + e*x))","A",1
1872,1,102,108,0.0518645,"\int \frac{1}{(d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{2 c^2 d^2 (d+e x)^2 \log (a e+c d x)+\left(c d^2-a e^2\right) \left(c d (3 d+2 e x)-a e^2\right)-2 c^2 d^2 (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left(c d^2-a e^2\right)^3}","\frac{c^2 d^2 \log (a e+c d x)}{\left(c d^2-a e^2\right)^3}-\frac{c^2 d^2 \log (d+e x)}{\left(c d^2-a e^2\right)^3}+\frac{c d}{(d+e x) \left(c d^2-a e^2\right)^2}+\frac{1}{2 (d+e x)^2 \left(c d^2-a e^2\right)}",1,"((c*d^2 - a*e^2)*(-(a*e^2) + c*d*(3*d + 2*e*x)) + 2*c^2*d^2*(d + e*x)^2*Log[a*e + c*d*x] - 2*c^2*d^2*(d + e*x)^2*Log[d + e*x])/(2*(c*d^2 - a*e^2)^3*(d + e*x)^2)","A",1
1873,1,135,139,0.0697566,"\int \frac{1}{(d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{\left(c d^2-a e^2\right) \left(2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left(11 d^2+15 d e x+6 e^2 x^2\right)\right)+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left(c d^2-a e^2\right)^4}","\frac{c^3 d^3 \log (a e+c d x)}{\left(c d^2-a e^2\right)^4}-\frac{c^3 d^3 \log (d+e x)}{\left(c d^2-a e^2\right)^4}+\frac{c^2 d^2}{(d+e x) \left(c d^2-a e^2\right)^3}+\frac{c d}{2 (d+e x)^2 \left(c d^2-a e^2\right)^2}+\frac{1}{3 (d+e x)^3 \left(c d^2-a e^2\right)}",1,"((c*d^2 - a*e^2)*(2*a^2*e^4 - a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(11*d^2 + 15*d*e*x + 6*e^2*x^2)) + 6*c^3*d^3*(d + e*x)^3*Log[a*e + c*d*x] - 6*c^3*d^3*(d + e*x)^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^4*(d + e*x)^3)","A",1
1874,1,339,219,0.1225694,"\int \frac{(d+e x)^8}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{-10 a^6 e^{12}+10 a^5 c d e^{10} (6 d+5 e x)-30 a^4 c^2 d^2 e^8 \left(5 d^2+8 d e x-e^2 x^2\right)+10 a^3 c^3 d^3 e^6 \left(20 d^3+45 d^2 e x-15 d e^2 x^2-e^3 x^3\right)-5 a^2 c^4 d^4 e^4 \left(30 d^4+80 d^3 e x-60 d^2 e^2 x^2-10 d e^3 x^3-e^4 x^4\right)+a c^5 d^5 e^2 \left(60 d^5+150 d^4 e x-300 d^3 e^2 x^2-100 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right)-60 e \left(a e^2-c d^2\right)^5 (a e+c d x) \log (a e+c d x)+c^6 d^6 \left(-10 d^6+150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4+15 d e^5 x^5+2 e^6 x^6\right)}{10 c^7 d^7 (a e+c d x)}","\frac{e^6 (a e+c d x)^5}{5 c^7 d^7}-\frac{\left(c d^2-a e^2\right)^6}{c^7 d^7 (a e+c d x)}+\frac{6 e \left(c d^2-a e^2\right)^5 \log (a e+c d x)}{c^7 d^7}+\frac{3 e^5 \left(c d^2-a e^2\right) (a e+c d x)^4}{2 c^7 d^7}+\frac{5 e^4 \left(c d^2-a e^2\right)^2 (a e+c d x)^3}{c^7 d^7}+\frac{10 e^3 \left(c d^2-a e^2\right)^3 (a e+c d x)^2}{c^7 d^7}+\frac{15 e^2 x \left(c d^2-a e^2\right)^4}{c^6 d^6}",1,"(-10*a^6*e^12 + 10*a^5*c*d*e^10*(6*d + 5*e*x) - 30*a^4*c^2*d^2*e^8*(5*d^2 + 8*d*e*x - e^2*x^2) + 10*a^3*c^3*d^3*e^6*(20*d^3 + 45*d^2*e*x - 15*d*e^2*x^2 - e^3*x^3) - 5*a^2*c^4*d^4*e^4*(30*d^4 + 80*d^3*e*x - 60*d^2*e^2*x^2 - 10*d*e^3*x^3 - e^4*x^4) + a*c^5*d^5*e^2*(60*d^5 + 150*d^4*e*x - 300*d^3*e^2*x^2 - 100*d^2*e^3*x^3 - 25*d*e^4*x^4 - 3*e^5*x^5) + c^6*d^6*(-10*d^6 + 150*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 50*d^2*e^4*x^4 + 15*d*e^5*x^5 + 2*e^6*x^6) - 60*e*(-(c*d^2) + a*e^2)^5*(a*e + c*d*x)*Log[a*e + c*d*x])/(10*c^7*d^7*(a*e + c*d*x))","A",1
1875,1,263,184,0.0831196,"\int \frac{(d+e x)^7}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left(4 d^2+6 d e x-e^2 x^2\right)-10 a^2 c^3 d^3 e^4 \left(12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right)+5 a c^4 d^4 e^2 \left(12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right)+60 e \left(c d^2-a e^2\right)^4 (a e+c d x) \log (a e+c d x)+c^5 d^5 \left(-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right)}{12 c^6 d^6 (a e+c d x)}","\frac{e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac{\left(c d^2-a e^2\right)^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left(c d^2-a e^2\right)^4 \log (a e+c d x)}{c^6 d^6}+\frac{5 e^4 \left(c d^2-a e^2\right) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left(c d^2-a e^2\right)^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left(c d^2-a e^2\right)^3}{c^5 d^5}",1,"(12*a^5*e^10 - 12*a^4*c*d*e^8*(5*d + 4*e*x) + 30*a^3*c^2*d^2*e^6*(4*d^2 + 6*d*e*x - e^2*x^2) - 10*a^2*c^3*d^3*e^4*(12*d^3 + 24*d^2*e*x - 12*d*e^2*x^2 - e^3*x^3) + 5*a*c^4*d^4*e^2*(12*d^4 + 24*d^3*e*x - 36*d^2*e^2*x^2 - 8*d*e^3*x^3 - e^4*x^4) + c^5*d^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*Log[a*e + c*d*x])/(12*c^6*d^6*(a*e + c*d*x))","A",1
1876,1,196,145,0.0734885,"\int \frac{(d+e x)^6}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{-3 a^4 e^8+3 a^3 c d e^6 (4 d+3 e x)-6 a^2 c^2 d^2 e^4 \left(3 d^2+4 d e x-e^2 x^2\right)+2 a c^3 d^3 e^2 \left(6 d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right)-12 e \left(a e^2-c d^2\right)^3 (a e+c d x) \log (a e+c d x)+c^4 d^4 \left(-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right)}{3 c^5 d^5 (a e+c d x)}","\frac{e^2 x \left(3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right)}{c^4 d^4}-\frac{\left(c d^2-a e^2\right)^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left(c d^2-a e^2\right)^3 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x^2 \left(2 c d^2-a e^2\right)}{c^3 d^3}+\frac{e^4 x^3}{3 c^2 d^2}",1,"(-3*a^4*e^8 + 3*a^3*c*d*e^6*(4*d + 3*e*x) - 6*a^2*c^2*d^2*e^4*(3*d^2 + 4*d*e*x - e^2*x^2) + 2*a*c^3*d^3*e^2*(6*d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) + c^4*d^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x))","A",1
1877,1,142,105,0.0483596,"\int \frac{(d+e x)^5}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{3 \left(a^2 e^5-2 a c d^2 e^3+c^2 d^4 e\right) \log (a e+c d x)}{c^4 d^4}+\frac{a^3 e^6-3 a^2 c d^2 e^4+3 a c^2 d^4 e^2-c^3 d^6}{c^4 d^4 (a e+c d x)}-\frac{e^2 x \left(2 a e^2-3 c d^2\right)}{c^3 d^3}+\frac{e^3 x^2}{2 c^2 d^2}","-\frac{\left(c d^2-a e^2\right)^3}{c^4 d^4 (a e+c d x)}+\frac{3 e \left(c d^2-a e^2\right)^2 \log (a e+c d x)}{c^4 d^4}+\frac{e^2 x \left(3 c d^2-2 a e^2\right)}{c^3 d^3}+\frac{e^3 x^2}{2 c^2 d^2}",1,"-((e^2*(-3*c*d^2 + 2*a*e^2)*x)/(c^3*d^3)) + (e^3*x^2)/(2*c^2*d^2) + (-(c^3*d^6) + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 + a^3*e^6)/(c^4*d^4*(a*e + c*d*x)) + (3*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*Log[a*e + c*d*x])/(c^4*d^4)","A",1
1878,1,65,74,0.0486681,"\int \frac{(d+e x)^4}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 \left(c d^2 e-a e^3\right) \log (a e+c d x)-\frac{\left(c d^2-a e^2\right)^2}{a e+c d x}+c d e^2 x}{c^3 d^3}","-\frac{\left(c d^2-a e^2\right)^2}{c^3 d^3 (a e+c d x)}+\frac{2 e \left(c d^2-a e^2\right) \log (a e+c d x)}{c^3 d^3}+\frac{e^2 x}{c^2 d^2}",1,"(c*d*e^2*x - (c*d^2 - a*e^2)^2/(a*e + c*d*x) + 2*(c*d^2*e - a*e^3)*Log[a*e + c*d*x])/(c^3*d^3)","A",1
1879,1,47,48,0.0131649,"\int \frac{(d+e x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{a e^2-c d^2}{c^2 d^2 (a e+c d x)}+\frac{e \log (a e+c d x)}{c^2 d^2}","\frac{e \log (a e+c d x)}{c^2 d^2}-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)}",1,"(-(c*d^2) + a*e^2)/(c^2*d^2*(a*e + c*d*x)) + (e*Log[a*e + c*d*x])/(c^2*d^2)","A",1
1880,1,18,18,0.0038631,"\int \frac{(d+e x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","-\frac{1}{c d (a e+c d x)}","-\frac{1}{c d (a e+c d x)}",1,"-(1/(c*d*(a*e + c*d*x)))","A",1
1881,1,74,75,0.0292966,"\int \frac{d+e x}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{1}{\left(a e^2-c d^2\right) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left(a e^2-c d^2\right)^2}+\frac{e \log (d+e x)}{\left(a e^2-c d^2\right)^2}","-\frac{1}{\left(c d^2-a e^2\right) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left(c d^2-a e^2\right)^2}+\frac{e \log (d+e x)}{\left(c d^2-a e^2\right)^2}",1,"1/((-(c*d^2) + a*e^2)*(a*e + c*d*x)) - (e*Log[a*e + c*d*x])/(-(c*d^2) + a*e^2)^2 + (e*Log[d + e*x])/(-(c*d^2) + a*e^2)^2","A",1
1882,1,86,114,0.0915137,"\int \frac{1}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-2),x]","\frac{\frac{\left(c d^2-a e^2\right) \left(a e^2+c d (d+2 e x)\right)}{(d+e x) (a e+c d x)}+2 c d e \log (a e+c d x)-2 c d e \log (d+e x)}{\left(a e^2-c d^2\right)^3}","-\frac{a e^2+c d^2+2 c d e x}{\left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)}-\frac{2 c d e \log (a e+c d x)}{\left(c d^2-a e^2\right)^3}+\frac{2 c d e \log (d+e x)}{\left(c d^2-a e^2\right)^3}",1,"(((c*d^2 - a*e^2)*(a*e^2 + c*d*(d + 2*e*x)))/((a*e + c*d*x)*(d + e*x)) + 2*c*d*e*Log[a*e + c*d*x] - 2*c*d*e*Log[d + e*x])/(-(c*d^2) + a*e^2)^3","A",1
1883,1,130,146,0.0932079,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]","\frac{\frac{2 c^2 d^2 \left(a e^2-c d^2\right)}{a e+c d x}-6 c^2 d^2 e \log (a e+c d x)+\frac{4 c d e \left(a e^2-c d^2\right)}{d+e x}-\frac{e \left(c d^2-a e^2\right)^2}{(d+e x)^2}+6 c^2 d^2 e \log (d+e x)}{2 \left(c d^2-a e^2\right)^4}","-\frac{c^2 d^2}{\left(c d^2-a e^2\right)^3 (a e+c d x)}-\frac{3 c^2 d^2 e \log (a e+c d x)}{\left(c d^2-a e^2\right)^4}+\frac{3 c^2 d^2 e \log (d+e x)}{\left(c d^2-a e^2\right)^4}-\frac{2 c d e}{(d+e x) \left(c d^2-a e^2\right)^3}-\frac{e}{2 (d+e x)^2 \left(c d^2-a e^2\right)^2}",1,"((2*c^2*d^2*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (e*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (4*c*d*e*(-(c*d^2) + a*e^2))/(d + e*x) - 6*c^2*d^2*e*Log[a*e + c*d*x] + 6*c^2*d^2*e*Log[d + e*x])/(2*(c*d^2 - a*e^2)^4)","A",1
1884,1,160,176,0.1364252,"\int \frac{1}{(d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]","\frac{12 c^3 d^3 e \log (a e+c d x)+\frac{3 c^3 d^3 \left(c d^2-a e^2\right)}{a e+c d x}+\frac{9 c^2 d^2 e \left(c d^2-a e^2\right)}{d+e x}+\frac{3 c d e \left(c d^2-a e^2\right)^2}{(d+e x)^2}-\frac{e \left(a e^2-c d^2\right)^3}{(d+e x)^3}-12 c^3 d^3 e \log (d+e x)}{3 \left(a e^2-c d^2\right)^5}","-\frac{c^3 d^3}{\left(c d^2-a e^2\right)^4 (a e+c d x)}-\frac{4 c^3 d^3 e \log (a e+c d x)}{\left(c d^2-a e^2\right)^5}+\frac{4 c^3 d^3 e \log (d+e x)}{\left(c d^2-a e^2\right)^5}-\frac{3 c^2 d^2 e}{(d+e x) \left(c d^2-a e^2\right)^4}-\frac{c d e}{(d+e x)^2 \left(c d^2-a e^2\right)^3}-\frac{e}{3 (d+e x)^3 \left(c d^2-a e^2\right)^2}",1,"((3*c^3*d^3*(c*d^2 - a*e^2))/(a*e + c*d*x) - (e*(-(c*d^2) + a*e^2)^3)/(d + e*x)^3 + (3*c*d*e*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (9*c^2*d^2*e*(c*d^2 - a*e^2))/(d + e*x) + 12*c^3*d^3*e*Log[a*e + c*d*x] - 12*c^3*d^3*e*Log[d + e*x])/(3*(-(c*d^2) + a*e^2)^5)","A",1
1885,1,337,221,0.1266165,"\int \frac{(d+e x)^9}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{22 a^6 e^{12}-4 a^5 c d e^{10} (27 d+4 e x)+2 a^4 c^2 d^2 e^8 \left(105 d^2+12 d e x-34 e^2 x^2\right)-4 a^3 c^3 d^3 e^6 \left(50 d^3-15 d^2 e x-63 d e^2 x^2+5 e^3 x^3\right)+5 a^2 c^4 d^4 e^4 \left(18 d^4-32 d^3 e x-66 d^2 e^2 x^2+16 d e^3 x^3+e^4 x^4\right)-2 a c^5 d^5 e^2 \left(6 d^5-60 d^4 e x-80 d^3 e^2 x^2+60 d^2 e^3 x^3+10 d e^4 x^4+e^5 x^5\right)+60 e^2 \left(c d^2-a e^2\right)^4 (a e+c d x)^2 \log (a e+c d x)+c^6 d^6 \left(-2 d^6-24 d^5 e x+80 d^3 e^3 x^3+30 d^2 e^4 x^4+8 d e^5 x^5+e^6 x^6\right)}{4 c^7 d^7 (a e+c d x)^2}","\frac{e^6 (a e+c d x)^4}{4 c^7 d^7}-\frac{6 e \left(c d^2-a e^2\right)^5}{c^7 d^7 (a e+c d x)}-\frac{\left(c d^2-a e^2\right)^6}{2 c^7 d^7 (a e+c d x)^2}+\frac{15 e^2 \left(c d^2-a e^2\right)^4 \log (a e+c d x)}{c^7 d^7}+\frac{2 e^5 \left(c d^2-a e^2\right) (a e+c d x)^3}{c^7 d^7}+\frac{15 e^4 \left(c d^2-a e^2\right)^2 (a e+c d x)^2}{2 c^7 d^7}+\frac{20 e^3 x \left(c d^2-a e^2\right)^3}{c^6 d^6}",1,"(22*a^6*e^12 - 4*a^5*c*d*e^10*(27*d + 4*e*x) + 2*a^4*c^2*d^2*e^8*(105*d^2 + 12*d*e*x - 34*e^2*x^2) - 4*a^3*c^3*d^3*e^6*(50*d^3 - 15*d^2*e*x - 63*d*e^2*x^2 + 5*e^3*x^3) + 5*a^2*c^4*d^4*e^4*(18*d^4 - 32*d^3*e*x - 66*d^2*e^2*x^2 + 16*d*e^3*x^3 + e^4*x^4) - 2*a*c^5*d^5*e^2*(6*d^5 - 60*d^4*e*x - 80*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 10*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-2*d^6 - 24*d^5*e*x + 80*d^3*e^3*x^3 + 30*d^2*e^4*x^4 + 8*d*e^5*x^5 + e^6*x^6) + 60*e^2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(4*c^7*d^7*(a*e + c*d*x)^2)","A",1
1886,1,262,185,0.0886175,"\int \frac{(d+e x)^8}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{-27 a^5 e^{10}+3 a^4 c d e^8 (35 d+2 e x)+3 a^3 c^2 d^2 e^6 \left(-50 d^2+10 d e x+21 e^2 x^2\right)+5 a^2 c^3 d^3 e^4 \left(18 d^3-24 d^2 e x-33 d e^2 x^2+4 e^3 x^3\right)-5 a c^4 d^4 e^2 \left(3 d^4-24 d^3 e x-24 d^2 e^2 x^2+12 d e^3 x^3+e^4 x^4\right)-60 e^2 \left(a e^2-c d^2\right)^3 (a e+c d x)^2 \log (a e+c d x)+c^5 d^5 \left(-3 d^5-30 d^4 e x+60 d^2 e^3 x^3+15 d e^4 x^4+2 e^5 x^5\right)}{6 c^6 d^6 (a e+c d x)^2}","\frac{e^3 x \left(6 a^2 e^4-15 a c d^2 e^2+10 c^2 d^4\right)}{c^5 d^5}-\frac{5 e \left(c d^2-a e^2\right)^4}{c^6 d^6 (a e+c d x)}-\frac{\left(c d^2-a e^2\right)^5}{2 c^6 d^6 (a e+c d x)^2}+\frac{10 e^2 \left(c d^2-a e^2\right)^3 \log (a e+c d x)}{c^6 d^6}+\frac{e^4 x^2 \left(5 c d^2-3 a e^2\right)}{2 c^4 d^4}+\frac{e^5 x^3}{3 c^3 d^3}",1,"(-27*a^5*e^10 + 3*a^4*c*d*e^8*(35*d + 2*e*x) + 3*a^3*c^2*d^2*e^6*(-50*d^2 + 10*d*e*x + 21*e^2*x^2) + 5*a^2*c^3*d^3*e^4*(18*d^3 - 24*d^2*e*x - 33*d*e^2*x^2 + 4*e^3*x^3) - 5*a*c^4*d^4*e^2*(3*d^4 - 24*d^3*e*x - 24*d^2*e^2*x^2 + 12*d*e^3*x^3 + e^4*x^4) + c^5*d^5*(-3*d^5 - 30*d^4*e*x + 60*d^2*e^3*x^3 + 15*d*e^4*x^4 + 2*e^5*x^5) - 60*e^2*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(6*c^6*d^6*(a*e + c*d*x)^2)","A",1
1887,1,191,142,0.0698249,"\int \frac{(d+e x)^7}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{7 a^4 e^8+2 a^3 c d e^6 (e x-10 d)+a^2 c^2 d^2 e^4 \left(18 d^2-16 d e x-11 e^2 x^2\right)-4 a c^3 d^3 e^2 \left(d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right)+12 e^2 \left(c d^2-a e^2\right)^2 (a e+c d x)^2 \log (a e+c d x)+c^4 d^4 \left(-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right)}{2 c^5 d^5 (a e+c d x)^2}","-\frac{4 e \left(c d^2-a e^2\right)^3}{c^5 d^5 (a e+c d x)}-\frac{\left(c d^2-a e^2\right)^4}{2 c^5 d^5 (a e+c d x)^2}+\frac{6 e^2 \left(c d^2-a e^2\right)^2 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x \left(4 c d^2-3 a e^2\right)}{c^4 d^4}+\frac{e^4 x^2}{2 c^3 d^3}",1,"(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*c^3*d^3*e^2*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + c^4*d^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)","A",1
1888,1,139,111,0.0532601,"\int \frac{(d+e x)^6}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{-5 a^3 e^6+a^2 c d e^4 (9 d-4 e x)+a c^2 d^2 e^2 \left(-3 d^2+12 d e x+4 e^2 x^2\right)-6 e^2 \left(a e^2-c d^2\right) (a e+c d x)^2 \log (a e+c d x)-c^3 \left(d^6+6 d^5 e x-2 d^3 e^3 x^3\right)}{2 c^4 d^4 (a e+c d x)^2}","-\frac{3 e \left(c d^2-a e^2\right)^2}{c^4 d^4 (a e+c d x)}-\frac{\left(c d^2-a e^2\right)^3}{2 c^4 d^4 (a e+c d x)^2}+\frac{3 e^2 \left(c d^2-a e^2\right) \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x}{c^3 d^3}",1,"(-5*a^3*e^6 + a^2*c*d*e^4*(9*d - 4*e*x) + a*c^2*d^2*e^2*(-3*d^2 + 12*d*e*x + 4*e^2*x^2) - c^3*(d^6 + 6*d^5*e*x - 2*d^3*e^3*x^3) - 6*e^2*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^4*d^4*(a*e + c*d*x)^2)","A",1
1889,1,65,85,0.0328847,"\int \frac{(d+e x)^5}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 \log (a e+c d x)-\frac{\left(c d^2-a e^2\right) \left(3 a e^2+c d (d+4 e x)\right)}{(a e+c d x)^2}}{2 c^3 d^3}","\frac{e^2 \log (a e+c d x)}{c^3 d^3}-\frac{2 e \left(c d^2-a e^2\right)}{c^3 d^3 (a e+c d x)}-\frac{\left(c d^2-a e^2\right)^2}{2 c^3 d^3 (a e+c d x)^2}",1,"(-(((c*d^2 - a*e^2)*(3*a*e^2 + c*d*(d + 4*e*x)))/(a*e + c*d*x)^2) + 2*e^2*Log[a*e + c*d*x])/(2*c^3*d^3)","A",1
1890,1,35,35,0.0132147,"\int \frac{(d+e x)^4}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","-\frac{a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2}","-\frac{(d+e x)^2}{2 \left(c d^2-a e^2\right) (a e+c d x)^2}",1,"-1/2*(a*e^2 + c*d*(d + 2*e*x))/(c^2*d^2*(a*e + c*d*x)^2)","A",1
1891,1,20,20,0.003695,"\int \frac{(d+e x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","-\frac{1}{2 c d (a e+c d x)^2}","-\frac{1}{2 c d (a e+c d x)^2}",1,"-1/2*1/(c*d*(a*e + c*d*x)^2)","A",1
1892,1,83,107,0.0741071,"\int \frac{(d+e x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","-\frac{\frac{\left(c d^2-a e^2\right) \left(c d (d-2 e x)-3 a e^2\right)}{(a e+c d x)^2}-2 e^2 \log (a e+c d x)+2 e^2 \log (d+e x)}{2 \left(c d^2-a e^2\right)^3}","\frac{e}{\left(c d^2-a e^2\right)^2 (a e+c d x)}-\frac{1}{2 \left(c d^2-a e^2\right) (a e+c d x)^2}+\frac{e^2 \log (a e+c d x)}{\left(c d^2-a e^2\right)^3}-\frac{e^2 \log (d+e x)}{\left(c d^2-a e^2\right)^3}",1,"-1/2*(((c*d^2 - a*e^2)*(-3*a*e^2 + c*d*(d - 2*e*x)))/(a*e + c*d*x)^2 - 2*e^2*Log[a*e + c*d*x] + 2*e^2*Log[d + e*x])/(c*d^2 - a*e^2)^3","A",1
1893,1,127,142,0.0998158,"\int \frac{d+e x}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{\frac{4 c d e \left(c d^2-a e^2\right)}{a e+c d x}-\frac{c d \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}+\frac{2 c d^2 e^2-2 a e^4}{d+e x}+6 c d e^2 \log (a e+c d x)-6 c d e^2 \log (d+e x)}{2 \left(c d^2-a e^2\right)^4}","\frac{e^2}{(d+e x) \left(c d^2-a e^2\right)^3}+\frac{2 c d e}{\left(c d^2-a e^2\right)^3 (a e+c d x)}-\frac{c d}{2 \left(c d^2-a e^2\right)^2 (a e+c d x)^2}+\frac{3 c d e^2 \log (a e+c d x)}{\left(c d^2-a e^2\right)^4}-\frac{3 c d e^2 \log (d+e x)}{\left(c d^2-a e^2\right)^4}",1,"(-((c*d*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2) + (4*c*d*e*(c*d^2 - a*e^2))/(a*e + c*d*x) + (2*c*d^2*e^2 - 2*a*e^4)/(d + e*x) + 6*c*d*e^2*Log[a*e + c*d*x] - 6*c*d*e^2*Log[d + e*x])/(2*(c*d^2 - a*e^2)^4)","A",1
1894,1,168,191,0.1438119,"\int \frac{1}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3),x]","\frac{\frac{6 c^2 d^2 e \left(a e^2-c d^2\right)}{a e+c d x}+\frac{c^2 d^2 \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}-12 c^2 d^2 e^2 \log (a e+c d x)-\frac{\left(c d^2 e-a e^3\right)^2}{(d+e x)^2}+\frac{6 c d e^2 \left(a e^2-c d^2\right)}{d+e x}+12 c^2 d^2 e^2 \log (d+e x)}{2 \left(a e^2-c d^2\right)^5}","\frac{6 c^2 d^2 e^2 \log (a e+c d x)}{\left(c d^2-a e^2\right)^5}-\frac{6 c^2 d^2 e^2 \log (d+e x)}{\left(c d^2-a e^2\right)^5}+\frac{3 c d e \left(a e^2+c d^2+2 c d e x\right)}{\left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)}-\frac{a e^2+c d^2+2 c d e x}{2 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^2}",1,"((c^2*d^2*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (6*c^2*d^2*e*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (c*d^2*e - a*e^3)^2/(d + e*x)^2 + (6*c*d*e^2*(-(c*d^2) + a*e^2))/(d + e*x) - 12*c^2*d^2*e^2*Log[a*e + c*d*x] + 12*c^2*d^2*e^2*Log[d + e*x])/(2*(-(c*d^2) + a*e^2)^5)","A",1
1895,1,201,223,0.1843014,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]","\frac{60 c^3 d^3 e^2 \log (a e+c d x)+\frac{24 c^3 d^3 e \left(c d^2-a e^2\right)}{a e+c d x}-\frac{3 c^3 d^3 \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}+\frac{36 c^2 d^2 e^2 \left(c d^2-a e^2\right)}{d+e x}+\frac{9 c d \left(c d^2 e-a e^3\right)^2}{(d+e x)^2}-\frac{2 e^2 \left(a e^2-c d^2\right)^3}{(d+e x)^3}-60 c^3 d^3 e^2 \log (d+e x)}{6 \left(c d^2-a e^2\right)^6}","\frac{4 c^3 d^3 e}{\left(c d^2-a e^2\right)^5 (a e+c d x)}-\frac{c^3 d^3}{2 \left(c d^2-a e^2\right)^4 (a e+c d x)^2}+\frac{10 c^3 d^3 e^2 \log (a e+c d x)}{\left(c d^2-a e^2\right)^6}-\frac{10 c^3 d^3 e^2 \log (d+e x)}{\left(c d^2-a e^2\right)^6}+\frac{6 c^2 d^2 e^2}{(d+e x) \left(c d^2-a e^2\right)^5}+\frac{3 c d e^2}{2 (d+e x)^2 \left(c d^2-a e^2\right)^4}+\frac{e^2}{3 (d+e x)^3 \left(c d^2-a e^2\right)^3}",1,"((-3*c^3*d^3*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (24*c^3*d^3*e*(c*d^2 - a*e^2))/(a*e + c*d*x) - (2*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^3 + (9*c*d*(c*d^2*e - a*e^3)^2)/(d + e*x)^2 + (36*c^2*d^2*e^2*(c*d^2 - a*e^2))/(d + e*x) + 60*c^3*d^3*e^2*Log[a*e + c*d*x] - 60*c^3*d^3*e^2*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)","A",1
1896,1,335,217,0.1349033,"\int \frac{(d+e x)^{10}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{-37 a^6 e^{12}+3 a^5 c d e^{10} (47 d-17 e x)+3 a^4 c^2 d^2 e^8 \left(-65 d^2+81 d e x+13 e^2 x^2\right)+a^3 c^3 d^3 e^6 \left(110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right)-3 a^2 c^4 d^4 e^4 \left(5 d^4-90 d^3 e x+45 d^2 e^2 x^2+63 d e^3 x^3-5 e^4 x^4\right)-3 a c^5 d^5 e^2 \left(d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right)-60 e^3 \left(a e^2-c d^2\right)^3 (a e+c d x)^3 \log (a e+c d x)+c^6 d^6 \left(-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right)}{3 c^7 d^7 (a e+c d x)^3}","\frac{e^4 x \left(10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right)}{c^6 d^6}-\frac{15 e^2 \left(c d^2-a e^2\right)^4}{c^7 d^7 (a e+c d x)}-\frac{3 e \left(c d^2-a e^2\right)^5}{c^7 d^7 (a e+c d x)^2}-\frac{\left(c d^2-a e^2\right)^6}{3 c^7 d^7 (a e+c d x)^3}+\frac{20 e^3 \left(c d^2-a e^2\right)^3 \log (a e+c d x)}{c^7 d^7}+\frac{e^5 x^2 \left(3 c d^2-2 a e^2\right)}{c^5 d^5}+\frac{e^6 x^3}{3 c^4 d^4}",1,"(-37*a^6*e^12 + 3*a^5*c*d*e^10*(47*d - 17*e*x) + 3*a^4*c^2*d^2*e^8*(-65*d^2 + 81*d*e*x + 13*e^2*x^2) + a^3*c^3*d^3*e^6*(110*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 73*e^3*x^3) - 3*a^2*c^4*d^4*e^4*(5*d^4 - 90*d^3*e*x + 45*d^2*e^2*x^2 + 63*d*e^3*x^3 - 5*e^4*x^4) - 3*a*c^5*d^5*e^2*(d^5 + 15*d^4*e*x - 60*d^3*e^2*x^2 - 45*d^2*e^3*x^3 + 15*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 45*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) - 60*e^3*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^7*d^7*(a*e + c*d*x)^3)","A",1
1897,1,259,179,0.0926034,"\int \frac{(d+e x)^9}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{47 a^5 e^{10}+a^4 c d e^8 (81 e x-130 d)+a^3 c^2 d^2 e^6 \left(110 d^2-270 d e x-9 e^2 x^2\right)-a^2 c^3 d^3 e^4 \left(20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right)-5 a c^4 d^4 e^2 \left(d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right)+60 e^3 \left(c d^2-a e^2\right)^2 (a e+c d x)^3 \log (a e+c d x)+c^5 d^5 \left(-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right)}{6 c^6 d^6 (a e+c d x)^3}","-\frac{10 e^2 \left(c d^2-a e^2\right)^3}{c^6 d^6 (a e+c d x)}-\frac{5 e \left(c d^2-a e^2\right)^4}{2 c^6 d^6 (a e+c d x)^2}-\frac{\left(c d^2-a e^2\right)^5}{3 c^6 d^6 (a e+c d x)^3}+\frac{10 e^3 \left(c d^2-a e^2\right)^2 \log (a e+c d x)}{c^6 d^6}+\frac{e^4 x \left(5 c d^2-4 a e^2\right)}{c^5 d^5}+\frac{e^5 x^2}{2 c^4 d^4}",1,"(47*a^5*e^10 + a^4*c*d*e^8*(-130*d + 81*e*x) + a^3*c^2*d^2*e^6*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*c^3*d^3*e^4*(20*d^3 - 270*d^2*e*x + 90*d*e^2*x^2 + 63*e^3*x^3) - 5*a*c^4*d^4*e^2*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 - 18*d*e^3*x^3 + 3*e^4*x^4) + c^5*d^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(6*c^6*d^6*(a*e + c*d*x)^3)","A",1
1898,1,194,146,0.0724747,"\int \frac{(d+e x)^8}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{-13 a^4 e^8+a^3 c d e^6 (22 d-27 e x)-3 a^2 c^2 d^2 e^4 \left(2 d^2-18 d e x+3 e^2 x^2\right)+a c^3 d^3 e^2 \left(-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right)-12 e^3 \left(a e^2-c d^2\right) (a e+c d x)^3 \log (a e+c d x)-c^4 \left(d^8+6 d^7 e x+18 d^6 e^2 x^2-3 d^4 e^4 x^4\right)}{3 c^5 d^5 (a e+c d x)^3}","-\frac{6 e^2 \left(c d^2-a e^2\right)^2}{c^5 d^5 (a e+c d x)}-\frac{2 e \left(c d^2-a e^2\right)^3}{c^5 d^5 (a e+c d x)^2}-\frac{\left(c d^2-a e^2\right)^4}{3 c^5 d^5 (a e+c d x)^3}+\frac{4 e^3 \left(c d^2-a e^2\right) \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x}{c^4 d^4}",1,"(-13*a^4*e^8 + a^3*c*d*e^6*(22*d - 27*e*x) - 3*a^2*c^2*d^2*e^4*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*c^3*d^3*e^2*(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - c^4*(d^8 + 6*d^7*e*x + 18*d^6*e^2*x^2 - 3*d^4*e^4*x^4) - 12*e^3*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x)^3)","A",1
1899,1,99,122,0.0515507,"\int \frac{(d+e x)^7}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{6 e^3 \log (a e+c d x)-\frac{\left(c d^2-a e^2\right) \left(11 a^2 e^4+a c d e^2 (5 d+27 e x)+c^2 d^2 \left(2 d^2+9 d e x+18 e^2 x^2\right)\right)}{(a e+c d x)^3}}{6 c^4 d^4}","\frac{e^3 \log (a e+c d x)}{c^4 d^4}-\frac{3 e^2 \left(c d^2-a e^2\right)}{c^4 d^4 (a e+c d x)}-\frac{3 e \left(c d^2-a e^2\right)^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left(c d^2-a e^2\right)^3}{3 c^4 d^4 (a e+c d x)^3}",1,"(-(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(5*d + 27*e*x) + c^2*d^2*(2*d^2 + 9*d*e*x + 18*e^2*x^2)))/(a*e + c*d*x)^3) + 6*e^3*Log[a*e + c*d*x])/(6*c^4*d^4)","A",1
1900,1,65,35,0.0291637,"\int \frac{(d+e x)^6}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","-\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left(d^2+3 d e x+3 e^2 x^2\right)}{3 c^3 d^3 (a e+c d x)^3}","-\frac{(d+e x)^3}{3 \left(c d^2-a e^2\right) (a e+c d x)^3}",1,"-1/3*(a^2*e^4 + a*c*d*e^2*(d + 3*e*x) + c^2*d^2*(d^2 + 3*d*e*x + 3*e^2*x^2))/(c^3*d^3*(a*e + c*d*x)^3)","A",1
1901,1,37,54,0.0164534,"\int \frac{(d+e x)^5}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","-\frac{a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3}","-\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2}",1,"-1/6*(a*e^2 + c*d*(2*d + 3*e*x))/(c^2*d^2*(a*e + c*d*x)^3)","A",1
1902,1,20,20,0.0045659,"\int \frac{(d+e x)^4}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","-\frac{1}{3 c d (a e+c d x)^3}","-\frac{1}{3 c d (a e+c d x)^3}",1,"-1/3*1/(c*d*(a*e + c*d*x)^3)","A",1
1903,1,117,139,0.0956653,"\int \frac{(d+e x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","-\frac{\frac{\left(c d^2-a e^2\right) \left(11 a^2 e^4+a c d e^2 (15 e x-7 d)+c^2 d^2 \left(2 d^2-3 d e x+6 e^2 x^2\right)\right)}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left(c d^2-a e^2\right)^4}","-\frac{e^2}{\left(c d^2-a e^2\right)^3 (a e+c d x)}+\frac{e}{2 \left(c d^2-a e^2\right)^2 (a e+c d x)^2}-\frac{1}{3 \left(c d^2-a e^2\right) (a e+c d x)^3}-\frac{e^3 \log (a e+c d x)}{\left(c d^2-a e^2\right)^4}+\frac{e^3 \log (d+e x)}{\left(c d^2-a e^2\right)^4}",1,"-1/6*(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(-7*d + 15*e*x) + c^2*d^2*(2*d^2 - 3*d*e*x + 6*e^2*x^2)))/(a*e + c*d*x)^3 + 6*e^3*Log[a*e + c*d*x] - 6*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4","A",1
1904,1,157,173,0.1428759,"\int \frac{(d+e x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{\frac{9 c d e^2 \left(c d^2-a e^2\right)}{a e+c d x}-\frac{3 c d e \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}+\frac{c d \left(c d^2-a e^2\right)^3}{(a e+c d x)^3}+\frac{3 c d^2 e^3-3 a e^5}{d+e x}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left(a e^2-c d^2\right)^5}","-\frac{3 c d e^2}{\left(c d^2-a e^2\right)^4 (a e+c d x)}+\frac{c d e}{\left(c d^2-a e^2\right)^3 (a e+c d x)^2}-\frac{c d}{3 \left(c d^2-a e^2\right)^2 (a e+c d x)^3}-\frac{e^3}{(d+e x) \left(c d^2-a e^2\right)^4}-\frac{4 c d e^3 \log (a e+c d x)}{\left(c d^2-a e^2\right)^5}+\frac{4 c d e^3 \log (d+e x)}{\left(c d^2-a e^2\right)^5}",1,"((c*d*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (3*c*d*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (9*c*d*e^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (3*c*d^2*e^3 - 3*a*e^5)/(d + e*x) + 12*c*d*e^3*Log[a*e + c*d*x] - 12*c*d*e^3*Log[d + e*x])/(3*(-(c*d^2) + a*e^2)^5)","A",1
1905,1,206,226,0.1876606,"\int \frac{d+e x}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{-60 c^2 d^2 e^3 \log (a e+c d x)+\frac{36 c^2 d^2 e^2 \left(a e^2-c d^2\right)}{a e+c d x}+\frac{9 c^2 d^2 e \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}+\frac{2 c^2 d^2 \left(a e^2-c d^2\right)^3}{(a e+c d x)^3}+\frac{24 c d e^3 \left(a e^2-c d^2\right)}{d+e x}-\frac{3 e^3 \left(c d^2-a e^2\right)^2}{(d+e x)^2}+60 c^2 d^2 e^3 \log (d+e x)}{6 \left(c d^2-a e^2\right)^6}","-\frac{6 c^2 d^2 e^2}{\left(c d^2-a e^2\right)^5 (a e+c d x)}+\frac{3 c^2 d^2 e}{2 \left(c d^2-a e^2\right)^4 (a e+c d x)^2}-\frac{c^2 d^2}{3 \left(c d^2-a e^2\right)^3 (a e+c d x)^3}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left(c d^2-a e^2\right)^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left(c d^2-a e^2\right)^6}-\frac{4 c d e^3}{(d+e x) \left(c d^2-a e^2\right)^5}-\frac{e^3}{2 (d+e x)^2 \left(c d^2-a e^2\right)^4}",1,"((2*c^2*d^2*(-(c*d^2) + a*e^2)^3)/(a*e + c*d*x)^3 + (9*c^2*d^2*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (36*c^2*d^2*e^2*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (3*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (24*c*d*e^3*(-(c*d^2) + a*e^2))/(d + e*x) - 60*c^2*d^2*e^3*Log[a*e + c*d*x] + 60*c^2*d^2*e^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)","A",1
1906,1,234,262,0.2508424,"\int \frac{1}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-4),x]","\frac{60 c^3 d^3 e^3 \log (a e+c d x)+\frac{30 c^3 d^3 e^2 \left(c d^2-a e^2\right)}{a e+c d x}-\frac{6 c^3 d^3 e \left(c d^2-a e^2\right)^2}{(a e+c d x)^2}+\frac{c^3 d^3 \left(c d^2-a e^2\right)^3}{(a e+c d x)^3}+\frac{30 c^2 d^2 e^3 \left(c d^2-a e^2\right)}{d+e x}+\frac{\left(c d^2 e-a e^3\right)^3}{(d+e x)^3}+\frac{6 c d e^3 \left(c d^2-a e^2\right)^2}{(d+e x)^2}-60 c^3 d^3 e^3 \log (d+e x)}{3 \left(a e^2-c d^2\right)^7}","-\frac{20 c^3 d^3 e^3 \log (a e+c d x)}{\left(c d^2-a e^2\right)^7}+\frac{20 c^3 d^3 e^3 \log (d+e x)}{\left(c d^2-a e^2\right)^7}-\frac{10 c^2 d^2 e^2 \left(a e^2+c d^2+2 c d e x\right)}{\left(c d^2-a e^2\right)^6 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)}+\frac{5 c d e \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^2}-\frac{a e^2+c d^2+2 c d e x}{3 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^3}",1,"((c^3*d^3*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (6*c^3*d^3*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (30*c^3*d^3*e^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (c*d^2*e - a*e^3)^3/(d + e*x)^3 + (6*c*d*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (30*c^2*d^2*e^3*(c*d^2 - a*e^2))/(d + e*x) + 60*c^3*d^3*e^3*Log[a*e + c*d*x] - 60*c^3*d^3*e^3*Log[d + e*x])/(3*(-(c*d^2) + a*e^2)^7)","A",1
1907,1,320,388,2.4322028,"\int (d+e x)^4 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{(a e+c d x) \sqrt{(d+e x) (a e+c d x)} \left(-\frac{315 c^{7/2} d^{7/2} \sqrt{c d} \left(c d^2-a e^2\right)^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{e^{3/2} (a e+c d x)^{3/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+1152 c^7 d^7 (d+e x)^3 \left(c d^2-a e^2\right)+1008 c^6 d^6 (d+e x)^2 \left(c d^2-a e^2\right)^2+840 c^5 d^5 (d+e x) \left(c d^2-a e^2\right)^3+\frac{315 c^4 d^4 \left(c d^2-a e^2\right)^5}{e (a e+c d x)}+630 \left(c^2 d^3-a c d e^2\right)^4+1280 c^8 d^8 (d+e x)^4\right)}{7680 c^9 d^9}","-\frac{21 \left(c d^2-a e^2\right)^6 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac{21 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac{7 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{64 c^4 d^4}+\frac{21 (d+e x) \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{160 c^3 d^3}+\frac{3 (d+e x)^2 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{20 c^2 d^2}+\frac{(d+e x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{6 c d}",1,"((a*e + c*d*x)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(630*(c^2*d^3 - a*c*d*e^2)^4 + (315*c^4*d^4*(c*d^2 - a*e^2)^5)/(e*(a*e + c*d*x)) + 840*c^5*d^5*(c*d^2 - a*e^2)^3*(d + e*x) + 1008*c^6*d^6*(c*d^2 - a*e^2)^2*(d + e*x)^2 + 1152*c^7*d^7*(c*d^2 - a*e^2)*(d + e*x)^3 + 1280*c^8*d^8*(d + e*x)^4 - (315*c^(7/2)*d^(7/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(3/2)*(a*e + c*d*x)^(3/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7680*c^9*d^9)","A",1
1908,1,291,328,1.9085571,"\int (d+e x)^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{(a e+c d x) \sqrt{(d+e x) (a e+c d x)} \left(-\frac{105 c^{5/2} d^{5/2} \sqrt{c d} \left(c d^2-a e^2\right)^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{e^{3/2} (a e+c d x)^{3/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+336 c^5 d^5 (d+e x)^2 \left(c d^2-a e^2\right)+280 c^4 d^4 (d+e x) \left(c d^2-a e^2\right)^2+\frac{105 c^3 d^3 \left(c d^2-a e^2\right)^4}{e (a e+c d x)}+210 \left(c^2 d^3-a c d e^2\right)^3+384 c^6 d^6 (d+e x)^3\right)}{1920 c^7 d^7}","-\frac{7 \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac{7 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 c^4 d^4 e}+\frac{7 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{48 c^3 d^3}+\frac{7 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{40 c^2 d^2}+\frac{(d+e x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 c d}",1,"((a*e + c*d*x)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(210*(c^2*d^3 - a*c*d*e^2)^3 + (105*c^3*d^3*(c*d^2 - a*e^2)^4)/(e*(a*e + c*d*x)) + 280*c^4*d^4*(c*d^2 - a*e^2)^2*(d + e*x) + 336*c^5*d^5*(c*d^2 - a*e^2)*(d + e*x)^2 + 384*c^6*d^6*(d + e*x)^3 - (105*c^(5/2)*d^(5/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(3/2)*(a*e + c*d*x)^(3/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(1920*c^7*d^7)","A",1
1909,1,316,268,0.95431,"\int (d+e x)^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{c d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(15 a^4 e^7+5 a^3 c d e^5 (e x-11 d)+a^2 c^2 d^2 e^3 \left(73 d^2-19 d e x-2 e^2 x^2\right)+a c^3 d^3 e \left(15 d^3+191 d^2 e x+172 d e^2 x^2+56 e^3 x^3\right)+c^4 d^4 x \left(15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right)\right)-15 \left(c d^2-a e^2\right)^{9/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{192 c^{9/2} d^{9/2} e^{3/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{5 \left(c d^2-a e^2\right)^4 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{7/2} d^{7/2} e^{3/2}}+\frac{5 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 c^3 d^3 e}+\frac{5 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 c^2 d^2}+\frac{(d+e x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 c d}",1,"(Sqrt[c*d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(15*a^4*e^7 + 5*a^3*c*d*e^5*(-11*d + e*x) + a^2*c^2*d^2*e^3*(73*d^2 - 19*d*e*x - 2*e^2*x^2) + c^4*d^4*x*(15*d^3 + 118*d^2*e*x + 136*d*e^2*x^2 + 48*e^3*x^3) + a*c^3*d^3*e*(15*d^3 + 191*d^2*e*x + 172*d*e^2*x^2 + 56*e^3*x^3)) - 15*(c*d^2 - a*e^2)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(192*c^(9/2)*d^(9/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1910,1,262,210,0.7098885,"\int (d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{c d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(-3 a^3 e^5+a^2 c d e^3 (8 d-e x)+a c^2 d^2 e \left(3 d^2+22 d e x+10 e^2 x^2\right)+c^3 d^3 x \left(3 d^2+14 d e x+8 e^2 x^2\right)\right)-3 \left(c d^2-a e^2\right)^{7/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{24 c^{7/2} d^{7/2} e^{3/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{5/2} d^{5/2} e^{3/2}}+\frac{\left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 c^2 d^2 e}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 c d}",1,"(Sqrt[c*d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(-3*a^3*e^5 + a^2*c*d*e^3*(8*d - e*x) + c^3*d^3*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2) + a*c^2*d^2*e*(3*d^2 + 22*d*e*x + 10*e^2*x^2)) - 3*(c*d^2 - a*e^2)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(24*c^(7/2)*d^(7/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1911,1,214,159,0.4882423,"\int \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{c} \sqrt{d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(a^2 e^3+a c d e (d+3 e x)+c^2 d^2 x (d+2 e x)\right)-\left(c d^2-a e^2\right)^{5/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{4 e^{3/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 c d e}-\frac{\left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{3/2} d^{3/2} e^{3/2}}",1,"(Sqrt[c]*Sqrt[d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(a^2*e^3 + c^2*d^2*x*(d + 2*e*x) + a*c*d*e*(d + 3*e*x)) - (c*d^2 - a*e^2)^(5/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(4*(c*d)^(5/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1912,1,155,131,0.7766802,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{d+e x} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{e}-\frac{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{(c d)^{3/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{e^{3/2}}","\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e}-\frac{\left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 \sqrt{c} \sqrt{d} e^{3/2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e] - (c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/((c*d)^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/e^(3/2)","A",1
1913,1,164,124,0.6953109,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^2,x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{c} \sqrt{d} \sqrt{c d} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{\sqrt{e}}{d+e x}\right)}{e^{3/2}}","\frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{e^{3/2}}-\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e (d+e x)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[e]/(d + e*x)) + (Sqrt[c]*Sqrt[d]*Sqrt[c*d]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/e^(3/2)","A",1
1914,1,43,54,0.0212106,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^3,x]","\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^3 \left(c d^2-a e^2\right)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 (d+e x)^3 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*(c*d^2 - a*e^2)*(d + e*x)^3)","A",1
1915,1,61,111,0.0309151,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^4,x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(c d (5 d+2 e x)-3 a e^2\right)}{15 (d+e x)^4 \left(c d^2-a e^2\right)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{15 (d+e x)^3 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 (d+e x)^4 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e^2 + c*d*(5*d + 2*e*x)))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^4)","A",1
1916,1,124,171,0.045,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^5,x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(15 a^3 e^5+3 a^2 c d e^3 (e x-14 d)+a c^2 d^2 e \left(35 d^2-14 d e x-4 e^2 x^2\right)+c^3 d^3 x \left(35 d^2+28 d e x+8 e^2 x^2\right)\right)}{105 (d+e x)^4 \left(c d^2-a e^2\right)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 (d+e x)^3 \left(c d^2-a e^2\right)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{35 (d+e x)^4 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{7 (d+e x)^5 \left(c d^2-a e^2\right)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(15*a^3*e^5 + 3*a^2*c*d*e^3*(-14*d + e*x) + a*c^2*d^2*e*(35*d^2 - 14*d*e*x - 4*e^2*x^2) + c^3*d^3*x*(35*d^2 + 28*d*e*x + 8*e^2*x^2)))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^4)","A",1
1917,1,180,231,0.0636761,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^6} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-35 a^4 e^7+5 a^3 c d e^5 (27 d-e x)+3 a^2 c^2 d^2 e^3 \left(-63 d^2+9 d e x+2 e^2 x^2\right)+a c^3 d^3 e \left(105 d^3-63 d^2 e x-36 d e^2 x^2-8 e^3 x^3\right)+c^4 d^4 x \left(105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right)\right)}{315 (d+e x)^5 \left(c d^2-a e^2\right)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{315 (d+e x)^3 \left(c d^2-a e^2\right)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 (d+e x)^4 \left(c d^2-a e^2\right)^3}+\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{21 (d+e x)^5 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{9 (d+e x)^6 \left(c d^2-a e^2\right)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-35*a^4*e^7 + 5*a^3*c*d*e^5*(27*d - e*x) + 3*a^2*c^2*d^2*e^3*(-63*d^2 + 9*d*e*x + 2*e^2*x^2) + a*c^3*d^3*e*(105*d^3 - 63*d^2*e*x - 36*d*e^2*x^2 - 8*e^3*x^3) + c^4*d^4*x*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3)))/(315*(c*d^2 - a*e^2)^4*(d + e*x)^5)","A",1
1918,1,1276,461,6.4180599,"\int (d+e x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \left(c d^2-a e^2\right)^5 (a e+c d x) ((a e+c d x) (d+e x))^{3/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{13/2} \left(\frac{5}{16} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{11}{14 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{33}{56 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{33}{80 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{33}{128 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}+\frac{33}{256 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^6}\right)-\frac{495 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3 \left(-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right)}{65536 c^3 d^3 e^3 (a e+c d x)^3 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^6}\right)}{5 c^6 d^6 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{11/2} (d+e x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{99 \left(c d^2-a e^2\right)^8 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{32768 c^{13/2} d^{13/2} e^{5/2}}-\frac{99 \left(c d^2-a e^2\right)^6 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16384 c^6 d^6 e^2}+\frac{33 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2048 c^5 d^5 e}+\frac{33 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{640 c^4 d^4}+\frac{33 (d+e x) \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{448 c^3 d^3}+\frac{11 (d+e x)^2 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{112 c^2 d^2}+\frac{(d+e x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{8 c d}",1,"(2*(c*d^2 - a*e^2)^5*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(3/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(13/2)*((5*(33/(256*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6) + 33/(128*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 33/(80*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 33/(56*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 11/(14*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/16 - (495*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(65536*c^3*d^3*e^3*(a*e + c*d*x)^3*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6)))/(5*c^6*d^6*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(11/2)*(d + e*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1919,1,1196,401,6.2509652,"\int (d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \left(c d^2-a e^2\right)^4 (a e+c d x) ((a e+c d x) (d+e x))^{3/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{11/2} \left(\frac{5}{14} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{3}{4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{21}{40 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{21}{64 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{21}{128 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}\right)-\frac{45 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3 \left(-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right)}{4096 c^3 d^3 e^3 (a e+c d x)^3 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}\right)}{5 c^5 d^5 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{9/2} (d+e x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{9 \left(c d^2-a e^2\right)^7 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2048 c^{11/2} d^{11/2} e^{5/2}}-\frac{9 \left(c d^2-a e^2\right)^5 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{128 c^4 d^4 e}+\frac{3 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{28 c^2 d^2}+\frac{(d+e x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 c d}",1,"(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(3/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(11/2)*((5*(21/(128*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 21/(64*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 21/(40*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 3/(4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/14 - (45*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(4096*c^3*d^3*e^3*(a*e + c*d*x)^3*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5)))/(5*c^5*d^5*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(9/2)*(d + e*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1920,1,328,341,3.2347189,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{(a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{105 c^{5/2} d^{5/2} \sqrt{c d} \left(c d^2-a e^2\right)^{11/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{e^{5/2} (a e+c d x)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+896 c^5 d^5 (d+e x)^2 \left(c d^2-a e^2\right)+560 c^4 d^4 (d+e x) \left(c d^2-a e^2\right)^2+\frac{70 c^3 d^3 \left(c d^2-a e^2\right)^4}{e (a e+c d x)}-\frac{105 c^3 d^3 \left(c d^2-a e^2\right)^5}{e^2 (a e+c d x)^2}+280 \left(c^2 d^3-a c d e^2\right)^3+1280 c^6 d^6 (d+e x)^3\right)}{7680 c^7 d^7}","\frac{7 \left(c d^2-a e^2\right)^6 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 c^{9/2} d^{9/2} e^{5/2}}-\frac{7 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{192 c^3 d^3 e}+\frac{7 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{6 c d}",1,"((a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(280*(c^2*d^3 - a*c*d*e^2)^3 - (105*c^3*d^3*(c*d^2 - a*e^2)^5)/(e^2*(a*e + c*d*x)^2) + (70*c^3*d^3*(c*d^2 - a*e^2)^4)/(e*(a*e + c*d*x)) + 560*c^4*d^4*(c*d^2 - a*e^2)^2*(d + e*x) + 896*c^5*d^5*(c*d^2 - a*e^2)*(d + e*x)^2 + 1280*c^6*d^6*(d + e*x)^3 + (105*c^(5/2)*d^(5/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(5/2)*(a*e + c*d*x)^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7680*c^7*d^7)","A",1
1921,1,299,283,2.2198009,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{(a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{15 c^{3/2} d^{3/2} \sqrt{c d} \left(c d^2-a e^2\right)^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{e^{5/2} (a e+c d x)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+80 c^3 d^3 (d+e x) \left(c d^2-a e^2\right)+40 \left(c^2 d^3-a c d e^2\right)^2-\frac{15 c^2 d^2 \left(c d^2-a e^2\right)^4}{e^2 (a e+c d x)^2}+\frac{10 c^2 d^2 \left(c d^2-a e^2\right)^3}{e (a e+c d x)}+128 c^4 d^4 (d+e x)^2\right)}{640 c^5 d^5}","\frac{3 \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 c^{7/2} d^{7/2} e^{5/2}}-\frac{3 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{16 c^2 d^2 e}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 c d}",1,"((a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(40*(c^2*d^3 - a*c*d*e^2)^2 - (15*c^2*d^2*(c*d^2 - a*e^2)^4)/(e^2*(a*e + c*d*x)^2) + (10*c^2*d^2*(c*d^2 - a*e^2)^3)/(e*(a*e + c*d*x)) + 80*c^3*d^3*(c*d^2 - a*e^2)*(d + e*x) + 128*c^4*d^4*(d + e*x)^2 + (15*c^(3/2)*d^(3/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(5/2)*(a*e + c*d*x)^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(640*c^5*d^5)","A",1
1922,1,317,232,0.9502594,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{\sqrt{c d} \left(3 \left(c d^2-a e^2\right)^{9/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(3 a^4 e^7+a^3 c d e^5 (e x-11 d)-a^2 c^2 d^2 e^3 \left(11 d^2+55 d e x+26 e^2 x^2\right)+a c^3 d^3 e \left(3 d^3-13 d^2 e x-68 d e^2 x^2-40 e^3 x^3\right)+c^4 d^4 x \left(3 d^3-2 d^2 e x-24 d e^2 x^2-16 e^3 x^3\right)\right)\right)}{64 c^{7/2} d^{7/2} e^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{3 \left(c d^2-a e^2\right)^4 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{5/2} d^{5/2} e^{5/2}}-\frac{3 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 c^2 d^2 e^2}+\frac{\left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{8 c d e}",1,"(Sqrt[c*d]*(-(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(3*a^4*e^7 + a^3*c*d*e^5*(-11*d + e*x) - a^2*c^2*d^2*e^3*(11*d^2 + 55*d*e*x + 26*e^2*x^2) + a*c^3*d^3*e*(3*d^3 - 13*d^2*e*x - 68*d*e^2*x^2 - 40*e^3*x^3) + c^4*d^4*x*(3*d^3 - 2*d^2*e*x - 24*d*e^2*x^2 - 16*e^3*x^3))) + 3*(c*d^2 - a*e^2)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(64*c^(7/2)*d^(7/2)*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1923,1,264,201,0.661885,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x),x]","\frac{\sqrt{c} \sqrt{d} \left(3 \left(c d^2-a e^2\right)^{7/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(-3 a^3 e^5-a^2 c d e^3 (8 d+17 e x)+a c^2 d^2 e \left(3 d^2-10 d e x-22 e^2 x^2\right)+c^3 d^3 x \left(3 d^2-2 d e x-8 e^2 x^2\right)\right)\right)}{24 e^{5/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e}+\frac{1}{8} \left(\frac{a}{c d}-\frac{d}{e^2}\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}",1,"(Sqrt[c]*Sqrt[d]*(-(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(-3*a^3*e^5 - a^2*c*d*e^3*(8*d + 17*e*x) + a*c^2*d^2*e*(3*d^2 - 10*d*e*x - 22*e^2*x^2) + c^3*d^3*x*(3*d^2 - 2*d*e*x - 8*e^2*x^2))) + 3*(c*d^2 - a*e^2)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(24*(c*d)^(5/2)*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1924,1,209,187,0.3808726,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^2,x]","\frac{\sqrt{e} (c d)^{3/2} (d+e x) \left(5 a^2 e^3+a c d e (7 e x-3 d)+c^2 d^2 x (2 e x-3 d)\right)+3 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^{5/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{4 e^{5/2} (c d)^{3/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 e (d+e x)}+\frac{3}{4} \left(a-\frac{c d^2}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}+\frac{3 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 \sqrt{c} \sqrt{d} e^{5/2}}",1,"((c*d)^(3/2)*Sqrt[e]*(d + e*x)*(5*a^2*e^3 + c^2*d^2*x*(-3*d + 2*e*x) + a*c*d*e*(-3*d + 7*e*x)) + 3*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(5/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(4*(c*d)^(3/2)*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1925,1,108,175,0.0760428,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^3,x]","\frac{2 c d (a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{5 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{e (d+e x)^2}+\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^2}-\frac{3 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 e^{5/2}}",1,"(2*c*d*(a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[3/2, 5/2, 7/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(5*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","C",1
1926,1,184,169,0.7052553,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{3 c^{3/2} d^{3/2} \sqrt{c d} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{\sqrt{e} \left(a e^2+c d (3 d+4 e x)\right)}{(d+e x)^2}\right)}{3 e^{5/2}}","\frac{c^{3/2} d^{3/2} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{e^{5/2}}-\frac{2 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^2 (d+e x)}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e (d+e x)^3}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(a*e^2 + c*d*(3*d + 4*e*x)))/(d + e*x)^2) + (3*c^(3/2)*d^(3/2)*Sqrt[c*d]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(3*e^(5/2))","A",1
1927,1,43,54,0.0315901,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^5,x]","\frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^5 \left(c d^2-a e^2\right)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 (d+e x)^5 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*(c*d^2 - a*e^2)*(d + e*x)^5)","A",1
1928,1,61,111,0.052785,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^6,x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(c d (7 d+2 e x)-5 a e^2\right)}{35 (d+e x)^6 \left(c d^2-a e^2\right)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{35 (d+e x)^5 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 (d+e x)^6 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-5*a*e^2 + c*d*(7*d + 2*e*x)))/(35*(c*d^2 - a*e^2)^2*(d + e*x)^6)","A",1
1929,1,94,171,0.0709859,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^7} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left(63 d^2+36 d e x+8 e^2 x^2\right)\right)}{315 (d+e x)^7 \left(c d^2-a e^2\right)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{315 (d+e x)^5 \left(c d^2-a e^2\right)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{63 (d+e x)^6 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{9 (d+e x)^7 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(35*a^2*e^4 - 10*a*c*d*e^2*(9*d + 2*e*x) + c^2*d^2*(63*d^2 + 36*d*e*x + 8*e^2*x^2)))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^7)","A",1
1930,1,138,231,0.0901123,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^8} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^8,x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left(99 d^2+44 d e x+8 e^2 x^2\right)+c^3 d^3 \left(231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right)\right)}{1155 (d+e x)^8 \left(c d^2-a e^2\right)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{1155 (d+e x)^5 \left(c d^2-a e^2\right)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{231 (d+e x)^6 \left(c d^2-a e^2\right)^3}+\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{33 (d+e x)^7 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{11 (d+e x)^8 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(11*d + 2*e*x) - 5*a*c^2*d^2*e^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2) + c^3*d^3*(231*d^3 + 198*d^2*e*x + 88*d*e^2*x^2 + 16*e^3*x^3)))/(1155*(c*d^2 - a*e^2)^4*(d + e*x)^8)","A",1
1931,1,1439,534,6.7908766,"\int (d+e x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 \left(c d^2-a e^2\right)^6 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{15/2} \left(\frac{1001 \left(c d^2-a e^2\right)^4 \left(\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^4}{524288 c^4 d^4 e^4 (a e+c d x)^4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^7}+\frac{7}{20} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{13}{18 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{143}{288 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{143}{448 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{143}{768 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}+\frac{143}{1536 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^6}+\frac{143}{4096 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^7}\right)\right)}{7 c^7 d^7 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{13/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{143 \left(c d^2-a e^2\right)^{10} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac{143 \left(c d^2-a e^2\right)^8 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac{143 \left(c d^2-a e^2\right)^6 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{49152 c^6 d^6 e^2}+\frac{143 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{15360 c^5 d^5 e}+\frac{143 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{4480 c^4 d^4}+\frac{143 (d+e x) \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{2880 c^3 d^3}+\frac{13 (d+e x)^2 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{10 c d}",1,"(2*(c*d^2 - a*e^2)^6*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(15/2)*((7*(143/(4096*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^7) + 143/(1536*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6) + 143/(768*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 143/(448*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 143/(288*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 13/(18*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/20 + (1001*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(524288*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^7)))/(7*c^7*d^7*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(13/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1932,1,1359,474,6.4971303,"\int (d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 \left(c d^2-a e^2\right)^5 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{13/2} \left(\frac{385 \left(c d^2-a e^2\right)^4 \left(\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^4}{131072 c^4 d^4 e^4 (a e+c d x)^4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^6}+\frac{7}{18} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{11}{16 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{99}{224 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{33}{128 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{33}{256 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}+\frac{99}{2048 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^6}\right)\right)}{7 c^6 d^6 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{11/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{55 \left(c d^2-a e^2\right)^9 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac{55 \left(c d^2-a e^2\right)^7 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32768 c^6 d^6 e^3}-\frac{55 \left(c d^2-a e^2\right)^5 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{12288 c^5 d^5 e^2}+\frac{11 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{768 c^4 d^4 e}+\frac{11 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{224 c^3 d^3}+\frac{11 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{144 c^2 d^2}+\frac{(d+e x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{9 c d}",1,"(2*(c*d^2 - a*e^2)^5*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(13/2)*((7*(99/(2048*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6) + 33/(256*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 33/(128*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 99/(224*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 11/(16*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/18 + (385*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(131072*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^6)))/(7*c^6*d^6*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(11/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1933,1,1279,414,6.3069788,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 \left(c d^2-a e^2\right)^4 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{11/2} \left(\frac{315 \left(c d^2-a e^2\right)^4 \left(\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^4}{65536 c^4 d^4 e^4 (a e+c d x)^4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}+\frac{7}{16} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{9}{14 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{3}{8 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{3}{16 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{9}{128 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^5}\right)\right)}{7 c^5 d^5 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{9/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{45 \left(c d^2-a e^2\right)^8 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac{45 \left(c d^2-a e^2\right)^6 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac{15 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{128 c^3 d^3 e}+\frac{9 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{8 c d}",1,"(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(11/2)*((7*(9/(128*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5) + 3/(16*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 3/(8*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 9/(14*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/16 + (315*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(65536*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^5)))/(7*c^5*d^5*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(9/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1934,1,1199,356,6.2176193,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 \left(c d^2-a e^2\right)^3 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{9/2} \left(\frac{35 \left(c d^2-a e^2\right)^4 \left(\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^4}{4096 c^4 d^4 e^4 (a e+c d x)^4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}+\frac{1}{2} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{7}{12 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{7}{24 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{7}{64 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^4}\right)\right)}{7 c^4 d^4 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{7/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{5 \left(c d^2-a e^2\right)^7 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac{5 \left(c d^2-a e^2\right)^5 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{1024 c^4 d^4 e^3}-\frac{5 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{384 c^3 d^3 e^2}+\frac{\left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{24 c^2 d^2 e}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{7 c d}",1,"(2*(c*d^2 - a*e^2)^3*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(9/2)*((7/(64*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 7/(24*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 7/(12*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1))/2 + (35*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(4096*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4)))/(7*c^4*d^4*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(7/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1935,1,1119,305,6.1122843,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 \left(c d^2-a e^2\right)^2 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^{7/2} \left(\frac{35 \left(c d^2-a e^2\right)^4 \left(\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left(c d^2-a e^2\right)^3 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left(c d^2-a e^2\right)^2 \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^2}+\frac{2 c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}}\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)^4}{2048 c^4 d^4 e^4 (a e+c d x)^4 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}+\frac{7}{12} \left(\frac{1}{\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1}+\frac{1}{2 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^2}+\frac{3}{16 \left(\frac{c d e (a e+c d x)}{\left(c d^2-a e^2\right) \left(\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right)}+1\right)^3}\right)\right)}{7 c^3 d^3 \left(\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right)^{5/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{5 \left(c d^2-a e^2\right)^6 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac{5 \left(c d^2-a e^2\right)^4 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac{5 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{192 c^2 d^2 e^2}+\frac{\left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{12 c d e}",1,"(2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(7/2)*((7*(3/(16*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 1/(2*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/12 + (35*(c*d^2 - a*e^2)^4*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])))/(2048*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3)))/(7*c^3*d^3*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(5/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","B",1
1936,1,384,274,1.2252971,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x),x]","\frac{\sqrt{c d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(-15 a^5 e^9+5 a^4 c d e^7 (14 d-e x)+2 a^3 c^2 d^2 e^5 \left(64 d^2+268 d e x+129 e^2 x^2\right)+2 a^2 c^3 d^3 e^3 \left(-35 d^3+87 d^2 e x+489 d e^2 x^2+292 e^3 x^3\right)+a c^4 d^4 e \left(15 d^4-80 d^3 e x+54 d^2 e^2 x^2+688 d e^3 x^3+464 e^4 x^4\right)+c^5 d^5 x \left(15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right)\right)-15 \left(c d^2-a e^2\right)^{11/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{640 c^{7/2} d^{7/2} e^{7/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{3 \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{3 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 e}+\frac{1}{16} \left(\frac{a}{c d}-\frac{d}{e^2}\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}",1,"(Sqrt[c*d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(-15*a^5*e^9 + 5*a^4*c*d*e^7*(14*d - e*x) + 2*a^3*c^2*d^2*e^5*(64*d^2 + 268*d*e*x + 129*e^2*x^2) + 2*a^2*c^3*d^3*e^3*(-35*d^3 + 87*d^2*e*x + 489*d*e^2*x^2 + 292*e^3*x^3) + c^5*d^5*x*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4) + a*c^4*d^4*e*(15*d^4 - 80*d^3*e*x + 54*d^2*e^2*x^2 + 688*d*e^3*x^3 + 464*e^4*x^4)) - 15*(c*d^2 - a*e^2)^(11/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(640*c^(7/2)*d^(7/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1937,1,316,261,0.8957373,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^2,x]","\frac{\sqrt{c} \sqrt{d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(15 a^4 e^7+a^3 c d e^5 (73 d+133 e x)+a^2 c^2 d^2 e^3 \left(-55 d^2+109 d e x+254 e^2 x^2\right)+a c^3 d^3 e \left(15 d^3-65 d^2 e x+44 d e^2 x^2+184 e^3 x^3\right)+c^4 d^4 x \left(15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right)\right)-15 \left(c d^2-a e^2\right)^{9/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{192 e^{7/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{5 \left(c d^2-a e^2\right)^4 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac{5}{24} \left(a-\frac{c d^2}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}+\frac{(a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 e}+\frac{5 \left(c d^2-a e^2\right)^2 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 c d e^3}",1,"(Sqrt[c]*Sqrt[d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(15*a^4*e^7 + a^3*c*d*e^5*(73*d + 133*e*x) + a^2*c^2*d^2*e^3*(-55*d^2 + 109*d*e*x + 254*e^2*x^2) + c^4*d^4*x*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3) + a*c^3*d^3*e*(15*d^3 - 65*d^2*e*x + 44*d*e^2*x^2 + 184*e^3*x^3)) - 15*(c*d^2 - a*e^2)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(192*(c*d)^(5/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1938,1,252,244,0.5023713,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^3,x]","\frac{\sqrt{e} (c d)^{3/2} (d+e x) \left(33 a^3 e^5+a^2 c d e^3 (59 e x-40 d)+a c^2 d^2 e \left(15 d^2-50 d e x+34 e^2 x^2\right)+c^3 d^3 x \left(15 d^2-10 d e x+8 e^2 x^2\right)\right)-15 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^{7/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{24 e^{7/2} (c d)^{3/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{e (d+e x)^2}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e^2}-\frac{5 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 \sqrt{c} \sqrt{d} e^{7/2}}+\frac{5 \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e^3}",1,"((c*d)^(3/2)*Sqrt[e]*(d + e*x)*(33*a^3*e^5 + a^2*c*d*e^3*(-40*d + 59*e*x) + c^3*d^3*x*(15*d^2 - 10*d*e*x + 8*e^2*x^2) + a*c^2*d^2*e*(15*d^2 - 50*d*e*x + 34*e^2*x^2)) - 15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(24*(c*d)^(3/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1939,1,108,235,0.0728738,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^4,x]","\frac{2 c d (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{e (d+e x)^3}+\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 e^2 (d+e x)}+\frac{15 c d \left(a-\frac{c d^2}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e}+\frac{15 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 e^{7/2}}",1,"(2*c*d*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[3/2, 7/2, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","C",1
1940,1,112,226,0.0724932,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^5,x]","\frac{2 c^2 d^2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 \left(c d^2-a e^2\right)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","-\frac{5 c^{3/2} d^{3/2} \left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 e^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^3}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3 e (d+e x)^4}-\frac{10 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e^2 (d+e x)^2}",1,"(2*c^2*d^2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[5/2, 7/2, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","C",1
1941,1,217,218,1.1221559,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^6,x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{15 c^{5/2} d^{5/2} \sqrt{c d} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{\sqrt{e} \left(3 a^2 e^4+a c d e^2 (5 d+11 e x)+c^2 d^2 \left(15 d^2+35 d e x+23 e^2 x^2\right)\right)}{(d+e x)^3}\right)}{15 e^{7/2}}","\frac{c^{5/2} d^{5/2} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{e^{7/2}}-\frac{2 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^3 (d+e x)}-\frac{2 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e^2 (d+e x)^3}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 e (d+e x)^5}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(3*a^2*e^4 + a*c*d*e^2*(5*d + 11*e*x) + c^2*d^2*(15*d^2 + 35*d*e*x + 23*e^2*x^2)))/(d + e*x)^3) + (15*c^(5/2)*d^(5/2)*Sqrt[c*d]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(15*e^(7/2))","A",1
1942,1,43,54,0.0358933,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^7} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^7,x]","\frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 (d+e x)^7 \left(c d^2-a e^2\right)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{7 (d+e x)^7 \left(c d^2-a e^2\right)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)","A",1
1943,1,71,111,0.0390019,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^8} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(c d (9 d+2 e x)-7 a e^2\right)}{63 (d+e x)^5 \left(c d^2-a e^2\right)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{63 (d+e x)^7 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{9 (d+e x)^8 \left(c d^2-a e^2\right)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-7*a*e^2 + c*d*(9*d + 2*e*x)))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^5)","A",1
1944,1,104,171,0.0579206,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^9} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^9,x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(63 a^2 e^4-14 a c d e^2 (11 d+2 e x)+c^2 d^2 \left(99 d^2+44 d e x+8 e^2 x^2\right)\right)}{693 (d+e x)^6 \left(c d^2-a e^2\right)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{693 (d+e x)^7 \left(c d^2-a e^2\right)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{99 (d+e x)^8 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{11 (d+e x)^9 \left(c d^2-a e^2\right)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^4 - 14*a*c*d*e^2*(11*d + 2*e*x) + c^2*d^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2)))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^6)","A",1
1945,1,148,231,0.0767665,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{10}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^10,x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(-231 a^3 e^6+63 a^2 c d e^4 (13 d+2 e x)-7 a c^2 d^2 e^2 \left(143 d^2+52 d e x+8 e^2 x^2\right)+c^3 d^3 \left(429 d^3+286 d^2 e x+104 d e^2 x^2+16 e^3 x^3\right)\right)}{3003 (d+e x)^7 \left(c d^2-a e^2\right)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{3003 (d+e x)^7 \left(c d^2-a e^2\right)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{429 (d+e x)^8 \left(c d^2-a e^2\right)^3}+\frac{12 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{143 (d+e x)^9 \left(c d^2-a e^2\right)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{13 (d+e x)^{10} \left(c d^2-a e^2\right)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^6 + 63*a^2*c*d*e^4*(13*d + 2*e*x) - 7*a*c^2*d^2*e^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2) + c^3*d^3*(429*d^3 + 286*d^2*e*x + 104*d*e^2*x^2 + 16*e^3*x^3)))/(3003*(c*d^2 - a*e^2)^4*(d + e*x)^7)","A",1
1946,1,214,255,0.5644075,"\int \frac{(d+e x)^3}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{c} \sqrt{d} \left(15 a^2 e^4-10 a c d e^2 (4 d+e x)+c^2 d^2 \left(33 d^2+26 d e x+8 e^2 x^2\right)\right)+\frac{15 \sqrt{c d} \left(c d^2-a e^2\right)^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{24 c^{7/2} d^{7/2}}","\frac{5 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{7/2} d^{7/2} \sqrt{e}}+\frac{5 \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 c^3 d^3}+\frac{5 (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 c^2 d^2}+\frac{(d+e x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c d}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*(15*a^2*e^4 - 10*a*c*d*e^2*(4*d + e*x) + c^2*d^2*(33*d^2 + 26*d*e*x + 8*e^2*x^2)) + (15*Sqrt[c*d]*(c*d^2 - a*e^2)^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(24*c^(7/2)*d^(7/2))","A",1
1947,1,182,195,0.5248481,"\int \frac{(d+e x)^2}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{3 \sqrt{c d} \left(c d^2-a e^2\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\sqrt{c} \sqrt{d} \left(c d (5 d+2 e x)-3 a e^2\right)\right)}{4 c^{5/2} d^{5/2}}","\frac{3 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{5/2} d^{5/2} \sqrt{e}}+\frac{3 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 c d}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*(-3*a*e^2 + c*d*(5*d + 2*e*x)) + (3*Sqrt[c*d]*(c*d^2 - a*e^2)^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(4*c^(5/2)*d^(5/2))","A",1
1948,1,203,134,0.2088897,"\int \frac{d+e x}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+\sqrt{c d} \sqrt{c d^2-a e^2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{c^{3/2} d^{3/2} \sqrt{e} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{\left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 c^{3/2} d^{3/2} \sqrt{e}}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(c^(3/2)*d^(3/2)*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","A",1
1949,1,148,82,0.0756698,"\int \frac{1}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{c^{3/2} d^{3/2} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}}","\frac{\tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{\sqrt{c} \sqrt{d} \sqrt{e}}",1,"(2*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(c^(3/2)*d^(3/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1950,1,42,52,0.0153436,"\int \frac{1}{(d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 (a e+c d x)}{\left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{(d+e x) \left(c d^2-a e^2\right)}",1,"(2*(a*e + c*d*x))/((c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1951,1,61,111,0.0410642,"\int \frac{1}{(d+e x)^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(c d (3 d+2 e x)-a e^2\right)}{3 (d+e x)^2 \left(c d^2-a e^2\right)^2}","\frac{4 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 (d+e x) \left(c d^2-a e^2\right)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 (d+e x)^2 \left(c d^2-a e^2\right)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(a*e^2) + c*d*(3*d + 2*e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^2)","A",1
1952,1,94,171,0.0458816,"\int \frac{1}{(d+e x)^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left(15 d^2+20 d e x+8 e^2 x^2\right)\right)}{15 (d+e x)^3 \left(c d^2-a e^2\right)^3}","\frac{16 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 (d+e x) \left(c d^2-a e^2\right)^3}+\frac{8 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 (d+e x)^2 \left(c d^2-a e^2\right)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 (d+e x)^3 \left(c d^2-a e^2\right)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^4 - 2*a*c*d*e^2*(5*d + 2*e*x) + c^2*d^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2)))/(15*(c*d^2 - a*e^2)^3*(d + e*x)^3)","A",1
1953,1,138,231,0.0671165,"\int \frac{1}{(d+e x)^4 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-5 a^3 e^6+3 a^2 c d e^4 (7 d+2 e x)-a c^2 d^2 e^2 \left(35 d^2+28 d e x+8 e^2 x^2\right)+c^3 d^3 \left(35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right)\right)}{35 (d+e x)^4 \left(c d^2-a e^2\right)^4}","\frac{32 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 (d+e x) \left(c d^2-a e^2\right)^4}+\frac{16 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 (d+e x)^2 \left(c d^2-a e^2\right)^3}+\frac{12 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 (d+e x)^3 \left(c d^2-a e^2\right)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{7 (d+e x)^4 \left(c d^2-a e^2\right)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^6 + 3*a^2*c*d*e^4*(7*d + 2*e*x) - a*c^2*d^2*e^2*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + c^3*d^3*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3)))/(35*(c*d^2 - a*e^2)^4*(d + e*x)^4)","A",1
1954,1,100,302,0.0695987,"\int \frac{(d+e x)^5}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \left(c d^2-a e^2\right)^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \, _2F_1\left(-\frac{7}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c^5 d^5 \sqrt{(d+e x) (a e+c d x)}}","\frac{35 \sqrt{e} \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{9/2} d^{9/2}}+\frac{35 e \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 c^4 d^4}+\frac{35 e (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 c^3 d^3}+\frac{7 e (d+e x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^2 d^2}-\frac{2 (d+e x)^4}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(c*d^2 - a*e^2)^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*Hypergeometric2F1[-7/2, -1/2, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^5*d^5*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
1955,1,100,241,0.0569128,"\int \frac{(d+e x)^4}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \left(c d^2-a e^2\right)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c^4 d^4 \sqrt{(d+e x) (a e+c d x)}}","\frac{15 \sqrt{e} \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{7/2} d^{7/2}}+\frac{15 e \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 c^3 d^3}+\frac{5 e (d+e x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 c^2 d^2}-\frac{2 (d+e x)^3}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(c*d^2 - a*e^2)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*Hypergeometric2F1[-5/2, -1/2, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^4*d^4*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
1956,1,93,180,0.0519519,"\int \frac{(d+e x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 (d+e x)^2 \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{3/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{3 \sqrt{e} \left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 c^{5/2} d^{5/2}}+\frac{3 e \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c^2 d^2}-\frac{2 (d+e x)^2}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(d + e*x)^2*Hypergeometric2F1[-3/2, -1/2, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
1957,1,164,125,0.3208826,"\int \frac{(d+e x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d^2-a e^2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-2 (c d)^{3/2} (d+e x)}{(c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\sqrt{e} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{c^{3/2} d^{3/2}}-\frac{2 (d+e x)}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(c*d)^(3/2)*(d + e*x) + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/((c*d)^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1958,1,39,50,0.0146634,"\int \frac{d+e x}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 (d+e x)}{\left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 (d+e x)}{\left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(d + e*x))/((c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1959,1,49,62,0.0230858,"\int \frac{1}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-3/2),x]","-\frac{2 \left(a e^2+c d (d+2 e x)\right)}{\left(c d^2-a e^2\right)^2 \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 \left(a e^2+c d^2+2 c d e x\right)}{\left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(a*e^2 + c*d*(d + 2*e*x)))/((c*d^2 - a*e^2)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1960,1,95,121,0.0505793,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{2 a^2 e^4-4 a c d e^2 (3 d+2 e x)-2 c^2 d^2 \left(3 d^2+12 d e x+8 e^2 x^2\right)}{3 (d+e x) \left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)}}","\frac{2}{3 (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{8 c d \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*a^2*e^4 - 4*a*c*d*e^2*(3*d + 2*e*x) - 2*c^2*d^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2))/(3*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1961,1,136,181,0.0613492,"\int \frac{1}{(d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \left(a^3 e^6-a^2 c d e^4 (5 d+2 e x)+a c^2 d^2 e^2 \left(15 d^2+20 d e x+8 e^2 x^2\right)+c^3 d^3 \left(5 d^3+30 d^2 e x+40 d e^2 x^2+16 e^3 x^3\right)\right)}{5 (d+e x)^2 \left(c d^2-a e^2\right)^4 \sqrt{(d+e x) (a e+c d x)}}","-\frac{16 c^2 d^2 \left(a e^2+c d^2+2 c d e x\right)}{5 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{4 c d}{5 (d+e x) \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{2}{5 (d+e x)^2 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(a^3*e^6 - a^2*c*d*e^4*(5*d + 2*e*x) + a*c^2*d^2*e^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + c^3*d^3*(5*d^3 + 30*d^2*e*x + 40*d*e^2*x^2 + 16*e^3*x^3)))/(5*(c*d^2 - a*e^2)^4*(d + e*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1962,1,193,241,0.0866462,"\int \frac{1}{(d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \left(-5 a^4 e^8+4 a^3 c d e^6 (7 d+2 e x)-2 a^2 c^2 d^2 e^4 \left(35 d^2+28 d e x+8 e^2 x^2\right)+4 a c^3 d^3 e^2 \left(35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right)+c^4 d^4 \left(35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right)\right)}{35 (d+e x)^3 \left(c d^2-a e^2\right)^5 \sqrt{(d+e x) (a e+c d x)}}","-\frac{128 c^3 d^3 \left(a e^2+c d^2+2 c d e x\right)}{35 \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{32 c^2 d^2}{35 (d+e x) \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{16 c d}{35 (d+e x)^2 \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{2}{7 (d+e x)^3 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(-5*a^4*e^8 + 4*a^3*c*d*e^6*(7*d + 2*e*x) - 2*a^2*c^2*d^2*e^4*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 4*a*c^3*d^3*e^2*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + c^4*d^4*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448*d*e^3*x^3 + 128*e^4*x^4)))/(35*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1963,1,258,301,0.113209,"\int \frac{1}{(d+e x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \left(7 a^5 e^{10}-5 a^4 c d e^8 (9 d+2 e x)+2 a^3 c^2 d^2 e^6 \left(63 d^2+36 d e x+8 e^2 x^2\right)-2 a^2 c^3 d^3 e^4 \left(105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right)+a c^4 d^4 e^2 \left(315 d^4+840 d^3 e x+1008 d^2 e^2 x^2+576 d e^3 x^3+128 e^4 x^4\right)+c^5 d^5 \left(63 d^5+630 d^4 e x+1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+1152 d e^4 x^4+256 e^5 x^5\right)\right)}{63 (d+e x)^4 \left(c d^2-a e^2\right)^6 \sqrt{(d+e x) (a e+c d x)}}","-\frac{256 c^4 d^4 \left(a e^2+c d^2+2 c d e x\right)}{63 \left(c d^2-a e^2\right)^6 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{64 c^3 d^3}{63 (d+e x) \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{32 c^2 d^2}{63 (d+e x)^2 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{20 c d}{63 (d+e x)^3 \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{2}{9 (d+e x)^4 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*(7*a^5*e^10 - 5*a^4*c*d*e^8*(9*d + 2*e*x) + 2*a^3*c^2*d^2*e^6*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 2*a^2*c^3*d^3*e^4*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + a*c^4*d^4*e^2*(315*d^4 + 840*d^3*e*x + 1008*d^2*e^2*x^2 + 576*d*e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(63*d^5 + 630*d^4*e*x + 1680*d^3*e^2*x^2 + 2016*d^2*e^3*x^3 + 1152*d*e^4*x^4 + 256*e^5*x^5)))/(63*(c*d^2 - a*e^2)^6*(d + e*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
1964,1,112,294,0.0599396,"\int \frac{(d+e x)^6}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 \left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(-\frac{7}{2},-\frac{3}{2};-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 c^4 d^4 (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{35 e^{3/2} \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{9/2} d^{9/2}}+\frac{35 e^2 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 c^4 d^4}+\frac{35 e^2 (d+e x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{6 c^3 d^3}-\frac{14 e (d+e x)^3}{3 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^5}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(c*d^2 - a*e^2)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-7/2, -3/2, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*c^4*d^4*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","C",1
1965,1,112,231,0.0601482,"\int \frac{(d+e x)^5}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 \left(c d^2-a e^2\right)^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 c^3 d^3 (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{5 e^{3/2} \left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 c^{7/2} d^{7/2}}+\frac{5 e^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c^3 d^3}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^4}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(c*d^2 - a*e^2)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-5/2, -3/2, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*c^3*d^3*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])","C",1
1966,1,98,172,0.0690271,"\int \frac{(d+e x)^4}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 c d (a e+c d x)^3 \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{3/2}}","\frac{e^{3/2} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{c^{5/2} d^{5/2}}-\frac{2 e (d+e x)}{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^3}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[-3/2, -3/2, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*c*d*(a*e + c*d*x)^3*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/2))","C",1
1967,1,46,54,0.0190293,"\int \frac{(d+e x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 \left(c d^2-a e^2\right) (a e+c d x)^3}","-\frac{2 (d+e x)^3}{3 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*(c*d^2 - a*e^2)*(a*e + c*d*x)^3)","A",1
1968,1,59,116,0.031356,"\int \frac{(d+e x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^2 \left(c d (d-2 e x)-3 a e^2\right)}{3 \left(c d^2-a e^2\right)^2 ((d+e x) (a e+c d x))^{3/2}}","\frac{2 e \left(a e^2+c d^2+2 c d e x\right)}{3 c d \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^2*(-3*a*e^2 + c*d*(d - 2*e*x)))/(3*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1969,1,90,118,0.0562215,"\int \frac{d+e x}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (d+e x) \left(3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^2 \left(-d^2+4 d e x+8 e^2 x^2\right)\right)}{3 \left(c d^2-a e^2\right)^3 ((d+e x) (a e+c d x))^{3/2}}","\frac{8 e \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)}{3 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(d + e*x)*(3*a^2*e^4 + 6*a*c*d*e^2*(d + 2*e*x) + c^2*d^2*(-d^2 + 4*d*e*x + 8*e^2*x^2)))/(3*(c*d^2 - a*e^2)^3*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1970,1,132,132,0.0643671,"\int \frac{1}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-5/2),x]","\frac{-2 a^3 e^6+6 a^2 c d e^4 (3 d+2 e x)+6 a c^2 d^2 e^2 \left(3 d^2+12 d e x+8 e^2 x^2\right)+2 c^3 d^3 \left(-d^3+6 d^2 e x+24 d e^2 x^2+16 e^3 x^3\right)}{3 \left(c d^2-a e^2\right)^4 ((d+e x) (a e+c d x))^{3/2}}","\frac{16 c d e \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 2*e*x) + 6*a*c^2*d^2*e^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2) + 2*c^3*d^3*(-d^3 + 6*d^2*e*x + 24*d*e^2*x^2 + 16*e^3*x^3))/(3*(c*d^2 - a*e^2)^4*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1971,1,193,192,0.090344,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 \left(3 a^4 e^8-4 a^3 c d e^6 (5 d+2 e x)+6 a^2 c^2 d^2 e^4 \left(15 d^2+20 d e x+8 e^2 x^2\right)+12 a c^3 d^3 e^2 \left(5 d^3+30 d^2 e x+40 d e^2 x^2+16 e^3 x^3\right)+c^4 d^4 \left(-5 d^4+40 d^3 e x+240 d^2 e^2 x^2+320 d e^3 x^3+128 e^4 x^4\right)\right)}{15 (d+e x) \left(c d^2-a e^2\right)^5 ((d+e x) (a e+c d x))^{3/2}}","\frac{128 c^2 d^2 e \left(a e^2+c d^2+2 c d e x\right)}{15 \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{16 c d \left(a e^2+c d^2+2 c d e x\right)}{15 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{2}{5 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(3*a^4*e^8 - 4*a^3*c*d*e^6*(5*d + 2*e*x) + 6*a^2*c^2*d^2*e^4*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + 12*a*c^3*d^3*e^2*(5*d^3 + 30*d^2*e*x + 40*d*e^2*x^2 + 16*e^3*x^3) + c^4*d^4*(-5*d^4 + 40*d^3*e*x + 240*d^2*e^2*x^2 + 320*d*e^3*x^3 + 128*e^4*x^4)))/(15*(c*d^2 - a*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1972,1,259,252,0.1184414,"\int \frac{1}{(d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 \left(-3 a^5 e^{10}+3 a^4 c d e^8 (7 d+2 e x)-2 a^3 c^2 d^2 e^6 \left(35 d^2+28 d e x+8 e^2 x^2\right)+6 a^2 c^3 d^3 e^4 \left(35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right)+3 a c^4 d^4 e^2 \left(35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right)+c^5 d^5 \left(-7 d^5+70 d^4 e x+560 d^3 e^2 x^2+1120 d^2 e^3 x^3+896 d e^4 x^4+256 e^5 x^5\right)\right)}{21 (d+e x)^2 \left(c d^2-a e^2\right)^6 ((d+e x) (a e+c d x))^{3/2}}","\frac{256 c^3 d^3 e \left(a e^2+c d^2+2 c d e x\right)}{21 \left(c d^2-a e^2\right)^6 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left(a e^2+c d^2+2 c d e x\right)}{21 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{4 c d}{7 (d+e x) \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{2}{7 (d+e x)^2 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(-3*a^5*e^10 + 3*a^4*c*d*e^8*(7*d + 2*e*x) - 2*a^3*c^2*d^2*e^6*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + 3*a*c^4*d^4*e^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448*d*e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(-7*d^5 + 70*d^4*e*x + 560*d^3*e^2*x^2 + 1120*d^2*e^3*x^3 + 896*d*e^4*x^4 + 256*e^5*x^5)))/(21*(c*d^2 - a*e^2)^6*(d + e*x)^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1973,1,336,312,0.1464377,"\int \frac{1}{(d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 \left(7 a^6 e^{12}-6 a^5 c d e^{10} (9 d+2 e x)+3 a^4 c^2 d^2 e^8 \left(63 d^2+36 d e x+8 e^2 x^2\right)-4 a^3 c^3 d^3 e^6 \left(105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right)+3 a^2 c^4 d^4 e^4 \left(315 d^4+840 d^3 e x+1008 d^2 e^2 x^2+576 d e^3 x^3+128 e^4 x^4\right)+6 a c^5 d^5 e^2 \left(63 d^5+630 d^4 e x+1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+1152 d e^4 x^4+256 e^5 x^5\right)+c^6 d^6 \left(-21 d^6+252 d^5 e x+2520 d^4 e^2 x^2+6720 d^3 e^3 x^3+8064 d^2 e^4 x^4+4608 d e^5 x^5+1024 e^6 x^6\right)\right)}{63 (d+e x)^3 \left(c d^2-a e^2\right)^7 ((d+e x) (a e+c d x))^{3/2}}","\frac{1024 c^4 d^4 e \left(a e^2+c d^2+2 c d e x\right)}{63 \left(c d^2-a e^2\right)^7 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left(a e^2+c d^2+2 c d e x\right)}{63 \left(c d^2-a e^2\right)^5 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{2}{9 (d+e x)^3 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(7*a^6*e^12 - 6*a^5*c*d*e^10*(9*d + 2*e*x) + 3*a^4*c^2*d^2*e^8*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 4*a^3*c^3*d^3*e^6*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + 3*a^2*c^4*d^4*e^4*(315*d^4 + 840*d^3*e*x + 1008*d^2*e^2*x^2 + 576*d*e^3*x^3 + 128*e^4*x^4) + 6*a*c^5*d^5*e^2*(63*d^5 + 630*d^4*e*x + 1680*d^3*e^2*x^2 + 2016*d^2*e^3*x^3 + 1152*d*e^4*x^4 + 256*e^5*x^5) + c^6*d^6*(-21*d^6 + 252*d^5*e*x + 2520*d^4*e^2*x^2 + 6720*d^3*e^3*x^3 + 8064*d^2*e^4*x^4 + 4608*d*e^5*x^5 + 1024*e^6*x^6)))/(63*(c*d^2 - a*e^2)^7*(d + e*x)^3*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
1974,1,88,1485,0.0579229,"\int \frac{d+e x}{\sqrt[3]{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3),x]","\frac{3 ((d+e x) (a e+c d x))^{2/3} \, _2F_1\left(-\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{2 c d \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{2/3}}","-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{4/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)} \left(c d^2-a e^2\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}{\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}\right)|-7-4 \sqrt{3}\right) \left(c d^2-a e^2\right)^{5/3}}{4 \sqrt[3]{2} c^{5/3} d^{5/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{2/3} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2}}+\frac{3^{3/4} \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{4/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)} \left(c d^2-a e^2\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}{\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}\right)|-7-4 \sqrt{3}\right) \left(c d^2-a e^2\right)^{5/3}}{2^{5/6} c^{5/3} d^{5/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{2/3} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2}}+\frac{3 \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2} \left(c d^2-a e^2\right)}{2 \sqrt[3]{2} c^{5/3} d^{5/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}+\frac{3 \left(c d e x^2+\left(c d^2+a e^2\right) x+a d e\right)^{2/3}}{4 c d}",1,"(3*((a*e + c*d*x)*(d + e*x))^(2/3)*Hypergeometric2F1[-2/3, 2/3, 5/3, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(2*c*d*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(2/3))","C",1
1975,1,95,1432,0.0414496,"\int \frac{1}{\sqrt[3]{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1/3),x]","\frac{3 \sqrt[3]{\frac{c d (d+e x)}{c d^2-a e^2}} ((d+e x) (a e+c d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{2 c d (d+e x)}","-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(c d^2-a e^2\right)^{2/3} \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{4/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)} \left(c d^2-a e^2\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}{\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{2/3} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2}}+\frac{\sqrt[6]{2} 3^{3/4} \left(c d^2-a e^2\right)^{2/3} \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{4/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)} \left(c d^2-a e^2\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} ((a e+c d x) (d+e x))^{2/3}}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}{\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}}\right)|-7-4 \sqrt{3}\right)}{c^{2/3} d^{2/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{2/3} \left(\left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}{\left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)^2}} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2}}+\frac{3 \sqrt{\left(c d^2+2 c e x d+a e^2\right)^2} \sqrt{\left(a e^2+c d (d+2 e x)\right)^2}}{\sqrt[3]{2} c^{2/3} d^{2/3} e^{2/3} \left(c d^2+2 c e x d+a e^2\right) \left(\left(1+\sqrt{3}\right) \left(c d^2-a e^2\right)^{2/3}+2^{2/3} \sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{e} \sqrt[3]{(a e+c d x) (d+e x)}\right)}",1,"(3*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/3)*((a*e + c*d*x)*(d + e*x))^(2/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(2*c*d*(d + e*x))","C",1
1976,1,34,43,0.0313254,"\int (d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 (d+e x)^{7/2} \left(9 a e^2+c d (7 e x-2 d)\right)}{63 e^2}","\frac{2}{7} (d+e x)^{7/2} \left(a-\frac{c d^2}{e^2}\right)+\frac{2 c d (d+e x)^{9/2}}{9 e^2}",1,"(2*(d + e*x)^(7/2)*(9*a*e^2 + c*d*(-2*d + 7*e*x)))/(63*e^2)","A",1
1977,1,34,43,0.0244999,"\int \sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 (d+e x)^{5/2} \left(7 a e^2+c d (5 e x-2 d)\right)}{35 e^2}","\frac{2}{5} (d+e x)^{5/2} \left(a-\frac{c d^2}{e^2}\right)+\frac{2 c d (d+e x)^{7/2}}{7 e^2}",1,"(2*(d + e*x)^(5/2)*(7*a*e^2 + c*d*(-2*d + 5*e*x)))/(35*e^2)","A",1
1978,1,34,43,0.0233823,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{\sqrt{d+e x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/Sqrt[d + e*x],x]","\frac{2 (d+e x)^{3/2} \left(5 a e^2+c d (3 e x-2 d)\right)}{15 e^2}","\frac{2}{3} (d+e x)^{3/2} \left(a-\frac{c d^2}{e^2}\right)+\frac{2 c d (d+e x)^{5/2}}{5 e^2}",1,"(2*(d + e*x)^(3/2)*(5*a*e^2 + c*d*(-2*d + 3*e*x)))/(15*e^2)","A",1
1979,1,33,41,0.0205793,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(3/2),x]","\frac{2 \sqrt{d+e x} \left(3 a e^2+c d (e x-2 d)\right)}{3 e^2}","2 \sqrt{d+e x} \left(a-\frac{c d^2}{e^2}\right)+\frac{2 c d (d+e x)^{3/2}}{3 e^2}",1,"(2*Sqrt[d + e*x]*(3*a*e^2 + c*d*(-2*d + e*x)))/(3*e^2)","A",1
1980,1,31,39,0.0193248,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(5/2),x]","\frac{2 c d (2 d+e x)-2 a e^2}{e^2 \sqrt{d+e x}}","\frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left(a-\frac{c d^2}{e^2}\right)}{\sqrt{d+e x}}",1,"(-2*a*e^2 + 2*c*d*(2*d + e*x))/(e^2*Sqrt[d + e*x])","A",1
1981,1,33,41,0.0214191,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]","-\frac{2 \left(a e^2+c d (2 d+3 e x)\right)}{3 e^2 (d+e x)^{3/2}}","-\frac{2 \left(a-\frac{c d^2}{e^2}\right)}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}",1,"(-2*(a*e^2 + c*d*(2*d + 3*e*x)))/(3*e^2*(d + e*x)^(3/2))","A",1
1982,1,34,43,0.0215098,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(9/2),x]","-\frac{2 \left(3 a e^2+c d (2 d+5 e x)\right)}{15 e^2 (d+e x)^{5/2}}","-\frac{2 \left(a-\frac{c d^2}{e^2}\right)}{5 (d+e x)^{5/2}}-\frac{2 c d}{3 e^2 (d+e x)^{3/2}}",1,"(-2*(3*a*e^2 + c*d*(2*d + 5*e*x)))/(15*e^2*(d + e*x)^(5/2))","A",1
1983,1,34,43,0.0230779,"\int \frac{a d e+\left(c d^2+a e^2\right) x+c d e x^2}{(d+e x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(11/2),x]","-\frac{2 \left(5 a e^2+c d (2 d+7 e x)\right)}{35 e^2 (d+e x)^{7/2}}","-\frac{2 \left(a-\frac{c d^2}{e^2}\right)}{7 (d+e x)^{7/2}}-\frac{2 c d}{5 e^2 (d+e x)^{5/2}}",1,"(-2*(5*a*e^2 + c*d*(2*d + 7*e*x)))/(35*e^2*(d + e*x)^(7/2))","A",1
1984,1,67,83,0.0561552,"\int \sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2 \, dx","Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 (d+e x)^{7/2} \left(99 a^2 e^4-22 a c d e^2 (2 d-7 e x)+c^2 d^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)}{693 e^3}","-\frac{4 c d (d+e x)^{9/2} \left(c d^2-a e^2\right)}{9 e^3}+\frac{2 (d+e x)^{7/2} \left(c d^2-a e^2\right)^2}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{11/2}}{11 e^3}",1,"(2*(d + e*x)^(7/2)*(99*a^2*e^4 - 22*a*c*d*e^2*(2*d - 7*e*x) + c^2*d^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)","A",1
1985,1,67,83,0.043668,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{\sqrt{d+e x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/Sqrt[d + e*x],x]","\frac{2 (d+e x)^{5/2} \left(63 a^2 e^4+18 a c d e^2 (5 e x-2 d)+c^2 d^2 \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)}{315 e^3}","-\frac{4 c d (d+e x)^{7/2} \left(c d^2-a e^2\right)}{7 e^3}+\frac{2 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}",1,"(2*(d + e*x)^(5/2)*(63*a^2*e^4 + 18*a*c*d*e^2*(-2*d + 5*e*x) + c^2*d^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)","A",1
1986,1,67,83,0.0396512,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(3/2),x]","\frac{2 (d+e x)^{3/2} \left(35 a^2 e^4+14 a c d e^2 (3 e x-2 d)+c^2 d^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)}{105 e^3}","-\frac{4 c d (d+e x)^{5/2} \left(c d^2-a e^2\right)}{5 e^3}+\frac{2 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}{3 e^3}+\frac{2 c^2 d^2 (d+e x)^{7/2}}{7 e^3}",1,"(2*(d + e*x)^(3/2)*(35*a^2*e^4 + 14*a*c*d*e^2*(-2*d + 3*e*x) + c^2*d^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)","A",1
1987,1,66,81,0.0390939,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(5/2),x]","\frac{2 \sqrt{d+e x} \left(15 a^2 e^4+10 a c d e^2 (e x-2 d)+c^2 d^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)}{15 e^3}","-\frac{4 c d (d+e x)^{3/2} \left(c d^2-a e^2\right)}{3 e^3}+\frac{2 \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3}",1,"(2*Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)","A",1
1988,1,65,79,0.035908,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(7/2),x]","\frac{2 \left(-3 a^2 e^4+6 a c d e^2 (2 d+e x)+c^2 d^2 \left(-8 d^2-4 d e x+e^2 x^2\right)\right)}{3 e^3 \sqrt{d+e x}}","-\frac{4 c d \sqrt{d+e x} \left(c d^2-a e^2\right)}{e^3}-\frac{2 \left(c d^2-a e^2\right)^2}{e^3 \sqrt{d+e x}}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3}",1,"(2*(-3*a^2*e^4 + 6*a*c*d*e^2*(2*d + e*x) + c^2*d^2*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])","A",1
1989,1,68,79,0.0400096,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]","\frac{-2 a^2 e^4-4 a c d e^2 (2 d+3 e x)+2 c^2 d^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)}{3 e^3 (d+e x)^{3/2}}","\frac{4 c d \left(c d^2-a e^2\right)}{e^3 \sqrt{d+e x}}-\frac{2 \left(c d^2-a e^2\right)^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 c^2 d^2 \sqrt{d+e x}}{e^3}",1,"(-2*a^2*e^4 - 4*a*c*d*e^2*(2*d + 3*e*x) + 2*c^2*d^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2))/(3*e^3*(d + e*x)^(3/2))","A",1
1990,1,67,81,0.0388257,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]","-\frac{2 \left(3 a^2 e^4+2 a c d e^2 (2 d+5 e x)+c^2 d^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)\right)}{15 e^3 (d+e x)^{5/2}}","\frac{4 c d \left(c d^2-a e^2\right)}{3 e^3 (d+e x)^{3/2}}-\frac{2 \left(c d^2-a e^2\right)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}}",1,"(-2*(3*a^2*e^4 + 2*a*c*d*e^2*(2*d + 5*e*x) + c^2*d^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))","A",1
1991,1,67,83,0.0392794,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2}{(d+e x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(13/2),x]","-\frac{2 \left(15 a^2 e^4+6 a c d e^2 (2 d+7 e x)+c^2 d^2 \left(8 d^2+28 d e x+35 e^2 x^2\right)\right)}{105 e^3 (d+e x)^{7/2}}","\frac{4 c d \left(c d^2-a e^2\right)}{5 e^3 (d+e x)^{5/2}}-\frac{2 \left(c d^2-a e^2\right)^2}{7 e^3 (d+e x)^{7/2}}-\frac{2 c^2 d^2}{3 e^3 (d+e x)^{3/2}}",1,"(-2*(15*a^2*e^4 + 6*a*c*d*e^2*(2*d + 7*e*x) + c^2*d^2*(8*d^2 + 28*d*e*x + 35*e^2*x^2)))/(105*e^3*(d + e*x)^(7/2))","A",1
1992,1,111,119,0.0892137,"\int \sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3 \, dx","Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 (d+e x)^{9/2} \left(715 a^3 e^6-195 a^2 c d e^4 (2 d-9 e x)+15 a c^2 d^2 e^2 \left(8 d^2-36 d e x+99 e^2 x^2\right)+c^3 d^3 \left(-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right)\right)}{6435 e^4}","-\frac{6 c^2 d^2 (d+e x)^{13/2} \left(c d^2-a e^2\right)}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left(c d^2-a e^2\right)^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left(c d^2-a e^2\right)^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}",1,"(2*(d + e*x)^(9/2)*(715*a^3*e^6 - 195*a^2*c*d*e^4*(2*d - 9*e*x) + 15*a*c^2*d^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + c^3*d^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3)))/(6435*e^4)","A",1
1993,1,98,119,0.077049,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{\sqrt{d+e x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/Sqrt[d + e*x],x]","\frac{2 (d+e x)^{7/2} \left(-819 c^2 d^2 (d+e x)^2 \left(c d^2-a e^2\right)+1001 c d (d+e x) \left(c d^2-a e^2\right)^2-429 \left(c d^2-a e^2\right)^3+231 c^3 d^3 (d+e x)^3\right)}{3003 e^4}","-\frac{6 c^2 d^2 (d+e x)^{11/2} \left(c d^2-a e^2\right)}{11 e^4}+\frac{2 c d (d+e x)^{9/2} \left(c d^2-a e^2\right)^2}{3 e^4}-\frac{2 (d+e x)^{7/2} \left(c d^2-a e^2\right)^3}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4}",1,"(2*(d + e*x)^(7/2)*(-429*(c*d^2 - a*e^2)^3 + 1001*c*d*(c*d^2 - a*e^2)^2*(d + e*x) - 819*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2 + 231*c^3*d^3*(d + e*x)^3))/(3003*e^4)","A",1
1994,1,98,119,0.0686505,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(3/2),x]","\frac{2 (d+e x)^{5/2} \left(-385 c^2 d^2 (d+e x)^2 \left(c d^2-a e^2\right)+495 c d (d+e x) \left(c d^2-a e^2\right)^2-231 \left(c d^2-a e^2\right)^3+105 c^3 d^3 (d+e x)^3\right)}{1155 e^4}","-\frac{2 c^2 d^2 (d+e x)^{9/2} \left(c d^2-a e^2\right)}{3 e^4}+\frac{6 c d (d+e x)^{7/2} \left(c d^2-a e^2\right)^2}{7 e^4}-\frac{2 (d+e x)^{5/2} \left(c d^2-a e^2\right)^3}{5 e^4}+\frac{2 c^3 d^3 (d+e x)^{11/2}}{11 e^4}",1,"(2*(d + e*x)^(5/2)*(-231*(c*d^2 - a*e^2)^3 + 495*c*d*(c*d^2 - a*e^2)^2*(d + e*x) - 385*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2 + 105*c^3*d^3*(d + e*x)^3))/(1155*e^4)","A",1
1995,1,98,119,0.065037,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(5/2),x]","\frac{2 (d+e x)^{3/2} \left(-135 c^2 d^2 (d+e x)^2 \left(c d^2-a e^2\right)+189 c d (d+e x) \left(c d^2-a e^2\right)^2-105 \left(c d^2-a e^2\right)^3+35 c^3 d^3 (d+e x)^3\right)}{315 e^4}","-\frac{6 c^2 d^2 (d+e x)^{7/2} \left(c d^2-a e^2\right)}{7 e^4}+\frac{6 c d (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}{5 e^4}-\frac{2 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}{3 e^4}+\frac{2 c^3 d^3 (d+e x)^{9/2}}{9 e^4}",1,"(2*(d + e*x)^(3/2)*(-105*(c*d^2 - a*e^2)^3 + 189*c*d*(c*d^2 - a*e^2)^2*(d + e*x) - 135*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2 + 35*c^3*d^3*(d + e*x)^3))/(315*e^4)","A",1
1996,1,110,115,0.0637833,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(7/2),x]","\frac{2 \sqrt{d+e x} \left(35 a^3 e^6+35 a^2 c d e^4 (e x-2 d)+7 a c^2 d^2 e^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)+c^3 d^3 \left(-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right)\right)}{35 e^4}","-\frac{6 c^2 d^2 (d+e x)^{5/2} \left(c d^2-a e^2\right)}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}{e^4}-\frac{2 \sqrt{d+e x} \left(c d^2-a e^2\right)^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4}",1,"(2*Sqrt[d + e*x]*(35*a^3*e^6 + 35*a^2*c*d*e^4*(-2*d + e*x) + 7*a*c^2*d^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c^3*d^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)))/(35*e^4)","A",1
1997,1,109,113,0.054095,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(9/2),x]","\frac{2 \left(-5 a^3 e^6+15 a^2 c d e^4 (2 d+e x)-5 a c^2 d^2 e^2 \left(8 d^2+4 d e x-e^2 x^2\right)+c^3 d^3 \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)\right)}{5 e^4 \sqrt{d+e x}}","-\frac{2 c^2 d^2 (d+e x)^{3/2} \left(c d^2-a e^2\right)}{e^4}+\frac{6 c d \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{e^4}+\frac{2 \left(c d^2-a e^2\right)^3}{e^4 \sqrt{d+e x}}+\frac{2 c^3 d^3 (d+e x)^{5/2}}{5 e^4}",1,"(2*(-5*a^3*e^6 + 15*a^2*c*d*e^4*(2*d + e*x) - 5*a*c^2*d^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + c^3*d^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(5*e^4*Sqrt[d + e*x])","A",1
1998,1,110,115,0.0571198,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(11/2),x]","-\frac{2 \left(a^3 e^6+3 a^2 c d e^4 (2 d+3 e x)-3 a c^2 d^2 e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)+c^3 d^3 \left(16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right)\right)}{3 e^4 (d+e x)^{3/2}}","-\frac{6 c^2 d^2 \sqrt{d+e x} \left(c d^2-a e^2\right)}{e^4}-\frac{6 c d \left(c d^2-a e^2\right)^2}{e^4 \sqrt{d+e x}}+\frac{2 \left(c d^2-a e^2\right)^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}",1,"(-2*(a^3*e^6 + 3*a^2*c*d*e^4*(2*d + 3*e*x) - 3*a*c^2*d^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^3*d^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)))/(3*e^4*(d + e*x)^(3/2))","A",1
1999,1,109,113,0.0537793,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3}{(d+e x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(13/2),x]","-\frac{2 \left(a^3 e^6+a^2 c d e^4 (2 d+5 e x)+a c^2 d^2 e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)-c^3 d^3 \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)\right)}{5 e^4 (d+e x)^{5/2}}","\frac{6 c^2 d^2 \left(c d^2-a e^2\right)}{e^4 \sqrt{d+e x}}-\frac{2 c d \left(c d^2-a e^2\right)^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left(c d^2-a e^2\right)^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4}",1,"(-2*(a^3*e^6 + a^2*c*d*e^4*(2*d + 5*e*x) + a*c^2*d^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - c^3*d^3*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)))/(5*e^4*(d + e*x)^(5/2))","A",1
2000,1,175,180,0.2845858,"\int \frac{(d+e x)^{9/2}}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 \left(c d^2-a e^2\right) \left(5 \left(c d^2-a e^2\right) \left(\sqrt{c} \sqrt{d} \sqrt{d+e x} \left(c d (4 d+e x)-3 a e^2\right)-3 \left(c d^2-a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)\right)+3 c^{5/2} d^{5/2} (d+e x)^{5/2}\right)}{15 c^{9/2} d^{9/2}}+\frac{2 (d+e x)^{7/2}}{7 c d}","-\frac{2 \left(c d^2-a e^2\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{9/2} d^{9/2}}+\frac{2 \sqrt{d+e x} \left(c d^2-a e^2\right)^3}{c^4 d^4}+\frac{2 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}{3 c^3 d^3}+\frac{2 (d+e x)^{5/2} \left(c d^2-a e^2\right)}{5 c^2 d^2}+\frac{2 (d+e x)^{7/2}}{7 c d}",1,"(2*(d + e*x)^(7/2))/(7*c*d) + (2*(c*d^2 - a*e^2)*(3*c^(5/2)*d^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]*(-3*a*e^2 + c*d*(4*d + e*x)) - 3*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])))/(15*c^(9/2)*d^(9/2))","A",1
2001,1,135,147,0.1142157,"\int \frac{(d+e x)^{7/2}}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 \sqrt{d+e x} \left(15 a^2 e^4-5 a c d e^2 (7 d+e x)+c^2 d^2 \left(23 d^2+11 d e x+3 e^2 x^2\right)\right)}{15 c^3 d^3}-\frac{2 \left(c d^2-a e^2\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{7/2} d^{7/2}}","-\frac{2 \left(c d^2-a e^2\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{7/2} d^{7/2}}+\frac{2 \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{c^3 d^3}+\frac{2 (d+e x)^{3/2} \left(c d^2-a e^2\right)}{3 c^2 d^2}+\frac{2 (d+e x)^{5/2}}{5 c d}",1,"(2*Sqrt[d + e*x]*(15*a^2*e^4 - 5*a*c*d*e^2*(7*d + e*x) + c^2*d^2*(23*d^2 + 11*d*e*x + 3*e^2*x^2)))/(15*c^3*d^3) - (2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(7/2)*d^(7/2))","A",1
2002,1,102,114,0.077879,"\int \frac{(d+e x)^{5/2}}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 \sqrt{d+e x} \left(c d (4 d+e x)-3 a e^2\right)}{3 c^2 d^2}-\frac{2 \left(c d^2-a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{5/2} d^{5/2}}","-\frac{2 \left(c d^2-a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{5/2} d^{5/2}}+\frac{2 \sqrt{d+e x} \left(c d^2-a e^2\right)}{c^2 d^2}+\frac{2 (d+e x)^{3/2}}{3 c d}",1,"(2*Sqrt[d + e*x]*(-3*a*e^2 + c*d*(4*d + e*x)))/(3*c^2*d^2) - (2*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(5/2)*d^(5/2))","A",1
2003,1,83,83,0.0346563,"\int \frac{(d+e x)^{3/2}}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{3/2} d^{3/2}}","\frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{3/2} d^{3/2}}",1,"(2*Sqrt[d + e*x])/(c*d) - (2*Sqrt[c*d^2 - a*e^2]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2))","A",1
2004,1,65,65,0.0161177,"\int \frac{\sqrt{d+e x}}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}}",1,"(-2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c]*Sqrt[d]*Sqrt[c*d^2 - a*e^2])","A",1
2005,1,55,91,0.0141792,"\int \frac{1}{\sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c d (d+e x)}{c d^2-a e^2}\right)}{\sqrt{d+e x} \left(a e^2-c d^2\right)}","\frac{2}{\sqrt{d+e x} \left(c d^2-a e^2\right)}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{3/2}}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((-(c*d^2) + a*e^2)*Sqrt[d + e*x])","C",1
2006,1,57,120,0.0120184,"\int \frac{1}{(d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c d (d+e x)}{c d^2-a e^2}\right)}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)}","-\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{5/2}}+\frac{2 c d}{\sqrt{d+e x} \left(c d^2-a e^2\right)^2}+\frac{2}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)}",1,"(2*Hypergeometric2F1[-3/2, 1, -1/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2))","C",1
2007,1,57,153,0.0122343,"\int \frac{1}{(d+e x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{2 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{c d (d+e x)}{c d^2-a e^2}\right)}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)}","-\frac{2 c^{5/2} d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{7/2}}+\frac{2 c^2 d^2}{\sqrt{d+e x} \left(c d^2-a e^2\right)^3}+\frac{2 c d}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}+\frac{2}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)}",1,"(2*Hypergeometric2F1[-5/2, 1, -3/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(5*(c*d^2 - a*e^2)*(d + e*x)^(5/2))","C",1
2008,1,57,186,0.0144453,"\int \frac{1}{(d+e x)^{7/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)} \, dx","Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]","\frac{2 \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\frac{c d (d+e x)}{c d^2-a e^2}\right)}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)}","-\frac{2 c^{7/2} d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{9/2}}+\frac{2 c^3 d^3}{\sqrt{d+e x} \left(c d^2-a e^2\right)^4}+\frac{2 c^2 d^2}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}+\frac{2 c d}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}+\frac{2}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)}",1,"(2*Hypergeometric2F1[-7/2, 1, -5/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(7*(c*d^2 - a*e^2)*(d + e*x)^(7/2))","C",1
2009,1,59,210,0.0253732,"\int \frac{(d+e x)^{13/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 e (d+e x)^{11/2} \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{11 \left(a e^2-c d^2\right)^2}","-\frac{9 e \left(c d^2-a e^2\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{11/2} d^{11/2}}+\frac{9 e \sqrt{d+e x} \left(c d^2-a e^2\right)^3}{c^5 d^5}+\frac{3 e (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}{c^4 d^4}+\frac{9 e (d+e x)^{5/2} \left(c d^2-a e^2\right)}{5 c^3 d^3}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}",1,"(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(11*(-(c*d^2) + a*e^2)^2)","C",1
2010,1,59,178,0.0176552,"\int \frac{(d+e x)^{11/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 e (d+e x)^{9/2} \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{9 \left(a e^2-c d^2\right)^2}","-\frac{7 e \left(c d^2-a e^2\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{9/2} d^{9/2}}+\frac{7 e \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{c^4 d^4}+\frac{7 e (d+e x)^{3/2} \left(c d^2-a e^2\right)}{3 c^3 d^3}-\frac{(d+e x)^{7/2}}{c d (a e+c d x)}+\frac{7 e (d+e x)^{5/2}}{5 c^2 d^2}",1,"(2*e*(d + e*x)^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(9*(-(c*d^2) + a*e^2)^2)","C",1
2011,1,59,144,0.0169455,"\int \frac{(d+e x)^{9/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 e (d+e x)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{7 \left(a e^2-c d^2\right)^2}","-\frac{5 e \left(c d^2-a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{7/2} d^{7/2}}+\frac{5 e \sqrt{d+e x} \left(c d^2-a e^2\right)}{c^3 d^3}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2}",1,"(2*e*(d + e*x)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(7*(-(c*d^2) + a*e^2)^2)","C",1
2012,1,59,112,0.0146756,"\int \frac{(d+e x)^{7/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{2 e (d+e x)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{5 \left(a e^2-c d^2\right)^2}","-\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{5/2} d^{5/2}}-\frac{(d+e x)^{3/2}}{c d (a e+c d x)}+\frac{3 e \sqrt{d+e x}}{c^2 d^2}",1,"(2*e*(d + e*x)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(5*(-(c*d^2) + a*e^2)^2)","C",1
2013,1,93,94,0.1041028,"\int \frac{(d+e x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{e \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right)}{c^{3/2} d^{3/2} \sqrt{a e^2-c d^2}}-\frac{\sqrt{d+e x}}{a c d e+c^2 d^2 x}","-\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)}",1,"-(Sqrt[d + e*x]/(a*c*d*e + c^2*d^2*x)) + (e*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(3/2)*d^(3/2)*Sqrt[-(c*d^2) + a*e^2])","A",1
2014,1,101,101,0.075795,"\int \frac{(d+e x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{e \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right)}{\sqrt{c} \sqrt{d} \left(a e^2-c d^2\right)^{3/2}}-\frac{\sqrt{d+e x}}{\left(c d^2-a e^2\right) (a e+c d x)}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^{3/2}}-\frac{\sqrt{d+e x}}{\left(c d^2-a e^2\right) (a e+c d x)}",1,"-(Sqrt[d + e*x]/((c*d^2 - a*e^2)*(a*e + c*d*x))) + (e*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(Sqrt[c]*Sqrt[d]*(-(c*d^2) + a*e^2)^(3/2))","A",1
2015,1,57,128,0.0137252,"\int \frac{\sqrt{d+e x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","-\frac{2 e \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{\sqrt{d+e x} \left(a e^2-c d^2\right)^2}","-\frac{3 e}{\sqrt{d+e x} \left(c d^2-a e^2\right)^2}-\frac{1}{\sqrt{d+e x} \left(c d^2-a e^2\right) (a e+c d x)}+\frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{5/2}}",1,"(-2*e*Hypergeometric2F1[-1/2, 2, 1/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^2*Sqrt[d + e*x])","C",1
2016,1,59,158,0.0159412,"\int \frac{1}{\sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]","-\frac{2 e \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{3 (d+e x)^{3/2} \left(a e^2-c d^2\right)^2}","\frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{7/2}}-\frac{5 c d e}{\sqrt{d+e x} \left(c d^2-a e^2\right)^3}-\frac{1}{(d+e x)^{3/2} \left(c d^2-a e^2\right) (a e+c d x)}-\frac{5 e}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}",1,"(-2*e*Hypergeometric2F1[-3/2, 2, -1/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(3*(-(c*d^2) + a*e^2)^2*(d + e*x)^(3/2))","C",1
2017,1,59,192,0.0178453,"\int \frac{1}{(d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]","-\frac{2 e \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{5 (d+e x)^{5/2} \left(a e^2-c d^2\right)^2}","\frac{7 c^{5/2} d^{5/2} e \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{9/2}}-\frac{7 c^2 d^2 e}{\sqrt{d+e x} \left(c d^2-a e^2\right)^4}-\frac{7 c d e}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}-\frac{1}{(d+e x)^{5/2} \left(c d^2-a e^2\right) (a e+c d x)}-\frac{7 e}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}",1,"(-2*e*Hypergeometric2F1[-5/2, 2, -3/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(5*(-(c*d^2) + a*e^2)^2*(d + e*x)^(5/2))","C",1
2018,1,61,222,0.0240808,"\int \frac{(d+e x)^{15/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(15/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 (d+e x)^{11/2} \, _2F_1\left(3,\frac{11}{2};\frac{13}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{11 \left(a e^2-c d^2\right)^3}","-\frac{63 e^2 \left(c d^2-a e^2\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 c^{11/2} d^{11/2}}+\frac{63 e^2 \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{4 c^5 d^5}+\frac{21 e^2 (d+e x)^{3/2} \left(c d^2-a e^2\right)}{4 c^4 d^4}-\frac{9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac{63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}",1,"(2*e^2*(d + e*x)^(11/2)*Hypergeometric2F1[3, 11/2, 13/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(11*(-(c*d^2) + a*e^2)^3)","C",1
2019,1,61,186,0.0197056,"\int \frac{(d+e x)^{13/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 (d+e x)^{9/2} \, _2F_1\left(3,\frac{9}{2};\frac{11}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{9 \left(a e^2-c d^2\right)^3}","-\frac{35 e^2 \left(c d^2-a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 c^{9/2} d^{9/2}}+\frac{35 e^2 \sqrt{d+e x} \left(c d^2-a e^2\right)}{4 c^4 d^4}-\frac{7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac{35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}",1,"(2*e^2*(d + e*x)^(9/2)*Hypergeometric2F1[3, 9/2, 11/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(9*(-(c*d^2) + a*e^2)^3)","C",1
2020,1,61,152,0.0174213,"\int \frac{(d+e x)^{11/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 (d+e x)^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{7 \left(a e^2-c d^2\right)^3}","-\frac{15 e^2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 c^{7/2} d^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 c^3 d^3}",1,"(2*e^2*(d + e*x)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(7*(-(c*d^2) + a*e^2)^3)","C",1
2021,1,118,130,0.1288675,"\int \frac{(d+e x)^{9/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{3 e^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right)}{4 c^{5/2} d^{5/2} \sqrt{a e^2-c d^2}}-\frac{\sqrt{d+e x} \left(3 a e^2+c d (2 d+5 e x)\right)}{4 c^2 d^2 (a e+c d x)^2}","-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2}",1,"-1/4*(Sqrt[d + e*x]*(3*a*e^2 + c*d*(2*d + 5*e*x)))/(c^2*d^2*(a*e + c*d*x)^2) + (3*e^2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(4*c^(5/2)*d^(5/2)*Sqrt[-(c*d^2) + a*e^2])","A",1
2022,1,61,144,0.0151122,"\int \frac{(d+e x)^{7/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 (d+e x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{3 \left(a e^2-c d^2\right)^3}","\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 c^{3/2} d^{3/2} \left(c d^2-a e^2\right)^{3/2}}-\frac{e \sqrt{d+e x}}{4 c d \left(c d^2-a e^2\right) (a e+c d x)}-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2}",1,"(2*e^2*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(3*(-(c*d^2) + a*e^2)^3)","C",1
2023,1,59,146,0.0126309,"\int \frac{(d+e x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{2 e^2 \sqrt{d+e x} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{\left(a e^2-c d^2\right)^3}","\frac{3 e \sqrt{d+e x}}{4 \left(c d^2-a e^2\right)^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left(c d^2-a e^2\right) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^{5/2}}",1,"(2*e^2*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 3, 3/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(-(c*d^2) + a*e^2)^3","C",1
2024,1,59,176,0.0162617,"\int \frac{(d+e x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","-\frac{2 e^2 \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{\sqrt{d+e x} \left(a e^2-c d^2\right)^3}","\frac{15 e^2}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^3}+\frac{5 e}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 (a e+c d x)}-\frac{1}{2 \sqrt{d+e x} \left(c d^2-a e^2\right) (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 \left(c d^2-a e^2\right)^{7/2}}",1,"(-2*e^2*Hypergeometric2F1[-1/2, 3, 1/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^3*Sqrt[d + e*x])","C",1
2025,1,61,208,0.0172227,"\int \frac{\sqrt{d+e x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","-\frac{2 e^2 \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{3 (d+e x)^{3/2} \left(a e^2-c d^2\right)^3}","-\frac{35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 \left(c d^2-a e^2\right)^{9/2}}+\frac{35 c d e^2}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^4}+\frac{35 e^2}{12 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}+\frac{7 e}{4 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{3/2} \left(c d^2-a e^2\right) (a e+c d x)^2}",1,"(-2*e^2*Hypergeometric2F1[-3/2, 3, -1/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(3*(-(c*d^2) + a*e^2)^3*(d + e*x)^(3/2))","C",1
2026,1,61,244,0.0203914,"\int \frac{1}{\sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]","-\frac{2 e^2 \, _2F_1\left(-\frac{5}{2},3;-\frac{3}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{5 (d+e x)^{5/2} \left(a e^2-c d^2\right)^3}","-\frac{63 c^{5/2} d^{5/2} e^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right)}{4 \left(c d^2-a e^2\right)^{11/2}}+\frac{63 c^2 d^2 e^2}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^5}+\frac{21 c d e^2}{4 (d+e x)^{3/2} \left(c d^2-a e^2\right)^4}+\frac{9 e}{4 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{5/2} \left(c d^2-a e^2\right) (a e+c d x)^2}+\frac{63 e^2}{20 (d+e x)^{5/2} \left(c d^2-a e^2\right)^3}",1,"(-2*e^2*Hypergeometric2F1[-5/2, 3, -3/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(5*(-(c*d^2) + a*e^2)^3*(d + e*x)^(5/2))","C",1
2027,1,187,295,0.1559134,"\int (d+e x)^{7/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(128 a^4 e^8-64 a^3 c d e^6 (11 d+3 e x)+48 a^2 c^2 d^2 e^4 \left(33 d^2+22 d e x+5 e^2 x^2\right)-8 a c^3 d^3 e^2 \left(231 d^3+297 d^2 e x+165 d e^2 x^2+35 e^3 x^3\right)+c^4 d^4 \left(1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right)\right)}{3465 c^5 d^5 (d+e x)^{3/2}}","\frac{256 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{11 c d}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(11*d + 3*e*x) + 48*a^2*c^2*d^2*e^4*(33*d^2 + 22*d*e*x + 5*e^2*x^2) - 8*a*c^3*d^3*e^2*(231*d^3 + 297*d^2*e*x + 165*d*e^2*x^2 + 35*e^3*x^3) + c^4*d^4*(1155*d^4 + 2772*d^3*e*x + 2970*d^2*e^2*x^2 + 1540*d*e^3*x^3 + 315*e^4*x^4)))/(3465*c^5*d^5*(d + e*x)^(3/2))","A",1
2028,1,131,233,0.1081777,"\int (d+e x)^{5/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(-16 a^3 e^6+24 a^2 c d e^4 (3 d+e x)-6 a c^2 d^2 e^2 \left(21 d^2+18 d e x+5 e^2 x^2\right)+c^3 d^3 \left(105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right)\right)}{315 c^4 d^4 (d+e x)^{3/2}}","\frac{32 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 c^3 d^3 \sqrt{d+e x}}+\frac{4 \sqrt{d+e x} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{21 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{9 c d}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-16*a^3*e^6 + 24*a^2*c*d*e^4*(3*d + e*x) - 6*a*c^2*d^2*e^2*(21*d^2 + 18*d*e*x + 5*e^2*x^2) + c^3*d^3*(105*d^3 + 189*d^2*e*x + 135*d*e^2*x^2 + 35*e^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2))","A",1
2029,1,88,171,0.0769902,"\int (d+e x)^{3/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(8 a^2 e^4-4 a c d e^2 (7 d+3 e x)+c^2 d^2 \left(35 d^2+42 d e x+15 e^2 x^2\right)\right)}{105 c^3 d^3 (d+e x)^{3/2}}","\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{7 c d}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(35*d^2 + 42*d*e*x + 15*e^2*x^2)))/(105*c^3*d^3*(d + e*x)^(3/2))","A",1
2030,1,55,109,0.0521431,"\int \sqrt{d+e x} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(c d (5 d+3 e x)-2 a e^2\right)}{15 c^2 d^2 (d+e x)^{3/2}}","\frac{4 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 c d \sqrt{d+e x}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*a*e^2 + c*d*(5*d + 3*e*x)))/(15*c^2*d^2*(d + e*x)^(3/2))","A",1
2031,1,37,48,0.0255323,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 c d (d+e x)^{3/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*c*d*(d + e*x)^(3/2))","A",1
2032,1,105,128,0.1207144,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(3/2),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\sqrt{e}-\frac{\sqrt{c d^2-a e^2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x}}\right)}{e^{3/2} \sqrt{d+e x}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{c d^2-a e^2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{3/2}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e] - (Sqrt[c*d^2 - a*e^2]*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/Sqrt[a*e + c*d*x]))/(e^(3/2)*Sqrt[d + e*x])","A",1
2033,1,112,129,0.1608842,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(5/2),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (d+e x) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{c d^2-a e^2} \sqrt{a e+c d x}}-\sqrt{e}\right)}{e^{3/2} (d+e x)^{3/2}}","\frac{c d \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e (d+e x)^{3/2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-Sqrt[e] + (c*d*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x])))/(e^(3/2)*(d + e*x)^(3/2))","A",1
2034,1,83,199,0.0370128,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(7/2),x]","\frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}","\frac{c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 e^{3/2} \left(c d^2-a e^2\right)^{3/2}}+\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left(c d^2-a e^2\right)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}",1,"(2*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2))","C",1
2035,1,83,264,0.0379742,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{(d+e x)^{9/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]","\frac{2 c^3 d^3 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^4}","\frac{c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 e^{3/2} \left(c d^2-a e^2\right)^{5/2}}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}+\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left(c d^2-a e^2\right)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}",1,"(2*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^4*(d + e*x)^(3/2))","C",1
2036,1,187,295,0.1671103,"\int (d+e x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(128 a^4 e^8-64 a^3 c d e^6 (13 d+5 e x)+16 a^2 c^2 d^2 e^4 \left(143 d^2+130 d e x+35 e^2 x^2\right)-8 a c^3 d^3 e^2 \left(429 d^3+715 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right)+c^4 d^4 \left(3003 d^4+8580 d^3 e x+10010 d^2 e^2 x^2+5460 d e^3 x^3+1155 e^4 x^4\right)\right)}{15015 c^5 d^5 (d+e x)^{5/2}}","\frac{256 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{13 c d}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(13*d + 5*e*x) + 16*a^2*c^2*d^2*e^4*(143*d^2 + 130*d*e*x + 35*e^2*x^2) - 8*a*c^3*d^3*e^2*(429*d^3 + 715*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(3003*d^4 + 8580*d^3*e*x + 10010*d^2*e^2*x^2 + 5460*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*c^5*d^5*(d + e*x)^(5/2))","A",1
2037,1,132,233,0.1223824,"\int (d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(-16 a^3 e^6+8 a^2 c d e^4 (11 d+5 e x)-2 a c^2 d^2 e^2 \left(99 d^2+110 d e x+35 e^2 x^2\right)+c^3 d^3 \left(231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right)\right)}{1155 c^4 d^4 (d+e x)^{5/2}}","\frac{32 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{11 c d}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(11*d + 5*e*x) - 2*a*c^2*d^2*e^2*(99*d^2 + 110*d*e*x + 35*e^2*x^2) + c^3*d^3*(231*d^3 + 495*d^2*e*x + 385*d*e^2*x^2 + 105*e^3*x^3)))/(1155*c^4*d^4*(d + e*x)^(5/2))","A",1
2038,1,88,171,0.0908688,"\int \sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(8 a^2 e^4-4 a c d e^2 (9 d+5 e x)+c^2 d^2 \left(63 d^2+90 d e x+35 e^2 x^2\right)\right)}{315 c^3 d^3 (d+e x)^{5/2}}","\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac{8 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{9 c d \sqrt{d+e x}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(9*d + 5*e*x) + c^2*d^2*(63*d^2 + 90*d*e*x + 35*e^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))","A",1
2039,1,55,109,0.0623058,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(c d (7 d+5 e x)-2 a e^2\right)}{35 c^2 d^2 (d+e x)^{5/2}}","\frac{4 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-2*a*e^2 + c*d*(7*d + 5*e*x)))/(35*c^2*d^2*(d + e*x)^(5/2))","A",1
2040,1,37,48,0.034475,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 c d (d+e x)^{5/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 c d (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*c*d*(d + e*x)^(5/2))","A",1
2041,1,137,181,0.2330574,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(5/2),x]","\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \left(3 \left(c d^2-a e^2\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)+\sqrt{e} \sqrt{a e+c d x} \left(4 a e^2+c d (e x-3 d)\right)\right)}{3 e^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e (d+e x)^{3/2}}+\frac{2 \left(a-\frac{c d^2}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x}}+\frac{2 \left(c d^2-a e^2\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{5/2}}",1,"(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(4*a*e^2 + c*d*(-3*d + e*x)) + 3*(c*d^2 - a*e^2)^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]]))/(3*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2042,1,79,175,0.0487319,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(7/2),x]","\frac{2 c d ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}","-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{e (d+e x)^{5/2}}+\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^2 \sqrt{d+e x}}-\frac{3 c d \sqrt{c d^2-a e^2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{5/2}}",1,"(2*c*d*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))","C",1
2043,1,176,185,0.1732816,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(9/2),x]","\frac{3 c^2 d^2 (d+e x)^2 \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)-\sqrt{e} \sqrt{c d^2-a e^2} \left(2 a^2 e^3+a c d e (3 d+7 e x)+c^2 d^2 x (3 d+5 e x)\right)}{4 e^{5/2} (d+e x)^{3/2} \sqrt{c d^2-a e^2} \sqrt{(d+e x) (a e+c d x)}}","\frac{3 c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 e^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 e (d+e x)^{7/2}}",1,"(-(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*(2*a^2*e^3 + c^2*d^2*x*(3*d + 5*e*x) + a*c*d*e*(3*d + 7*e*x))) + 3*c^2*d^2*Sqrt[a*e + c*d*x]*(d + e*x)^2*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(4*e^(5/2)*Sqrt[c*d^2 - a*e^2]*(d + e*x)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2044,1,83,250,0.0492738,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(11/2),x]","\frac{2 c^3 d^3 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)^4}","\frac{c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 e^{5/2} \left(c d^2-a e^2\right)^{3/2}}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e^2 (d+e x)^{3/2} \left(c d^2-a e^2\right)}-\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e^2 (d+e x)^{5/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e (d+e x)^{9/2}}",1,"(2*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(5*(c*d^2 - a*e^2)^4*(d + e*x)^(5/2))","C",1
2045,1,83,315,0.0541895,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(13/2),x]","\frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{5 (d+e x)^{5/2} \left(c d^2-a e^2\right)^5}","\frac{3 c^4 d^4 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{64 e^{5/2} \left(c d^2-a e^2\right)^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 e^2 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32 e^2 (d+e x)^{5/2} \left(c d^2-a e^2\right)}-\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e^2 (d+e x)^{7/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 e (d+e x)^{11/2}}",1,"(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(5*(c*d^2 - a*e^2)^5*(d + e*x)^(5/2))","C",1
2046,1,197,295,0.1431449,"\int (d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(128 a^4 e^8-64 a^3 c d e^6 (15 d+7 e x)+48 a^2 c^2 d^2 e^4 \left(65 d^2+70 d e x+21 e^2 x^2\right)-8 a c^3 d^3 e^2 \left(715 d^3+1365 d^2 e x+945 d e^2 x^2+231 e^3 x^3\right)+c^4 d^4 \left(6435 d^4+20020 d^3 e x+24570 d^2 e^2 x^2+13860 d e^3 x^3+3003 e^4 x^4\right)\right)}{45045 c^5 d^5 \sqrt{d+e x}}","\frac{256 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac{128 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{15 c d}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^8 - 64*a^3*c*d*e^6*(15*d + 7*e*x) + 48*a^2*c^2*d^2*e^4*(65*d^2 + 70*d*e*x + 21*e^2*x^2) - 8*a*c^3*d^3*e^2*(715*d^3 + 1365*d^2*e*x + 945*d*e^2*x^2 + 231*e^3*x^3) + c^4*d^4*(6435*d^4 + 20020*d^3*e*x + 24570*d^2*e^2*x^2 + 13860*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*c^5*d^5*Sqrt[d + e*x])","A",1
2047,1,142,233,0.1155416,"\int \sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2} \, dx","Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(-16 a^3 e^6+8 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left(143 d^2+182 d e x+63 e^2 x^2\right)+c^3 d^3 \left(429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right)\right)}{3003 c^4 d^4 \sqrt{d+e x}}","\frac{32 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{12 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{13 c d \sqrt{d+e x}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(13*d + 7*e*x) - 2*a*c^2*d^2*e^2*(143*d^2 + 182*d*e*x + 63*e^2*x^2) + c^3*d^3*(429*d^3 + 1001*d^2*e*x + 819*d*e^2*x^2 + 231*e^3*x^3)))/(3003*c^4*d^4*Sqrt[d + e*x])","A",1
2048,1,98,171,0.0805735,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{\sqrt{d+e x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/Sqrt[d + e*x],x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(8 a^2 e^4-4 a c d e^2 (11 d+7 e x)+c^2 d^2 \left(99 d^2+154 d e x+63 e^2 x^2\right)\right)}{693 c^3 d^3 \sqrt{d+e x}}","\frac{16 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac{8 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{11 c d (d+e x)^{3/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2*(11*d + 7*e*x) + c^2*d^2*(99*d^2 + 154*d*e*x + 63*e^2*x^2)))/(693*c^3*d^3*Sqrt[d + e*x])","A",1
2049,1,65,109,0.0575407,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(c d (9 d+7 e x)-2 a e^2\right)}{63 c^2 d^2 \sqrt{d+e x}}","\frac{4 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{9 c d (d+e x)^{5/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(9*d + 7*e*x)))/(63*c^2*d^2*Sqrt[d + e*x])","A",1
2050,1,37,48,0.039752,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2),x]","\frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 c d (d+e x)^{7/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{7 c d (d+e x)^{7/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*c*d*(d + e*x)^(7/2))","A",1
2051,1,152,240,0.3090494,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2),x]","\frac{((d+e x) (a e+c d x))^{5/2} \left(\frac{10 \left(a e^2-c d^2\right) \left(4 a e^2+c d (e x-3 d)\right)}{3 e^2 (a e+c d x)^2}-\frac{10 \left(c d^2-a e^2\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)}{e^{5/2} (a e+c d x)^{5/2}}+2\right)}{5 e (d+e x)^{5/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 e (d+e x)^{5/2}}+\frac{2 \left(a-\frac{c d^2}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 (d+e x)^{3/2}}-\frac{2 \left(c d^2-a e^2\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{7/2}}+\frac{2 \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^3 \sqrt{d+e x}}",1,"(((a*e + c*d*x)*(d + e*x))^(5/2)*(2 + (10*(-(c*d^2) + a*e^2)*(4*a*e^2 + c*d*(-3*d + e*x)))/(3*e^2*(a*e + c*d*x)^2) - (10*(c*d^2 - a*e^2)^(5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(e^(5/2)*(a*e + c*d*x)^(5/2))))/(5*e*(d + e*x)^(5/2))","A",1
2052,1,79,233,0.0666268,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(9/2),x]","\frac{2 c d ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)^2}","-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{e (d+e x)^{7/2}}+\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{5 c d \left(a-\frac{c d^2}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e \sqrt{d+e x}}+\frac{5 c d \left(c d^2-a e^2\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{e^{7/2}}",1,"(2*c*d*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^2*(d + e*x)^(7/2))","C",1
2053,1,83,236,0.0685213,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2),x]","\frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)^3}","-\frac{15 c^2 d^2 \sqrt{c d^2-a e^2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 e^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e^3 \sqrt{d+e x}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{2 e (d+e x)^{9/2}}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 e^2 (d+e x)^{5/2}}",1,"(2*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^3*(d + e*x)^(7/2))","C",1
2054,1,220,236,0.2485685,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x]","\frac{15 c^3 d^3 (d+e x)^3 \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)-\sqrt{e} \sqrt{c d^2-a e^2} \left(8 a^3 e^5+2 a^2 c d e^3 (5 d+17 e x)+a c^2 d^2 e \left(15 d^2+50 d e x+59 e^2 x^2\right)+c^3 d^3 x \left(15 d^2+40 d e x+33 e^2 x^2\right)\right)}{24 e^{7/2} (d+e x)^{5/2} \sqrt{c d^2-a e^2} \sqrt{(d+e x) (a e+c d x)}}","\frac{5 c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 e^{7/2} \sqrt{c d^2-a e^2}}-\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3 e (d+e x)^{11/2}}",1,"(-(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*(8*a^3*e^5 + 2*a^2*c*d*e^3*(5*d + 17*e*x) + c^3*d^3*x*(15*d^2 + 40*d*e*x + 33*e^2*x^2) + a*c^2*d^2*e*(15*d^2 + 50*d*e*x + 59*e^2*x^2))) + 15*c^3*d^3*Sqrt[a*e + c*d*x]*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(24*e^(7/2)*Sqrt[c*d^2 - a*e^2]*(d + e*x)^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2055,1,83,301,0.0700068,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{15/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x]","\frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)^5}","\frac{5 c^4 d^4 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{64 e^{7/2} \left(c d^2-a e^2\right)^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 e^3 (d+e x)^{3/2} \left(c d^2-a e^2\right)}-\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32 e^3 (d+e x)^{5/2}}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4 e (d+e x)^{13/2}}",1,"(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^5*(d + e*x)^(7/2))","C",1
2056,1,83,366,0.0692343,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{17/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x]","\frac{2 c^5 d^5 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{7 (d+e x)^{7/2} \left(c d^2-a e^2\right)^6}","\frac{3 c^5 d^5 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{128 e^{7/2} \left(c d^2-a e^2\right)^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}+\frac{c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left(c d^2-a e^2\right)}-\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac{c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 e (d+e x)^{15/2}}",1,"(2*c^5*d^5*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^6*(d + e*x)^(7/2))","C",1
2057,1,131,233,0.0987938,"\int \frac{(d+e x)^{7/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-16 a^3 e^6+8 a^2 c d e^4 (7 d+e x)-2 a c^2 d^2 e^2 \left(35 d^2+14 d e x+3 e^2 x^2\right)+c^3 d^3 \left(35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3\right)\right)}{35 c^4 d^4 \sqrt{d+e x}}","\frac{32 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 c^4 d^4 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 c^3 d^3}+\frac{12 (d+e x)^{3/2} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 c^2 d^2}+\frac{2 (d+e x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{7 c d}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(7*d + e*x) - 2*a*c^2*d^2*e^2*(35*d^2 + 14*d*e*x + 3*e^2*x^2) + c^3*d^3*(35*d^3 + 35*d^2*e*x + 21*d*e^2*x^2 + 5*e^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])","A",1
2058,1,87,171,0.0654455,"\int \frac{(d+e x)^{5/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(8 a^2 e^4-4 a c d e^2 (5 d+e x)+c^2 d^2 \left(15 d^2+10 d e x+3 e^2 x^2\right)\right)}{15 c^3 d^3 \sqrt{d+e x}}","\frac{16 \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 c^3 d^3 \sqrt{d+e x}}+\frac{8 \sqrt{d+e x} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 c^2 d^2}+\frac{2 (d+e x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 c d}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2*(5*d + e*x) + c^2*d^2*(15*d^2 + 10*d*e*x + 3*e^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])","A",1
2059,1,54,109,0.0428172,"\int \frac{(d+e x)^{3/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(c d (3 d+e x)-2 a e^2\right)}{3 c^2 d^2 \sqrt{d+e x}}","\frac{4 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c d}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(3*d + e*x)))/(3*c^2*d^2*Sqrt[d + e*x])","A",1
2060,1,35,46,0.0169578,"\int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)}}{c d \sqrt{d+e x}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)])/(c*d*Sqrt[d + e*x])","A",1
2061,1,97,84,0.0311708,"\int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{c d^2-a e^2} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{c d^2-a e^2}}",1,"(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2062,1,135,139,0.0751659,"\int \frac{1}{(d+e x)^{3/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{e} \sqrt{c d^2-a e^2} (a e+c d x)+c d (d+e x) \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{d+e x} \left(c d^2-a e^2\right)^{3/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{(d+e x)^{3/2} \left(c d^2-a e^2\right)}+\frac{c d \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \left(c d^2-a e^2\right)^{3/2}}",1,"(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x) + c*d*Sqrt[a*e + c*d*x]*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[e]*(c*d^2 - a*e^2)^(3/2)*Sqrt[d + e*x]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2063,1,81,207,0.0303533,"\int \frac{1}{(d+e x)^{5/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 c^2 d^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\sqrt{d+e x} \left(c d^2-a e^2\right)^3}","\frac{3 c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 \sqrt{e} \left(c d^2-a e^2\right)^{5/2}}+\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left(c d^2-a e^2\right)}",1,"(2*c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 3, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^3*Sqrt[d + e*x])","C",1
2064,1,81,269,0.033773,"\int \frac{1}{(d+e x)^{7/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 c^3 d^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\sqrt{d+e x} \left(c d^2-a e^2\right)^4}","\frac{5 c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 \sqrt{e} \left(c d^2-a e^2\right)^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3}+\frac{5 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left(c d^2-a e^2\right)}",1,"(2*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 4, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^4*Sqrt[d + e*x])","C",1
2065,1,87,171,0.0556042,"\int \frac{(d+e x)^{7/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} \left(8 a^2 e^4+4 a c d e^2 (e x-3 d)+c^2 d^2 \left(3 d^2-6 d e x-e^2 x^2\right)\right)}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}}","-\frac{16 \sqrt{d+e x} \left(c d^2-a e^2\right)^2}{3 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{8 (d+e x)^{3/2} \left(c d^2-a e^2\right)}{3 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x]*(8*a^2*e^4 + 4*a*c*d*e^2*(-3*d + e*x) + c^2*d^2*(3*d^2 - 6*d*e*x - e^2*x^2)))/(3*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2066,1,51,105,0.034679,"\int \frac{(d+e x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} \left(c d (d-e x)-2 a e^2\right)}{c^2 d^2 \sqrt{(d+e x) (a e+c d x)}}","\frac{2 (d+e x)^{3/2}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{4 \sqrt{d+e x} \left(c d^2-a e^2\right)}{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x]*(-2*a*e^2 + c*d*(d - e*x)))/(c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2067,1,35,46,0.0124042,"\int \frac{(d+e x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x}}{c d \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 \sqrt{d+e x}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x])/(c*d*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
2068,1,75,139,0.0190882,"\int \frac{\sqrt{d+e x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 \sqrt{d+e x}}{\left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{3/2}}",1,"(-2*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 1, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
2069,1,77,196,0.0250855,"\int \frac{1}{\sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c d \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\left(c d^2-a e^2\right)^2 \sqrt{(d+e x) (a e+c d x)}}","-\frac{3 c d \sqrt{d+e x}}{\left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{1}{\sqrt{d+e x} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{3 c d \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{5/2}}",1,"(-2*c*d*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 2, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
2070,1,81,269,0.0249426,"\int \frac{1}{(d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c^2 d^2 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)}}","-\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 \left(c d^2-a e^2\right)^{7/2}}+\frac{5 c d}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{1}{2 (d+e x)^{3/2} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*c^2*d^2*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 3, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
2071,1,81,331,0.0258531,"\int \frac{1}{(d+e x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c^3 d^3 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},4;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\left(c d^2-a e^2\right)^4 \sqrt{(d+e x) (a e+c d x)}}","-\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 \left(c d^2-a e^2\right)^{9/2}}+\frac{35 c^2 d^2}{24 \sqrt{d+e x} \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{7 c d}{12 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{1}{3 (d+e x)^{5/2} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*c^3*d^3*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 4, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^4*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
2072,1,81,393,0.0269535,"\int \frac{1}{(d+e x)^{7/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c^4 d^4 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},5;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{\left(c d^2-a e^2\right)^5 \sqrt{(d+e x) (a e+c d x)}}","-\frac{315 c^4 d^4 \sqrt{d+e x}}{64 \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{315 c^4 d^4 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{64 \left(c d^2-a e^2\right)^{11/2}}+\frac{105 c^3 d^3}{64 \sqrt{d+e x} \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{21 c^2 d^2}{32 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3 c d}{8 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{1}{4 (d+e x)^{7/2} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*c^4*d^4*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 5, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^5*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
2073,1,53,107,0.043814,"\int \frac{(d+e x)^{7/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} \left(2 a e^2+c d (d+3 e x)\right)}{3 c^2 d^2 ((d+e x) (a e+c d x))^{3/2}}","\frac{4 (d+e x)^{3/2} \left(c d^2-a e^2\right)}{3 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}-\frac{2 (d+e x)^{5/2}}{c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2)*(2*a*e^2 + c*d*(d + 3*e*x)))/(3*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
2074,1,37,48,0.0281711,"\int \frac{(d+e x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2}}{3 c d ((d+e x) (a e+c d x))^{3/2}}","-\frac{2 (d+e x)^{3/2}}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2))/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
2075,1,77,251,0.0239021,"\int \frac{(d+e x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(a e^2-c d^2\right) ((d+e x) (a e+c d x))^{3/2}}","\frac{2 e \sqrt{d+e x}}{\left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 e}{3 c d \sqrt{d+e x} \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{5/2}}",1,"(2*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 1, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(-(c*d^2) + a*e^2)*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2076,1,79,257,0.0309073,"\int \frac{\sqrt{d+e x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 c d (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(c d^2-a e^2\right)^2 ((d+e x) (a e+c d x))^{3/2}}","\frac{5 c d e \sqrt{d+e x}}{\left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{5 e}{3 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{5 c d e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{\left(c d^2-a e^2\right)^{7/2}}",1,"(-2*c*d*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 2, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2077,1,83,329,0.0322934,"\int \frac{1}{\sqrt{d+e x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c^2 d^2 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(c d^2-a e^2\right)^3 ((d+e x) (a e+c d x))^{3/2}}","\frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{35 c^2 d^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{4 \left(c d^2-a e^2\right)^{9/2}}-\frac{35 c d e}{12 \sqrt{d+e x} \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{7 c d \sqrt{d+e x}}{6 \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{1}{2 \sqrt{d+e x} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*c^2*d^2*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 3, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^3*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2078,1,83,395,0.0348818,"\int \frac{1}{(d+e x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c^3 d^3 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},4;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(c d^2-a e^2\right)^4 ((d+e x) (a e+c d x))^{3/2}}","\frac{105 c^3 d^3 e \sqrt{d+e x}}{8 \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{105 c^3 d^3 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{8 \left(c d^2-a e^2\right)^{11/2}}-\frac{35 c^2 d^2 e}{8 \sqrt{d+e x} \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{7 c^2 d^2 \sqrt{d+e x}}{4 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{3 c d}{4 \sqrt{d+e x} \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{1}{3 (d+e x)^{3/2} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*c^3*d^3*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 4, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^4*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2079,1,83,457,0.0366647,"\int \frac{1}{(d+e x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c^4 d^4 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},5;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(c d^2-a e^2\right)^5 ((d+e x) (a e+c d x))^{3/2}}","\frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left(c d^2-a e^2\right)^6 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{1155 c^4 d^4 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{64 \left(c d^2-a e^2\right)^{13/2}}-\frac{385 c^3 d^3 e}{64 \sqrt{d+e x} \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{77 c^3 d^3 \sqrt{d+e x}}{32 \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{33 c^2 d^2}{32 \sqrt{d+e x} \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{11 c d}{24 (d+e x)^{3/2} \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{1}{4 (d+e x)^{5/2} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*c^4*d^4*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 5, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2080,1,83,519,0.0412529,"\int \frac{1}{(d+e x)^{7/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c^5 d^5 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},6;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{3 \left(c d^2-a e^2\right)^6 ((d+e x) (a e+c d x))^{3/2}}","\frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left(c d^2-a e^2\right)^7 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right)}{128 \left(c d^2-a e^2\right)^{15/2}}-\frac{1001 c^4 d^4 e}{128 \sqrt{d+e x} \left(c d^2-a e^2\right)^6 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left(c d^2-a e^2\right)^5 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{429 c^3 d^3}{320 \sqrt{d+e x} \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{143 c^2 d^2}{240 (d+e x)^{3/2} \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{13 c d}{40 (d+e x)^{5/2} \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{1}{5 (d+e x)^{7/2} \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*c^5*d^5*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 6, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2 - a*e^2)^6*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
2081,1,52,52,0.0502187,"\int \frac{1}{\sqrt{d+e x} \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d} e}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right)}{\sqrt{d} e}",1,"-((Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[d]*e))","A",1
2082,1,53,53,0.0625761,"\int \frac{1}{\sqrt{-d+e x} \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(Sqrt[-d + e*x]*Sqrt[d^2 - e^2*x^2]),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{e x-d}}\right)}{\sqrt{d} e}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{e x-d}}\right)}{\sqrt{d} e}",1,"(Sqrt[2]*ArcTan[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[-d + e*x])])/(Sqrt[d]*e)","A",1
2083,1,95,566,0.0579469,"\int \frac{(d+e x)^{2/3}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(2/3)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d \sqrt[3]{d+e x} \sqrt[6]{\frac{c d (d+e x)}{c d^2-a e^2}}}","\frac{3^{3/4} (d+e x)^{2/3} \left(c d^2-a e^2\right)^{2/3} \sqrt{a d e+c d^2 x} \left(\sqrt[3]{c d^2-a e^2}-\sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}\right) \sqrt{\frac{\left(c d^2-a e^2\right)^{2/3}+\sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1} \sqrt[3]{c d^2-a e^2}+c^{2/3} d^{4/3} \left(\frac{e x}{d}+1\right)^{2/3}}{\left(\sqrt[3]{c d^2-a e^2}-\left(1+\sqrt{3}\right) \sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{c d^2-a e^2}-\left(1-\sqrt{3}\right) \sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}}{\sqrt[3]{c d^2-a e^2}-\left(1+\sqrt{3}\right) \sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{4 c d e \sqrt{d (a e+c d x)} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \sqrt{-\frac{\sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1} \left(\sqrt[3]{c d^2-a e^2}-\sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}\right)}{\left(\sqrt[3]{c d^2-a e^2}-\left(1+\sqrt{3}\right) \sqrt[3]{c} d^{2/3} \sqrt[3]{\frac{e x}{d}+1}\right)^2}}}+\frac{3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-1/6, 1/2, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(d + e*x)^(1/3)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/6))","C",1
2084,1,114,130,0.0940028,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{(d+e x)^{m+4} \left(-\frac{3 c^2 d^2 (d+e x)^2 \left(c d^2-a e^2\right)}{m+6}+\frac{3 c d (d+e x) \left(c d^2-a e^2\right)^2}{m+5}-\frac{\left(c d^2-a e^2\right)^3}{m+4}+\frac{c^3 d^3 (d+e x)^3}{m+7}\right)}{e^4}","-\frac{3 c^2 d^2 \left(c d^2-a e^2\right) (d+e x)^{m+6}}{e^4 (m+6)}-\frac{\left(c d^2-a e^2\right)^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c d \left(c d^2-a e^2\right)^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac{c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)}",1,"((d + e*x)^(4 + m)*(-((c*d^2 - a*e^2)^3/(4 + m)) + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x))/(5 + m) - (3*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^2)/(6 + m) + (c^3*d^3*(d + e*x)^3)/(7 + m)))/e^4","A",1
2085,1,79,90,0.0820717,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{(d+e x)^{m+3} \left(-\frac{2 c d (d+e x) \left(c d^2-a e^2\right)}{m+4}+\frac{\left(c d^2-a e^2\right)^2}{m+3}+\frac{c^2 d^2 (d+e x)^2}{m+5}\right)}{e^3}","\frac{\left(c d^2-a e^2\right)^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac{2 c d \left(c d^2-a e^2\right) (d+e x)^{m+4}}{e^3 (m+4)}+\frac{c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)}",1,"((d + e*x)^(3 + m)*((c*d^2 - a*e^2)^2/(3 + m) - (2*c*d*(c*d^2 - a*e^2)*(d + e*x))/(4 + m) + (c^2*d^2*(d + e*x)^2)/(5 + m)))/e^3","A",1
2086,1,45,52,0.0377173,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right) \, dx","Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","\frac{(d+e x)^{m+2} \left(a e^2 (m+3)+c d (e (m+2) x-d)\right)}{e^2 (m+2) (m+3)}","\frac{c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac{\left(c d^2-a e^2\right) (d+e x)^{m+2}}{e^2 (m+2)}",1,"((d + e*x)^(2 + m)*(a*e^2*(3 + m) + c*d*(-d + e*(2 + m)*x)))/(e^2*(2 + m)*(3 + m))","A",1
2087,1,54,54,0.0109735,"\int \frac{(d+e x)^m}{a d e+\left(c d^2+a e^2\right) x+c d e x^2} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]","-\frac{(d+e x)^m \, _2F_1\left(1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{m \left(c d^2-a e^2\right)}","-\frac{(d+e x)^m \, _2F_1\left(1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{m \left(c d^2-a e^2\right)}",1,"-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)*m))","A",1
2088,1,59,61,0.0250182,"\int \frac{(d+e x)^m}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]","\frac{e (d+e x)^{m-1} \, _2F_1\left(2,m-1;m;-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{(m-1) \left(a e^2-c d^2\right)^2}","-\frac{e (d+e x)^{m-1} \, _2F_1\left(2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(1-m) \left(c d^2-a e^2\right)^2}",1,"(e*(d + e*x)^(-1 + m)*Hypergeometric2F1[2, -1 + m, m, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^2*(-1 + m))","A",1
2089,1,63,64,0.0218189,"\int \frac{(d+e x)^m}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^3} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]","\frac{e^2 (d+e x)^{m-2} \, _2F_1\left(3,m-2;m-1;-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{(m-2) \left(a e^2-c d^2\right)^3}","\frac{e^2 (d+e x)^{m-2} \, _2F_1\left(3,m-2;m-1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(2-m) \left(c d^2-a e^2\right)^3}",1,"(e^2*(d + e*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^3*(-2 + m))","A",1
2090,1,63,65,0.026993,"\int \frac{(d+e x)^m}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^4} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]","\frac{e^3 (d+e x)^{m-3} \, _2F_1\left(4,m-3;m-2;-\frac{c d (d+e x)}{a e^2-c d^2}\right)}{(m-3) \left(a e^2-c d^2\right)^4}","-\frac{e^3 (d+e x)^{m-3} \, _2F_1\left(4,m-3;m-2;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(3-m) \left(c d^2-a e^2\right)^4}",1,"(e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^4*(-3 + m))","A",1
2091,1,107,100,0.0529802,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{p+1} \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-m-p} \, _2F_1\left(-m-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d (p+1)}","-\frac{(d+e x)^{m+1} (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,m+2 p+2;m+p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(m+p+1) \left(c d^2-a e^2\right)}",1,"((d + e*x)^(-1 + m)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(-m - p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*Hypergeometric2F1[-m - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(1 + p))","A",1
2092,1,112,95,0.0533255,"\int (d+e x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{\left(c d^2-a e^2\right)^3 (a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(-p-3,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c^4 d^4 (p+1)}","-\frac{(d+e x)^4 (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,2 p+5;p+5;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(p+4) \left(c d^2-a e^2\right)}",1,"((c*d^2 - a*e^2)^3*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-3 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^4*d^4*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)","A",1
2093,1,112,95,0.0450802,"\int (d+e x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{\left(c d^2-a e^2\right)^2 (a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(-p-2,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c^3 d^3 (p+1)}","-\frac{(d+e x)^3 (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,2 (p+2);p+4;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(p+3) \left(c d^2-a e^2\right)}",1,"((c*d^2 - a*e^2)^2*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-2 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^3*d^3*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)","A",1
2094,1,94,95,0.0537106,"\int (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{\left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p-1} ((d+e x) (a e+c d x))^{p+1} \, _2F_1\left(-p-1,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d (p+1)}","-\frac{(d+e x)^2 (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,2 p+3;p+3;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(p+2) \left(c d^2-a e^2\right)}",1,"(((c*d*(d + e*x))/(c*d^2 - a*e^2))^(-1 - p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*Hypergeometric2F1[-1 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(1 + p))","A",1
2095,1,96,113,0.0266339,"\int \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d (p+1)}","-\frac{\left(-\frac{e (a e+c d x)}{c d^2-a e^2}\right)^{-p-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1} \, _2F_1\left(-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(p+1) \left(c d^2-a e^2\right)}",1,"((a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)","A",1
2096,1,81,93,0.0269686,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x),x]","\frac{\left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{e p}","\frac{\left(-\frac{e (a e+c d x)}{c d^2-a e^2}\right)^{-p} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{e p}",1,"(((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, p, 1 + p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e*p*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^p)","A",1
2097,1,108,92,0.0389435,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^2,x]","\frac{c d (a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(2-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{(p+1) \left(c d^2-a e^2\right)^2}","\frac{(a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,2 p;p;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(1-p) (d+e x) \left(c d^2-a e^2\right)}",1,"(c*d*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^2*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)","A",1
2098,1,112,96,0.0420872,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^3,x]","\frac{c^2 d^2 (a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(3-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{(p+1) \left(c d^2-a e^2\right)^3}","\frac{(a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(1,2 p-1;p-1;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{(2-p) (d+e x)^2 \left(c d^2-a e^2\right)}",1,"(c^2*d^2*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)^3*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)","A",1
2099,1,97,113,0.0437079,"\int (d+e x)^{-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^(2*p),x]","\frac{(d+e x)^{-2 p-1} \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^p ((d+e x) (a e+c d x))^{p+1} \, _2F_1\left(p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right)}{c d (p+1)}","\frac{(d+e x)^{-2 p} (a e+c d x) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^p \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right)}{c d (p+1)}",1,"((d + e*x)^(-1 - 2*p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p*((a*e + c*d*x)*(d + e*x))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(1 + p))","A",1
2100,1,95,107,0.0212334,"\int (d+e x)^{-1-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^(-1 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","-\frac{(d+e x)^{-2 p} \left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left(-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{e p}","-\frac{(d+e x)^{-2 p} \left(-\frac{e (a e+c d x)}{c d^2-a e^2}\right)^{-p} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^p \, _2F_1\left(-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right)}{e p}",1,"-((((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, -p, 1 - p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e*p*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^p*(d + e*x)^(2*p)))","A",1
2101,1,49,60,0.0283265,"\int (d+e x)^{-2-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^(-2 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(d+e x)^{-2 (p+1)} ((d+e x) (a e+c d x))^{p+1}}{(p+1) \left(c d^2-a e^2\right)}","\frac{(d+e x)^{-2 (p+1)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+1) \left(c d^2-a e^2\right)}",1,"((a*e + c*d*x)*(d + e*x))^(1 + p)/((c*d^2 - a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p)))","A",1
2102,1,76,128,0.0539952,"\int (d+e x)^{-3-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^(-3 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(d+e x)^{-2 p-3} ((d+e x) (a e+c d x))^{p+1} \left(c d (d (p+2)+e x)-a e^2 (p+1)\right)}{(p+1) (p+2) \left(c d^2-a e^2\right)^2}","\frac{(d+e x)^{-2 p-3} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+2) \left(c d^2-a e^2\right)}+\frac{c d (d+e x)^{-2 (p+1)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+1) (p+2) \left(c d^2-a e^2\right)^2}",1,"((d + e*x)^(-3 - 2*p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*(-(a*e^2*(1 + p)) + c*d*(d*(2 + p) + e*x)))/((c*d^2 - a*e^2)^2*(1 + p)*(2 + p))","A",1
2103,1,131,206,0.0875235,"\int (d+e x)^{-4-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(d+e x)^{-2 (p+2)} ((d+e x) (a e+c d x))^{p+1} \left(a^2 e^4 \left(p^2+3 p+2\right)-2 a c d e^2 (p+1) (d (p+3)+e x)+c^2 d^2 \left(d^2 \left(p^2+5 p+6\right)+2 d e (p+3) x+2 e^2 x^2\right)\right)}{(p+1) (p+2) (p+3) \left(c d^2-a e^2\right)^3}","\frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+1) (p+2) (p+3) \left(c d^2-a e^2\right)^3}+\frac{2 c d (d+e x)^{-2 p-3} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+2) (p+3) \left(c d^2-a e^2\right)^2}+\frac{(d+e x)^{-2 (p+2)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+3) \left(c d^2-a e^2\right)}",1,"(((a*e + c*d*x)*(d + e*x))^(1 + p)*(a^2*e^4*(2 + 3*p + p^2) - 2*a*c*d*e^2*(1 + p)*(d*(3 + p) + e*x) + c^2*d^2*(d^2*(6 + 5*p + p^2) + 2*d*e*(3 + p)*x + 2*e^2*x^2)))/((c*d^2 - a*e^2)^3*(1 + p)*(2 + p)*(3 + p)*(d + e*x)^(2*(2 + p)))","A",1
2104,1,217,288,0.1448042,"\int (d+e x)^{-5-2 p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]","\frac{(d+e x)^{-2 p-5} ((d+e x) (a e+c d x))^{p+1} \left(-a^3 e^6 \left(p^3+6 p^2+11 p+6\right)+3 a^2 c d e^4 \left(p^2+3 p+2\right) (d (p+4)+e x)-3 a c^2 d^2 e^2 (p+1) \left(d^2 \left(p^2+7 p+12\right)+2 d e (p+4) x+2 e^2 x^2\right)+c^3 d^3 \left(d^3 \left(p^3+9 p^2+26 p+24\right)+3 d^2 e \left(p^2+7 p+12\right) x+6 d e^2 (p+4) x^2+6 e^3 x^3\right)\right)}{(p+1) (p+2) (p+3) (p+4) \left(c d^2-a e^2\right)^4}","\frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left(c d^2-a e^2\right)^4}+\frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+2) (p+3) (p+4) \left(c d^2-a e^2\right)^3}+\frac{(d+e x)^{-2 p-5} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+4) \left(c d^2-a e^2\right)}+\frac{3 c d (d+e x)^{-2 (p+2)} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{(p+3) (p+4) \left(c d^2-a e^2\right)^2}",1,"((d + e*x)^(-5 - 2*p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*(-(a^3*e^6*(6 + 11*p + 6*p^2 + p^3)) + 3*a^2*c*d*e^4*(2 + 3*p + p^2)*(d*(4 + p) + e*x) - 3*a*c^2*d^2*e^2*(1 + p)*(d^2*(12 + 7*p + p^2) + 2*d*e*(4 + p)*x + 2*e^2*x^2) + c^3*d^3*(d^3*(24 + 26*p + 9*p^2 + p^3) + 3*d^2*e*(12 + 7*p + p^2)*x + 6*d*e^2*(4 + p)*x^2 + 6*e^3*x^3)))/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)*(3 + p)*(4 + p))","A",1
2105,1,42,54,0.0266476,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","-\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)}","\frac{(d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c d (1-m)}",1,"-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m))/(c*d*(-1 + m)))","A",1
2106,1,41,52,0.0205348,"\int (d+e x)^{-p} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^p \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p/(d + e*x)^p,x]","\frac{(d+e x)^{-p-1} ((d+e x) (a e+c d x))^{p+1}}{c d (p+1)}","\frac{(d+e x)^{-p-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{p+1}}{c d (p+1)}",1,"((d + e*x)^(-1 - p)*((a*e + c*d*x)*(d + e*x))^(1 + p))/(c*d*(1 + p))","A",1
2107,1,135,69,0.036702,"\int (d+e x)^4 \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^4*(a + b*x + c*x^2),x]","\frac{1}{5} e^2 x^5 \left(a e^2+4 b d e+6 c d^2\right)+\frac{1}{2} d e x^4 \left(2 a e^2+3 b d e+2 c d^2\right)+\frac{1}{3} d^2 x^3 \left(6 a e^2+4 b d e+c d^2\right)+\frac{1}{2} d^3 x^2 (4 a e+b d)+a d^4 x+\frac{1}{6} e^3 x^6 (b e+4 c d)+\frac{1}{7} c e^4 x^7","\frac{(d+e x)^5 \left(a e^2-b d e+c d^2\right)}{5 e^3}-\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{c (d+e x)^7}{7 e^3}",1,"a*d^4*x + (d^3*(b*d + 4*a*e)*x^2)/2 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^3)/3 + (d*e*(2*c*d^2 + 3*b*d*e + 2*a*e^2)*x^4)/2 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^5)/5 + (e^3*(4*c*d + b*e)*x^6)/6 + (c*e^4*x^7)/7","A",1
2108,1,104,69,0.0236614,"\int (d+e x)^3 \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2),x]","\frac{1}{4} e x^4 \left(a e^2+3 b d e+3 c d^2\right)+\frac{1}{3} d x^3 \left(3 a e^2+3 b d e+c d^2\right)+\frac{1}{2} d^2 x^2 (3 a e+b d)+a d^3 x+\frac{1}{5} e^2 x^5 (b e+3 c d)+\frac{1}{6} c e^3 x^6","\frac{(d+e x)^4 \left(a e^2-b d e+c d^2\right)}{4 e^3}-\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{c (d+e x)^6}{6 e^3}",1,"a*d^3*x + (d^2*(b*d + 3*a*e)*x^2)/2 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^3)/3 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x^4)/4 + (e^2*(3*c*d + b*e)*x^5)/5 + (c*e^3*x^6)/6","A",1
2109,1,73,69,0.018734,"\int (d+e x)^2 \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2),x]","\frac{1}{3} x^3 \left(a e^2+2 b d e+c d^2\right)+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 x+\frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{5} c e^2 x^5","\frac{(d+e x)^3 \left(a e^2-b d e+c d^2\right)}{3 e^3}-\frac{(d+e x)^4 (2 c d-b e)}{4 e^3}+\frac{c (d+e x)^5}{5 e^3}",1,"a*d^2*x + (d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + 2*b*d*e + a*e^2)*x^3)/3 + (e*(2*c*d + b*e)*x^4)/4 + (c*e^2*x^5)/5","A",1
2110,1,42,42,0.0132854,"\int (d+e x) \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2),x]","\frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4","\frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4",1,"a*d*x + ((b*d + a*e)*x^2)/2 + ((c*d + b*e)*x^3)/3 + (c*e*x^4)/4","A",1
2111,1,20,20,0.0000824,"\int \left(a+b x+c x^2\right) \, dx","Integrate[a + b*x + c*x^2,x]","a x+\frac{b x^2}{2}+\frac{c x^3}{3}","a x+\frac{b x^2}{2}+\frac{c x^3}{3}",1,"a*x + (b*x^2)/2 + (c*x^3)/3","A",1
2112,1,48,52,0.0168238,"\int \frac{a+b x+c x^2}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x),x]","\frac{2 \log (d+e x) \left(e (a e-b d)+c d^2\right)+e x (2 b e-2 c d+c e x)}{2 e^3}","\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)}{e^3}-\frac{x (c d-b e)}{e^2}+\frac{c x^2}{2 e}",1,"(e*x*(-2*c*d + 2*b*e + c*e*x) + 2*(c*d^2 + e*(-(b*d) + a*e))*Log[d + e*x])/(2*e^3)","A",1
2113,1,49,55,0.0257933,"\int \frac{a+b x+c x^2}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^2,x]","\frac{-\frac{a e^2-b d e+c d^2}{d+e x}+(b e-2 c d) \log (d+e x)+c e x}{e^3}","-\frac{a e^2-b d e+c d^2}{e^3 (d+e x)}-\frac{(2 c d-b e) \log (d+e x)}{e^3}+\frac{c x}{e^2}",1,"(c*e*x - (c*d^2 - b*d*e + a*e^2)/(d + e*x) + (-2*c*d + b*e)*Log[d + e*x])/e^3","A",1
2114,1,57,62,0.0198642,"\int \frac{a+b x+c x^2}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^3,x]","\frac{-e (a e+b d+2 b e x)+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2}","-\frac{a e^2-b d e+c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d-b e}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3}",1,"(c*d*(3*d + 4*e*x) - e*(b*d + a*e + 2*b*e*x) + 2*c*(d + e*x)^2*Log[d + e*x])/(2*e^3*(d + e*x)^2)","A",1
2115,1,50,67,0.0177206,"\int \frac{a+b x+c x^2}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^4,x]","-\frac{e (2 a e+b (d+3 e x))+2 c \left(d^2+3 d e x+3 e^2 x^2\right)}{6 e^3 (d+e x)^3}","-\frac{a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)}",1,"-1/6*(2*c*(d^2 + 3*d*e*x + 3*e^2*x^2) + e*(2*a*e + b*(d + 3*e*x)))/(e^3*(d + e*x)^3)","A",1
2116,1,49,69,0.0177773,"\int \frac{a+b x+c x^2}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^5,x]","-\frac{e (3 a e+b (d+4 e x))+c \left(d^2+4 d e x+6 e^2 x^2\right)}{12 e^3 (d+e x)^4}","-\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2}",1,"-1/12*(c*(d^2 + 4*d*e*x + 6*e^2*x^2) + e*(3*a*e + b*(d + 4*e*x)))/(e^3*(d + e*x)^4)","A",1
2117,1,51,69,0.0207206,"\int \frac{a+b x+c x^2}{(d+e x)^6} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^6,x]","-\frac{3 e (4 a e+b (d+5 e x))+2 c \left(d^2+5 d e x+10 e^2 x^2\right)}{60 e^3 (d+e x)^5}","-\frac{a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3}",1,"-1/60*(2*c*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*e*(4*a*e + b*(d + 5*e*x)))/(e^3*(d + e*x)^5)","A",1
2118,1,283,156,0.0727925,"\int (d+e x)^4 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^4*(a + b*x + c*x^2)^2,x]","\frac{1}{5} x^5 \left(e^2 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)+4 c d^2 e (3 a e+2 b d)+c^2 d^4\right)+\frac{1}{2} d x^4 \left(2 a^2 e^3+6 a b d e^2+4 a c d^2 e+2 b^2 d^2 e+b c d^3\right)+a^2 d^4 x+\frac{1}{7} e^2 x^7 \left(2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right)+\frac{1}{3} d^2 x^3 \left(8 a b d e+2 a \left(3 a e^2+c d^2\right)+b^2 d^2\right)+\frac{1}{3} e x^6 \left(2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right)+a d^3 x^2 (2 a e+b d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9","\frac{(d+e x)^7 \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{7 e^5}-\frac{(d+e x)^6 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{3 e^5}+\frac{(d+e x)^5 \left(a e^2-b d e+c d^2\right)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}",1,"a^2*d^4*x + a*d^3*(b*d + 2*a*e)*x^2 + (d^2*(b^2*d^2 + 8*a*b*d*e + 2*a*(c*d^2 + 3*a*e^2))*x^3)/3 + (d*(b*c*d^3 + 2*b^2*d^2*e + 4*a*c*d^2*e + 6*a*b*d*e^2 + 2*a^2*e^3)*x^4)/2 + ((c^2*d^4 + 4*c*d^2*e*(2*b*d + 3*a*e) + e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^5)/5 + (e*(2*c^2*d^3 + b*e^2*(2*b*d + a*e) + 2*c*d*e*(3*b*d + 2*a*e))*x^6)/3 + (e^2*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(4*b*d + a*e))*x^7)/7 + (c*e^3*(2*c*d + b*e)*x^8)/4 + (c^2*e^4*x^9)/9","A",1
2119,1,223,156,0.06226,"\int (d+e x)^3 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^2,x]","\frac{1}{4} x^4 \left(a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right)+a^2 d^3 x+\frac{1}{6} e x^6 \left(2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)+\frac{1}{3} d x^3 \left(6 a b d e+a \left(3 a e^2+2 c d^2\right)+b^2 d^2\right)+\frac{1}{5} x^5 \left(6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right)+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8","\frac{(d+e x)^6 \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{6 e^5}-\frac{2 (d+e x)^5 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{5 e^5}+\frac{(d+e x)^4 \left(a e^2-b d e+c d^2\right)^2}{4 e^5}-\frac{2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}",1,"a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^2)/2 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^3)/3 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^4)/4 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^5)/5 + (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8","A",1
2120,1,153,156,0.0418475,"\int (d+e x)^2 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^2,x]","\frac{1}{3} x^3 \left(a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right)+a^2 d^2 x+\frac{1}{5} x^5 \left(2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right)+\frac{1}{2} x^4 \left(a b e^2+2 a c d e+b^2 d e+b c d^2\right)+a d x^2 (a e+b d)+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{7} c^2 e^2 x^7","\frac{(d+e x)^5 \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{2 e^5}+\frac{(d+e x)^3 \left(a e^2-b d e+c d^2\right)^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}",1,"a^2*d^2*x + a*d*(b*d + a*e)*x^2 + ((b^2*d^2 + 2*a*c*d^2 + 4*a*b*d*e + a^2*e^2)*x^3)/3 + ((b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3 + (c^2*e^2*x^7)/7","A",1
2121,1,96,96,0.0206959,"\int (d+e x) \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^2,x]","a^2 d x+\frac{1}{4} x^4 \left(2 a c e+b^2 e+2 b c d\right)+\frac{1}{3} x^3 \left(2 a b e+2 a c d+b^2 d\right)+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6","a^2 d x+\frac{1}{4} x^4 \left(2 a c e+b^2 e+2 b c d\right)+\frac{1}{3} x^3 \left(2 a b e+2 a c d+b^2 d\right)+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6",1,"a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6","A",1
2122,1,46,46,0.0063511,"\int \left(a+b x+c x^2\right)^2 \, dx","Integrate[(a + b*x + c*x^2)^2,x]","a^2 x+\frac{1}{3} x^3 \left(2 a c+b^2\right)+a b x^2+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}","a^2 x+\frac{1}{3} x^3 \left(2 a c+b^2\right)+a b x^2+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5}",1,"a^2*x + a*b*x^2 + ((b^2 + 2*a*c)*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5","A",1
2123,1,128,129,0.0615532,"\int \frac{\left(a+b x+c x^2\right)^2}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x),x]","\frac{e x \left(4 c e \left(3 a e (e x-2 d)+b \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)+6 b e^2 (4 a e-2 b d+b e x)+c^2 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+12 \log (d+e x) \left(e (a e-b d)+c d^2\right)^2}{12 e^5}","\frac{x^2 \left(-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right)}{2 e^3}-\frac{x (c d-b e) \left(c d^2-e (b d-2 a e)\right)}{e^4}+\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)^2}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e}",1,"(e*x*(6*b*e^2*(-2*b*d + 4*a*e + b*e*x) + c^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 4*c*e*(3*a*e*(-2*d + e*x) + b*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + 12*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(12*e^5)","A",1
2124,1,127,131,0.1085448,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^2,x]","\frac{3 e x \left(2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right)-\frac{3 \left(e (a e-b d)+c d^2\right)^2}{d+e x}-6 (2 c d-b e) \log (d+e x) \left(e (a e-b d)+c d^2\right)+3 c e^2 x^2 (b e-c d)+c^2 e^3 x^3}{3 e^5}","\frac{x \left(-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right)}{e^4}-\frac{\left(a e^2-b d e+c d^2\right)^2}{e^5 (d+e x)}-\frac{2 (2 c d-b e) \log (d+e x) \left(a e^2-b d e+c d^2\right)}{e^5}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2}",1,"(3*e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e))*x + 3*c*e^2*(-(c*d) + b*e)*x^2 + c^2*e^3*x^3 - (3*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) - 6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*Log[d + e*x])/(3*e^5)","A",1
2125,1,176,138,0.0751563,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^3,x]","\frac{2 (d+e x)^2 \log (d+e x) \left(2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right)+2 c e \left(a d e (3 d+4 e x)+b \left(-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)\right)+e^2 (b d-a e) (a e+3 b d+4 b e x)+c^2 \left(7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right)}{2 e^5 (d+e x)^2}","\frac{\log (d+e x) \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{e^5}-\frac{\left(a e^2-b d e+c d^2\right)^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{e^5 (d+e x)}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^2}{2 e^3}",1,"(e^2*(b*d - a*e)*(3*b*d + a*e + 4*b*e*x) + c^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 2*c*e*(a*d*e*(3*d + 4*e*x) + b*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 2*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*(d + e*x)^2*Log[d + e*x])/(2*e^5*(d + e*x)^2)","A",1
2126,1,176,139,0.0822101,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^4,x]","\frac{-e^2 \left(a^2 e^2+a b e (d+3 e x)+b^2 \left(d^2+3 d e x+3 e^2 x^2\right)\right)+c e \left(b d \left(11 d^2+27 d e x+18 e^2 x^2\right)-2 a e \left(d^2+3 d e x+3 e^2 x^2\right)\right)-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left(-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right)}{3 e^5 (d+e x)^3}","-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)}{e^5 (d+e x)^2}-\frac{\left(a e^2-b d e+c d^2\right)^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4}",1,"(c^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - e^2*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) + c*e*(-2*a*e*(d^2 + 3*d*e*x + 3*e^2*x^2) + b*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) - 6*c*(2*c*d - b*e)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)","A",1
2127,1,170,150,0.0687638,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^5,x]","\frac{-e^2 \left(3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left(d^2+4 d e x+6 e^2 x^2\right)\right)-2 c e \left(a e \left(d^2+4 d e x+6 e^2 x^2\right)+3 b \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)\right)+c^2 d \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}","-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{3 e^5 (d+e x)^3}-\frac{\left(a e^2-b d e+c d^2\right)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}",1,"(c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - e^2*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) - 2*c*e*(a*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*b*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)","A",1
2128,1,160,151,0.0690862,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^6} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^6,x]","-\frac{e^2 \left(6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left(d^2+5 d e x+10 e^2 x^2\right)\right)+c e \left(2 a e \left(d^2+5 d e x+10 e^2 x^2\right)+3 b \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)\right)+6 c^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)}{30 e^5 (d+e x)^5}","-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)}{2 e^5 (d+e x)^4}-\frac{\left(a e^2-b d e+c d^2\right)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}",1,"-1/30*(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))/(e^5*(d + e*x)^5)","A",1
2129,1,159,156,0.0628047,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^7} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^7,x]","-\frac{e^2 \left(10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left(d^2+6 d e x+15 e^2 x^2\right)\right)+2 c e \left(a e \left(d^2+6 d e x+15 e^2 x^2\right)+b \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)\right)+2 c^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)}{60 e^5 (d+e x)^6}","-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{4 e^5 (d+e x)^4}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{5 e^5 (d+e x)^5}-\frac{\left(a e^2-b d e+c d^2\right)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{c^2}{2 e^5 (d+e x)^2}",1,"-1/60*(2*c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + e^2*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 2*c*e*(a*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)))/(e^5*(d + e*x)^6)","A",1
2130,1,161,156,0.0688436,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^8} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^8,x]","-\frac{2 e^2 \left(15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left(d^2+7 d e x+21 e^2 x^2\right)\right)+c e \left(4 a e \left(d^2+7 d e x+21 e^2 x^2\right)+3 b \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)\right)+2 c^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)}{210 e^5 (d+e x)^7}","-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)}{3 e^5 (d+e x)^6}-\frac{\left(a e^2-b d e+c d^2\right)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}",1,"-1/210*(2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 2*e^2*(15*a^2*e^2 + 5*a*b*e*(d + 7*e*x) + b^2*(d^2 + 7*d*e*x + 21*e^2*x^2)) + c*e*(4*a*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)))/(e^5*(d + e*x)^7)","A",1
2131,1,497,272,0.1508697,"\int (d+e x)^4 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^4*(a + b*x + c*x^2)^3,x]","a^3 d^4 x+\frac{1}{7} x^7 \left(3 c e^2 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)+b^2 e^3 (3 a e+4 b d)+6 c^2 d^2 e (3 a e+2 b d)+c^3 d^4\right)+\frac{1}{5} x^5 \left(a \left(a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right)+3 b^2 \left(6 a d^2 e^2+c d^4\right)+12 a b d e \left(a e^2+2 c d^2\right)+4 b^3 d^3 e\right)+\frac{1}{2} x^6 \left(b \left(a^2 e^4+12 a c d^2 e^2+c^2 d^4\right)+4 b^2 \left(a d e^3+c d^3 e\right)+4 a c d e \left(a e^2+c d^2\right)+2 b^3 d^2 e^2\right)+\frac{1}{4} d x^4 \left(4 a^2 e \left(a e^2+3 c d^2\right)+12 a b^2 d^2 e+6 a b d \left(3 a e^2+c d^2\right)+b^3 d^3\right)+\frac{1}{2} a^2 d^3 x^2 (4 a e+3 b d)+\frac{1}{8} e x^8 \left(6 c^2 d e (2 a e+3 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+4 c^3 d^3\right)+\frac{1}{3} c e^2 x^9 \left(c e (a e+4 b d)+b^2 e^2+2 c^2 d^2\right)+a d^2 x^3 \left(4 a b d e+a \left(2 a e^2+c d^2\right)+b^2 d^2\right)+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11}","\frac{c (d+e x)^9 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{3 e^7}-\frac{(d+e x)^8 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{8 e^7}+\frac{3 (d+e x)^7 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{2 e^7}+\frac{(d+e x)^5 \left(a e^2-b d e+c d^2\right)^3}{5 e^7}-\frac{3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7}",1,"a^3*d^4*x + (a^2*d^3*(3*b*d + 4*a*e)*x^2)/2 + a*d^2*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2))*x^3 + (d*(b^3*d^3 + 12*a*b^2*d^2*e + 4*a^2*e*(3*c*d^2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2))*x^4)/4 + ((4*b^3*d^3*e + 12*a*b*d*e*(2*c*d^2 + a*e^2) + 3*b^2*(c*d^4 + 6*a*d^2*e^2) + a*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + ((2*b^3*d^2*e^2 + 4*a*c*d*e*(c*d^2 + a*e^2) + 4*b^2*(c*d^3*e + a*d*e^3) + b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x^6)/2 + ((c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3*c*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^7)/7 + (e*(4*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e))*x^8)/8 + (c*e^2*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e))*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/10 + (c^3*e^4*x^11)/11","A",1
2132,1,372,272,0.1250119,"\int (d+e x)^3 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^3,x]","a^3 d^3 x+\frac{1}{4} x^4 \left(a^2 e \left(a e^2+9 c d^2\right)+9 a b^2 d^2 e+3 a b d \left(3 a e^2+2 c d^2\right)+b^3 d^3\right)+\frac{3}{2} a^2 d^2 x^2 (a e+b d)+\frac{1}{7} x^7 \left(9 c^2 d e (a e+b d)+3 b c e^2 (2 a e+3 b d)+b^3 e^3+c^3 d^3\right)+\frac{3}{8} c e x^8 \left(c e (a e+3 b d)+b^2 e^2+c^2 d^2\right)+a d x^3 \left(3 a b d e+a \left(a e^2+c d^2\right)+b^2 d^2\right)+\frac{1}{2} x^6 \left(b^2 \left(a e^3+3 c d^2 e\right)+b c d \left(6 a e^2+c d^2\right)+a c e \left(a e^2+3 c d^2\right)+b^3 d e^2\right)+\frac{3}{5} x^5 \left(b^2 \left(3 a d e^2+c d^3\right)+a b e \left(a e^2+6 c d^2\right)+a c d \left(3 a e^2+c d^2\right)+b^3 d^2 e\right)+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}","\frac{3 c (d+e x)^8 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{8 e^7}-\frac{(d+e x)^7 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{7 e^7}+\frac{(d+e x)^6 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{2 e^7}-\frac{3 (d+e x)^5 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{5 e^7}+\frac{(d+e x)^4 \left(a e^2-b d e+c d^2\right)^3}{4 e^7}-\frac{c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac{c^3 (d+e x)^{10}}{10 e^7}",1,"a^3*d^3*x + (3*a^2*d^2*(b*d + a*e)*x^2)/2 + a*d*(b^2*d^2 + 3*a*b*d*e + a*(c*d^2 + a*e^2))*x^3 + ((b^3*d^3 + 9*a*b^2*d^2*e + a^2*e*(9*c*d^2 + a*e^2) + 3*a*b*d*(2*c*d^2 + 3*a*e^2))*x^4)/4 + (3*(b^3*d^2*e + a*b*e*(6*c*d^2 + a*e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*x^5)/5 + ((b^3*d*e^2 + a*c*e*(3*c*d^2 + a*e^2) + b*c*d*(c*d^2 + 6*a*e^2) + b^2*(3*c*d^2*e + a*e^3))*x^6)/2 + ((c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(b*d + a*e) + 3*b*c*e^2*(3*b*d + 2*a*e))*x^7)/7 + (3*c*e*(c^2*d^2 + b^2*e^2 + c*e*(3*b*d + a*e))*x^8)/8 + (c^2*e^2*(c*d + b*e)*x^9)/3 + (c^3*e^3*x^10)/10","A",1
2133,1,282,272,0.0932149,"\int (d+e x)^2 \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^3,x]","a^3 d^2 x+\frac{1}{4} x^4 \left(6 a^2 c d e+6 a b^2 d e+3 a b \left(a e^2+2 c d^2\right)+b^3 d^2\right)+\frac{1}{2} a^2 d x^2 (2 a e+3 b d)+\frac{1}{7} c x^7 \left(3 c e (a e+2 b d)+3 b^2 e^2+c^2 d^2\right)+\frac{1}{3} a x^3 \left(6 a b d e+a \left(a e^2+3 c d^2\right)+3 b^2 d^2\right)+\frac{1}{6} x^6 \left(3 b c \left(2 a e^2+c d^2\right)+6 a c^2 d e+b^3 e^2+6 b^2 c d e\right)+\frac{1}{5} x^5 \left(3 b^2 \left(a e^2+c d^2\right)+12 a b c d e+3 a c \left(a e^2+c d^2\right)+2 b^3 d e\right)+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9","\frac{3 c (d+e x)^7 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{6 e^7}+\frac{3 (d+e x)^5 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7}-\frac{3 (d+e x)^4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{4 e^7}+\frac{(d+e x)^3 \left(a e^2-b d e+c d^2\right)^3}{3 e^7}-\frac{3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}",1,"a^3*d^2*x + (a^2*d*(3*b*d + 2*a*e)*x^2)/2 + (a*(3*b^2*d^2 + 6*a*b*d*e + a*(3*c*d^2 + a*e^2))*x^3)/3 + ((b^3*d^2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b*(2*c*d^2 + a*e^2))*x^4)/4 + ((2*b^3*d*e + 12*a*b*c*d*e + 3*b^2*(c*d^2 + a*e^2) + 3*a*c*(c*d^2 + a*e^2))*x^5)/5 + ((6*b^2*c*d*e + 6*a*c^2*d*e + b^3*e^2 + 3*b*c*(c*d^2 + 2*a*e^2))*x^6)/6 + (c*(c^2*d^2 + 3*b^2*e^2 + 3*c*e*(2*b*d + a*e))*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9","A",1
2134,1,161,161,0.0391998,"\int (d+e x) \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^3,x]","a^3 d x+\frac{1}{4} x^4 \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{2} c x^6 \left(a c e+b^2 e+b c d\right)+a x^3 \left(a b e+a c d+b^2 d\right)+\frac{1}{5} x^5 \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8","a^3 d x+\frac{1}{4} x^4 \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{2} c x^6 \left(a c e+b^2 e+b c d\right)+a x^3 \left(a b e+a c d+b^2 d\right)+\frac{1}{5} x^5 \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8",1,"a^3*d*x + (a^2*(3*b*d + a*e)*x^2)/2 + a*(b^2*d + a*c*d + a*b*e)*x^3 + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^4)/4 + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^5)/5 + (c*(b*c*d + b^2*e + a*c*e)*x^6)/2 + (c^2*(c*d + 3*b*e)*x^7)/7 + (c^3*e*x^8)/8","A",1
2135,1,81,81,0.012794,"\int \left(a+b x+c x^2\right)^3 \, dx","Integrate[(a + b*x + c*x^2)^3,x]","a^3 x+\frac{3}{2} a^2 b x^2+\frac{3}{5} c x^5 \left(a c+b^2\right)+\frac{1}{4} b x^4 \left(6 a c+b^2\right)+a x^3 \left(a c+b^2\right)+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7}","a^3 x+\frac{3}{2} a^2 b x^2+\frac{3}{5} c x^5 \left(a c+b^2\right)+\frac{1}{4} b x^4 \left(6 a c+b^2\right)+a x^3 \left(a c+b^2\right)+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7}",1,"a^3*x + (3*a^2*b*x^2)/2 + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^4)/4 + (3*c*(b^2 + a*c)*x^5)/5 + (b*c^2*x^6)/2 + (c^3*x^7)/7","A",1
2136,1,308,260,0.1583667,"\int \frac{\left(a+b x+c x^2\right)^3}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x),x]","\frac{e x \left(15 c e^2 \left(6 a^2 e^2 (e x-2 d)+4 a b e \left(6 d^2-3 d e x+2 e^2 x^2\right)+b^2 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+10 b e^3 \left(18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)+3 c^2 e \left(5 a e \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+b \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)\right)+c^3 \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)+60 \log (d+e x) \left(e (a e-b d)+c d^2\right)^3}{60 e^7}","\frac{3 c (d+e x)^4 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{4 e^7}-\frac{(d+e x)^3 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{3 e^7}+\frac{3 (d+e x)^2 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{2 e^7}+\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)^3}{e^7}-\frac{3 x (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^6}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7}",1,"(e*x*(c^3*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + 10*b*e^3*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x) + 4*a*b*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c^2*e*(5*a*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4))) + 60*(c*d^2 + e*(-(b*d) + a*e))^3*Log[d + e*x])/(60*e^7)","A",1
2137,1,255,256,0.0986486,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^2,x]","\frac{20 e x \left(3 c e^2 \left(a^2 e^2-4 a b d e+3 b^2 d^2\right)+b^2 e^3 (3 a e-2 b d)+3 c^2 d^2 e (3 a e-4 b d)+5 c^3 d^4\right)+10 e^2 x^2 (b e-c d) \left(c e (6 a e-5 b d)+b^2 e^2+4 c^2 d^2\right)+20 c e^3 x^3 \left(c e (a e-2 b d)+b^2 e^2+c^2 d^2\right)-\frac{20 \left(e (a e-b d)+c d^2\right)^3}{d+e x}-60 (2 c d-b e) \log (d+e x) \left(e (a e-b d)+c d^2\right)^2+5 c^2 e^4 x^4 (3 b e-2 c d)+4 c^3 e^5 x^5}{20 e^7}","\frac{c (d+e x)^3 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{2 e^7}+\frac{3 x \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^6}-\frac{\left(a e^2-b d e+c d^2\right)^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left(a e^2-b d e+c d^2\right)^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7}",1,"(20*e*(5*c^3*d^4 + 3*c^2*d^2*e*(-4*b*d + 3*a*e) + b^2*e^3*(-2*b*d + 3*a*e) + 3*c*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*x + 10*e^2*(-(c*d) + b*e)*(4*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + 6*a*e))*x^2 + 20*c*e^3*(c^2*d^2 + b^2*e^2 + c*e*(-2*b*d + a*e))*x^3 + 5*c^2*e^4*(-2*c*d + 3*b*e)*x^4 + 4*c^3*e^5*x^5 - (20*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x) - 60*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(20*e^7)","A",1
2138,1,265,255,0.126058,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^3,x]","\frac{12 \log (d+e x) \left(c e^2 \left(a^2 e^2-6 a b d e+6 b^2 d^2\right)+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right)+4 e x \left(9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right)+6 c e^2 x^2 \left(c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right)+\frac{12 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^2}{d+e x}-\frac{2 \left(e (a e-b d)+c d^2\right)^3}{(d+e x)^2}+4 c^2 e^3 x^3 (b e-c d)+c^3 e^4 x^4}{4 e^7}","-\frac{x \left(-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right)}{e^6}+\frac{3 \log (d+e x) \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}+\frac{3 c x^2 \left(-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right)}{2 e^5}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7 (d+e x)}-\frac{\left(a e^2-b d e+c d^2\right)^3}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{c^3 x^4}{4 e^3}",1,"(4*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e))*x + 6*c*e^2*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e))*x^2 + 4*c^2*e^3*(-(c*d) + b*e)*x^3 + c^3*e^4*x^4 - (2*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 + (12*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*Log[d + e*x])/(4*e^7)","A",1
2139,1,260,251,0.1041284,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^4,x]","\frac{-\frac{18 \left(c e^2 \left(a^2 e^2-6 a b d e+6 b^2 d^2\right)+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right)}{d+e x}+6 c e x \left(3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right)-6 (2 c d-b e) \log (d+e x) \left(2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right)-\frac{2 \left(e (a e-b d)+c d^2\right)^3}{(d+e x)^3}+\frac{9 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^2}{(d+e x)^2}+3 c^2 e^2 x^2 (3 b e-4 c d)+2 c^3 e^3 x^3}{6 e^7}","-\frac{3 \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right) \left(a e^2-b d e+c d^2\right)}{e^7 (d+e x)}-\frac{(2 c d-b e) \log (d+e x) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{e^7}+\frac{c x \left(-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right)}{e^6}-\frac{\left(a e^2-b d e+c d^2\right)^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{2 e^7 (d+e x)^2}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac{c^3 x^3}{3 e^4}",1,"(6*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e))*x + 3*c^2*e^2*(-4*c*d + 3*b*e)*x^2 + 2*c^3*e^3*x^3 - (2*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 + (9*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (18*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))/(d + e*x) - 6*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*Log[d + e*x])/(6*e^7)","A",1
2140,1,402,251,0.1748175,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^5,x]","\frac{-c e^2 \left(a^2 e^2 \left(d^2+4 d e x+6 e^2 x^2\right)+6 a b e \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)+b^2 (-d) \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)\right)-e^3 \left(a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left(d^2+4 d e x+6 e^2 x^2\right)+b^3 \left(d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right)\right)+12 c (d+e x)^4 \log (d+e x) \left(c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right)+c^2 e \left(a d e \left(25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right)-b \left(77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right)\right)+c^3 \left(57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right)}{4 e^7 (d+e x)^4}","\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{e^7 (d+e x)}-\frac{3 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7 (d+e x)^3}-\frac{\left(a e^2-b d e+c d^2\right)^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5}",1,"(c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - e^3*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) - c*e^2*(a^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*a*b*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - b^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + c^2*e*(a*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - b*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + 12*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)","A",1
2141,1,396,256,0.2514921,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^6} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^6,x]","-\frac{2 c e^2 \left(a^2 e^2 \left(d^2+5 d e x+10 e^2 x^2\right)+3 a b e \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)+6 b^2 \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)\right)+e^3 \left(4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left(d^2+5 d e x+10 e^2 x^2\right)+b^3 \left(d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right)\right)+c^2 e \left(12 a e \left(d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right)-b d \left(137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right)\right)+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left(87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right)}{20 e^7 (d+e x)^5}","-\frac{3 c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 (d+e x)}+\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{2 e^7 (d+e x)^2}-\frac{\left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{4 e^7 (d+e x)^4}-\frac{\left(a e^2-b d e+c d^2\right)^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6}",1,"-1/20*(2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) + 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(12*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)","A",1
2142,1,385,266,0.1631079,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^7} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^7,x]","\frac{-3 c e^2 \left(a^2 e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+2 a b e \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+2 b^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)\right)-e^3 \left(10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left(d^2+6 d e x+15 e^2 x^2\right)+b^3 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)\right)-6 c^2 e \left(a e \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)+5 b \left(d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right)\right)+c^3 d \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}","-\frac{3 c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{3 e^7 (d+e x)^3}-\frac{3 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{4 e^7 (d+e x)^4}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{5 e^7 (d+e x)^5}-\frac{\left(a e^2-b d e+c d^2\right)^3}{6 e^7 (d+e x)^6}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}",1,"(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) - 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) - 6*c^2*e*(a*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)","A",1
2143,1,377,268,0.1396196,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^8} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^8,x]","-\frac{2 c e^2 \left(2 a^2 e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+3 a b e \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+2 b^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)\right)+e^3 \left(20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left(d^2+7 d e x+21 e^2 x^2\right)+b^3 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)\right)+2 c^2 e \left(2 a e \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+5 b \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)\right)+20 c^3 \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)}{140 e^7 (d+e x)^7}","-\frac{c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{4 e^7 (d+e x)^4}-\frac{3 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{2 e^7 (d+e x)^6}-\frac{\left(a e^2-b d e+c d^2\right)^3}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)}",1,"-1/140*(20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) + e^3*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4*a*b^2*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + b^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 2*c*e^2*(2*a^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*b^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + 2*c^2*e*(2*a*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*b*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^7*(d + e*x)^7)","A",1
2144,1,375,269,0.1497641,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^9} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^9,x]","-\frac{c e^2 \left(5 a^2 e^2 \left(d^2+8 d e x+28 e^2 x^2\right)+6 a b e \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+3 b^2 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)\right)+e^3 \left(35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left(d^2+8 d e x+28 e^2 x^2\right)+b^3 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)\right)+c^2 e \left(3 a e \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+5 b \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)\right)+5 c^3 \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)}{280 e^7 (d+e x)^8}","-\frac{3 c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{5 e^7 (d+e x)^5}-\frac{\left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{2 e^7 (d+e x)^6}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{7 e^7 (d+e x)^7}-\frac{\left(a e^2-b d e+c d^2\right)^3}{8 e^7 (d+e x)^8}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{c^3}{2 e^7 (d+e x)^2}",1,"-1/280*(5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6) + e^3*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + c*e^2*(5*a^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a*b*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c^2*e*(3*a*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^7*(d + e*x)^8)","A",1
2145,1,378,272,0.1467503,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^10,x]","-\frac{6 c e^2 \left(5 a^2 e^2 \left(d^2+9 d e x+36 e^2 x^2\right)+5 a b e \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+2 b^2 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)\right)+5 e^3 \left(56 a^3 e^3+21 a^2 b e^2 (d+9 e x)+6 a b^2 e \left(d^2+9 d e x+36 e^2 x^2\right)+b^3 \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)\right)+3 c^2 e \left(4 a e \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+5 b \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)\right)+10 c^3 \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)}{2520 e^7 (d+e x)^9}","-\frac{3 c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{6 e^7 (d+e x)^6}-\frac{3 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{7 e^7 (d+e x)^7}+\frac{3 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{8 e^7 (d+e x)^8}-\frac{\left(a e^2-b d e+c d^2\right)^3}{9 e^7 (d+e x)^9}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{c^3}{3 e^7 (d+e x)^3}",1,"-1/2520*(10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 5*e^3*(56*a^3*e^3 + 21*a^2*b*e^2*(d + 9*e*x) + 6*a*b^2*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + b^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 6*c*e^2*(5*a^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 2*b^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 3*c^2*e*(4*a*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)))/(e^7*(d + e*x)^9)","A",1
2146,1,766,443,0.2481501,"\int (d+e x)^4 \left(a+b x+c x^2\right)^4 \, dx","Integrate[(d + e*x)^4*(a + b*x + c*x^2)^4,x]","a^4 d^4 x+2 a^3 d^3 x^2 (a e+b d)+\frac{2}{3} a^2 d^2 x^3 \left(8 a b d e+a \left(3 a e^2+2 c d^2\right)+3 b^2 d^2\right)+\frac{1}{9} x^9 \left(6 c^2 e^2 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)+4 b^2 c e^3 (3 a e+4 b d)+8 c^3 d^2 e (3 a e+2 b d)+b^4 e^4+c^4 d^4\right)+a d x^4 \left(a^2 e \left(a e^2+4 c d^2\right)+6 a b^2 d^2 e+3 a b d \left(2 a e^2+c d^2\right)+b^3 d^3\right)+\frac{1}{2} x^8 \left(b c \left(3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right)+b^3 \left(a e^4+6 c d^2 e^2\right)+6 b^2 c d e \left(2 a e^2+c d^2\right)+2 a c^2 d e \left(3 a e^2+2 c d^2\right)+b^4 d e^3\right)+\frac{2}{3} x^6 \left(a b \left(a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right)+2 a^2 c d e \left(2 a e^2+3 c d^2\right)+b^3 \left(6 a d^2 e^2+c d^4\right)+6 a b^2 d e \left(a e^2+2 c d^2\right)+b^4 d^3 e\right)+\frac{1}{5} x^5 \left(16 a^2 b d e \left(a e^2+3 c d^2\right)+a^2 \left(a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right)+16 a b^3 d^3 e+12 a b^2 d^2 \left(3 a e^2+c d^2\right)+b^4 d^4\right)+\frac{2}{7} x^7 \left(3 b^2 \left(a^2 e^4+12 a c d^2 e^2+c^2 d^4\right)+2 a c \left(a^2 e^4+9 a c d^2 e^2+c^2 d^4\right)+8 b^3 \left(a d e^3+c d^3 e\right)+24 a b c d e \left(a e^2+c d^2\right)+3 b^4 d^2 e^2\right)+\frac{2}{5} c e x^{10} \left(2 c^2 d e (2 a e+3 b d)+3 b c e^2 (a e+2 b d)+b^3 e^3+c^3 d^3\right)+\frac{2}{11} c^2 e^2 x^{11} \left(2 c e (a e+4 b d)+3 b^2 e^2+3 c^2 d^2\right)+\frac{1}{3} c^3 e^3 x^{12} (b e+c d)+\frac{1}{13} c^4 e^4 x^{13}","\frac{(d+e x)^9 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{9 e^9}+\frac{2 c^2 (d+e x)^{11} \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{11 e^9}-\frac{2 c (d+e x)^{10} (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{5 e^9}-\frac{(d+e x)^8 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{2 e^9}+\frac{2 (d+e x)^7 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{7 e^9}-\frac{2 (d+e x)^6 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{3 e^9}+\frac{(d+e x)^5 \left(a e^2-b d e+c d^2\right)^4}{5 e^9}-\frac{c^3 (d+e x)^{12} (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^{13}}{13 e^9}",1,"a^4*d^4*x + 2*a^3*d^3*(b*d + a*e)*x^2 + (2*a^2*d^2*(3*b^2*d^2 + 8*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^3)/3 + a*d*(b^3*d^3 + 6*a*b^2*d^2*e + a^2*e*(4*c*d^2 + a*e^2) + 3*a*b*d*(c*d^2 + 2*a*e^2))*x^4 + ((b^4*d^4 + 16*a*b^3*d^3*e + 16*a^2*b*d*e*(3*c*d^2 + a*e^2) + 12*a*b^2*d^2*(c*d^2 + 3*a*e^2) + a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + (2*(b^4*d^3*e + 6*a*b^2*d*e*(2*c*d^2 + a*e^2) + 2*a^2*c*d*e*(3*c*d^2 + 2*a*e^2) + b^3*(c*d^4 + 6*a*d^2*e^2) + a*b*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^6)/3 + (2*(3*b^4*d^2*e^2 + 24*a*b*c*d*e*(c*d^2 + a*e^2) + 8*b^3*(c*d^3*e + a*d*e^3) + 2*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4) + 3*b^2*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x^7)/7 + ((b^4*d*e^3 + 6*b^2*c*d*e*(c*d^2 + 2*a*e^2) + 2*a*c^2*d*e*(2*c*d^2 + 3*a*e^2) + b^3*(6*c*d^2*e^2 + a*e^4) + b*c*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^8)/2 + ((c^4*d^4 + b^4*e^4 + 8*c^3*d^2*e*(2*b*d + 3*a*e) + 4*b^2*c*e^3*(4*b*d + 3*a*e) + 6*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^9)/9 + (2*c*e*(c^3*d^3 + b^3*e^3 + 3*b*c*e^2*(2*b*d + a*e) + 2*c^2*d*e*(3*b*d + 2*a*e))*x^10)/5 + (2*c^2*e^2*(3*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(4*b*d + a*e))*x^11)/11 + (c^3*e^3*(c*d + b*e)*x^12)/3 + (c^4*e^4*x^13)/13","A",1
2147,1,611,443,0.1919999,"\int (d+e x)^3 \left(a+b x+c x^2\right)^4 \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^4,x]","a^4 d^3 x+\frac{1}{2} a^3 d^2 x^2 (3 a e+4 b d)+\frac{1}{3} a^2 d x^3 \left(12 a b d e+a \left(3 a e^2+4 c d^2\right)+6 b^2 d^2\right)+\frac{1}{4} a x^4 \left(a^2 e \left(a e^2+12 c d^2\right)+18 a b^2 d^2 e+12 a b d \left(a e^2+c d^2\right)+4 b^3 d^3\right)+\frac{1}{6} x^6 \left(2 a^2 c e \left(2 a e^2+9 c d^2\right)+4 b^3 \left(3 a d e^2+c d^3\right)+6 a b^2 e \left(a e^2+6 c d^2\right)+12 a b c d \left(3 a e^2+c d^2\right)+3 b^4 d^2 e\right)+\frac{1}{5} x^5 \left(4 a^2 b e \left(a e^2+9 c d^2\right)+6 a^2 c d \left(2 a e^2+c d^2\right)+12 a b^3 d^2 e+6 a b^2 d \left(3 a e^2+2 c d^2\right)+b^4 d^3\right)+\frac{1}{9} c x^9 \left(12 c^2 d e (a e+b d)+6 b c e^2 (2 a e+3 b d)+4 b^3 e^3+c^3 d^3\right)+\frac{1}{10} c^2 e x^{10} \left(4 c e (a e+3 b d)+6 b^2 e^2+3 c^2 d^2\right)+\frac{1}{8} x^8 \left(6 b^2 c e \left(2 a e^2+3 c d^2\right)+4 b c^2 d \left(9 a e^2+c d^2\right)+6 a c^2 e \left(a e^2+2 c d^2\right)+b^4 e^3+12 b^3 c d e^2\right)+\frac{1}{7} x^7 \left(4 b^3 \left(a e^3+3 c d^2 e\right)+6 b^2 c d \left(6 a e^2+c d^2\right)+12 a b c e \left(a e^2+3 c d^2\right)+2 a c^2 d \left(9 a e^2+2 c d^2\right)+3 b^4 d e^2\right)+\frac{1}{11} c^3 e^2 x^{11} (4 b e+3 c d)+\frac{1}{12} c^4 e^3 x^{12}","\frac{(d+e x)^8 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{8 e^9}+\frac{c^2 (d+e x)^{10} \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{5 e^9}-\frac{4 c (d+e x)^9 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{9 e^9}-\frac{4 (d+e x)^7 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{7 e^9}+\frac{(d+e x)^6 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{3 e^9}-\frac{4 (d+e x)^5 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{5 e^9}+\frac{(d+e x)^4 \left(a e^2-b d e+c d^2\right)^4}{4 e^9}-\frac{4 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^9}+\frac{c^4 (d+e x)^{12}}{12 e^9}",1,"a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + (a^2*d*(6*b^2*d^2 + 12*a*b*d*e + a*(4*c*d^2 + 3*a*e^2))*x^3)/3 + (a*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a*b*d*(c*d^2 + a*e^2) + a^2*e*(12*c*d^2 + a*e^2))*x^4)/4 + ((b^4*d^3 + 12*a*b^3*d^2*e + 4*a^2*b*e*(9*c*d^2 + a*e^2) + 6*a^2*c*d*(c*d^2 + 2*a*e^2) + 6*a*b^2*d*(2*c*d^2 + 3*a*e^2))*x^5)/5 + ((3*b^4*d^2*e + 6*a*b^2*e*(6*c*d^2 + a*e^2) + 2*a^2*c*e*(9*c*d^2 + 2*a*e^2) + 12*a*b*c*d*(c*d^2 + 3*a*e^2) + 4*b^3*(c*d^3 + 3*a*d*e^2))*x^6)/6 + ((3*b^4*d*e^2 + 12*a*b*c*e*(3*c*d^2 + a*e^2) + 6*b^2*c*d*(c*d^2 + 6*a*e^2) + 2*a*c^2*d*(2*c*d^2 + 9*a*e^2) + 4*b^3*(3*c*d^2*e + a*e^3))*x^7)/7 + ((12*b^3*c*d*e^2 + b^4*e^3 + 6*a*c^2*e*(2*c*d^2 + a*e^2) + 6*b^2*c*e*(3*c*d^2 + 2*a*e^2) + 4*b*c^2*d*(c*d^2 + 9*a*e^2))*x^8)/8 + (c*(c^3*d^3 + 4*b^3*e^3 + 12*c^2*d*e*(b*d + a*e) + 6*b*c*e^2*(3*b*d + 2*a*e))*x^9)/9 + (c^2*e*(3*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(3*b*d + a*e))*x^10)/10 + (c^3*e^2*(3*c*d + 4*b*e)*x^11)/11 + (c^4*e^3*x^12)/12","A",1
2148,1,428,441,0.1410658,"\int (d+e x)^2 \left(a+b x+c x^2\right)^4 \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^4,x]","a^4 d^2 x+a^3 d x^2 (a e+2 b d)+\frac{1}{3} a^2 x^3 \left(8 a b d e+a \left(a e^2+4 c d^2\right)+6 b^2 d^2\right)+a x^4 \left(2 a^2 c d e+3 a b^2 d e+a b \left(a e^2+3 c d^2\right)+b^3 d^2\right)+\frac{1}{3} x^6 \left(6 a^2 c^2 d e+2 b^3 \left(a e^2+c d^2\right)+12 a b^2 c d e+6 a b c \left(a e^2+c d^2\right)+b^4 d e\right)+\frac{1}{5} x^5 \left(24 a^2 b c d e+2 a^2 c \left(2 a e^2+3 c d^2\right)+8 a b^3 d e+6 a b^2 \left(a e^2+2 c d^2\right)+b^4 d^2\right)+\frac{1}{9} c^2 x^9 \left(4 c e (a e+2 b d)+6 b^2 e^2+c^2 d^2\right)+\frac{1}{2} c x^8 \left(b c \left(3 a e^2+c d^2\right)+2 a c^2 d e+b^3 e^2+3 b^2 c d e\right)+\frac{1}{7} x^7 \left(6 b^2 c \left(2 a e^2+c d^2\right)+24 a b c^2 d e+2 a c^2 \left(3 a e^2+2 c d^2\right)+b^4 e^2+8 b^3 c d e\right)+\frac{1}{5} c^3 e x^{10} (2 b e+c d)+\frac{1}{11} c^4 e^2 x^{11}","\frac{(d+e x)^7 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{7 e^9}+\frac{2 c^2 (d+e x)^9 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{9 e^9}-\frac{c (d+e x)^8 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{2 e^9}-\frac{2 (d+e x)^6 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9}+\frac{2 (d+e x)^5 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{5 e^9}-\frac{(d+e x)^4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{e^9}+\frac{(d+e x)^3 \left(a e^2-b d e+c d^2\right)^4}{3 e^9}-\frac{2 c^3 (d+e x)^{10} (2 c d-b e)}{5 e^9}+\frac{c^4 (d+e x)^{11}}{11 e^9}",1,"a^4*d^2*x + a^3*d*(2*b*d + a*e)*x^2 + (a^2*(6*b^2*d^2 + 8*a*b*d*e + a*(4*c*d^2 + a*e^2))*x^3)/3 + a*(b^3*d^2 + 3*a*b^2*d*e + 2*a^2*c*d*e + a*b*(3*c*d^2 + a*e^2))*x^4 + ((b^4*d^2 + 8*a*b^3*d*e + 24*a^2*b*c*d*e + 6*a*b^2*(2*c*d^2 + a*e^2) + 2*a^2*c*(3*c*d^2 + 2*a*e^2))*x^5)/5 + ((b^4*d*e + 12*a*b^2*c*d*e + 6*a^2*c^2*d*e + 2*b^3*(c*d^2 + a*e^2) + 6*a*b*c*(c*d^2 + a*e^2))*x^6)/3 + ((8*b^3*c*d*e + 24*a*b*c^2*d*e + b^4*e^2 + 6*b^2*c*(c*d^2 + 2*a*e^2) + 2*a*c^2*(2*c*d^2 + 3*a*e^2))*x^7)/7 + (c*(3*b^2*c*d*e + 2*a*c^2*d*e + b^3*e^2 + b*c*(c*d^2 + 3*a*e^2))*x^8)/2 + (c^2*(c^2*d^2 + 6*b^2*e^2 + 4*c*e*(2*b*d + a*e))*x^9)/9 + (c^3*e*(c*d + 2*b*e)*x^10)/5 + (c^4*e^2*x^11)/11","A",1
2149,1,268,268,0.0674945,"\int (d+e x) \left(a+b x+c x^2\right)^4 \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^4,x]","a^4 d x+\frac{1}{2} a^3 x^2 (a e+4 b d)+\frac{2}{3} a^2 x^3 \left(2 a b e+2 a c d+3 b^2 d\right)+\frac{1}{2} a x^4 \left(2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right)+\frac{1}{6} x^6 \left(6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+b^4 e+4 b^3 c d\right)+\frac{1}{5} x^5 \left(12 a^2 b c e+6 a^2 c^2 d+4 a b^3 e+12 a b^2 c d+b^4 d\right)+\frac{1}{4} c^2 x^8 \left(2 a c e+3 b^2 e+2 b c d\right)+\frac{2}{7} c x^7 \left(6 a b c e+2 a c^2 d+2 b^3 e+3 b^2 c d\right)+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10}","a^4 d x+\frac{1}{2} a^3 x^2 (a e+4 b d)+\frac{2}{3} a^2 x^3 \left(2 a b e+2 a c d+3 b^2 d\right)+\frac{1}{2} a x^4 \left(2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right)+\frac{1}{6} x^6 \left(6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+b^4 e+4 b^3 c d\right)+\frac{1}{5} x^5 \left(12 a^2 b c e+6 a^2 c^2 d+4 a b^3 e+12 a b^2 c d+b^4 d\right)+\frac{1}{4} c^2 x^8 \left(2 a c e+3 b^2 e+2 b c d\right)+\frac{2}{7} c x^7 \left(6 a b c e+2 a c^2 d+2 b^3 e+3 b^2 c d\right)+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10}",1,"a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d + 3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e + 6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10","A",1
2150,1,133,133,0.020826,"\int \left(a+b x+c x^2\right)^4 \, dx","Integrate[(a + b*x + c*x^2)^4,x]","a^4 x+2 a^3 b x^2+\frac{2}{3} a^2 x^3 \left(2 a c+3 b^2\right)+\frac{1}{5} x^5 \left(6 a^2 c^2+12 a b^2 c+b^4\right)+\frac{2}{7} c^2 x^7 \left(2 a c+3 b^2\right)+\frac{2}{3} b c x^6 \left(3 a c+b^2\right)+a b x^4 \left(3 a c+b^2\right)+\frac{1}{2} b c^3 x^8+\frac{c^4 x^9}{9}","a^4 x+2 a^3 b x^2+\frac{2}{3} a^2 x^3 \left(2 a c+3 b^2\right)+\frac{1}{5} x^5 \left(6 a^2 c^2+12 a b^2 c+b^4\right)+\frac{2}{7} c^2 x^7 \left(2 a c+3 b^2\right)+\frac{2}{3} b c x^6 \left(3 a c+b^2\right)+a b x^4 \left(3 a c+b^2\right)+\frac{1}{2} b c^3 x^8+\frac{c^4 x^9}{9}",1,"a^4*x + 2*a^3*b*x^2 + (2*a^2*(3*b^2 + 2*a*c)*x^3)/3 + a*b*(b^2 + 3*a*c)*x^4 + ((b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^5)/5 + (2*b*c*(b^2 + 3*a*c)*x^6)/3 + (2*c^2*(3*b^2 + 2*a*c)*x^7)/7 + (b*c^3*x^8)/2 + (c^4*x^9)/9","A",1
2151,1,616,428,0.3929309,"\int \frac{\left(a+b x+c x^2\right)^4}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x),x]","\frac{x \left(84 c^2 e^2 \left(5 a^2 e^2 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+2 a b e \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)+b^2 \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)+56 c e^3 \left(30 a^3 e^3 (e x-2 d)+30 a^2 b e^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)+15 a b^2 e \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+b^3 \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)\right)+70 b e^4 \left(48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left(6 d^2-3 d e x+2 e^2 x^2\right)+b^3 \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+8 c^3 e \left(7 a e \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)+b \left(420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right)\right)+c^4 \left(-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right)\right)}{840 e^8}+\frac{\log (d+e x) \left(e (a e-b d)+c d^2\right)^4}{e^9}","\frac{(d+e x)^4 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{4 e^9}+\frac{c^2 (d+e x)^6 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{3 e^9}-\frac{4 c (d+e x)^5 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{5 e^9}-\frac{4 (d+e x)^3 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9}+\frac{(d+e x)^2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9}+\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)^4}{e^9}-\frac{4 x (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{e^8}-\frac{4 c^3 (d+e x)^7 (2 c d-b e)}{7 e^9}+\frac{c^4 (d+e x)^8}{8 e^9}",1,"(x*(c^4*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + 70*b*e^4*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 56*c*e^3*(30*a^3*e^3*(-2*d + e*x) + 30*a^2*b*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 15*a*b^2*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^3*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 84*c^2*e^2*(5*a^2*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*a*b*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^2*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 8*c^3*e*(7*a*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + b*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6))))/(840*e^8) + ((c*d^2 + e*(-(b*d) + a*e))^4*Log[d + e*x])/e^9","A",1
2152,1,780,426,0.3482206,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^2,x]","\frac{21 c^2 e^2 \left(10 a^2 e^2 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+5 a b e \left(12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right)+b^2 \left(-30 d^6+150 d^5 e x+90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4-9 d e^5 x^5+6 e^6 x^6\right)\right)+35 c e^3 \left(12 a^3 e^3 \left(-d^2+d e x+e^2 x^2\right)+18 a^2 b e^2 \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)+12 a b^2 e \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+b^3 \left(12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right)\right)+35 e^4 \left(-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left(-d^2+d e x+e^2 x^2\right)+6 a b^3 e \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)+b^4 \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)\right)+7 c^3 e \left(6 a e \left(-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right)+b \left(60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right)\right)-420 (d+e x) (2 c d-b e) \log (d+e x) \left(e (a e-b d)+c d^2\right)^3+c^4 \left(-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right)}{105 e^9 (d+e x)}","\frac{(d+e x)^3 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{3 e^9}+\frac{2 c^2 (d+e x)^5 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{5 e^9}-\frac{c (d+e x)^4 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9}-\frac{2 (d+e x)^2 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9}+\frac{2 x \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^8}-\frac{\left(a e^2-b d e+c d^2\right)^4}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left(a e^2-b d e+c d^2\right)^3}{e^9}-\frac{2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^7}{7 e^9}",1,"(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 35*e^4*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 35*c*e^3*(12*a^3*e^3*(-d^2 + d*e*x + e^2*x^2) + 18*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 12*a*b^2*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^3*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 21*c^2*e^2*(10*a^2*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^2*(-30*d^6 + 150*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*d*e^5*x^5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)*Log[d + e*x])/(105*e^9*(d + e*x))","A",1
2153,1,440,430,0.1995379,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^3,x]","\frac{15 e^2 x^2 \left(6 c^2 e^2 \left(a^2 e^2-6 a b d e+6 b^2 d^2\right)-12 b^2 c e^3 (b d-a e)-8 c^3 d^2 e (5 b d-3 a e)+b^4 e^4+15 c^4 d^4\right)+30 e x \left(-6 c^2 d e^2 \left(3 a^2 e^2-12 a b d e+10 b^2 d^2\right)+12 b c e^3 \left(a^2 e^2-3 a b d e+2 b^2 d^2\right)+b^3 e^4 (4 a e-3 b d)+20 c^3 d^3 e (3 b d-2 a e)-21 c^4 d^5\right)+60 \log (d+e x) \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^2+15 c^2 e^4 x^4 \left(2 c e (a e-3 b d)+3 b^2 e^2+3 c^2 d^2\right)+20 c e^3 x^3 (b e-c d) \left(c e (6 a e-7 b d)+2 b^2 e^2+5 c^2 d^2\right)+\frac{120 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^3}{d+e x}-\frac{15 \left(e (a e-b d)+c d^2\right)^4}{(d+e x)^2}+6 c^3 e^5 x^5 (4 b e-3 c d)+5 c^4 e^6 x^6}{30 e^9}","\frac{(d+e x)^2 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{2 e^9}+\frac{c^2 (d+e x)^4 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{2 e^9}-\frac{4 c (d+e x)^3 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9}+\frac{2 \log (d+e x) \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9}-\frac{4 x (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^8}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{e^9 (d+e x)}-\frac{\left(a e^2-b d e+c d^2\right)^4}{2 e^9 (d+e x)^2}-\frac{4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac{c^4 (d+e x)^6}{6 e^9}",1,"(30*e*(-21*c^4*d^5 + 20*c^3*d^3*e*(3*b*d - 2*a*e) + b^3*e^4*(-3*b*d + 4*a*e) + 12*b*c*e^3*(2*b^2*d^2 - 3*a*b*d*e + a^2*e^2) - 6*c^2*d*e^2*(10*b^2*d^2 - 12*a*b*d*e + 3*a^2*e^2))*x + 15*e^2*(15*c^4*d^4 + b^4*e^4 - 8*c^3*d^2*e*(5*b*d - 3*a*e) - 12*b^2*c*e^3*(b*d - a*e) + 6*c^2*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*x^2 + 20*c*e^3*(-(c*d) + b*e)*(5*c^2*d^2 + 2*b^2*e^2 + c*e*(-7*b*d + 6*a*e))*x^3 + 15*c^2*e^4*(3*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-3*b*d + a*e))*x^4 + 6*c^3*e^5*(-3*c*d + 4*b*e)*x^5 + 5*c^4*e^6*x^6 - (15*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^2 + (120*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x) + 60*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(30*e^9)","A",1
2154,1,425,417,0.2156497,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^4,x]","\frac{-60 (2 c d-b e) \log (d+e x) \left(c e^2 \left(3 a^2 e^2-10 a b d e+8 b^2 d^2\right)+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right)+15 e x \left(6 c^2 e^2 \left(a^2 e^2-8 a b d e+10 b^2 d^2\right)-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (a e-2 b d)+b^4 e^4+35 c^4 d^4\right)+30 c e^2 x^2 \left(2 c^2 d e (5 b d-2 a e)+3 b c e^2 (a e-2 b d)+b^3 e^3-5 c^3 d^3\right)-\frac{30 \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^2}{d+e x}+10 c^2 e^3 x^3 \left(2 c e (a e-4 b d)+3 b^2 e^2+5 c^2 d^2\right)+\frac{30 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^3}{(d+e x)^2}-\frac{5 \left(e (a e-b d)+c d^2\right)^4}{(d+e x)^3}+15 c^3 e^4 x^4 (b e-c d)+3 c^4 e^5 x^5}{15 e^9}","\frac{x \left(6 c^2 e^2 \left(a^2 e^2-8 a b d e+10 b^2 d^2\right)-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right)}{e^8}-\frac{2 c x^2 \left(-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right)}{e^7}-\frac{2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9}+\frac{2 c^2 x^3 \left(-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right)}{3 e^6}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{e^9 (d+e x)^2}-\frac{\left(a e^2-b d e+c d^2\right)^4}{3 e^9 (d+e x)^3}-\frac{c^3 x^4 (c d-b e)}{e^5}+\frac{c^4 x^5}{5 e^4}",1,"(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d + a*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (30*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) - 60*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2))*Log[d + e*x])/(15*e^9)","A",1
2155,1,430,426,0.1955915,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^5,x]","\frac{\frac{48 (2 c d-b e) \left(c e^2 \left(3 a^2 e^2-10 a b d e+8 b^2 d^2\right)+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right)}{d+e x}+12 \log (d+e x) \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)+12 c e x \left(20 c^2 d e (3 b d-a e)-6 b c e^2 (5 b d-2 a e)+4 b^3 e^3-35 c^3 d^3\right)+6 c^2 e^2 x^2 \left(4 c e (a e-5 b d)+6 b^2 e^2+15 c^2 d^2\right)-\frac{12 \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^2}{(d+e x)^2}-\frac{3 \left(e (a e-b d)+c d^2\right)^4}{(d+e x)^4}+\frac{16 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^3}{(d+e x)^3}+4 c^3 e^3 x^3 (4 b e-5 c d)+3 c^4 e^4 x^4}{12 e^9}","\frac{\log (d+e x) \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{e^9}-\frac{c x \left(-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right)}{e^8}-\frac{\left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right) \left(a e^2-b d e+c d^2\right)^2}{e^9 (d+e x)^2}+\frac{4 (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right) \left(a e^2-b d e+c d^2\right)}{e^9 (d+e x)}+\frac{c^2 x^2 \left(-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right)}{2 e^7}-\frac{\left(a e^2-b d e+c d^2\right)^4}{4 e^9 (d+e x)^4}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{3 e^9 (d+e x)^3}-\frac{c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac{c^4 x^4}{4 e^5}",1,"(12*c*e*(-35*c^3*d^3 + 4*b^3*e^3 - 6*b*c*e^2*(5*b*d - 2*a*e) + 20*c^2*d*e*(3*b*d - a*e))*x + 6*c^2*e^2*(15*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-5*b*d + a*e))*x^2 + 4*c^3*e^3*(-5*c*d + 4*b*e)*x^3 + 3*c^4*e^4*x^4 - (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^4 + (16*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (12*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (48*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x) + 12*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*Log[d + e*x])/(12*e^9)","A",1
2156,1,419,414,0.2156427,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^6} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^6,x]","\frac{\frac{30 (2 c d-b e) \left(c e^2 \left(3 a^2 e^2-10 a b d e+8 b^2 d^2\right)+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right)}{(d+e x)^2}-\frac{15 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{d+e x}+15 c^2 e x \left(4 c e (a e-6 b d)+6 b^2 e^2+21 c^2 d^2\right)-\frac{10 \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^2}{(d+e x)^3}-60 c (2 c d-b e) \log (d+e x) \left(c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right)+\frac{15 (2 c d-b e) \left(e (a e-b d)+c d^2\right)^3}{(d+e x)^4}-\frac{3 \left(e (a e-b d)+c d^2\right)^4}{(d+e x)^5}+15 c^3 e^2 x^2 (2 b e-3 c d)+5 c^4 e^3 x^3}{15 e^9}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (d+e x)^2}-\frac{2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{3 e^9 (d+e x)^3}-\frac{4 c (2 c d-b e) \log (d+e x) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9}+\frac{c^2 x \left(-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right)}{e^8}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{e^9 (d+e x)^4}-\frac{\left(a e^2-b d e+c d^2\right)^4}{5 e^9 (d+e x)^5}-\frac{c^3 x^2 (3 c d-2 b e)}{e^7}+\frac{c^4 x^3}{3 e^6}",1,"(15*c^2*e*(21*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-6*b*d + a*e))*x + 15*c^3*e^2*(-3*c*d + 2*b*e)*x^2 + 5*c^4*e^3*x^3 - (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^5 + (15*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 - (10*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^3 + (30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x)^2 - (15*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2)))/(d + e*x) - 60*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*Log[d + e*x])/(15*e^9)","A",1
2157,1,764,426,0.340989,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^7} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^7,x]","\frac{-3 c^2 e^2 \left(2 a^2 e^2 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)+20 a b e \left(d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right)+b^2 (-d) \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)\right)-2 c e^3 \left(a^3 e^3 \left(d^2+6 d e x+15 e^2 x^2\right)+3 a^2 b e^2 \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+6 a b^2 e \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)+10 b^3 \left(d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right)\right)-e^4 \left(5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left(d^2+6 d e x+15 e^2 x^2\right)+2 a b^3 e \left(d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right)+b^4 \left(d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right)\right)+60 c^2 (d+e x)^6 \log (d+e x) \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right)+2 c^3 e \left(a d e \left(147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right)-b \left(669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right)\right)+c^4 \left(1023 d^8+5298 d^7 e x+10725 d^6 e^2 x^2+10100 d^5 e^3 x^3+3375 d^4 e^4 x^4-1170 d^3 e^5 x^5-1035 d^2 e^6 x^6-120 d e^7 x^7+15 e^8 x^8\right)}{30 e^9 (d+e x)^6}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{2 e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (d+e x)}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9 (d+e x)^3}-\frac{\left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{2 e^9 (d+e x)^4}+\frac{2 c^2 \log (d+e x) \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{5 e^9 (d+e x)^5}-\frac{\left(a e^2-b d e+c d^2\right)^4}{6 e^9 (d+e x)^6}-\frac{c^3 x (7 c d-4 b e)}{e^8}+\frac{c^4 x^2}{2 e^7}",1,"(c^4*(1023*d^8 + 5298*d^7*e*x + 10725*d^6*e^2*x^2 + 10100*d^5*e^3*x^3 + 3375*d^4*e^4*x^4 - 1170*d^3*e^5*x^5 - 1035*d^2*e^6*x^6 - 120*d*e^7*x^7 + 15*e^8*x^8) - e^4*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) - 2*c*e^3*(a^3*e^3*(d^2 + 6*d*e*x + 15*e^2*x^2) + 3*a^2*b*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 6*a*b^2*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 10*b^3*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) - 3*c^2*e^2*(2*a^2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 20*a*b*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - b^2*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + 2*c^3*e*(a*d*e*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - b*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*c^2*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(d + e*x)^6*Log[d + e*x])/(30*e^9*(d + e*x)^6)","A",1
2158,1,748,424,0.5129153,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^8} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^8,x]","-\frac{6 c^2 e^2 \left(a^2 e^2 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+5 a b e \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)+15 b^2 \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)\right)+c e^3 \left(4 a^3 e^3 \left(d^2+7 d e x+21 e^2 x^2\right)+9 a^2 b e^2 \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+12 a b^2 e \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)+10 b^3 \left(d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right)\right)+e^4 \left(15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left(d^2+7 d e x+21 e^2 x^2\right)+3 a b^3 e \left(d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right)+b^4 \left(d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right)\right)+c^3 e \left(60 a e \left(d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right)-b d \left(1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right)\right)+420 c^3 (d+e x)^7 (2 c d-b e) \log (d+e x)+c^4 \left(1443 d^8+9261 d^7 e x+24843 d^6 e^2 x^2+35525 d^5 e^3 x^3+28175 d^4 e^4 x^4+11025 d^3 e^5 x^5+735 d^2 e^6 x^6-735 d e^7 x^7-105 e^8 x^8\right)}{105 e^9 (d+e x)^7}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac{2 c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9 (d+e x)}+\frac{2 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (d+e x)^2}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (d+e x)^4}-\frac{2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{5 e^9 (d+e x)^5}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{3 e^9 (d+e x)^6}-\frac{\left(a e^2-b d e+c d^2\right)^4}{7 e^9 (d+e x)^7}-\frac{4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac{c^4 x}{e^8}",1,"-1/105*(c^4*(1443*d^8 + 9261*d^7*e*x + 24843*d^6*e^2*x^2 + 35525*d^5*e^3*x^3 + 28175*d^4*e^4*x^4 + 11025*d^3*e^5*x^5 + 735*d^2*e^6*x^6 - 735*d*e^7*x^7 - 105*e^8*x^8) + e^4*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + c*e^3*(4*a^3*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 9*a^2*b*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 12*a*b^2*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 10*b^3*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 6*c^2*e^2*(a^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*a*b*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 15*b^2*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + c^3*e*(60*a*e*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - b*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*c^3*(2*c*d - b*e)*(d + e*x)^7*Log[d + e*x])/(e^9*(d + e*x)^7)","A",1
2159,1,740,435,0.3467673,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^9} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^9,x]","\frac{-6 c^2 e^2 \left(3 a^2 e^2 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+10 a b e \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)+15 b^2 \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)\right)-4 c e^3 \left(5 a^3 e^3 \left(d^2+8 d e x+28 e^2 x^2\right)+9 a^2 b e^2 \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+9 a b^2 e \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)+5 b^3 \left(d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right)\right)-3 e^4 \left(35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left(d^2+8 d e x+28 e^2 x^2\right)+4 a b^3 e \left(d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right)+b^4 \left(d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right)\right)-60 c^3 e \left(a e \left(d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right)+7 b \left(d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right)\right)+c^4 d \left(2283 d^7+17424 d^6 e x+57624 d^5 e^2 x^2+107408 d^4 e^3 x^3+122500 d^3 e^4 x^4+86240 d^2 e^5 x^5+35280 d e^6 x^6+6720 e^7 x^7\right)+840 c^4 (d+e x)^8 \log (d+e x)}{840 e^9 (d+e x)^8}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{4 e^9 (d+e x)^4}-\frac{c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9 (d+e x)^3}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{5 e^9 (d+e x)^5}-\frac{\left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{3 e^9 (d+e x)^6}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{7 e^9 (d+e x)^7}-\frac{\left(a e^2-b d e+c d^2\right)^4}{8 e^9 (d+e x)^8}+\frac{4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac{c^4 \log (d+e x)}{e^9}",1,"(c^4*d*(2283*d^7 + 17424*d^6*e*x + 57624*d^5*e^2*x^2 + 107408*d^4*e^3*x^3 + 122500*d^3*e^4*x^4 + 86240*d^2*e^5*x^5 + 35280*d*e^6*x^6 + 6720*e^7*x^7) - 3*e^4*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) - 4*c*e^3*(5*a^3*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 9*a^2*b*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 9*a*b^2*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b^3*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) - 6*c^2*e^2*(3*a^2*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 10*a*b*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 15*b^2*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) - 60*c^3*e*(a*e*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6) + 7*b*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)) + 840*c^4*(d + e*x)^8*Log[d + e*x])/(840*e^9*(d + e*x)^8)","A",1
2160,1,730,436,0.3102738,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^{10}} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^10,x]","-\frac{3 c^2 e^2 \left(2 a^2 e^2 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+5 a b e \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)+5 b^2 \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)\right)+c e^3 \left(10 a^3 e^3 \left(d^2+9 d e x+36 e^2 x^2\right)+15 a^2 b e^2 \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+12 a b^2 e \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)+5 b^3 \left(d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right)\right)+e^4 \left(70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left(d^2+9 d e x+36 e^2 x^2\right)+5 a b^3 e \left(d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right)+b^4 \left(d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right)\right)+5 c^3 e \left(2 a e \left(d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right)+7 b \left(d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right)\right)+70 c^4 \left(d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right)}{630 e^9 (d+e x)^9}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{5 e^9 (d+e x)^5}-\frac{2 c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{3 e^9 (d+e x)^3}+\frac{c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (d+e x)^4}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9 (d+e x)^6}-\frac{2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{7 e^9 (d+e x)^7}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{2 e^9 (d+e x)^8}-\frac{\left(a e^2-b d e+c d^2\right)^4}{9 e^9 (d+e x)^9}+\frac{2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac{c^4}{e^9 (d+e x)}",1,"-1/630*(70*c^4*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8) + e^4*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + c*e^3*(10*a^3*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2) + 15*a^2*b*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*a*b^2*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b^3*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 3*c^2*e^2*(2*a^2*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*a*b*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 5*b^2*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 5*c^3*e*(2*a*e*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*b*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^9*(d + e*x)^9)","A",1
2161,1,731,443,0.2914273,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^{11}} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^11,x]","-\frac{3 c^2 e^2 \left(2 a^2 e^2 \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)+4 a b e \left(d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right)+3 b^2 \left(d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right)\right)+2 c e^3 \left(7 a^3 e^3 \left(d^2+10 d e x+45 e^2 x^2\right)+9 a^2 b e^2 \left(d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right)+6 a b^2 e \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)+2 b^3 \left(d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right)\right)+e^4 \left(126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left(d^2+10 d e x+45 e^2 x^2\right)+6 a b^3 e \left(d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right)+b^4 \left(d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right)\right)+2 c^3 e \left(3 a e \left(d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right)+7 b \left(d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right)\right)+14 c^4 \left(d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right)}{1260 e^9 (d+e x)^{10}}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac{c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{2 e^9 (d+e x)^4}+\frac{4 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{5 e^9 (d+e x)^5}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{7 e^9 (d+e x)^7}-\frac{\left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{4 e^9 (d+e x)^8}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{9 e^9 (d+e x)^9}-\frac{\left(a e^2-b d e+c d^2\right)^4}{10 e^9 (d+e x)^{10}}+\frac{4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac{c^4}{2 e^9 (d+e x)^2}",1,"-1/1260*(14*c^4*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + e^4*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 2*c*e^3*(7*a^3*e^3*(d^2 + 10*d*e*x + 45*e^2*x^2) + 9*a^2*b*e^2*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 6*a*b^2*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 2*b^3*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)) + 3*c^2*e^2*(2*a^2*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*b^2*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + 2*c^3*e*(3*a*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6) + 7*b*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)))/(e^9*(d + e*x)^10)","A",1
2162,1,731,440,0.3093817,"\int \frac{\left(a+b x+c x^2\right)^4}{(d+e x)^{12}} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x)^12,x]","-\frac{6 c^2 e^2 \left(3 a^2 e^2 \left(d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right)+5 a b e \left(d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right)+3 b^2 \left(d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right)\right)+c e^3 \left(56 a^3 e^3 \left(d^2+11 d e x+55 e^2 x^2\right)+63 a^2 b e^2 \left(d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right)+36 a b^2 e \left(d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right)+10 b^3 \left(d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right)\right)+3 e^4 \left(210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left(d^2+11 d e x+55 e^2 x^2\right)+7 a b^3 e \left(d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right)+b^4 \left(d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right)\right)+3 c^3 e \left(4 a e \left(d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right)+7 b \left(d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right)\right)+14 c^4 \left(d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right)}{6930 e^9 (d+e x)^{11}}","-\frac{6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac{2 c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{5 e^9 (d+e x)^5}+\frac{2 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{3 e^9 (d+e x)^6}+\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{2 e^9 (d+e x)^8}-\frac{2 \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{9 e^9 (d+e x)^9}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^3}{5 e^9 (d+e x)^{10}}-\frac{\left(a e^2-b d e+c d^2\right)^4}{11 e^9 (d+e x)^{11}}+\frac{c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac{c^4}{3 e^9 (d+e x)^3}",1,"-1/6930*(14*c^4*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 3*e^4*(210*a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a^2*b^2*e^2*(d^2 + 11*d*e*x + 55*e^2*x^2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + b^4*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + c*e^3*(56*a^3*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) + 63*a^2*b*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 36*a*b^2*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 10*b^3*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)) + 6*c^2*e^2*(3*a^2*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 3*b^2*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + 3*c^3*e*(4*a*e*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*b*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)))/(e^9*(d + e*x)^11)","A",1
2163,1,32,32,0.001057,"\int x^4 \left(3-4 x+x^2\right)^2 \, dx","Integrate[x^4*(3 - 4*x + x^2)^2,x]","\frac{x^9}{9}-x^8+\frac{22 x^7}{7}-4 x^6+\frac{9 x^5}{5}","\frac{x^9}{9}-x^8+\frac{22 x^7}{7}-4 x^6+\frac{9 x^5}{5}",1,"(9*x^5)/5 - 4*x^6 + (22*x^7)/7 - x^8 + x^9/9","A",1
2164,1,36,36,0.0006707,"\int x^3 \left(3-4 x+x^2\right)^2 \, dx","Integrate[x^3*(3 - 4*x + x^2)^2,x]","\frac{x^8}{8}-\frac{8 x^7}{7}+\frac{11 x^6}{3}-\frac{24 x^5}{5}+\frac{9 x^4}{4}","\frac{x^8}{8}-\frac{8 x^7}{7}+\frac{11 x^6}{3}-\frac{24 x^5}{5}+\frac{9 x^4}{4}",1,"(9*x^4)/4 - (24*x^5)/5 + (11*x^6)/3 - (8*x^7)/7 + x^8/8","A",1
2165,1,32,32,0.0008694,"\int x^2 \left(3-4 x+x^2\right)^2 \, dx","Integrate[x^2*(3 - 4*x + x^2)^2,x]","\frac{x^7}{7}-\frac{4 x^6}{3}+\frac{22 x^5}{5}-6 x^4+3 x^3","\frac{x^7}{7}-\frac{4 x^6}{3}+\frac{22 x^5}{5}-6 x^4+3 x^3",1,"3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7","A",1
2166,1,34,34,0.0011036,"\int x \left(3-4 x+x^2\right)^2 \, dx","Integrate[x*(3 - 4*x + x^2)^2,x]","\frac{x^6}{6}-\frac{8 x^5}{5}+\frac{11 x^4}{2}-8 x^3+\frac{9 x^2}{2}","\frac{x^6}{6}-\frac{8 x^5}{5}+\frac{11 x^4}{2}-8 x^3+\frac{9 x^2}{2}",1,"(9*x^2)/2 - 8*x^3 + (11*x^4)/2 - (8*x^5)/5 + x^6/6","A",1
2167,1,28,28,0.000691,"\int \left(3-4 x+x^2\right)^2 \, dx","Integrate[(3 - 4*x + x^2)^2,x]","\frac{x^5}{5}-2 x^4+\frac{22 x^3}{3}-12 x^2+9 x","-\frac{1}{5} (3-x)^5-\frac{4}{3} (3-x)^3+(x-3)^4",1,"9*x - 12*x^2 + (22*x^3)/3 - 2*x^4 + x^5/5","A",1
2168,1,27,27,0.0007573,"\int \frac{\left(3-4 x+x^2\right)^2}{x} \, dx","Integrate[(3 - 4*x + x^2)^2/x,x]","\frac{x^4}{4}-\frac{8 x^3}{3}+11 x^2-24 x+9 \log (x)","\frac{x^4}{4}-\frac{8 x^3}{3}+11 x^2-24 x+9 \log (x)",1,"-24*x + 11*x^2 - (8*x^3)/3 + x^4/4 + 9*Log[x]","A",1
2169,1,25,25,0.0008129,"\int \frac{\left(3-4 x+x^2\right)^2}{x^2} \, dx","Integrate[(3 - 4*x + x^2)^2/x^2,x]","\frac{x^3}{3}-4 x^2+22 x-\frac{9}{x}-24 \log (x)","\frac{x^3}{3}-4 x^2+22 x-\frac{9}{x}-24 \log (x)",1,"-9/x + 22*x - 4*x^2 + x^3/3 - 24*Log[x]","A",1
2170,1,27,27,0.0008118,"\int \frac{\left(3-4 x+x^2\right)^2}{x^3} \, dx","Integrate[(3 - 4*x + x^2)^2/x^3,x]","\frac{x^2}{2}-\frac{9}{2 x^2}-8 x+\frac{24}{x}+22 \log (x)","\frac{x^2}{2}-\frac{9}{2 x^2}-8 x+\frac{24}{x}+22 \log (x)",1,"-9/(2*x^2) + 24/x - 8*x + x^2/2 + 22*Log[x]","A",1
2171,1,21,21,0.000686,"\int \frac{\left(3-4 x+x^2\right)^2}{x^4} \, dx","Integrate[(3 - 4*x + x^2)^2/x^4,x]","-\frac{3}{x^3}+\frac{12}{x^2}+x-\frac{22}{x}-8 \log (x)","-\frac{3}{x^3}+\frac{12}{x^2}+x-\frac{22}{x}-8 \log (x)",1,"-3/x^3 + 12/x^2 - 22/x + x - 8*Log[x]","A",1
2172,1,25,25,0.0008621,"\int \frac{\left(3-4 x+x^2\right)^2}{x^5} \, dx","Integrate[(3 - 4*x + x^2)^2/x^5,x]","-\frac{9}{4 x^4}+\frac{8}{x^3}-\frac{11}{x^2}+\frac{8}{x}+\log (x)","-\frac{9}{4 x^4}+\frac{8}{x^3}-\frac{11}{x^2}+\frac{8}{x}+\log (x)",1,"-9/(4*x^4) + 8/x^3 - 11/x^2 + 8/x + Log[x]","A",1
2173,1,30,30,0.0007493,"\int \frac{\left(3-4 x+x^2\right)^2}{x^6} \, dx","Integrate[(3 - 4*x + x^2)^2/x^6,x]","-\frac{9}{5 x^5}+\frac{6}{x^4}-\frac{22}{3 x^3}+\frac{4}{x^2}-\frac{1}{x}","-\frac{9}{5 x^5}+\frac{6}{x^4}-\frac{22}{3 x^3}+\frac{4}{x^2}-\frac{1}{x}",1,"-9/(5*x^5) + 6/x^4 - 22/(3*x^3) + 4/x^2 - x^(-1)","A",1
2174,1,36,36,0.0007093,"\int \frac{\left(3-4 x+x^2\right)^2}{x^7} \, dx","Integrate[(3 - 4*x + x^2)^2/x^7,x]","-\frac{3}{2 x^6}+\frac{24}{5 x^5}-\frac{11}{2 x^4}+\frac{8}{3 x^3}-\frac{1}{2 x^2}","-\frac{3}{2 x^6}+\frac{24}{5 x^5}-\frac{11}{2 x^4}+\frac{8}{3 x^3}-\frac{1}{2 x^2}",1,"-3/(2*x^6) + 24/(5*x^5) - 11/(2*x^4) + 8/(3*x^3) - 1/(2*x^2)","A",1
2175,1,15,14,0.0038332,"\int \frac{2+2 x+x^2}{2+x} \, dx","Integrate[(2 + 2*x + x^2)/(2 + x),x]","\frac{1}{2} \left(x^2+4 \log (x+2)-4\right)","\frac{x^2}{2}+2 \log (x+2)",1,"(-4 + x^2 + 4*Log[2 + x])/2","A",1
2176,1,18,19,0.0035522,"\int \frac{5+4 x+x^2}{-2+x} \, dx","Integrate[(5 + 4*x + x^2)/(-2 + x),x]","\frac{x^2}{2}+6 x+17 \log (x-2)-14","\frac{x^2}{2}+6 x+17 \log (2-x)",1,"-14 + 6*x + x^2/2 + 17*Log[-2 + x]","A",1
2177,1,14,14,0.0054139,"\int \frac{2+2 x+x^2}{(1+x)^3} \, dx","Integrate[(2 + 2*x + x^2)/(1 + x)^3,x]","\log (x+1)-\frac{1}{2 (x+1)^2}","\log (x+1)-\frac{1}{2 (x+1)^2}",1,"-1/2*1/(1 + x)^2 + Log[1 + x]","A",1
2178,1,19,19,0.007672,"\int \frac{3+3 x+2 x^2}{(1+x)^3} \, dx","Integrate[(3 + 3*x + 2*x^2)/(1 + x)^3,x]","\frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1)","\frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1)",1,"-(1 + x)^(-2) + (1 + x)^(-1) + 2*Log[1 + x]","A",1
2179,1,11,11,0.0007428,"\int \frac{1+x+x^2}{x} \, dx","Integrate[(1 + x + x^2)/x,x]","\frac{x^2}{2}+x+\log (x)","\frac{x^2}{2}+x+\log (x)",1,"x + x^2/2 + Log[x]","A",1
2180,1,11,11,0.0011783,"\int \frac{9+6 x+x^2}{x^2} \, dx","Integrate[(9 + 6*x + x^2)/x^2,x]","x-\frac{9}{x}+6 \log (x)","x-\frac{9}{x}+6 \log (x)",1,"-9/x + x + 6*Log[x]","A",1
2181,1,18,18,0.0010835,"\int \frac{1+2 x+x^2}{x^4} \, dx","Integrate[(1 + 2*x + x^2)/x^4,x]","-\frac{1}{3 x^3}-\frac{1}{x^2}-\frac{1}{x}","-\frac{1}{3 x^3}-\frac{1}{x^2}-\frac{1}{x}",1,"-1/3*1/x^3 - x^(-2) - x^(-1)","A",1
2182,1,240,243,0.1941472,"\int \frac{(d+e x)^4}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2),x]","\frac{\frac{6 \left(2 c^2 e^2 \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+6 c e^2 x \left(-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right)+3 e (2 c d-b e) \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \log (a+x (b+c x))+3 c^2 e^3 x^2 (4 c d-b e)+2 c^3 e^4 x^3}{6 c^4}","-\frac{\left(2 c^2 e^2 \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \log \left(a+b x+c x^2\right)}{2 c^4}+\frac{e^2 x \left(-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right)}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}+\frac{e^4 x^3}{3 c}",1,"(6*c*e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x + 3*c^2*e^3*(4*c*d - b*e)*x^2 + 2*c^3*e^4*x^3 + (6*(2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 3*e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Log[a + x*(b + c*x)])/(6*c^4)","A",1
2183,1,148,151,0.1209103,"\int \frac{(d+e x)^3}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2),x]","\frac{e \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \log (a+x (b+c x))+\frac{2 (2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+2 c e^2 x (3 c d-b e)+c^2 e^3 x^2}{2 c^3}","\frac{e \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \log \left(a+b x+c x^2\right)}{2 c^3}-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c}",1,"(2*c*e^2*(3*c*d - b*e)*x + c^2*e^3*x^2 + (2*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + x*(b + c*x)])/(2*c^3)","A",1
2184,1,101,101,0.0685189,"\int \frac{(d+e x)^2}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2),x]","\frac{\frac{2 \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+e (2 c d-b e) \log (a+x (b+c x))+2 c e^2 x}{2 c^2}","-\frac{\left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{e^2 x}{c}",1,"(2*c*e^2*x + (2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*(2*c*d - b*e)*Log[a + x*(b + c*x)])/(2*c^2)","A",1
2185,1,66,66,0.0549982,"\int \frac{d+e x}{a+b x+c x^2} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2),x]","\frac{e \log (a+x (b+c x))-\frac{2 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{2 c}","\frac{e \log \left(a+b x+c x^2\right)}{2 c}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c}}",1,"((-2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*Log[a + x*(b + c*x)])/(2*c)","A",1
2186,1,38,34,0.0064998,"\int \frac{1}{a+b x+c x^2} \, dx","Integrate[(a + b*x + c*x^2)^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"(2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]","A",1
2187,1,105,122,0.079928,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)),x]","\frac{e \sqrt{4 a c-b^2} (\log (a+x (b+c x))-2 \log (d+e x))+(2 b e-4 c d) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{2 \sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)}","-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}-\frac{e \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)}+\frac{e \log (d+e x)}{a e^2-b d e+c d^2}",1,"((-4*c*d + 2*b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*e*(-2*Log[d + e*x] + Log[a + x*(b + c*x)]))/(2*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e)))","A",1
2188,1,151,186,0.2041668,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)),x]","\frac{\frac{2 \left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{2 e \left(e (a e-b d)+c d^2\right)}{d+e x}+e (b e-2 c d) \log (a+x (b+c x))-2 e (b e-2 c d) \log (d+e x)}{2 \left(e (a e-b d)+c d^2\right)^2}","-\frac{\left(-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)^2}-\frac{e (2 c d-b e) \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^2}-\frac{e}{(d+e x) \left(a e^2-b d e+c d^2\right)}+\frac{e (2 c d-b e) \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^2}",1,"((-2*e*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x) + (2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*e*(-2*c*d + b*e)*Log[d + e*x] + e*(-2*c*d + b*e)*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
2189,1,272,272,0.3127296,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)),x]","\frac{e \log (d+e x) \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)}{\left(e (a e-b d)+c d^2\right)^3}+\frac{e \left(c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right) \log (a+x (b+c x))}{2 \left(e (a e-b d)+c d^2\right)^3}+\frac{(b e-2 c d) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)^3}+\frac{e (b e-2 c d)}{(d+e x) \left(e (a e-b d)+c d^2\right)^2}-\frac{e}{2 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}","-\frac{e \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^3}+\frac{e \log (d+e x) \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right)}{\left(a e^2-b d e+c d^2\right)^3}-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)^3}-\frac{e (2 c d-b e)}{(d+e x) \left(a e^2-b d e+c d^2\right)^2}-\frac{e}{2 (d+e x)^2 \left(a e^2-b d e+c d^2\right)}",1,"-1/2*e/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (e*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + ((-2*c*d + b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e))^3) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e*(-3*c^2*d^2 - b^2*e^2 + c*e*(3*b*d + a*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)","A",1
2190,1,422,374,0.6643074,"\int \frac{(d+e x)^5}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^5/(a + b*x + c*x^2)^2,x]","\frac{\frac{2 (b e-2 c d) \left(2 c^2 e^2 \left(15 a^2 e^2+10 a b d e+b^2 d^2\right)-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\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(-2 b^2 c e^2 \left(2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right)+b c^2 \left(5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right)+2 c^2 \left(a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right)+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left(a e (d+e x)-2 c d^2 x\right)+b^5 e^5 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+e^3 \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right) \log (a+x (b+c x))+2 c e^4 x (5 c d-2 b e)+c^2 e^5 x^2}{2 c^4}","\frac{(2 c d-b e) \left(-2 c^2 e^2 \left(15 a^2 e^2+10 a b d e+b^2 d^2\right)+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \left(b^2-4 a c\right)^{3/2}}+\frac{e^3 x^2 \left(-2 c e (4 a e+5 b d)+3 b^2 e^2+16 c^2 d^2\right)}{2 c^2 \left(b^2-4 a c\right)}+\frac{e^3 \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right) \log \left(a+b x+c x^2\right)}{2 c^4}+\frac{e^4 x^3 (2 c d-b e)}{c \left(b^2-4 a c\right)}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{e^2 x \left(-10 c^2 d e (3 a e+b d)+b c e^2 (11 a e+10 b d)-3 b^3 e^3+12 c^3 d^3\right)}{c^3 \left(b^2-4 a c\right)}",1,"(2*c*e^4*(5*c*d - 2*b*e)*x + c^2*e^5*x^2 + (2*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + x*(b + c*x)])/(2*c^4)","A",1
2191,1,298,260,0.4593456,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^2,x]","\frac{\frac{-b c \left(-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right)-2 c^2 \left(a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right)+b^3 e^3 (4 c d x-a e)+2 b^2 c e^2 \left(2 a e (d+e x)-3 c d^2 x\right)-b^4 e^4 x}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{2 \left(-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\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}}+e^3 (2 c d-b e) \log (a+x (b+c x))+c e^4 x}{c^3}","\frac{2 e^2 x \left(-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right)}{c^2 \left(b^2-4 a c\right)}+\frac{e^3 x^2 (2 c d-b e)}{c \left(b^2-4 a c\right)}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{2 \left(2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{e^3 (2 c d-b e) \log \left(a+b x+c x^2\right)}{c^3}",1,"(c*e^4*x + (-(b^4*e^4*x) + b^3*e^3*(-(a*e) + 4*c*d*x) + 2*b^2*c*e^2*(-3*c*d^2*x + 2*a*e*(d + e*x)) - b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) - 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d + a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*(2*c*d - b*e)*Log[a + x*(b + c*x)])/c^3","A",1
2192,1,201,173,0.2909732,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^2,x]","\frac{\frac{2 \left(-2 c \left(a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right)+b^2 e^2 (a e-3 c d x)-b c \left(3 a e^2 (d+e x)+c d^2 (d-3 e x)\right)+b^3 e^3 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 (b e-2 c d) \left(2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\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}}+e^3 \log (a+x (b+c x))}{2 c^2}","\frac{(2 c d-b e) \left(-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \left(b^2-4 a c\right)^{3/2}}+\frac{e^2 x (2 c d-b e)}{c \left(b^2-4 a c\right)}-\frac{(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{e^3 \log \left(a+b x+c x^2\right)}{2 c^2}",1,"((2*(b^3*e^3*x + b^2*e^2*(a*e - 3*c*d*x) - 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) - b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(-2*c*d + b*e)*(-2*c^2*d^2 + b^2*e^2 + 2*c*e*(b*d - 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e^3*Log[a + x*(b + c*x)])/(2*c^2)","A",1
2193,1,128,99,0.1176519,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^2,x]","\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))}+\frac{4 \left(e (a e-b d)+c d^2\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{4 \left(a e^2-b d e+c d^2\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}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
2194,1,88,87,0.0770696,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^2,x]","\frac{\frac{2 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{2 a e-b d+b e x-2 c d x}{a+x (b+c x)}}{b^2-4 a c}","\frac{2 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{-2 a e+x (2 c d-b e)+b d}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"((-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(a + x*(b + c*x)) + (2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c)","A",1
2195,1,70,66,0.0841569,"\int \frac{1}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(a + b*x + c*x^2)^(-2),x]","-\frac{\frac{4 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c}","\frac{4 c \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{b+2 c x}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c))","A",1
2196,1,223,224,0.308303,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^2),x]","\frac{(b e-2 c d) \left(2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\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} \left(e (a e-b d)+c d^2\right)^2}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (b d-a e)-c d^2\right)}+\frac{e^3 \log (d+e x)}{\left(e (a e-b d)+c d^2\right)^2}-\frac{e^3 \log (a+x (b+c x))}{2 \left(e (a e-b d)+c d^2\right)^2}","\frac{(2 c d-b e) \left(-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\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(a e^2-b d e+c d^2\right)^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{e^3 \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^2}+\frac{e^3 \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^2}",1,"(-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) + ((-2*c*d + b*e)*(-2*c^2*d^2 + b^2*e^2 + 2*c*e*(b*d - 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2) + (e^3*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^2 - (e^3*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
2197,1,339,344,0.6675959,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^2),x]","-\frac{2 \left(2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\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} \left(e (b d-a e)-c d^2\right)^3}+\frac{b c \left(3 a e^2-c d (d-2 e x)\right)-2 c^2 \left(a e (2 d-e x)+c d^2 x\right)+b^3 \left(-e^2\right)+b^2 c e (2 d-e x)}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (a e-b d)+c d^2\right)^2}-\frac{e^3}{(d+e x) \left(e (a e-b d)+c d^2\right)^2}-\frac{2 e^3 (b e-2 c d) \log (d+e x)}{\left(e (a e-b d)+c d^2\right)^3}+\frac{e^3 (b e-2 c d) \log (a+x (b+c x))}{\left(e (a e-b d)+c d^2\right)^3}","-\frac{2 e \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\left(b^2-4 a c\right) (d+e x) \left(a e^2-b d e+c d^2\right)^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left(b^2-4 a c\right) (d+e x) \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}+\frac{2 \left(2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\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(a e^2-b d e+c d^2\right)^3}-\frac{e^3 (2 c d-b e) \log \left(a+b x+c x^2\right)}{\left(a e^2-b d e+c d^2\right)^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^3}",1,"-(e^3/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))) + (-(b^3*e^2) + b^2*c*e*(2*d - e*x) + b*c*(3*a*e^2 - c*d*(d - 2*e*x)) - 2*c^2*(c*d^2*x + a*e*(2*d - e*x)))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - (2*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(-(c*d^2) + e*(b*d - a*e))^3) - (2*e^3*(-2*c*d + b*e)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e^3*(-2*c*d + b*e)*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3","A",1
2198,1,489,485,1.1198702,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]","\frac{(b e-2 c d) \left(2 c^2 e^2 \left(15 a^2 e^2+10 a b d e+b^2 d^2\right)-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\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} \left(e (a e-b d)+c d^2\right)^4}+\frac{2 c^2 \left(-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right)+b^2 c e \left(4 a e^2-3 c d (d-e x)\right)+b c^2 \left(3 a e^2 (e x-3 d)+c d^2 (d-3 e x)\right)+b^4 \left(-e^3\right)+b^3 c e^2 (3 d-e x)}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (b d-a e)-c d^2\right)^3}+\frac{e^3 \log (d+e x) \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right)}{\left(e (a e-b d)+c d^2\right)^4}-\frac{e^3 \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right) \log (a+x (b+c x))}{2 \left(e (a e-b d)+c d^2\right)^4}+\frac{2 e^3 (b e-2 c d)}{(d+e x) \left(e (a e-b d)+c d^2\right)^3}-\frac{e^3}{2 (d+e x)^2 \left(e (a e-b d)+c d^2\right)^2}","\frac{(2 c d-b e) \left(-2 c^2 e^2 \left(15 a^2 e^2+10 a b d e+b^2 d^2\right)+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\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(a e^2-b d e+c d^2\right)^4}-\frac{e (2 c d-b e) \left(-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right)}{\left(b^2-4 a c\right) (d+e x) \left(a e^2-b d e+c d^2\right)^3}-\frac{e \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right)}{2 \left(b^2-4 a c\right) (d+e x)^2 \left(a e^2-b d e+c d^2\right)^2}-\frac{e^3 \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right) \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^4}+\frac{e^3 \log (d+e x) \left(-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right)}{\left(a e^2-b d e+c d^2\right)^4}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left(b^2-4 a c\right) (d+e x)^2 \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}",1,"-1/2*e^3/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^2) + (2*e^3*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)) + (-(b^4*e^3) + b^3*c*e^2*(3*d - e*x) + b^2*c*e*(4*a*e^2 - 3*c*d*(d - e*x)) + 2*c^2*(-(a^2*e^3) + c^2*d^3*x + 3*a*c*d*e*(d - e*x)) + b*c^2*(c*d^2*(d - 3*e*x) + 3*a*e^2*(-3*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))^3*(a + x*(b + c*x))) + ((-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^4)","A",1
2199,1,299,294,0.4221362,"\int \frac{x^7}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[x^7/(a + b*x + c*x^2)^3,x]","\frac{\frac{6 b c \left(70 a^3 c^3-70 a^2 b^2 c^2+21 a b^4 c-2 b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{-2 a^4 c^3+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (7 c x-3 b)+a b^5 (b-7 c x)+b^7 x}{\left(b^2-4 a c\right) (a+x (b+c x))^2}-\frac{48 a^4 c^4-153 a^3 b^2 c^3+126 a^3 b c^4 x+88 a^2 b^4 c^2-182 a^2 b^3 c^3 x-17 a b^6 c+70 a b^5 c^2 x+b^8-8 b^7 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}-3 c \left(a c-2 b^2\right) \log (a+x (b+c x))-6 b c^2 x+c^3 x^2}{2 c^6}","\frac{3 x^2 \left(16 a^2 c^2-13 a b^2 c+2 b^4\right)}{2 c^3 \left(b^2-4 a c\right)^2}+\frac{3 b \left(-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^5 \left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(2 b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 c^5}-\frac{3 b x \left(2 b^2-9 a c\right) \left(b^2-3 a c\right)}{c^4 \left(b^2-4 a c\right)^2}-\frac{b x^3 \left(2 b^2-11 a c\right)}{c^2 \left(b^2-4 a c\right)^2}+\frac{x^6 (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{3 x^4 \left(b x \left(b^2-6 a c\right)+a \left(b^2-8 a c\right)\right)}{2 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}",1,"(-6*b*c^2*x + c^3*x^2 - (b^8 - 17*a*b^6*c + 88*a^2*b^4*c^2 - 153*a^3*b^2*c^3 + 48*a^4*c^4 - 8*b^7*c*x + 70*a*b^5*c^2*x - 182*a^2*b^3*c^3*x + 126*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-2*a^4*c^3 + b^7*x + a*b^5*(b - 7*c*x) + a^3*b*c^2*(9*b - 7*c*x) + 2*a^2*b^3*c*(-3*b + 7*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*b*c*(-2*b^6 + 21*a*b^4*c - 70*a^2*b^2*c^2 + 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 3*c*(-2*b^2 + a*c)*Log[a + x*(b + c*x)])/(2*c^6)","A",1
2200,1,260,238,0.3648604,"\int \frac{x^6}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[x^6/(a + b*x + c*x^2)^3,x]","\frac{\frac{a^3 c^2 (2 c x-5 b)+a^2 b^2 c (5 b-9 c x)-a b^4 (b-6 c x)+b^6 (-x)}{\left(b^2-4 a c\right) (a+x (b+c x))^2}+\frac{6 c \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{-78 a^3 b c^3+36 a^3 c^4 x+61 a^2 b^3 c^2-102 a^2 b^2 c^3 x-14 a b^5 c+48 a b^4 c^2 x+b^7-6 b^6 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}-3 b c \log (a+x (b+c x))+2 c^2 x}{2 c^5}","\frac{3 x \left(10 a^2 c^2-7 a b^2 c+b^4\right)}{c^3 \left(b^2-4 a c\right)^2}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \left(b^2-4 a c\right)^{5/2}}-\frac{3 b x^2 \left(b^2-6 a c\right)}{2 c^2 \left(b^2-4 a c\right)^2}+\frac{x^5 (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{x^3 \left(b x \left(b^2-7 a c\right)+a \left(b^2-10 a c\right)\right)}{c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 b \log \left(a+b x+c x^2\right)}{2 c^4}",1,"(2*c^2*x + (b^7 - 14*a*b^5*c + 61*a^2*b^3*c^2 - 78*a^3*b*c^3 - 6*b^6*c*x + 48*a*b^4*c^2*x - 102*a^2*b^2*c^3*x + 36*a^3*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^6*x) + a^2*b^2*c*(5*b - 9*c*x) - a*b^4*(b - 6*c*x) + a^3*c^2*(-5*b + 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*c*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 3*b*c*Log[a + x*(b + c*x)])/(2*c^5)","A",1
2201,1,628,388,1.0664237,"\int \frac{(d+e x)^5}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^5/(a + b*x + c*x^2)^3,x]","\frac{\frac{2 c (2 c d-b e) \left(2 c^2 e^2 \left(15 a^2 e^2-10 a b d e+2 b^2 d^2\right)+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+b^4 e^4+6 c^4 d^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{-2 b^2 c e^2 \left(2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right)+b c^2 \left(5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right)+2 c^2 \left(a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right)+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left(a e (d+e x)-2 c d^2 x\right)+b^5 e^5 x}{\left(b^2-4 a c\right) (a+x (b+c x))^2}+\frac{b^2 c^2 e \left(-39 a^2 e^4+10 a c d e^2 (5 d+8 e x)-5 c^2 d^3 (3 d-4 e x)\right)+2 b c^3 \left(5 a^2 e^4 (11 d+5 e x)+10 a c d^2 e^2 (d-3 e x)+3 c^2 d^4 (d-5 e x)\right)+4 c^3 \left(8 a^3 e^5-5 a^2 c d e^3 (8 d+5 e x)+10 a c^2 d^3 e^2 x+3 c^3 d^5 x\right)+b^4 c e^3 \left(11 a e^2-10 c d (d+e x)\right)+10 b^3 c^2 e^2 \left(c d^3-a e^2 (4 d+3 e x)\right)+b^6 \left(-e^5\right)+b^5 c e^4 (5 d+4 e x)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+c e^5 \log (a+x (b+c x))}{2 c^4}","-\frac{\left(20 c^3 d e^2 \left(3 a^2 e^2-3 a b d e+b^2 d^2\right)-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{5/2}}-\frac{e^2 x (2 c d-b e) \left(-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right)}{c^2 \left(b^2-4 a c\right)^2}-\frac{(d+e x)^2 \left(-x (2 c d-b e) \left(-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right)+b^2 \left(7 c d^2 e-a e^3\right)-6 b c d \left(3 a e^2+c d^2\right)+8 a c e \left(2 a e^2+c d^2\right)\right)}{2 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{e^5 \log \left(a+b x+c x^2\right)}{2 c^3}",1,"((b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*e^5) + b^5*c*e^4*(5*d + 4*e*x) + b^4*c*e^3*(11*a*e^2 - 10*c*d*(d + e*x)) + 10*b^3*c^2*e^2*(c*d^3 - a*e^2*(4*d + 3*e*x)) + 4*c^3*(8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(8*d + 5*e*x)) + 2*b*c^3*(3*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(11*d + 5*e*x)) + b^2*c^2*e*(-39*a^2*e^4 - 5*c^2*d^3*(3*d - 4*e*x) + 10*a*c*d*e^2*(5*d + 8*e*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(2*c*d - b*e)*(6*c^4*d^4 + b^4*e^4 + 2*b^2*c*e^3*(b*d - 5*a*e) - 4*c^3*d^2*e*(3*b*d - 5*a*e) + 2*c^2*e^2*(2*b^2*d^2 - 10*a*b*d*e + 15*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*e^5*Log[a + x*(b + c*x)])/(2*c^4)","A",1
2202,1,413,169,0.5531565,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^3,x]","\frac{1}{2} \left(\frac{b c \left(-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right)+2 c^2 \left(a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right)+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left(3 c d^2 x-2 a e (d+e x)\right)+b^4 e^4 x}{c^3 \left(4 a c-b^2\right) (a+x (b+c x))^2}+\frac{2 b c^2 \left(11 a^2 e^4+6 a c d e^2 (d-2 e x)+3 c^2 d^3 (d-4 e x)\right)+4 c^3 \left(-a^2 e^3 (16 d+5 e x)+6 a c d^2 e^2 x+3 c^2 d^4 x\right)+2 b^3 c e^2 \left(3 c d^2-4 a e^2\right)+4 b^2 c^2 e \left(a e^2 (5 d+4 e x)-3 c d^2 (d-e x)\right)+b^5 e^4-2 b^4 c e^3 (2 d+e x)}{c^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{24 \left(e (a e-b d)+c d^2\right)^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}\right)","\frac{3 (d+e x) \left(a e^2-b d e+c d^2\right) (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{12 \left(a e^2-b d e+c d^2\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"((b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (b^5*e^4 + 2*b^3*c*e^2*(3*c*d^2 - 4*a*e^2) - 2*b^4*c*e^3*(2*d + e*x) + 2*b*c^2*(11*a^2*e^4 + 3*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(-3*c*d^2*(d - e*x) + a*e^2*(5*d + 4*e*x)) + 4*c^3*(3*c^2*d^4*x + 6*a*c*d^2*e^2*x - a^2*e^3*(16*d + 5*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (24*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","B",1
2203,1,308,158,0.402705,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^3,x]","\frac{1}{2} \left(\frac{2 c \left(a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right)+b^2 e^2 (3 c d x-a e)+b c \left(3 a e^2 (d+e x)+c d^2 (d-3 e x)\right)-b^3 e^3 x}{c^2 \left(4 a c-b^2\right) (a+x (b+c x))^2}+\frac{4 c^2 \left(-4 a^2 e^3+3 a c d e^2 x+3 c^2 d^3 x\right)+b^2 c e \left(5 a e^2-9 c d^2+6 c d e x\right)+6 b c^2 \left(a e^2 (d-e x)+c d^2 (d-3 e x)\right)+b^4 \left(-e^3\right)+3 b^3 c d e^2}{c^2 \left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{12 (2 c d-b e) \left(e (a e-b d)+c d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}\right)","-\frac{6 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{(b+2 c x) (d+e x)^3}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{3 (d+e x) (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}",1,"((3*b^3*c*d*e^2 - b^4*e^3 + b^2*c*e*(-9*c*d^2 + 5*a*e^2 + 6*c*d*e*x) + 4*c^2*(-4*a^2*e^3 + 3*c^2*d^3*x + 3*a*c*d*e^2*x) + 6*b*c^2*(c*d^2*(d - 3*e*x) + a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x)))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (12*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","A",1
2204,1,203,198,0.2873703,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^3,x]","\frac{1}{2} \left(\frac{(b+2 c x) \left(2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))^2}+\frac{4 \left(2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}\right)","-\frac{-x \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)-3 b \left(a e^2+c d^2\right)+4 a c d e+2 b^2 d e}{\left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{2 \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(((6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (4*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","A",1
2205,1,128,131,0.1306395,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^3,x]","\frac{\frac{\left(b^2-4 a c\right) (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^2}-\frac{12 c (b e-2 c d) \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 d-b e)}{a+x (b+c x)}}{2 \left(b^2-4 a c\right)^2}","\frac{3 (b+2 c x) (2 c d-b e)}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{-2 a e+x (2 c d-b e)+b d}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{6 c (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}",1,"(((b^2 - 4*a*c)*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^2 + (3*(2*c*d - b*e)*(b + 2*c*x))/(a + x*(b + c*x)) - (12*c*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2)","A",1
2206,1,97,101,0.0958611,"\int \frac{1}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(a + b*x + c*x^2)^(-3),x]","\frac{\frac{24 c^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left(-2 c \left(5 a+3 c x^2\right)+b^2-6 b c x\right)}{(a+x (b+c x))^2}}{2 \left(b^2-4 a c\right)^2}","-\frac{12 c^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 c (b+2 c x)}{\left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{b+2 c x}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(-(((b + 2*c*x)*(b^2 - 6*b*c*x - 2*c*(5*a + 3*c*x^2)))/(a + x*(b + c*x))^2) + (24*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2)","A",1
2207,1,429,429,1.1688952,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^3),x]","\frac{1}{2} \left(\frac{2 \left(-20 c^3 d e^2 \left(3 a^2 e^2-3 a b d e+b^2 d^2\right)+30 a^2 b c^2 e^5-10 a b^3 c e^5+10 c^4 d^3 e (3 b d-4 a e)+b^5 e^5-12 c^5 d^5\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2} \left(e (b d-a e)-c d^2\right)^3}+\frac{4 c^2 \left(4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right)+b^2 c e \left(c d (2 e x-9 d)-15 a e^2\right)+2 b c^2 \left(7 a e^2 (d-e x)+3 c d^2 (d-3 e x)\right)+2 b^4 e^3+b^3 c e^2 (d+2 e x)}{\left(b^2-4 a c\right)^2 (a+x (b+c x)) \left(e (a e-b d)+c d^2\right)^2}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left(b^2-4 a c\right) (a+x (b+c x))^2 \left(e (b d-a e)-c d^2\right)}+\frac{2 e^5 \log (d+e x)}{\left(e (a e-b d)+c d^2\right)^3}-\frac{e^5 \log (a+x (b+c x))}{\left(e (a e-b d)+c d^2\right)^3}\right)","-\frac{\left(20 c^3 d e^2 \left(3 a^2 e^2-3 a b d e+b^2 d^2\right)-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2} \left(a e^2-b d e+c d^2\right)^3}-\frac{-2 c x (2 c d-b e) \left(-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right)+3 a c e (2 c d-b e)^2}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2 \left(a e^2-b d e+c d^2\right)}-\frac{e^5 \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^3}+\frac{e^5 \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^3}",1,"((-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^2) + (2*b^4*e^3 + b^3*c*e^2*(d + 2*e*x) + 4*c^2*(4*a^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x) + 2*b*c^2*(3*c*d^2*(d - 3*e*x) + 7*a*e^2*(d - e*x)) + b^2*c*e*(-15*a*e^2 + c*d*(-9*d + 2*e*x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) + (2*(-12*c^5*d^5 + b^5*e^5 - 10*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5 + 10*c^4*d^3*e*(3*b*d - 4*a*e) - 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(5/2)*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^5*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - (e^5*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2","A",1
2208,1,221,239,0.450277,"\int \frac{1}{x^2 \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/(x^2*(a + b*x + c*x^2)^3),x]","\frac{\frac{a^2 \left(-3 a b c-2 a c^2 x+b^3+b^2 c x\right)}{\left(4 a c-b^2\right) (a+x (b+c x))^2}-\frac{a \left(46 a^2 b c^2+28 a^2 c^3 x-29 a b^3 c-26 a b^2 c^2 x+4 b^5+4 b^4 c x\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{6 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+3 b \log (a+x (b+c x))-\frac{2 a}{x}-6 b \log (x)}{2 a^4}","\frac{3 b \log \left(a+b x+c x^2\right)}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left(b^2-5 a c\right) \left(b^2-2 a c\right)}{a^3 x \left(b^2-4 a c\right)^2}+\frac{20 a^2 c^2+3 b c x \left(b^2-6 a c\right)-20 a b^2 c+3 b^4}{2 a^2 x \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"((-2*a)/x + (a^2*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x - 26*a*b^2*c^2*x + 28*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (6*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 6*b*Log[x] + 3*b*Log[a + x*(b + c*x)])/(2*a^4)","A",1
2209,1,269,306,0.5197481,"\int \frac{1}{x^3 \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/(x^3*(a + b*x + c*x^2)^3),x]","\frac{\frac{a^2 \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))^2}-\frac{a^2}{x^2}-\frac{6 b \left(-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{a \left(-32 a^3 c^3+97 a^2 b^2 c^2+66 a^2 b c^3 x-47 a b^4 c-42 a b^3 c^2 x+6 b^6+6 b^5 c x\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+6 \log (x) \left(2 b^2-a c\right)+3 \left(a c-2 b^2\right) \log (a+x (b+c x))+\frac{6 a b}{x}}{2 a^5}","-\frac{3 \left(2 b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 a^5}+\frac{3 \log (x) \left(2 b^2-a c\right)}{a^5}+\frac{3 b \left(2 b^2-9 a c\right) \left(b^2-3 a c\right)}{a^4 x \left(b^2-4 a c\right)^2}+\frac{24 a^2 c^2+2 b c x \left(2 b^2-11 a c\right)-25 a b^2 c+4 b^4}{2 a^2 x^2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 \left(16 a^2 c^2-13 a b^2 c+2 b^4\right)}{2 a^3 x^2 \left(b^2-4 a c\right)^2}+\frac{3 b \left(-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^5 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x^2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(-(a^2/x^2) + (6*a*b)/x + (a^2*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (a*(6*b^6 - 47*a*b^4*c + 97*a^2*b^2*c^2 - 32*a^3*c^3 + 6*b^5*c*x - 42*a*b^3*c^2*x + 66*a^2*b*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (6*b*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + 6*(2*b^2 - a*c)*Log[x] + 3*(-2*b^2 + a*c)*Log[a + x*(b + c*x)])/(2*a^5)","A",1
2210,1,435,349,0.7657188,"\int \frac{x^8}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[x^8/(a + b*x + c*x^2)^4,x]","\frac{\frac{a^4 c^3 (7 b-2 c x)-2 a^3 b^2 c^2 (7 b-8 c x)+a^2 b^4 c (7 b-20 c x)-a b^6 (b-8 c x)+b^8 (-x)}{\left(b^2-4 a c\right) (a+x (b+c x))^3}-\frac{12 c^2 \left(70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}-\frac{6 c \left(-163 a^4 b c^4+58 a^4 c^5 x+198 a^3 b^3 c^3-212 a^3 b^2 c^4 x-83 a^2 b^5 c^2+146 a^2 b^4 c^3 x+15 a b^7 c-36 a b^6 c^2 x-b^9+3 b^8 c x\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))}+\frac{125 a^4 b c^4-38 a^4 c^5 x-202 a^3 b^3 c^3+220 a^3 b^2 c^4 x+95 a^2 b^5 c^2-212 a^2 b^4 c^3 x-17 a b^7 c+68 a b^6 c^2 x+b^9-7 b^8 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))^2}-6 b c^2 \log (a+x (b+c x))+3 c^3 x}{3 c^7}","\frac{x^3 \left(b x \left(122 a^2 c^2-39 a b^2 c+4 b^4\right)+4 a \left(35 a^2 c^2-9 a b^2 c+b^4\right)\right)}{3 c^2 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{2 b x^2 \left(29 a^2 c^2-10 a b^2 c+b^4\right)}{c^3 \left(b^2-4 a c\right)^3}+\frac{4 x \left(-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right)}{c^4 \left(b^2-4 a c\right)^3}-\frac{4 \left(70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^5 \left(b^2-4 a c\right)^{7/2}}+\frac{x^7 (2 a+b x)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{x^5 \left(b x \left(b^2-9 a c\right)+a \left(b^2-14 a c\right)\right)}{3 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}-\frac{2 b \log \left(a+b x+c x^2\right)}{c^5}",1,"(3*c^3*x + (-(b^8*x) + a^2*b^4*c*(7*b - 20*c*x) - a*b^6*(b - 8*c*x) - 2*a^3*b^2*c^2*(7*b - 8*c*x) + a^4*c^3*(7*b - 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) + (b^9 - 17*a*b^7*c + 95*a^2*b^5*c^2 - 202*a^3*b^3*c^3 + 125*a^4*b*c^4 - 7*b^8*c*x + 68*a*b^6*c^2*x - 212*a^2*b^4*c^3*x + 220*a^3*b^2*c^4*x - 38*a^4*c^5*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*c*(-b^9 + 15*a*b^7*c - 83*a^2*b^5*c^2 + 198*a^3*b^3*c^3 - 163*a^4*b*c^4 + 3*b^8*c*x - 36*a*b^6*c^2*x + 146*a^2*b^4*c^3*x - 212*a^3*b^2*c^4*x + 58*a^4*c^5*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (12*c^2*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) - 6*b*c^2*Log[a + x*(b + c*x)])/(3*c^7)","A",1
2211,1,386,291,0.5731252,"\int \frac{x^7}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[x^7/(a + b*x + c*x^2)^4,x]","\frac{\frac{6 b c^2 \left(-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+\frac{2 \left(-2 a^4 c^3+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (7 c x-3 b)+a b^5 (b-7 c x)+b^7 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))^3}-\frac{3 c \left(192 a^4 c^4-374 a^3 b^2 c^3+308 a^3 b c^4 x+191 a^2 b^4 c^2-266 a^2 b^3 c^3 x-40 a b^6 c+70 a b^5 c^2 x+3 b^8-6 b^7 c x\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))}+\frac{-72 a^4 c^4+233 a^3 b^2 c^3-182 a^3 b c^4 x-139 a^2 b^4 c^2+259 a^2 b^3 c^3 x+29 a b^6 c-98 a b^5 c^2 x-2 b^8+11 b^7 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))^2}+3 c^2 \log (a+x (b+c x))}{6 c^6}","\frac{x^2 \left(b x \left(140 a^2 c^2-32 a b^2 c+3 b^4\right)+3 a \left(64 a^2 c^2-10 a b^2 c+b^4\right)\right)}{6 c^2 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{b x \left(38 a^2 c^2-11 a b^2 c+b^4\right)}{c^3 \left(b^2-4 a c\right)^3}+\frac{b \left(-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \left(b^2-4 a c\right)^{7/2}}+\frac{x^6 (2 a+b x)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{x^4 \left(b x \left(b^2-14 a c\right)+a \left(b^2-24 a c\right)\right)}{6 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}+\frac{\log \left(a+b x+c x^2\right)}{2 c^4}",1,"((-2*b^8 + 29*a*b^6*c - 139*a^2*b^4*c^2 + 233*a^3*b^2*c^3 - 72*a^4*c^4 + 11*b^7*c*x - 98*a*b^5*c^2*x + 259*a^2*b^3*c^3*x - 182*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*c*(3*b^8 - 40*a*b^6*c + 191*a^2*b^4*c^2 - 374*a^3*b^2*c^3 + 192*a^4*c^4 - 6*b^7*c*x + 70*a*b^5*c^2*x - 266*a^2*b^3*c^3*x + 308*a^3*b*c^4*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (2*(-2*a^4*c^3 + b^7*x + a*b^5*(b - 7*c*x) + a^3*b*c^2*(9*b - 7*c*x) + 2*a^2*b^3*c*(-3*b + 7*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) + (6*b*c^2*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) + 3*c^2*Log[a + x*(b + c*x)])/(6*c^6)","A",1
2212,1,314,146,0.2012981,"\int \frac{x^6}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[x^6/(a + b*x + c*x^2)^4,x]","\frac{40 a^3 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+\frac{a^3 c^2 (5 b-2 c x)+a^2 b^2 c (9 c x-5 b)+a b^4 (b-6 c x)+b^6 x}{3 c^5 \left(4 a c-b^2\right) (a+x (b+c x))^3}+\frac{74 a^3 b c^3-44 a^3 c^4 x-48 a^2 b^3 c^2+48 a^2 b^2 c^3 x+12 a b^5 c-12 a b^4 c^2 x-b^7+b^6 c x}{c^4 \left(4 a c-b^2\right)^3 (a+x (b+c x))}+\frac{-59 a^3 b c^3+26 a^3 c^4 x+48 a^2 b^3 c^2-72 a^2 b^2 c^3 x-12 a b^5 c+33 a b^4 c^2 x+b^7-4 b^6 c x}{3 c^5 \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}","\frac{40 a^3 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}+\frac{10 a^2 x (2 a+b x)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}+\frac{x^5 (2 a+b x)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}-\frac{5 a x^3 (2 a+b x)}{3 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}",1,"(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 59*a^3*b*c^3 - 4*b^6*c*x + 33*a*b^4*c^2*x - 72*a^2*b^2*c^3*x + 26*a^3*c^4*x)/(3*c^5*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (-b^7 + 12*a*b^5*c - 48*a^2*b^3*c^2 + 74*a^3*b*c^3 + b^6*c*x - 12*a*b^4*c^2*x + 48*a^2*b^2*c^3*x - 44*a^3*c^4*x)/(c^4*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (b^6*x + a*b^4*(b - 6*c*x) + a^3*c^2*(5*b - 2*c*x) + a^2*b^2*c*(-5*b + 9*c*x))/(3*c^5*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (40*a^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2)","B",1
2213,1,266,145,0.2744894,"\int \frac{x^5}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[x^5/(a + b*x + c*x^2)^4,x]","\frac{1}{6} \left(-\frac{120 a^2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+\frac{3 \left(-64 a^3 c^3+38 a^2 b^2 c^2-20 a^2 b c^3 x-12 a b^4 c+b^6\right)}{c^3 \left(4 a c-b^2\right)^3 (a+x (b+c x))}-\frac{2 \left(2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x\right)}{c^4 \left(4 a c-b^2\right) (a+x (b+c x))^3}+\frac{48 a^3 c^3-61 a^2 b^2 c^2+70 a^2 b c^3 x+19 a b^4 c-40 a b^3 c^2 x-2 b^6+5 b^5 c x}{c^4 \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}\right)","-\frac{20 a^2 b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{5 a b x (2 a+b x)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{x^5 (b+2 c x)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{5 b x^3 (2 a+b x)}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}",1,"((-2*b^6 + 19*a*b^4*c - 61*a^2*b^2*c^2 + 48*a^3*c^3 + 5*b^5*c*x - 40*a*b^3*c^2*x + 70*a^2*b*c^3*x)/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (3*(b^6 - 12*a*b^4*c + 38*a^2*b^2*c^2 - 64*a^3*c^3 - 20*a^2*b*c^3*x))/(c^3*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) - (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*x)))/(c^4*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (120*a^2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/6","A",1
2214,1,572,259,1.0566825,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^4,x]","\frac{1}{3} \left(\frac{6 (b+2 c x) \left(c e^2 \left(a^2 e^2-6 a b d e+6 b^2 d^2\right)+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right)}{c \left(4 a c-b^2\right)^3 (a+x (b+c x))}+\frac{24 \left(c e^2 \left(a^2 e^2-6 a b d e+6 b^2 d^2\right)+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+\frac{b c \left(-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right)+2 c^2 \left(a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right)+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left(3 c d^2 x-2 a e (d+e x)\right)+b^4 e^4 x}{c^3 \left(4 a c-b^2\right) (a+x (b+c x))^3}+\frac{b c^2 \left(17 a^2 e^4+6 a c d e^2 (d-2 e x)+5 c^2 d^3 (d-4 e x)\right)+2 c^3 \left(-a^2 e^3 (24 d+7 e x)+6 a c d^2 e^2 x+5 c^2 d^4 x\right)+b^3 c e^2 \left(2 c d (3 d-e x)-7 a e^2\right)+2 b^2 c^2 e \left(a e^2 (9 d+5 e x)+c d^2 (6 e x-5 d)\right)+b^5 e^4-b^4 c e^3 (4 d+e x)}{c^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}\right)","-\frac{2 (d+e x) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}+\frac{8 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{(b+2 c x) (d+e x)^4}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{(d+e x)^3 \left(-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right)}{3 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}",1,"((6*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (b^5*e^4 - b^4*c*e^3*(4*d + e*x) + b*c^2*(17*a^2*e^4 + 5*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + b^3*c*e^2*(-7*a*e^2 + 2*c*d*(3*d - e*x)) + 2*b^2*c^2*e*(a*e^2*(9*d + 5*e*x) + c*d^2*(-5*d + 6*e*x)) + 2*c^3*(5*c^2*d^4*x + 6*a*c*d^2*e^2*x - a^2*e^3*(24*d + 7*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (24*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3","B",1
2215,1,401,306,0.7796739,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^4,x]","\frac{1}{6} \left(\frac{2 \left(2 c \left(a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right)+b^2 e^2 (3 c d x-a e)+b c \left(3 a e^2 (d+e x)+c d^2 (d-3 e x)\right)-b^3 e^3 x\right)}{c^2 \left(4 a c-b^2\right) (a+x (b+c x))^3}+\frac{4 c^2 \left(-6 a^2 e^3+3 a c d e^2 x+5 c^2 d^3 x\right)+3 b^2 c e \left(3 a e^2+c d (4 e x-5 d)\right)+2 b c^2 \left(3 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right)-2 b^4 e^3+b^3 c e^2 (6 d-e x)}{c^2 \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}+\frac{3 (b+2 c x) (2 c d-b e) \left(2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right)}{c \left(4 a c-b^2\right)^3 (a+x (b+c x))}-\frac{12 (b e-2 c d) \left(2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}\right)","\frac{2 (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{(b+2 c x) (d+e x)^3}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{(d+e x)^2 \left(-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right)}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}-\frac{2 x (2 c d-b e) \left(-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right)-b^2 \left(11 a e^3+25 c d^2 e\right)+6 b c d \left(13 a e^2+5 c d^2\right)-16 a c e \left(a e^2+5 c d^2\right)+3 b^3 d e^2}{3 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}",1,"((3*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (2*(-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (-2*b^4*e^3 + b^3*c*e^2*(6*d - e*x) + 4*c^2*(-6*a^2*e^3 + 5*c^2*d^3*x + 3*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x)) + 3*b^2*c*e*(3*a*e^2 + c*d*(-5*d + 4*e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (12*(-2*c*d + b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/6","A",1
2216,1,258,260,0.5494171,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^4,x]","\frac{1}{3} \left(-\frac{6 (b+2 c x) \left(c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))}+\frac{(b+2 c x) \left(c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}+\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))^3}+\frac{24 c \left(c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}\right)","-\frac{2 (b+2 c x) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{-2 x \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)-5 b \left(a e^2+c d^2\right)+8 a c d e+3 b^2 d e}{3 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}+\frac{8 c \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}",1,"(((5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (24*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3","A",1
2217,1,168,173,0.2075998,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^4,x]","\frac{\frac{120 c^2 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}+\frac{2 \left(b^2-4 a c\right)^2 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^3}+\frac{30 c (b+2 c x) (b e-2 c d)}{a+x (b+c x)}}{6 \left(b^2-4 a c\right)^3}","\frac{20 c^2 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{5 c (b+2 c x) (2 c d-b e)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}+\frac{5 (b+2 c x) (2 c d-b e)}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}-\frac{-2 a e+x (2 c d-b e)+b d}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}",1,"((2*(b^2 - 4*a*c)^2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^3 - (5*(b^2 - 4*a*c)*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (30*c*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(6*(b^2 - 4*a*c)^3)","A",1
2218,1,134,136,0.1524589,"\int \frac{1}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(a + b*x + c*x^2)^(-4),x]","-\frac{\frac{120 c^3 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{5 c \left(b^2-4 a c\right) (b+2 c x)}{(a+x (b+c x))^2}+\frac{\left(b^2-4 a c\right)^2 (b+2 c x)}{(a+x (b+c x))^3}+\frac{30 c^2 (b+2 c x)}{a+x (b+c x)}}{3 \left(b^2-4 a c\right)^3}","\frac{40 c^3 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}-\frac{10 c^2 (b+2 c x)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}+\frac{5 c (b+2 c x)}{3 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}-\frac{b+2 c x}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}",1,"-1/3*(((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x))^3 - (5*c*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (30*c^2*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c)^3","A",1
2219,1,769,771,3.2882576,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^4} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^4),x]","\frac{1}{6} \left(\frac{4 c^2 \left(6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right)+b^2 c e \left(c d (4 e x-15 d)-23 a e^2\right)+2 b c^2 \left(11 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right)+3 b^4 e^3+b^3 c e^2 (2 d+3 e x)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))^2 \left(e (a e-b d)+c d^2\right)^2}+\frac{6 \left(-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left(10 a^2 e^2-15 a b d e+6 b^2 d^2\right)-70 c^4 d e^3 \left(-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right)-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2} \left(e (a e-b d)+c d^2\right)^4}+\frac{3 \left(2 b^2 c^2 e \left(-43 a^2 e^4+2 a c d e^2 (5 e x-24 d)+c^2 d^3 (34 e x-25 d)\right)+4 b c^3 \left(19 a^2 e^4 (d-e x)+16 a c d^2 e^2 (d-3 e x)+5 c^2 d^4 (d-5 e x)\right)+8 c^3 \left(8 a^3 e^5+19 a^2 c d e^4 x+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right)-b^4 c e^3 \left(c d (d+2 e x)-23 a e^2\right)+2 b^3 c^2 e^2 \left(a e^2 (5 d+11 e x)+c d^2 (17 d-e x)\right)-2 b^6 e^5-b^5 c e^4 (d+2 e x)\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x)) \left(e (b d-a e)-c d^2\right)^3}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{\left(b^2-4 a c\right) (a+x (b+c x))^3 \left(e (b d-a e)-c d^2\right)}-\frac{3 e^7 \log (a+x (b+c x))}{\left(e (a e-b d)+c d^2\right)^4}+\frac{6 e^7 \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^4}\right)","\frac{\left(-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left(10 a^2 e^2-15 a b d e+6 b^2 d^2\right)-70 c^4 d e^3 \left(-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right)-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2} \left(a e^2-b d e+c d^2\right)^4}+\frac{-64 a^3 c^3 e^5+2 b^2 c^2 e \left(43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right)-2 c x (2 c d-b e) \left(c^2 e^2 \left(38 a^2 e^2-32 a b d e+7 b^2 d^2\right)+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right)-4 b c^3 d \left(19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right)+b^4 c e^3 \left(c d^2-23 a e^2\right)-2 b^3 c^2 d e^2 \left(5 a e^2+17 c d^2\right)+2 b^6 e^5+b^5 c d e^4}{2 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)^3}-\frac{-c x (2 c d-b e) \left(-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right)+5 a c e (2 c d-b e)^2}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2 \left(a e^2-b d e+c d^2\right)^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3 \left(a e^2-b d e+c d^2\right)}-\frac{e^7 \log \left(a+b x+c x^2\right)}{2 \left(a e^2-b d e+c d^2\right)^4}+\frac{e^7 \log (d+e x)}{\left(a e^2-b d e+c d^2\right)^4}",1,"((-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^3) + (3*b^4*e^3 + b^3*c*e^2*(2*d + 3*e*x) + 4*c^2*(6*a^2*e^3 + 5*c^2*d^3*x + 11*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 11*a*e^2*(d - e*x)) + b^2*c*e*(-23*a*e^2 + c*d*(-15*d + 4*e*x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^2) + (3*(-2*b^6*e^5 - b^5*c*e^4*(d + 2*e*x) + 8*c^3*(8*a^3*e^5 + 5*c^3*d^5*x + 16*a*c^2*d^3*e^2*x + 19*a^2*c*d*e^4*x) + 4*b*c^3*(5*c^2*d^4*(d - 5*e*x) + 16*a*c*d^2*e^2*(d - 3*e*x) + 19*a^2*e^4*(d - e*x)) - b^4*c*e^3*(-23*a*e^2 + c*d*(d + 2*e*x)) + 2*b^3*c^2*e^2*(c*d^2*(17*d - e*x) + a*e^2*(5*d + 11*e*x)) + 2*b^2*c^2*e*(-43*a^2*e^4 + 2*a*c*d*e^2*(-24*d + 5*e*x) + c^2*d^3*(-25*d + 34*e*x))))/((b^2 - 4*a*c)^3*(-(c*d^2) + e*(b*d - a*e))^3*(a + x*(b + c*x))) + (6*(40*c^7*d^7 + b^7*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*c^2*e^7 - 140*a^3*b*c^3*e^7 - 28*c^6*d^5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b^2*d^2 - 15*a*b*d*e + 10*a^2*e^2) - 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 4*a^3*e^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(7/2)*(c*d^2 + e*(-(b*d) + a*e))^4) + (6*e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (3*e^7*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^4)/6","A",1
2220,1,329,352,0.8967946,"\int \frac{1}{x^2 \left(a+b x+c x^2\right)^4} \, dx","Integrate[1/(x^2*(a + b*x + c*x^2)^4),x]","\frac{\frac{a^3 \left(-3 a b c-2 a c^2 x+b^3+b^2 c x\right)}{\left(4 a c-b^2\right) (a+x (b+c x))^3}-\frac{a^2 \left(35 a^2 b c^2+22 a^2 c^3 x-22 a b^3 c-20 a b^2 c^2 x+3 b^5+3 b^4 c x\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))^2}+\frac{3 a \left(134 a^3 b c^3+76 a^3 c^4 x-124 a^2 b^3 c^2-104 a^2 b^2 c^3 x+34 a b^5 c+32 a b^4 c^2 x-3 b^7-3 b^6 c x\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))}-\frac{12 \left(70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+6 b \log (a+x (b+c x))-\frac{3 a}{x}-12 b \log (x)}{3 a^5}","\frac{2 b \log \left(a+b x+c x^2\right)}{a^5}-\frac{4 b \log (x)}{a^5}+\frac{2 \left(7 a^2 c^2-7 a b^2 c+b^4\right)+b c x \left(2 b^2-13 a c\right)}{3 a^2 x \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}+\frac{2 \left(-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left(29 a^2 c^2-10 a b^2 c+b^4\right)-32 a b^4 c+3 b^6\right)}{3 a^3 x \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{4 \left(-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right)}{a^4 x \left(b^2-4 a c\right)^3}-\frac{4 \left(70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^5 \left(b^2-4 a c\right)^{7/2}}+\frac{-2 a c+b^2+b c x}{3 a x \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}",1,"((-3*a)/x + (a^3*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (a^2*(3*b^5 - 22*a*b^3*c + 35*a^2*b*c^2 + 3*b^4*c*x - 20*a*b^2*c^2*x + 22*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (3*a*(-3*b^7 + 34*a*b^5*c - 124*a^2*b^3*c^2 + 134*a^3*b*c^3 - 3*b^6*c*x + 32*a*b^4*c^2*x - 104*a^2*b^2*c^3*x + 76*a^3*c^4*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (12*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) - 12*b*Log[x] + 6*b*Log[a + x*(b + c*x)])/(3*a^5)","A",1
2221,1,985,388,2.5079476,"\int \frac{(d+e x)^5}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(d + e*x)^5/(a + b*x + c*x^2)^5,x]","\frac{1}{12} \left(\frac{30 (2 c d-b e) \left(7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left(8 b^2 d^2-10 a b e d+3 a^2 e^2\right)\right) (b+2 c x)}{c \left(b^2-4 a c\right)^4 (a+x (b+c x))}+\frac{3 e^5 b^6+5 c d e^4 b^5+c e^3 \left(10 c d (e x-5 d)-41 a e^2\right) b^4+10 c^2 e^2 \left(5 c (3 d-2 e x) d^2+a e^2 (6 d-e x)\right) b^3+c^2 e \left(129 a^2 e^4-30 a c d (5 d-4 e x) e^2-25 c^2 d^3 (7 d-12 e x)\right) b^2+10 c^3 \left(7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+3 a^2 e^4 (d-e x)\right) b+4 c^3 \left(35 c^3 x d^5+50 a c^2 e^2 x d^3+15 a^2 c e^4 x d-48 a^3 e^5\right)}{c^3 \left(4 a c-b^2\right)^3 (a+x (b+c x))^2}-\frac{3 \left(b^5 x e^5+b^4 (a e-5 c d x) e^4-5 b^3 c \left(a e (d+e x)-2 c d^2 x\right) e^3-2 b^2 c \left(5 c^2 x d^3-5 a c e (d+2 e x) d+2 a^2 e^3\right) e^2+2 c^2 \left(-c^3 x d^5+5 a c^2 e (d+2 e x) d^3-5 a^2 c e^3 (2 d+e x) d+a^3 e^5\right)+b c^2 \left(-c^2 (d-5 e x) d^4-10 a c e^2 (d+3 e x) d^2+5 a^2 e^4 (3 d+e x)\right)\right)}{c^4 \left(4 a c-b^2\right) (a+x (b+c x))^4}+\frac{-3 e^5 b^6+3 c e^4 (5 d+2 e x) b^5+c e^3 \left(27 a e^2-10 c d (3 d+e x)\right) b^4-10 c^2 e^2 \left(c (2 e x-3 d) d^2+5 a e^2 (2 d+e x)\right) b^3+c^2 e \left(-83 a^2 e^4+10 a c d (13 d+12 e x) e^2+5 c^2 d^3 (12 e x-7 d)\right) b^2+2 c^3 \left(7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+5 a^2 e^4 (23 d+9 e x)\right) b+4 c^3 \left(7 c^3 x d^5+10 a c^2 e^2 x d^3-5 a^2 c e^3 (16 d+9 e x) d+16 a^3 e^5\right)}{c^4 \left(b^2-4 a c\right)^2 (a+x (b+c x))^3}+\frac{120 (2 c d-b e) \left(7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left(8 b^2 d^2-10 a b e d+3 a^2 e^2\right)\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{9/2}}\right)","\frac{5 (d+e x) (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{10 (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}-\frac{(b+2 c x) (d+e x)^5}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}+\frac{(d+e x)^4 \left(-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right)}{12 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}+\frac{(d+e x)^3 \left(-c x \left(-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right)-10 b c \left(3 a e^2+7 c d^2\right)+28 a c^2 d e-10 b^3 e^2+63 b^2 c d e\right)}{12 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}",1,"((30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (5*b^5*c*d*e^4 + 3*b^6*e^5 + 4*c^3*(-48*a^3*e^5 + 35*c^3*d^5*x + 50*a*c^2*d^3*e^2*x + 15*a^2*c*d*e^4*x) + b^2*c^2*e*(129*a^2*e^4 - 25*c^2*d^3*(7*d - 12*e*x) - 30*a*c*d*e^2*(5*d - 4*e*x)) + 10*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 3*a^2*e^4*(d - e*x)) + 10*b^3*c^2*e^2*(5*c*d^2*(3*d - 2*e*x) + a*e^2*(6*d - e*x)) + b^4*c*e^3*(-41*a*e^2 + 10*c*d*(-5*d + e*x)))/(c^3*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) - (3*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x))))/(c^4*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (-3*b^6*e^5 + 3*b^5*c*e^4*(5*d + 2*e*x) + b^4*c*e^3*(27*a*e^2 - 10*c*d*(3*d + e*x)) - 10*b^3*c^2*e^2*(5*a*e^2*(2*d + e*x) + c*d^2*(-3*d + 2*e*x)) + 4*c^3*(16*a^3*e^5 + 7*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(16*d + 9*e*x)) + 2*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(23*d + 9*e*x)) + b^2*c^2*e*(-83*a^2*e^4 + 5*c^2*d^3*(-7*d + 12*e*x) + 10*a*c*d*e^2*(13*d + 12*e*x)))/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (120*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12","B",1
2222,1,713,545,1.8910312,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^5,x]","\frac{1}{12} \left(\frac{6 (b+2 c x) \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{c \left(b^2-4 a c\right)^4 (a+x (b+c x))}+\frac{(b+2 c x) \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{c^2 \left(4 a c-b^2\right)^3 (a+x (b+c x))^2}+\frac{24 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{9/2}}+\frac{3 \left(b c \left(-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right)+2 c^2 \left(a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right)+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left(3 c d^2 x-2 a e (d+e x)\right)+b^4 e^4 x\right)}{c^3 \left(4 a c-b^2\right) (a+x (b+c x))^4}+\frac{2 b c^2 \left(23 a^2 e^4+6 a c d e^2 (d-2 e x)+7 c^2 d^3 (d-4 e x)\right)-4 c^3 \left(a^2 e^3 (32 d+9 e x)-6 a c d^2 e^2 x-7 c^2 d^4 x\right)+2 b^3 c e^2 \left(c d (9 d-4 e x)-10 a e^2\right)+4 b^2 c^2 e \left(a e^2 (13 d+6 e x)+c d^2 (9 e x-7 d)\right)+3 b^5 e^4-2 b^4 c e^3 (6 d+e x)}{c^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^3}\right)","-\frac{2 \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}-\frac{-x \left(2 c^2 e^2 \left(-18 a^2 e^2-40 a b d e+305 b^2 d^2\right)-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right)-10 b c \left(11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right)-5 b^3 \left(5 a e^4+19 c d^2 e^2\right)+4 b^2 c d e \left(83 a e^2+70 c d^2\right)+16 a c^2 d e \left(16 a e^2+35 c d^2\right)+6 b^4 d e^3}{6 \left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{(b+2 c x) (d+e x)^4}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}+\frac{(d+e x)^3 \left(-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right)}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}+\frac{(d+e x)^2 \left(-c x \left(-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right)-b c \left(23 a e^2+35 c d^2\right)+28 a c^2 d e-3 b^3 e^2+28 b^2 c d e\right)}{6 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}",1,"(((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c^2*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) + (6*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (3*b^5*e^4 - 2*b^4*c*e^3*(6*d + e*x) + 2*b^3*c*e^2*(-10*a*e^2 + c*d*(9*d - 4*e*x)) + 2*b*c^2*(23*a^2*e^4 + 7*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(a*e^2*(13*d + 6*e*x) + c*d^2*(-7*d + 9*e*x)) - 4*c^3*(-7*c^2*d^4*x - 6*a*c*d^2*e^2*x + a^2*e^3*(32*d + 9*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (24*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12","A",1
2223,1,467,378,1.2762943,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^5,x]","\frac{1}{12} \left(\frac{3 \left(2 c \left(a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right)+b^2 e^2 (3 c d x-a e)+b c \left(3 a e^2 (d+e x)+c d^2 (d-3 e x)\right)-b^3 e^3 x\right)}{c^2 \left(4 a c-b^2\right) (a+x (b+c x))^4}+\frac{4 c^2 \left(-8 a^2 e^3+3 a c d e^2 x+7 c^2 d^3 x\right)+b^2 c e \left(13 a e^2-3 c d (7 d-6 e x)\right)+2 b c^2 \left(3 a e^2 (d-e x)+7 c d^2 (d-3 e x)\right)-3 b^4 e^3+b^3 c e^2 (9 d-2 e x)}{c^2 \left(b^2-4 a c\right)^2 (a+x (b+c x))^3}+\frac{30 (b+2 c x) (2 c d-b e) \left(c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right)}{\left(b^2-4 a c\right)^4 (a+x (b+c x))}+\frac{5 (b+2 c x) (2 c d-b e) \left(c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right)}{c \left(4 a c-b^2\right)^3 (a+x (b+c x))^2}+\frac{120 c (2 c d-b e) \left(c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{9/2}}\right)","\frac{5 (b+2 c x) (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{2 \left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{10 c (2 c d-b e) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}-\frac{(b+2 c x) (d+e x)^3}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}+\frac{(d+e x)^2 \left(-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right)}{12 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}-\frac{2 x (2 c d-b e) \left(-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right)-b^2 \left(27 a e^3+49 c d^2 e\right)+2 b c d \left(99 a e^2+35 c d^2\right)-32 a c e \left(a e^2+7 c d^2\right)+3 b^3 d e^2}{12 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}",1,"((5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) + (30*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (-3*b^4*e^3 + b^3*c*e^2*(9*d - 2*e*x) + 4*c^2*(-8*a^2*e^3 + 7*c^2*d^3*x + 3*a*c*d*e^2*x) + b^2*c*e*(13*a*e^2 - 3*c*d*(7*d - 6*e*x)) + 2*b*c^2*(7*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (3*(-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (120*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12","A",1
2224,1,326,330,0.8918789,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^5,x]","\frac{1}{12} \left(\frac{30 c (b+2 c x) \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right)}{\left(b^2-4 a c\right)^4 (a+x (b+c x))}-\frac{5 (b+2 c x) \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))^2}+\frac{3 \left(a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right)}{c \left(4 a c-b^2\right) (a+x (b+c x))^4}+\frac{(b+2 c x) \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))^3}+\frac{120 c^2 \left(2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{9/2}}\right)","\frac{5 c (b+2 c x) \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{2 \left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{5 (b+2 c x) \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{12 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}-\frac{-x \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)-7 b \left(a e^2+c d^2\right)+12 a c d e+4 b^2 d e}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}-\frac{10 c^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}",1,"(((14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) - (5*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))^2) + (30*c*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (120*c^2*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12","A",1
2225,1,209,219,0.306625,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^5,x]","\frac{-\frac{840 c^3 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{35 c \left(b^2-4 a c\right) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}-\frac{7 \left(b^2-4 a c\right)^2 (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^3}+\frac{3 \left(b^2-4 a c\right)^3 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^4}+\frac{210 c^2 (b+2 c x) (2 c d-b e)}{a+x (b+c x)}}{12 \left(b^2-4 a c\right)^4}","-\frac{70 c^3 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}+\frac{35 c^2 (b+2 c x) (2 c d-b e)}{2 \left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{35 c (b+2 c x) (2 c d-b e)}{12 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}+\frac{7 (b+2 c x) (2 c d-b e)}{12 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}-\frac{-2 a e+x (2 c d-b e)+b d}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}",1,"((3*(b^2 - 4*a*c)^3*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^4 - (7*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^3 + (35*c*(b^2 - 4*a*c)*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (210*c^2*(2*c*d - b*e)*(b + 2*c*x))/(a + x*(b + c*x)) - (840*c^3*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)","A",1
2226,1,167,171,0.1759858,"\int \frac{1}{\left(a+b x+c x^2\right)^5} \, dx","Integrate[(a + b*x + c*x^2)^(-5),x]","\frac{\frac{1680 c^4 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{70 c^2 \left(b^2-4 a c\right) (b+2 c x)}{(a+x (b+c x))^2}+\frac{14 c \left(b^2-4 a c\right)^2 (b+2 c x)}{(a+x (b+c x))^3}-\frac{3 \left(b^2-4 a c\right)^3 (b+2 c x)}{(a+x (b+c x))^4}+\frac{420 c^3 (b+2 c x)}{a+x (b+c x)}}{12 \left(b^2-4 a c\right)^4}","-\frac{140 c^4 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2}}+\frac{35 c^3 (b+2 c x)}{\left(b^2-4 a c\right)^4 \left(a+b x+c x^2\right)}-\frac{35 c^2 (b+2 c x)}{6 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^2}+\frac{7 c (b+2 c x)}{6 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^3}-\frac{b+2 c x}{4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^4}",1,"((-3*(b^2 - 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x))^4 + (14*c*(b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x))^3 - (70*c^2*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (420*c^3*(b + 2*c*x))/(a + x*(b + c*x)) + (1680*c^4*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)","A",1
2227,1,1601,1324,6.5838356,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^5} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^5),x]","\frac{\log (d+e x) e^9}{\left(c d^2-b e d+a e^2\right)^5}-\frac{\log \left(c x^2+b x+a\right) e^9}{2 \left(c d^2-b e d+a e^2\right)^5}+\frac{\left(-140 c^9 d^9+630 b c^8 e d^8-720 a c^8 e^2 d^7-1080 b^2 c^7 e^2 d^7+2520 a b c^7 e^3 d^6+840 b^3 c^6 e^3 d^6-1512 a^2 c^7 e^4 d^5-3024 a b^2 c^6 e^4 d^5-252 b^4 c^5 e^4 d^5+3780 a^2 b c^6 e^5 d^4+1260 a b^3 c^5 e^5 d^4-1680 a^3 c^6 e^6 d^3-2520 a^2 b^2 c^5 e^6 d^3+2520 a^3 b c^5 e^7 d^2-1260 a^4 c^5 e^8 d+b^9 e^9+630 a^4 b c^4 e^9-420 a^3 b^3 c^3 e^9+126 a^2 b^5 c^2 e^9-18 a b^7 c e^9\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(b^2-4 a c\right)^4 \sqrt{4 a c-b^2} \left(-c d^2+b e d-a e^2\right)^5}+\frac{2 e^7 b^8+c d e^6 b^7+2 c e^7 x b^7-31 a c e^7 b^6+c^2 d^2 e^5 b^6+2 c^2 d e^6 x b^6-14 a c^2 d e^6 b^5+c^3 d^3 e^4 b^5-30 a c^2 e^7 x b^5+2 c^3 d^2 e^5 x b^5+178 a^2 c^2 e^7 b^4-13 a c^3 d^2 e^5 b^4-125 c^4 d^4 e^3 b^4-28 a c^3 d e^6 x b^4+2 c^4 d^3 e^4 x b^4+69 a^2 c^3 d e^6 b^3+492 a c^4 d^3 e^4 b^3+295 c^5 d^5 e^2 b^3+164 a^2 c^3 e^7 x b^3-26 a c^4 d^2 e^5 x b^3-250 c^5 d^4 e^3 x b^3-443 a^3 c^3 e^7 b^2-699 a^2 c^4 d^2 e^5 b^2-725 a c^5 d^4 e^3 b^2-245 c^6 d^6 e b^2+138 a^2 c^4 d e^6 x b^2+984 a c^5 d^3 e^4 x b^2+590 c^6 d^5 e^2 x b^2+70 c^7 d^7 b+374 a^3 c^4 d e^6 b+466 a^2 c^5 d^3 e^4 b+290 a c^6 d^5 e^2 b-374 a^3 c^4 e^7 x b-1398 a^2 c^5 d^2 e^5 x b-1450 a c^6 d^4 e^3 x b-490 c^7 d^6 e x b+256 a^4 c^4 e^7+140 c^8 d^7 x+748 a^3 c^5 d e^6 x+932 a^2 c^6 d^3 e^4 x+580 a c^7 d^5 e^2 x}{2 \left(4 a c-b^2\right)^4 \left(c d^2-b e d+a e^2\right)^4 \left(c x^2+b x+a\right)}+\frac{-6 e^5 b^6-4 c d e^4 b^5-6 c e^5 x b^5+70 a c e^5 b^4-5 c^2 d^2 e^3 b^4-8 c^2 d e^4 x b^4+42 a c^2 d e^4 b^3+120 c^3 d^3 e^2 b^3+68 a c^2 e^5 x b^3-10 c^3 d^2 e^3 x b^3-267 a^2 c^2 e^5 b^2-330 a c^3 d^2 e^3 b^2-175 c^4 d^4 e b^2+84 a c^3 d e^4 x b^2+240 c^4 d^3 e^2 x b^2+70 c^5 d^5 b+246 a^2 c^3 d e^4 b+220 a c^4 d^3 e^2 b-246 a^2 c^3 e^5 x b-660 a c^4 d^2 e^3 x b-350 c^5 d^4 e x b+192 a^3 c^3 e^5+140 c^6 d^5 x+492 a^2 c^4 d e^4 x+440 a c^5 d^3 e^2 x}{12 \left(4 a c-b^2\right)^3 \left(c d^2-b e d+a e^2\right)^3 \left(c x^2+b x+a\right)^2}+\frac{4 e^3 b^4+3 c d e^2 b^3+4 c e^3 x b^3-31 a c e^3 b^2-21 c^2 d^2 e b^2+6 c^2 d e^2 x b^2+14 c^3 d^3 b+30 a c^2 d e^2 b-30 a c^2 e^3 x b-42 c^3 d^2 e x b+32 a^2 c^2 e^3+28 c^4 d^3 x+60 a c^3 d e^2 x}{12 \left(4 a c-b^2\right)^2 \left(c d^2-b e d+a e^2\right)^2 \left(c x^2+b x+a\right)^3}+\frac{-e b^2+c d b-c e x b+2 a c e+2 c^2 d x}{4 \left(4 a c-b^2\right) \left(c d^2-b e d+a e^2\right) \left(c x^2+b x+a\right)^4}","\frac{\log (d+e x) e^9}{\left(c d^2-b e d+a e^2\right)^5}-\frac{\log \left(c x^2+b x+a\right) e^9}{2 \left(c d^2-b e d+a e^2\right)^5}-\frac{\left(140 c^9 d^9-90 c^8 e (7 b d-8 a e) d^7+72 c^7 e^2 \left(15 b^2 d^2-35 a b e d+21 a^2 e^2\right) d^5-84 c^6 e^3 \left(10 b^3 d^3-36 a b^2 e d^2+45 a^2 b e^2 d-20 a^3 e^3\right) d^3+252 c^5 e^4 \left(b^4 d^4-5 a b^3 e d^3+10 a^2 b^2 e^2 d^2-10 a^3 b e^3 d+5 a^4 e^4\right) d-b^9 e^9-630 a^4 b c^4 e^9+420 a^3 b^3 c^3 e^9-126 a^2 b^5 c^2 e^9+18 a b^7 c e^9\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2} \left(c d^2-e (b d-a e)\right)^5}+\frac{2 e^7 b^8+c d e^6 b^7+c e^5 \left(c d^2-31 a e^2\right) b^6+c^2 d e^4 \left(c d^2-14 a e^2\right) b^5-c^2 e^3 \left(125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right) b^4+c^3 d e^2 \left(295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right) b^3-c^3 e \left(245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right) b^2+2 c^4 d \left(35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left(35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left(95 b^2 d^2-290 a b e d+233 a^2 e^2\right) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left(7 b^2 d^2-44 a b e d+82 a^2 e^2\right)-c^3 e^3 \left(15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right)\right) x}{2 \left(b^2-4 a c\right)^4 \left(c d^2-b e d+a e^2\right)^4 \left(c x^2+b x+a\right)}+\frac{5 a c e \left(7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right) (2 c d-b e)^2-2 c \left(35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left(25 b^2 d^2-110 a b e d+123 a^2 e^2\right)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left(5 b^2 d^2-25 a b e d+32 a^2 e^2\right)\right)}{12 \left(b^2-4 a c\right)^3 \left(c d^2-b e d+a e^2\right)^3 \left(c x^2+b x+a\right)^2}-\frac{7 a c e (2 c d-b e)^2-2 c \left(7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right)}{12 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 \left(c x^2+b x+a\right)^3}-\frac{-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \left(c x^2+b x+a\right)^4}",1,"(b*c*d - b^2*e + 2*a*c*e + 2*c^2*d*x - b*c*e*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^4) + (14*b*c^3*d^3 - 21*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 + 30*a*b*c^2*d*e^2 + 4*b^4*e^3 - 31*a*b^2*c*e^3 + 32*a^2*c^2*e^3 + 28*c^4*d^3*x - 42*b*c^3*d^2*e*x + 6*b^2*c^2*d*e^2*x + 60*a*c^3*d*e^2*x + 4*b^3*c*e^3*x - 30*a*b*c^2*e^3*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^3) + (70*b*c^5*d^5 - 175*b^2*c^4*d^4*e + 120*b^3*c^3*d^3*e^2 + 220*a*b*c^4*d^3*e^2 - 5*b^4*c^2*d^2*e^3 - 330*a*b^2*c^3*d^2*e^3 - 4*b^5*c*d*e^4 + 42*a*b^3*c^2*d*e^4 + 246*a^2*b*c^3*d*e^4 - 6*b^6*e^5 + 70*a*b^4*c*e^5 - 267*a^2*b^2*c^2*e^5 + 192*a^3*c^3*e^5 + 140*c^6*d^5*x - 350*b*c^5*d^4*e*x + 240*b^2*c^4*d^3*e^2*x + 440*a*c^5*d^3*e^2*x - 10*b^3*c^3*d^2*e^3*x - 660*a*b*c^4*d^2*e^3*x - 8*b^4*c^2*d*e^4*x + 84*a*b^2*c^3*d*e^4*x + 492*a^2*c^4*d*e^4*x - 6*b^5*c*e^5*x + 68*a*b^3*c^2*e^5*x - 246*a^2*b*c^3*e^5*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)^2) + (70*b*c^7*d^7 - 245*b^2*c^6*d^6*e + 295*b^3*c^5*d^5*e^2 + 290*a*b*c^6*d^5*e^2 - 125*b^4*c^4*d^4*e^3 - 725*a*b^2*c^5*d^4*e^3 + b^5*c^3*d^3*e^4 + 492*a*b^3*c^4*d^3*e^4 + 466*a^2*b*c^5*d^3*e^4 + b^6*c^2*d^2*e^5 - 13*a*b^4*c^3*d^2*e^5 - 699*a^2*b^2*c^4*d^2*e^5 + b^7*c*d*e^6 - 14*a*b^5*c^2*d*e^6 + 69*a^2*b^3*c^3*d*e^6 + 374*a^3*b*c^4*d*e^6 + 2*b^8*e^7 - 31*a*b^6*c*e^7 + 178*a^2*b^4*c^2*e^7 - 443*a^3*b^2*c^3*e^7 + 256*a^4*c^4*e^7 + 140*c^8*d^7*x - 490*b*c^7*d^6*e*x + 590*b^2*c^6*d^5*e^2*x + 580*a*c^7*d^5*e^2*x - 250*b^3*c^5*d^4*e^3*x - 1450*a*b*c^6*d^4*e^3*x + 2*b^4*c^4*d^3*e^4*x + 984*a*b^2*c^5*d^3*e^4*x + 932*a^2*c^6*d^3*e^4*x + 2*b^5*c^3*d^2*e^5*x - 26*a*b^3*c^4*d^2*e^5*x - 1398*a^2*b*c^5*d^2*e^5*x + 2*b^6*c^2*d*e^6*x - 28*a*b^4*c^3*d*e^6*x + 138*a^2*b^2*c^4*d*e^6*x + 748*a^3*c^5*d*e^6*x + 2*b^7*c*e^7*x - 30*a*b^5*c^2*e^7*x + 164*a^2*b^3*c^3*e^7*x - 374*a^3*b*c^4*e^7*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x + c*x^2)) + ((-140*c^9*d^9 + 630*b*c^8*d^8*e - 1080*b^2*c^7*d^7*e^2 - 720*a*c^8*d^7*e^2 + 840*b^3*c^6*d^6*e^3 + 2520*a*b*c^7*d^6*e^3 - 252*b^4*c^5*d^5*e^4 - 3024*a*b^2*c^6*d^5*e^4 - 1512*a^2*c^7*d^5*e^4 + 1260*a*b^3*c^5*d^4*e^5 + 3780*a^2*b*c^6*d^4*e^5 - 2520*a^2*b^2*c^5*d^3*e^6 - 1680*a^3*c^6*d^3*e^6 + 2520*a^3*b*c^5*d^2*e^7 - 1260*a^4*c^5*d*e^8 + b^9*e^9 - 18*a*b^7*c*e^9 + 126*a^2*b^5*c^2*e^9 - 420*a^3*b^3*c^3*e^9 + 630*a^4*b*c^4*e^9)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*c)^4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^5)","A",1
2228,1,2147,1761,7.0642579,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^5} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^5),x]","\text{Result too large to show}","\frac{5 (2 c d-b e) \log (d+e x) e^9}{\left(c d^2-b e d+a e^2\right)^6}-\frac{5 (2 c d-b e) \log \left(c x^2+b x+a\right) e^9}{2 \left(c d^2-b e d+a e^2\right)^6}+\frac{5 \left(14 c^8 d^8-4 c^7 e (14 b d-19 a e) d^6+c^6 e^2 \left(79 b^2 d^2-228 a b e d+176 a^2 e^2\right) d^4-c^5 e^3 \left(41 b^3 d^3-197 a b^2 e d^2+352 a^2 b e^2 d-244 a^3 e^3\right) d^2-b^8 e^8+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left(b^2 d^2-16 a b e d-82 a^2 e^2\right)+b^2 c^3 e^5 \left(b^3 d^3-15 a b^2 e d^2+95 a^2 b e^2 d+187 a^3 e^3\right)+c^4 e^4 \left(b^4 d^4-14 a b^3 e d^3+81 a^2 b^2 e^2 d^2-244 a^3 b e^3 d-126 a^4 e^4\right)\right) e}{\left(b^2-4 a c\right)^4 \left(c d^2-b e d+a e^2\right)^5 (d+e x)}-\frac{5 \left(28 c^{10} d^{10}-20 c^9 e (7 b d-9 a e) d^8+18 c^8 e^2 \left(15 b^2 d^2-40 a b e d+28 a^2 e^2\right) d^6-24 c^7 e^3 \left(10 b^3 d^3-42 a b^2 e d^2+63 a^2 b e^2 d-35 a^3 e^3\right) d^4+84 c^6 e^4 \left(b^4 d^4-6 a b^3 e d^3+15 a^2 b^2 e^2 d^2-20 a^3 b e^3 d+15 a^4 e^4\right) d^2+b^{10} e^{10}-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{9/2} \left(c d^2-e (b d-a e)\right)^6}-\frac{5 \left(2 a c e \left(14 c^4 d^4-4 c^3 e (7 b d-12 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left(3 b^2 d^2-16 a b e d+22 a^2 e^2\right)\right) (2 c d-b e)^2-3 c \left(14 c^6 d^6-2 c^5 e (21 b d-31 a e) d^4+c^4 e^2 \left(37 b^2 d^2-124 a b e d+114 a^2 e^2\right) d^2-b^6 e^6-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left(3 b^2 d^2-26 a b e d+69 a^2 e^2\right)-2 c^3 e^3 \left(2 b^3 d^3-18 a b^2 e d^2+57 a^2 b e^2 d-65 a^3 e^3\right)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(42 c^6 d^6-2 c^5 e (49 b d-65 a e) d^4+c^4 e^2 \left(55 b^2 d^2-164 a b e d+150 a^2 e^2\right) d^2-3 b^6 e^6-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left(b^2 d^2+16 a b e d-123 a^2 e^2\right)+6 c^3 e^3 \left(b^3 d^3-4 a b^2 e d^2-3 a^2 b e^2 d+21 a^3 e^3\right)\right)\right)}{6 \left(b^2-4 a c\right)^4 \left(c d^2-b e d+a e^2\right)^4 (d+e x) \left(c x^2+b x+a\right)}+\frac{3 a c e \left(14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right) (2 c d-b e)^2-5 c \left(14 c^4 d^4-4 c^3 e (7 b d-12 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left(3 b^2 d^2-16 a b e d+22 a^2 e^2\right)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(70 c^4 d^4-2 c^3 e (49 b d-78 a e) d^2+10 b^4 e^4+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left(b^2 d^2-18 a b e d+42 a^2 e^2\right)\right)}{12 \left(b^2-4 a c\right)^3 \left(c d^2-b e d+a e^2\right)^3 (d+e x) \left(c x^2+b x+a\right)^2}-\frac{8 a c e (2 c d-b e)^2-c \left(14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right)}{12 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 (d+e x) \left(c x^2+b x+a\right)^3}-\frac{-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) (d+e x) \left(c x^2+b x+a\right)^4}",1,"-(e^9/((c*d^2 - b*d*e + a*e^2)^5*(d + e*x))) + (b*c^2*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 - 3*a*b*c*e^2 + 2*c^3*d^2*x - 2*b*c^2*d*e*x + b^2*c*e^2*x - 2*a*c^2*e^2*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^4) + (14*b*c^4*d^4 - 28*b^2*c^3*d^3*e + 9*b^3*c^2*d^2*e^2 + 48*a*b*c^3*d^2*e^2 + 13*b^4*c*d*e^3 - 112*a*b^2*c^2*d*e^3 + 128*a^2*c^3*d*e^3 - 8*b^5*e^4 + 59*a*b^3*c*e^4 - 94*a^2*b*c^2*e^4 + 28*c^5*d^4*x - 56*b*c^4*d^3*e*x + 18*b^2*c^3*d^2*e^2*x + 96*a*c^4*d^2*e^2*x + 10*b^3*c^2*d*e^3*x - 96*a*b*c^3*d*e^3*x - 8*b^4*c*e^4*x + 54*a*b^2*c^2*e^4*x - 60*a^2*c^3*e^4*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)^3) + (70*b*c^6*d^6 - 210*b^2*c^5*d^5*e + 185*b^3*c^4*d^4*e^2 + 310*a*b*c^5*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 620*a*b^2*c^4*d^3*e^3 - 15*b^5*c^2*d^2*e^4 + 180*a*b^3*c^3*d^2*e^4 + 570*a^2*b*c^4*d^2*e^4 - 28*b^6*c*d*e^5 + 346*a*b^4*c^2*d*e^5 - 1434*a^2*b^2*c^3*d*e^5 + 1152*a^3*c^4*d*e^5 + 18*b^7*e^6 - 206*a*b^5*c*e^6 + 759*a^2*b^3*c^2*e^6 - 822*a^3*b*c^3*e^6 + 140*c^7*d^6*x - 420*b*c^6*d^5*e*x + 370*b^2*c^5*d^4*e^2*x + 620*a*c^6*d^4*e^2*x - 40*b^3*c^4*d^3*e^3*x - 1240*a*b*c^5*d^3*e^3*x - 30*b^4*c^3*d^2*e^4*x + 360*a*b^2*c^4*d^2*e^4*x + 1140*a^2*c^5*d^2*e^4*x - 20*b^5*c^2*d*e^5*x + 260*a*b^3*c^3*d*e^5*x - 1140*a^2*b*c^4*d*e^5*x + 18*b^6*c*e^6*x - 196*a*b^4*c^2*e^6*x + 654*a^2*b^2*c^3*e^6*x - 492*a^3*c^4*e^6*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x + c*x^2)^2) + (70*b*c^8*d^8 - 280*b^2*c^7*d^7*e + 395*b^3*c^6*d^6*e^2 + 380*a*b*c^7*d^6*e^2 - 205*b^4*c^5*d^5*e^3 - 1140*a*b^2*c^6*d^5*e^3 + 5*b^5*c^4*d^4*e^4 + 985*a*b^3*c^5*d^4*e^4 + 880*a^2*b*c^6*d^4*e^4 + 5*b^6*c^3*d^3*e^5 - 70*a*b^4*c^4*d^3*e^5 - 1760*a^2*b^2*c^5*d^3*e^5 + 5*b^7*c^2*d^2*e^6 - 75*a*b^5*c^3*d^2*e^6 + 405*a^2*b^3*c^4*d^2*e^6 + 1220*a^3*b*c^5*d^2*e^6 + 13*b^8*c*d*e^7 - 208*a*b^6*c^2*d*e^7 + 1243*a^2*b^4*c^3*d*e^7 - 3268*a^3*b^2*c^4*d*e^7 + 2048*a^4*c^5*d*e^7 - 8*b^9*e^8 + 123*a*b^7*c*e^8 - 698*a^2*b^5*c^2*e^8 + 1703*a^3*b^3*c^3*e^8 - 1398*a^4*b*c^4*e^8 + 140*c^9*d^8*x - 560*b*c^8*d^7*e*x + 790*b^2*c^7*d^6*e^2*x + 760*a*c^8*d^6*e^2*x - 410*b^3*c^6*d^5*e^3*x - 2280*a*b*c^7*d^5*e^3*x + 10*b^4*c^5*d^4*e^4*x + 1970*a*b^2*c^6*d^4*e^4*x + 1760*a^2*c^7*d^4*e^4*x + 10*b^5*c^4*d^3*e^5*x - 140*a*b^3*c^5*d^3*e^5*x - 3520*a^2*b*c^6*d^3*e^5*x + 10*b^6*c^3*d^2*e^6*x - 150*a*b^4*c^4*d^2*e^6*x + 810*a^2*b^2*c^5*d^2*e^6*x + 2440*a^3*c^6*d^2*e^6*x + 10*b^7*c^2*d*e^7*x - 160*a*b^5*c^3*d*e^7*x + 950*a^2*b^3*c^4*d*e^7*x - 2440*a^3*b*c^5*d*e^7*x - 8*b^8*c*e^8*x + 118*a*b^6*c^2*e^8*x - 628*a^2*b^4*c^3*e^8*x + 1358*a^3*b^2*c^4*e^8*x - 748*a^4*c^5*e^8*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(a + b*x + c*x^2)) + (5*(28*c^10*d^10 - 140*b*c^9*d^9*e + 270*b^2*c^8*d^8*e^2 + 180*a*c^9*d^8*e^2 - 240*b^3*c^7*d^7*e^3 - 720*a*b*c^8*d^7*e^3 + 84*b^4*c^6*d^6*e^4 + 1008*a*b^2*c^7*d^6*e^4 + 504*a^2*c^8*d^6*e^4 - 504*a*b^3*c^6*d^5*e^5 - 1512*a^2*b*c^7*d^5*e^5 + 1260*a^2*b^2*c^6*d^4*e^6 + 840*a^3*c^7*d^4*e^6 - 1680*a^3*b*c^6*d^3*e^7 + 1260*a^4*c^6*d^2*e^8 - 2*b^9*c*d*e^9 + 36*a*b^7*c^2*d*e^9 - 252*a^2*b^5*c^3*d*e^9 + 840*a^3*b^3*c^4*d*e^9 - 1260*a^4*b*c^5*d*e^9 + b^10*e^10 - 18*a*b^8*c*e^10 + 126*a^2*b^6*c^2*e^10 - 420*a^3*b^4*c^3*e^10 + 630*a^4*b^2*c^4*e^10 - 252*a^5*c^5*e^10)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*c)^4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^6) + (5*(2*c*d*e^9 - b*e^10)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*(2*c*d*e^9 - b*e^10)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)","A",1
2229,1,78,89,0.1065785,"\int \frac{1}{(1+2 x) \left(2+3 x+5 x^2\right)^3} \, dx","Integrate[1/((1 + 2*x)*(2 + 3*x + 5*x^2)^3),x]","\frac{8 \left(-59582 \log \left(4 \left(5 x^2+3 x+2\right)\right)+\frac{217 \left(45800 x^3+79660 x^2+53968 x+28901\right)}{16 \left(5 x^2+3 x+2\right)^2}+119164 \log (2 x+1)+15703 \sqrt{31} \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)\right)}{10218313}","\frac{20 x+37}{434 \left(5 x^2+3 x+2\right)^2}+\frac{2 (2290 x+2609)}{47089 \left(5 x^2+3 x+2\right)}-\frac{16}{343} \log \left(5 x^2+3 x+2\right)+\frac{32}{343} \log (2 x+1)+\frac{125624 \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)}{329623 \sqrt{31}}",1,"(8*((217*(28901 + 53968*x + 79660*x^2 + 45800*x^3))/(16*(2 + 3*x + 5*x^2)^2) + 15703*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 119164*Log[1 + 2*x] - 59582*Log[4*(2 + 3*x + 5*x^2)]))/10218313","A",1
2230,1,98,114,0.0750162,"\int \frac{1}{(1+2 x)^2 \left(2+3 x+5 x^2\right)^3} \, dx","Integrate[1/((1 + 2*x)^2*(2 + 3*x + 5*x^2)^3),x]","\frac{4 \left(-\frac{47089 (270 x-43)}{8 \left(5 x^2+3 x+2\right)^2}-\frac{217 (51910 x-15179)}{4 \left(5 x^2+3 x+2\right)}-1429968 \log \left(4 \left(5 x^2+3 x+2\right)\right)-\frac{1668296}{2 x+1}+2859936 \log (2 x+1)-266253 \sqrt{31} \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)\right)}{71528191}","\frac{20 x+37}{434 (2 x+1) \left(5 x^2+3 x+2\right)^2}+\frac{5820 x+6427}{47089 (2 x+1) \left(5 x^2+3 x+2\right)}-\frac{192 \log \left(5 x^2+3 x+2\right)}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)}{2307361 \sqrt{31}}",1,"(4*(-1668296/(1 + 2*x) - (47089*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^2) - (217*(-15179 + 51910*x))/(4*(2 + 3*x + 5*x^2)) - 266253*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 2859936*Log[1 + 2*x] - 1429968*Log[4*(2 + 3*x + 5*x^2)]))/71528191","A",1
2231,1,88,110,0.1304103,"\int \frac{1}{(1+2 x) \left(2+3 x+5 x^2\right)^4} \, dx","Integrate[1/((1 + 2*x)*(2 + 3*x + 5*x^2)^4),x]","\frac{16 \left(-11082252 \log \left(4 \left(5 x^2+3 x+2\right)\right)+\frac{217 \left(60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right)}{16 \left(5 x^2+3 x+2\right)^3}+22164504 \log (2 x+1)+3563883 \sqrt{31} \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)\right)}{6652121763}","\frac{20 x+37}{651 \left(5 x^2+3 x+2\right)^3}+\frac{4 (203230 x+180133)}{10218313 \left(5 x^2+3 x+2\right)}+\frac{4 (1805 x+1983)}{141267 \left(5 x^2+3 x+2\right)^2}-\frac{64 \log \left(5 x^2+3 x+2\right)}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)}{71528191 \sqrt{31}}",1,"(16*((217*(13831165 + 44933184*x + 105257844*x^2 + 143405620*x^3 + 127202700*x^4 + 60969000*x^5))/(16*(2 + 3*x + 5*x^2)^3) + 3563883*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 22164504*Log[1 + 2*x] - 11082252*Log[4*(2 + 3*x + 5*x^2)]))/6652121763","A",1
2232,1,119,142,0.1013037,"\int \frac{1}{(1+2 x)^2 \left(2+3 x+5 x^2\right)^4} \, dx","Integrate[1/((1 + 2*x)^2*(2 + 3*x + 5*x^2)^4),x]","\frac{8 \left(-\frac{10218313 (270 x-43)}{8 \left(5 x^2+3 x+2\right)^3}-\frac{651 (3736330 x-1739037)}{4 \left(5 x^2+3 x+2\right)}-\frac{141267 (27530 x-7117)}{8 \left(5 x^2+3 x+2\right)^2}-354632064 \log \left(4 \left(5 x^2+3 x+2\right)\right)-\frac{310303056}{2 x+1}+709264128 \log (2 x+1)-43521369 \sqrt{31} \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)\right)}{46564852341}","\frac{20 x+37}{651 (2 x+1) \left(5 x^2+3 x+2\right)^3}+\frac{2 (603620 x+504757)}{10218313 (2 x+1) \left(5 x^2+3 x+2\right)}+\frac{2820 x+3047}{47089 (2 x+1) \left(5 x^2+3 x+2\right)^2}-\frac{1024 \log \left(5 x^2+3 x+2\right)}{16807}-\frac{6802312}{71528191 (2 x+1)}+\frac{2048 \log (2 x+1)}{16807}-\frac{116056984 \tan ^{-1}\left(\frac{10 x+3}{\sqrt{31}}\right)}{500697337 \sqrt{31}}",1,"(8*(-310303056/(1 + 2*x) - (10218313*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^3) - (141267*(-7117 + 27530*x))/(8*(2 + 3*x + 5*x^2)^2) - (651*(-1739037 + 3736330*x))/(4*(2 + 3*x + 5*x^2)) - 43521369*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 709264128*Log[1 + 2*x] - 354632064*Log[4*(2 + 3*x + 5*x^2)]))/46564852341","A",1
2233,1,47,47,0.0385864,"\int \frac{7-3 x}{-5+2 x+x^2} \, dx","Integrate[(7 - 3*x)/(-5 + 2*x + x^2),x]","\frac{1}{6} \left(5 \sqrt{6}-9\right) \log \left(-x+\sqrt{6}-1\right)+\frac{1}{6} \left(-9-5 \sqrt{6}\right) \log \left(x+\sqrt{6}+1\right)","-\frac{1}{6} \left(9-5 \sqrt{6}\right) \log \left(x-\sqrt{6}+1\right)-\frac{1}{6} \left(9+5 \sqrt{6}\right) \log \left(x+\sqrt{6}+1\right)",1,"((-9 + 5*Sqrt[6])*Log[-1 + Sqrt[6] - x])/6 + ((-9 - 5*Sqrt[6])*Log[1 + Sqrt[6] + x])/6","A",1
2234,1,41,41,0.0110566,"\int \frac{1}{(-1+x) \left(1+x+x^2\right)} \, dx","Integrate[1/((-1 + x)*(1 + x + x^2)),x]","-\frac{1}{6} \log \left(x^2+x+1\right)+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{6} \log \left(x^2+x+1\right)+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-(ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[1 - x]/3 - Log[1 + x + x^2]/6","A",1
2235,1,111,114,0.084997,"\int \frac{2 \left(\left(\frac{a}{b}\right)^{\frac{1}{n}}-x \cos \left(\frac{(-1+2 k) \pi }{n}\right)\right)}{\left(\frac{a}{b}\right)^{2/n}+x^2-2 \left(\frac{a}{b}\right)^{\frac{1}{n}} x \cos \left(\frac{(-1+2 k) \pi }{n}\right)} \, dx","Integrate[(2*((a/b)^n^(-1) - x*Cos[((-1 + 2*k)*Pi)/n]))/((a/b)^(2/n) + x^2 - 2*(a/b)^n^(-1)*x*Cos[((-1 + 2*k)*Pi)/n]),x]","2 \left(\sin \left(\frac{\pi  (2 k-1)}{n}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{\pi  (2 k-1)}{2 n}\right) \left(\left(\frac{a}{b}\right)^{\frac{1}{n}}+x\right)}{\left(\frac{a}{b}\right)^{\frac{1}{n}}-x}\right)-\frac{1}{2} \cos \left(\frac{\pi  (2 k-1)}{n}\right) \log \left(-2 x \left(\frac{a}{b}\right)^{\frac{1}{n}} \cos \left(\frac{\pi  (2 k-1)}{n}\right)+\left(\frac{a}{b}\right)^{2/n}+x^2\right)\right)","2 \sin \left(\frac{\pi -2 \pi  k}{n}\right) \tan ^{-1}\left(\left(\frac{a}{b}\right)^{-1/n} \csc \left(\frac{\pi -2 \pi  k}{n}\right) \left(x-\left(\frac{a}{b}\right)^{\frac{1}{n}} \cos \left(\frac{\pi -2 \pi  k}{n}\right)\right)\right)-\cos \left(\frac{\pi -2 \pi  k}{n}\right) \log \left(-2 x \left(\frac{a}{b}\right)^{\frac{1}{n}} \cos \left(\frac{\pi -2 \pi  k}{n}\right)+\left(\frac{a}{b}\right)^{2/n}+x^2\right)",1,"2*(-1/2*(Cos[((-1 + 2*k)*Pi)/n]*Log[(a/b)^(2/n) + x^2 - 2*(a/b)^n^(-1)*x*Cos[((-1 + 2*k)*Pi)/n]]) + ArcTan[(((a/b)^n^(-1) + x)*Tan[((-1 + 2*k)*Pi)/(2*n)])/((a/b)^n^(-1) - x)]*Sin[((-1 + 2*k)*Pi)/n])","A",1
2236,1,40,40,0.0053075,"\int \frac{x^4}{2+13 x+15 x^2} \, dx","Integrate[x^4/(2 + 13*x + 15*x^2),x]","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}",1,"(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375","A",1
2237,1,33,33,0.0042942,"\int \frac{x^3}{2+13 x+15 x^2} \, dx","Integrate[x^3/(2 + 13*x + 15*x^2),x]","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)",1,"(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875","A",1
2238,1,26,26,0.0037629,"\int \frac{x^2}{2+13 x+15 x^2} \, dx","Integrate[x^2/(2 + 13*x + 15*x^2),x]","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)",1,"x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175","A",1
2239,1,21,21,0.0031551,"\int \frac{x}{2+13 x+15 x^2} \, dx","Integrate[x/(2 + 13*x + 15*x^2),x]","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)",1,"(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35","A",1
2240,1,21,21,0.0025482,"\int \frac{1}{2+13 x+15 x^2} \, dx","Integrate[(2 + 13*x + 15*x^2)^(-1),x]","\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)","\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)",1,"-1/7*Log[2 + 3*x] + Log[1 + 5*x]/7","A",1
2241,1,27,27,0.0045886,"\int \frac{1}{x \left(2+13 x+15 x^2\right)} \, dx","Integrate[1/(x*(2 + 13*x + 15*x^2)),x]","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)",1,"Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7","A",1
2242,1,34,34,0.0041454,"\int \frac{1}{x^2 \left(2+13 x+15 x^2\right)} \, dx","Integrate[1/(x^2*(2 + 13*x + 15*x^2)),x]","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)",1,"-1/2*1/x - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7","A",1
2243,1,41,41,0.0048057,"\int \frac{1}{x^3 \left(2+13 x+15 x^2\right)} \, dx","Integrate[1/(x^3*(2 + 13*x + 15*x^2)),x]","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)",1,"-1/4*1/x^2 + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x])/7","A",1
2244,1,48,48,0.0047813,"\int \frac{1}{x^4 \left(2+13 x+15 x^2\right)} \, dx","Integrate[1/(x^4*(2 + 13*x + 15*x^2)),x]","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)",1,"-1/6*1/x^3 + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7","A",1
2245,1,40,40,0.0048945,"\int \frac{x^5}{2 x+13 x^2+15 x^3} \, dx","Integrate[x^5/(2*x + 13*x^2 + 15*x^3),x]","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}",1,"(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375","A",1
2246,1,33,33,0.0043638,"\int \frac{x^4}{2 x+13 x^2+15 x^3} \, dx","Integrate[x^4/(2*x + 13*x^2 + 15*x^3),x]","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)",1,"(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875","A",1
2247,1,26,26,0.0035785,"\int \frac{x^3}{2 x+13 x^2+15 x^3} \, dx","Integrate[x^3/(2*x + 13*x^2 + 15*x^3),x]","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)",1,"x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175","A",1
2248,1,21,21,0.0031448,"\int \frac{x^2}{2 x+13 x^2+15 x^3} \, dx","Integrate[x^2/(2*x + 13*x^2 + 15*x^3),x]","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)",1,"(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35","A",1
2249,1,21,21,0.0029216,"\int \frac{x}{2 x+13 x^2+15 x^3} \, dx","Integrate[x/(2*x + 13*x^2 + 15*x^3),x]","\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)","\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)",1,"-1/7*Log[2 + 3*x] + Log[1 + 5*x]/7","A",1
2250,1,27,27,0.0036919,"\int \frac{1}{2 x+13 x^2+15 x^3} \, dx","Integrate[(2*x + 13*x^2 + 15*x^3)^(-1),x]","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)",1,"Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7","A",1
2251,1,34,34,0.0038386,"\int \frac{1}{x \left(2 x+13 x^2+15 x^3\right)} \, dx","Integrate[1/(x*(2*x + 13*x^2 + 15*x^3)),x]","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)",1,"-1/2*1/x - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7","A",1
2252,1,41,41,0.0045655,"\int \frac{1}{x^2 \left(2 x+13 x^2+15 x^3\right)} \, dx","Integrate[1/(x^2*(2*x + 13*x^2 + 15*x^3)),x]","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)",1,"-1/4*1/x^2 + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x])/7","A",1
2253,1,48,48,0.0044998,"\int \frac{1}{x^3 \left(2 x+13 x^2+15 x^3\right)} \, dx","Integrate[1/(x^3*(2*x + 13*x^2 + 15*x^3)),x]","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)",1,"-1/6*1/x^3 + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7","A",1
2254,1,12,12,0.0039082,"\int \frac{x}{4+4 x+x^2} \, dx","Integrate[x/(4 + 4*x + x^2),x]","\frac{2}{x+2}+\log (x+2)","\frac{2}{x+2}+\log (x+2)",1,"2/(2 + x) + Log[2 + x]","A",1
2255,1,26,26,0.0038469,"\int \frac{x}{5+2 x+x^2} \, dx","Integrate[x/(5 + 2*x + x^2),x]","\frac{1}{2} \log \left(x^2+2 x+5\right)-\frac{1}{2} \tan ^{-1}\left(\frac{x+1}{2}\right)","\frac{1}{2} \log \left(x^2+2 x+5\right)-\frac{1}{2} \tan ^{-1}\left(\frac{x+1}{2}\right)",1,"-1/2*ArcTan[(1 + x)/2] + Log[5 + 2*x + x^2]/2","A",1
2256,1,17,17,0.0035596,"\int \frac{x}{6-5 x+x^2} \, dx","Integrate[x/(6 - 5*x + x^2),x]","3 \log (3-x)-2 \log (2-x)","3 \log (3-x)-2 \log (2-x)",1,"-2*Log[2 - x] + 3*Log[3 - x]","A",1
2257,1,28,26,0.0120102,"\int \frac{x}{\left(2+2 x+x^2\right)^2} \, dx","Integrate[x/(2 + 2*x + x^2)^2,x]","\frac{-x-2}{2 \left(x^2+2 x+2\right)}-\frac{1}{2} \tan ^{-1}(x+1)","-\frac{x+2}{2 \left(x^2+2 x+2\right)}-\frac{1}{2} \tan ^{-1}(x+1)",1,"(-2 - x)/(2*(2 + 2*x + x^2)) - ArcTan[1 + x]/2","A",1
2258,1,49,54,0.027132,"\int \frac{x}{\left(1+x+x^2\right)^3} \, dx","Integrate[x/(1 + x + x^2)^3,x]","\frac{1}{18} \left(-\frac{3 \left(2 x^3+3 x^2+4 x+3\right)}{\left(x^2+x+1\right)^2}-4 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{x+2}{6 \left(x^2+x+1\right)^2}-\frac{2 x+1}{6 \left(x^2+x+1\right)}-\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"((-3*(3 + 4*x + 3*x^2 + 2*x^3))/(1 + x + x^2)^2 - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]])/18","A",1
2259,1,32,32,0.0083898,"\int \frac{x^2}{1+x+x^2} \, dx","Integrate[x^2/(1 + x + x^2),x]","-\frac{1}{2} \log \left(x^2+x+1\right)+x-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{2} \log \left(x^2+x+1\right)+x-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"x - ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3] - Log[1 + x + x^2]/2","A",1
2260,1,18,18,0.0035379,"\int \frac{x^2}{2-3 x+x^2} \, dx","Integrate[x^2/(2 - 3*x + x^2),x]","x-\log (1-x)+4 \log (2-x)","x-\log (1-x)+4 \log (2-x)",1,"x - Log[1 - x] + 4*Log[2 - x]","A",1
2261,1,20,20,0.0033647,"\int \frac{x^2}{-6+x+x^2} \, dx","Integrate[x^2/(-6 + x + x^2),x]","x+\frac{4}{5} \log (2-x)-\frac{9}{5} \log (x+3)","x+\frac{4}{5} \log (2-x)-\frac{9}{5} \log (x+3)",1,"x + (4*Log[2 - x])/5 - (9*Log[3 + x])/5","A",1
2262,1,15,15,0.0084139,"\int \frac{x^2}{\left(2+2 x+x^2\right)^2} \, dx","Integrate[x^2/(2 + 2*x + x^2)^2,x]","\frac{1}{x^2+2 x+2}+\tan ^{-1}(x+1)","\frac{1}{x^2+2 x+2}+\tan ^{-1}(x+1)",1,"(2 + 2*x + x^2)^(-1) + ArcTan[1 + x]","A",1
2263,1,27,27,0.0038134,"\int \frac{x^3}{2-3 x+x^2} \, dx","Integrate[x^3/(2 - 3*x + x^2),x]","\frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x)","\frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x)",1,"3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]","A",1
2264,1,26,22,0.0094173,"\int \frac{x^3}{1+2 x+x^2} \, dx","Integrate[x^3/(1 + 2*x + x^2),x]","\frac{1}{2} (x+1)^2-3 (x+1)+\frac{1}{x+1}+3 \log (x+1)","\frac{x^2}{2}-2 x+\frac{1}{x+1}+3 \log (x+1)",1,"(1 + x)^(-1) - 3*(1 + x) + (1 + x)^2/2 + 3*Log[1 + x]","A",1
2265,1,25,26,0.0077456,"\int \frac{x^3}{1-2 x+x^2} \, dx","Integrate[x^3/(1 - 2*x + x^2),x]","\frac{1}{2} \left(x^2+4 x-\frac{2}{x-1}+6 \log (x-1)-5\right)","\frac{x^2}{2}+2 x+\frac{1}{1-x}+3 \log (1-x)",1,"(-5 - 2/(-1 + x) + 4*x + x^2 + 6*Log[-1 + x])/2","A",1
2266,1,30,29,0.0099529,"\int \frac{x^4}{4+4 x+x^2} \, dx","Integrate[x^4/(4 + 4*x + x^2),x]","\frac{1}{3} \left(x^3-6 x^2+36 x-\frac{48}{x+2}-96 \log (x+2)+104\right)","\frac{x^3}{3}-2 x^2+12 x-\frac{16}{x+2}-32 \log (x+2)",1,"(104 + 36*x - 6*x^2 + x^3 - 48/(2 + x) - 96*Log[2 + x])/3","A",1
2267,1,33,33,0.0086472,"\int \frac{1}{x \left(1+x+x^2\right)} \, dx","Integrate[1/(x*(1 + x + x^2)),x]","-\frac{1}{2} \log \left(x^2+x+1\right)+\log (x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{2} \log \left(x^2+x+1\right)+\log (x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-(ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[x] - Log[1 + x + x^2]/2","A",1
2268,1,55,75,0.0617928,"\int (d+e x)^{5/2} \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2),x]","\frac{2 (d+e x)^{7/2} \left(11 e (9 a e-2 b d+7 b e x)+c \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)}{693 e^3}","\frac{2 (d+e x)^{7/2} \left(a e^2-b d e+c d^2\right)}{7 e^3}-\frac{2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}",1,"(2*(d + e*x)^(7/2)*(11*e*(-2*b*d + 9*a*e + 7*b*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)","A",1
2269,1,55,75,0.0454763,"\int (d+e x)^{3/2} \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2),x]","\frac{2 (d+e x)^{5/2} \left(9 e (7 a e-2 b d+5 b e x)+c \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)}{315 e^3}","\frac{2 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)}{5 e^3}-\frac{2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}",1,"(2*(d + e*x)^(5/2)*(9*e*(-2*b*d + 7*a*e + 5*b*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)","A",1
2270,1,55,75,0.0412519,"\int \sqrt{d+e x} \left(a+b x+c x^2\right) \, dx","Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2),x]","\frac{2 (d+e x)^{3/2} \left(7 e (5 a e-2 b d+3 b e x)+c \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)}{105 e^3}","\frac{2 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}",1,"(2*(d + e*x)^(3/2)*(7*e*(-2*b*d + 5*a*e + 3*b*e*x) + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)","A",1
2271,1,54,73,0.0453722,"\int \frac{a+b x+c x^2}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(5 e (3 a e-2 b d+b e x)+c \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)}{15 e^3}","\frac{2 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}{e^3}-\frac{2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}",1,"(2*Sqrt[d + e*x]*(5*e*(-2*b*d + 3*a*e + b*e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)","A",1
2272,1,54,71,0.0489664,"\int \frac{a+b x+c x^2}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^(3/2),x]","\frac{6 e (-a e+2 b d+b e x)+2 c \left(-8 d^2-4 d e x+e^2 x^2\right)}{3 e^3 \sqrt{d+e x}}","-\frac{2 \left(a e^2-b d e+c d^2\right)}{e^3 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3}",1,"(6*e*(2*b*d - a*e + b*e*x) + 2*c*(-8*d^2 - 4*d*e*x + e^2*x^2))/(3*e^3*Sqrt[d + e*x])","A",1
2273,1,55,71,0.0445515,"\int \frac{a+b x+c x^2}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^(5/2),x]","\frac{2 c \left(8 d^2+12 d e x+3 e^2 x^2\right)-2 e (a e+2 b d+3 b e x)}{3 e^3 (d+e x)^{3/2}}","-\frac{2 \left(a e^2-b d e+c d^2\right)}{3 e^3 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c \sqrt{d+e x}}{e^3}",1,"(-2*e*(2*b*d + a*e + 3*b*e*x) + 2*c*(8*d^2 + 12*d*e*x + 3*e^2*x^2))/(3*e^3*(d + e*x)^(3/2))","A",1
2274,1,54,73,0.0456941,"\int \frac{a+b x+c x^2}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)/(d + e*x)^(7/2),x]","-\frac{2 \left(e (3 a e+2 b d+5 b e x)+c \left(8 d^2+20 d e x+15 e^2 x^2\right)\right)}{15 e^3 (d+e x)^{5/2}}","-\frac{2 \left(a e^2-b d e+c d^2\right)}{5 e^3 (d+e x)^{5/2}}+\frac{2 (2 c d-b e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 c}{e^3 \sqrt{d+e x}}",1,"(-2*(e*(2*b*d + 3*a*e + 5*b*e*x) + c*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))","A",1
2275,1,173,166,0.1583809,"\int (d+e x)^{5/2} \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{7/2} \left(65 e^2 \left(99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)-10 c e \left(3 b \left(16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right)-13 a e \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)+c^2 \left(128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right)\right)}{45045 e^5}","\frac{2 (d+e x)^{11/2} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{11 e^5}-\frac{4 (d+e x)^{9/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{9 e^5}+\frac{2 (d+e x)^{7/2} \left(a e^2-b d e+c d^2\right)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}",1,"(2*(d + e*x)^(7/2)*(c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 65*e^2*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 10*c*e*(-13*a*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*b*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3))))/(45045*e^5)","A",1
2276,1,174,166,0.1425924,"\int (d+e x)^{3/2} \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{5/2} \left(143 e^2 \left(63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)-26 c e \left(3 b \left(16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right)-11 a e \left(8 d^2-20 d e x+35 e^2 x^2\right)\right)+3 c^2 \left(128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right)\right)}{45045 e^5}","\frac{2 (d+e x)^{9/2} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{9 e^5}-\frac{4 (d+e x)^{7/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{7 e^5}+\frac{2 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}",1,"(2*(d + e*x)^(5/2)*(3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + 143*e^2*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 26*c*e*(-11*a*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*b*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(45045*e^5)","A",1
2277,1,172,166,0.1429597,"\int \sqrt{d+e x} \left(a+b x+c x^2\right)^2 \, dx","Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^2,x]","\frac{2 (d+e x)^{3/2} \left(33 e^2 \left(35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)-22 c e \left(b \left(16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right)-3 a e \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)+c^2 \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)}{3465 e^5}","\frac{2 (d+e x)^{7/2} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{7 e^5}-\frac{4 (d+e x)^{5/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{5 e^5}+\frac{2 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}",1,"(2*(d + e*x)^(3/2)*(c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 33*e^2*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 22*c*e*(-3*a*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3))))/(3465*e^5)","A",1
2278,1,172,164,0.1593271,"\int \frac{\left(a+b x+c x^2\right)^2}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)^2/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} \left(21 e^2 \left(15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)-6 c e \left(3 b \left(16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right)-7 a e \left(8 d^2-4 d e x+3 e^2 x^2\right)\right)+c^2 \left(128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right)\right)}{315 e^5}","\frac{2 (d+e x)^{5/2} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{5 e^5}-\frac{4 (d+e x)^{3/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{3 e^5}+\frac{2 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}",1,"(2*Sqrt[d + e*x]*(c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 21*e^2*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 6*c*e*(-7*a*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 3*b*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3))))/(315*e^5)","A",1
2279,1,171,162,0.1289115,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^(3/2),x]","\frac{-70 e^2 \left(3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left(8 d^2+4 d e x-e^2 x^2\right)\right)+28 c e \left(5 a e \left(-8 d^2-4 d e x+e^2 x^2\right)+3 b \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)\right)-6 c^2 \left(128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right)}{105 e^5 \sqrt{d+e x}}","\frac{2 (d+e x)^{3/2} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{3 e^5}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{e^5}-\frac{2 \left(a e^2-b d e+c d^2\right)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}",1,"(-6*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) - 70*e^2*(3*a^2*e^2 - 6*a*b*e*(2*d + e*x) + b^2*(8*d^2 + 4*d*e*x - e^2*x^2)) + 28*c*e*(5*a*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*b*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(105*e^5*Sqrt[d + e*x])","A",1
2280,1,170,162,0.1283998,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^(5/2),x]","\frac{2 \left(-5 e^2 \left(a^2 e^2+2 a b e (2 d+3 e x)-\left(b^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)\right)\right)+10 c e \left(a e \left(8 d^2+12 d e x+3 e^2 x^2\right)+b \left(-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right)\right)+c^2 \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{3/2}}","\frac{2 \sqrt{d+e x} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{e^5}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{e^5 \sqrt{d+e x}}-\frac{2 \left(a e^2-b d e+c d^2\right)^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}",1,"(2*(c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) - 5*e^2*(a^2*e^2 + 2*a*b*e*(2*d + 3*e*x) - b^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2)) + 10*c*e*(a*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3))))/(15*e^5*(d + e*x)^(3/2))","A",1
2281,1,172,162,0.1317848,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x)^(7/2),x]","-\frac{2 \left(3 a^2 e^4+2 a b e^3 (2 d+5 e x)+2 a c e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)+b^2 e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)-6 b c e \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)+c^2 \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)\right)}{15 e^5 (d+e x)^{5/2}}","-\frac{2 \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left(a e^2-b d e+c d^2\right)}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left(a e^2-b d e+c d^2\right)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}",1,"(-2*(3*a^2*e^4 + 2*a*b*e^3*(2*d + 5*e*x) + b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 2*a*c*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))","A",1
2282,1,321,286,0.8681186,"\int (d+e x)^{5/2} \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^3,x]","\frac{2 \left((d+e x)^{7/2} (a+x (b+c x))^3-\frac{2 (d+e x)^{9/2} \left(-646 c e^2 \left(65 a^2 e^2 (2 d-9 e x)-15 a b e \left(8 d^2-36 d e x+99 e^2 x^2\right)+2 b^2 \left(16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right)\right)+1615 b e^3 \left(143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left(8 d^2-36 d e x+99 e^2 x^2\right)\right)+19 c^2 e \left(68 a e \left(-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right)+5 b \left(128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right)\right)-10 c^3 \left(256 d^5-1152 d^4 e x+3168 d^3 e^2 x^2-6864 d^2 e^3 x^3+12870 d e^4 x^4-21879 e^5 x^5\right)\right)}{692835 e^6}\right)}{7 e}","\frac{2 c (d+e x)^{15/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{13 e^7}+\frac{6 (d+e x)^{11/2} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left(a e^2-b d e+c d^2\right)^3}{7 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}",1,"(2*((d + e*x)^(7/2)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(9/2)*(-10*c^3*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x^2 - 6864*d^2*e^3*x^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5) + 1615*b*e^3*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 646*c*e^2*(65*a^2*e^2*(2*d - 9*e*x) - 15*a*b*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 2*b^2*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 - 429*e^3*x^3)) + 19*c^2*e*(68*a*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 5*b*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4))))/(692835*e^6)))/(7*e)","A",1
2283,1,320,286,0.812997,"\int (d+e x)^{3/2} \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]","\frac{2 \left((d+e x)^{5/2} (a+x (b+c x))^3-\frac{2 (d+e x)^{7/2} \left(-34 c e^2 \left(143 a^2 e^2 (2 d-7 e x)-39 a b e \left(8 d^2-28 d e x+63 e^2 x^2\right)+6 b^2 \left(16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right)\right)+221 b e^3 \left(99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left(8 d^2-28 d e x+63 e^2 x^2\right)\right)+17 c^2 e \left(12 a e \left(-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right)+b \left(128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right)\right)-2 c^3 \left(256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right)\right)}{51051 e^6}\right)}{5 e}","\frac{6 c (d+e x)^{13/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{11 e^7}+\frac{2 (d+e x)^{9/2} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{3 e^7}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)^3}{5 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}",1,"(2*((d + e*x)^(5/2)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(7/2)*(-2*c^3*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5) + 221*b*e^3*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 34*c*e^2*(143*a^2*e^2*(2*d - 7*e*x) - 39*a*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 6*b^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 17*c^2*e*(12*a*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(51051*e^6)))/(5*e)","A",1
2284,1,396,286,0.3724818,"\int \sqrt{d+e x} \left(a+b x+c x^2\right)^3 \, dx","Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^3,x]","\frac{2 (d+e x)^{3/2} \left(39 c e^2 \left(33 a^2 e^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)+22 a b e \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)+b^2 \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)+143 e^3 \left(105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left(8 d^2-12 d e x+15 e^2 x^2\right)+b^3 \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)\right)-3 c^2 e \left(5 b \left(256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right)-13 a e \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)+c^3 \left(1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right)\right)}{45045 e^7}","\frac{6 c (d+e x)^{11/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{9 e^7}+\frac{6 (d+e x)^{7/2} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{7 e^7}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)^3}{3 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}",1,"(2*(d + e*x)^(3/2)*(c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6) + 143*e^3*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) + 9*a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)) + 39*c*e^2*(33*a^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*a*b*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + b^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) - 3*c^2*e*(-13*a*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 5*b*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e^5*x^5))))/(45045*e^7)","A",1
2285,1,317,282,0.5462469,"\int \frac{\left(a+b x+c x^2\right)^3}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)^3/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} (a+x (b+c x))^3}{e}-\frac{4 (d+e x)^{3/2} \left(-286 c e^2 \left(21 a^2 e^2 (2 d-3 e x)-9 a b e \left(8 d^2-12 d e x+15 e^2 x^2\right)+b^2 \left(32 d^3-48 d^2 e x+60 d e^2 x^2-70 e^3 x^3\right)\right)+429 b e^3 \left(35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left(8 d^2-12 d e x+15 e^2 x^2\right)\right)+13 c^2 e \left(44 a e \left(-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right)+5 b \left(128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right)\right)-10 c^3 \left(256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right)\right)}{15015 e^7}","\frac{2 c (d+e x)^{9/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{7 e^7}+\frac{6 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7}+\frac{2 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^3}{e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}",1,"(2*Sqrt[d + e*x]*(a + x*(b + c*x))^3)/e - (4*(d + e*x)^(3/2)*(-10*c^3*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e^5*x^5) + 429*b*e^3*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 286*c*e^2*(21*a^2*e^2*(2*d - 3*e*x) - 9*a*b*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^2*(32*d^3 - 48*d^2*e*x + 60*d*e^2*x^2 - 70*e^3*x^3)) + 13*c^2*e*(44*a*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4))))/(15015*e^7)","A",1
2286,1,394,280,0.4089467,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(3/2),x]","\frac{-66 c e^2 \left(35 a^2 e^2 \left(8 d^2+4 d e x-e^2 x^2\right)-42 a b e \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)+3 b^2 \left(128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right)\right)+462 e^3 \left(-5 a^3 e^3+15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left(-8 d^2-4 d e x+e^2 x^2\right)+b^3 \left(16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right)\right)+22 c^2 e \left(9 a e \left(-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right)+5 b \left(256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right)\right)-10 c^3 \left(1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right)}{1155 e^7 \sqrt{d+e x}}","\frac{6 c (d+e x)^{7/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{5 e^7}+\frac{2 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7}-\frac{2 \left(a e^2-b d e+c d^2\right)^3}{e^7 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}",1,"(-10*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6) + 462*e^3*(-5*a^3*e^3 + 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + b^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 66*c*e^2*(35*a^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*a*b*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*b^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 22*c^2*e*(9*a*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*b*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(1155*e^7*Sqrt[d + e*x])","A",1
2287,1,395,282,0.370412,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(5/2),x]","\frac{2 \left(63 c e^2 \left(5 a^2 e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)+10 a b e \left(-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right)+b^2 \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)\right)-105 e^3 \left(a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left(8 d^2+12 d e x+3 e^2 x^2\right)+b^3 \left(16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right)\right)-9 c^2 e \left(5 b \left(256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right)-7 a e \left(128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right)\right)+5 c^3 \left(1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right)\right)}{315 e^7 (d+e x)^{3/2}}","\frac{6 c (d+e x)^{5/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{3 e^7}+\frac{6 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}+\frac{6 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7 \sqrt{d+e x}}-\frac{2 \left(a e^2-b d e+c d^2\right)^3}{3 e^7 (d+e x)^{3/2}}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}",1,"(2*(5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6) - 105*e^3*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + b^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 63*c*e^2*(5*a^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*a*b*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) - 9*c^2*e*(-7*a*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*b*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5))))/(315*e^7*(d + e*x)^(3/2))","A",1
2288,1,391,278,0.371519,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]","-\frac{2 \left(7 c e^2 \left(a^2 e^2 \left(8 d^2+20 d e x+15 e^2 x^2\right)-6 a b e \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)+b^2 \left(128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right)\right)+7 e^3 \left(a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left(8 d^2+20 d e x+15 e^2 x^2\right)-\left(b^3 \left(16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right)\right)\right)-7 c^2 e \left(a e \left(-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right)+b \left(256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right)\right)+c^3 \left(1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right)\right)}{35 e^7 (d+e x)^{5/2}}","\frac{2 c (d+e x)^{3/2} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{e^7}-\frac{6 \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left(a e^2-b d e+c d^2\right)^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}",1,"(-2*(c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6) + 7*e^3*(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - b^3*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 7*c*e^2*(a^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*b*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + b^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)) - 7*c^2*e*(a*e*(-128*d^4 - 320*d^3*e*x - 240*d^2*e^2*x^2 - 40*d*e^3*x^3 + 5*e^4*x^4) + b*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(35*e^7*(d + e*x)^(5/2))","A",1
2289,1,455,459,1.0488625,"\int \frac{(d+e x)^{5/2}}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2),x]","-\frac{\sqrt{2} \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}-2 a e-b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}-b\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 e \sqrt{d+e x} (b e-2 c d)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}","-\frac{\sqrt{2} \left(-3 c^2 d e \left(-d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)-b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}",1,"(-2*e*(-2*c*d + b*e)*Sqrt[d + e*x])/c^2 + (2*e*(d + e*x)^(3/2))/(3*c) - (Sqrt[2]*(2*c^3*d^3 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
2290,1,317,322,0.7623029,"\int \frac{(d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]","\frac{\frac{\sqrt{2} \left(2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)-2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+2 \sqrt{c} e \sqrt{d+e x}}{c^{3/2}}","-\frac{\sqrt{2} \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 e \sqrt{d+e x}}{c}",1,"(2*Sqrt[c]*e*Sqrt[d + e*x] + (Sqrt[2]*(-2*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/c^(3/2)","A",1
2291,1,175,198,0.4579402,"\int \frac{\sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2),x]","\frac{\sqrt{2} \left(\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)-\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)\right)}{\sqrt{c} \sqrt{b^2-4 a c}}","\frac{\sqrt{2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c}}",1,"(Sqrt[2]*(-(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]]) + Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
2292,1,176,199,0.5259077,"\int \frac{1}{\sqrt{d+e x} \left(a+b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)),x]","\frac{2 \sqrt{2} \sqrt{c} \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}\right)}{\sqrt{b^2-4 a c}}","\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}",1,"(2*Sqrt[2]*Sqrt[c]*(-(ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]]/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]]/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c]","A",1
2293,1,273,310,0.3893355,"\int \frac{1}{(d+e x)^{3/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]","\frac{2 \left(-\frac{\sqrt{c} \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{c} \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{e}{\sqrt{d+e x}}\right)}{e (b d-a e)-c d^2}","-\frac{\sqrt{2} \sqrt{c} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{2} \sqrt{c} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}-\frac{2 e}{\sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}",1,"(2*(e/Sqrt[d + e*x] - (Sqrt[c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(-(c*d^2) + e*(b*d - a*e))","A",1
2294,1,377,414,1.2539162,"\int \frac{1}{(d+e x)^{5/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)),x]","\frac{2 \left(-\frac{3 \sqrt{c} \left(\frac{\left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}+\frac{3 e (b e-2 c d)}{\sqrt{d+e x} \left(e (a e-b d)+c d^2\right)}-\frac{e}{(d+e x)^{3/2}}\right)}{3 \left(e (a e-b d)+c d^2\right)}","-\frac{\sqrt{2} \sqrt{c} \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)^2}+\frac{\sqrt{2} \sqrt{c} \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 e (2 c d-b e)}{\sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 e}{3 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}",1,"(2*(-(e/(d + e*x)^(3/2)) + (3*e*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x]) - (3*Sqrt[c]*(-(((2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e)))))/(3*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2295,1,1116,691,6.5014105,"\int \frac{(d+e x)^{7/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x]","-\frac{\left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right) (d+e x)^{9/2}}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \left(c x^2+b x+a\right)}-\frac{\frac{2 \left(\frac{2 \left(\frac{2 \left(\frac{4 \left(\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left(\frac{105}{32} c^2 e (2 c d-b e) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right)-\frac{2 c \left(\frac{105}{32} c^2 e \left(c d^2-b e d+a e^2\right) \left(4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right)-\frac{105}{32} c^2 d e (2 c d-b e) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right)\right)-\frac{105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right)}{\sqrt{b^2-4 a c} e}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right)}{\sqrt{2} \sqrt{c} \left(-2 c d+b e+\sqrt{b^2-4 a c} e\right)}+\frac{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left(\frac{105}{32} e (2 c d-b e) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right) c^2+\frac{2 c \left(\frac{105}{32} c^2 e \left(c d^2-b e d+a e^2\right) \left(4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right)-\frac{105}{32} c^2 d e (2 c d-b e) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right)\right)-\frac{105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left(c d^2-b e d+a e^2\right) \left(c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right)}{\sqrt{b^2-4 a c} e}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right)}{\sqrt{2} \sqrt{c} \left(-2 c d+b e-\sqrt{b^2-4 a c} e\right)}\right)}{c}-\frac{105}{8} c e \left(c d^2-b e d+a e^2\right) \left(2 c^2 d^2-2 b c e d+3 b^2 e^2-10 a c e^2\right) \sqrt{d+e x}\right)}{3 c}-\frac{35}{4} c e (2 c d-b e) \left(c d^2-e (b d-a e)\right) (d+e x)^{3/2}\right)}{5 c}-7 c e \left(c d^2-e (b d-a e)\right) (d+e x)^{5/2}\right)}{7 c}-e (2 c d-b e) (d+e x)^{7/2}}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)}","\frac{e \sqrt{d+e x} \left(-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right)}{c^2 \left(b^2-4 a c\right)}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{c \left(b^2-4 a c\right)}+\frac{\left(-2 c^3 d^2 e \left(-d \sqrt{b^2-4 a c}-18 a e+8 b d\right)+c^2 e^2 \left(-3 b d \left(d \sqrt{b^2-4 a c}+12 a e\right)+2 a e \left(13 d \sqrt{b^2-4 a c}-10 a e\right)+3 b^2 d^2\right)+b c e^3 \left(-5 b d \sqrt{b^2-4 a c}-13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right)-3 b^3 e^4 \left(b-\sqrt{b^2-4 a c}\right)+8 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{5/2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(-2 c^3 d^2 e \left(d \sqrt{b^2-4 a c}-18 a e+8 b d\right)+c^2 e^2 \left(3 b d \left(d \sqrt{b^2-4 a c}-12 a e\right)-2 a e \left(13 d \sqrt{b^2-4 a c}+10 a e\right)+3 b^2 d^2\right)+b c e^3 \left(5 b d \sqrt{b^2-4 a c}+13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right)-3 b^3 e^4 \left(\sqrt{b^2-4 a c}+b\right)+8 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{5/2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}",1,"-(((d + e*x)^(9/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (-(e*(2*c*d - b*e)*(d + e*x)^(7/2)) + (2*(-7*c*e*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(5/2) + (2*((-35*c*e*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(3/2))/4 + (2*((-105*c*e*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2 - 10*a*c*e^2)*Sqrt[d + e*x])/8 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 - ((-105*c^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + 2*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (105*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4 - 10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + ((-105*c^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + 2*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (105*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4 - 10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c)))/(5*c)))/(7*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))","A",1
2296,1,549,504,5.2666409,"\int \frac{(d+e x)^{5/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]","\frac{\frac{1}{2} \left(e (a e-b d)+c d^2\right) \left(\frac{\sqrt{2} \left(\frac{\left(2 c^2 d e \left(d \sqrt{b^2-4 a c}+8 a e-6 b d\right)+2 c e^2 \left(-b \left(d \sqrt{b^2-4 a c}+4 a e\right)+3 a e \sqrt{b^2-4 a c}+b^2 d\right)+b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(-2 c^2 d e \left(d \sqrt{b^2-4 a c}-8 a e+6 b d\right)+2 c e^2 \left(b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c}}-\frac{2 e \sqrt{d+e x} (b e-2 c d)}{c}+4 e (d+e x)^{3/2}\right)+\frac{(d+e x)^{7/2} \left(-2 c (a e+c d x)+b^2 e+b c (e x-d)\right)}{a+x (b+c x)}+e (d+e x)^{5/2} (2 c d-b e)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{\left(-2 c^2 d e \left(-d \sqrt{b^2-4 a c}-8 a e+6 b d\right)+2 c e^2 \left(-b \left(d \sqrt{b^2-4 a c}+4 a e\right)+3 a e \sqrt{b^2-4 a c}+b^2 d\right)+b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(-2 c^2 d e \left(d \sqrt{b^2-4 a c}-8 a e+6 b d\right)+2 c e^2 \left(b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{3/2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{e \sqrt{d+e x} (2 c d-b e)}{c \left(b^2-4 a c\right)}",1,"(e*(2*c*d - b*e)*(d + e*x)^(5/2) + ((d + e*x)^(7/2)*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x)) + ((c*d^2 + e*(-(b*d) + a*e))*((-2*e*(-2*c*d + b*e)*Sqrt[d + e*x])/c + 4*e*(d + e*x)^(3/2) + (Sqrt[2]*(((8*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 8*a*e) + 2*c*e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(c^(3/2)*Sqrt[b^2 - 4*a*c])))/2)/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2297,1,418,363,2.8727368,"\int \frac{(d+e x)^{3/2}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^2,x]","\frac{\frac{1}{2} \left(e (a e-b d)+c d^2\right) \left(\frac{\sqrt{2} \left(\frac{\left(2 c e \left(d \sqrt{b^2-4 a c}+2 a e-4 b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}-2 a e+4 b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c}}+4 e \sqrt{d+e x}\right)+\frac{(d+e x)^{5/2} \left(-2 c (a e+c d x)+b^2 e+b c (e x-d)\right)}{a+x (b+c x)}+e (d+e x)^{3/2} (2 c d-b e)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{\left(-2 c e \left(-d \sqrt{b^2-4 a c}-2 a e+4 b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}-2 a e+4 b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(e*(2*c*d - b*e)*(d + e*x)^(3/2) + ((d + e*x)^(5/2)*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x)) + ((c*d^2 + e*(-(b*d) + a*e))*(4*e*Sqrt[d + e*x] + (Sqrt[2]*(((8*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c])))/2)/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2298,1,283,287,0.594733,"\int \frac{\sqrt{d+e x}}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^2,x]","\frac{-\frac{\sqrt{2} \sqrt{c} \left(e \left(\sqrt{b^2-4 a c}-2 b\right)+4 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \sqrt{c} \left(4 c d-e \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{(b+2 c x) \sqrt{d+e x}}{a+x (b+c x)}}{4 a c-b^2}","-\frac{(b+2 c x) \sqrt{d+e x}}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{\sqrt{2} \sqrt{c} \left(4 c d-e \left(2 b-\sqrt{b^2-4 a c}\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \sqrt{c} \left(4 c d-e \left(\sqrt{b^2-4 a c}+2 b\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}",1,"(((b + 2*c*x)*Sqrt[d + e*x])/(a + x*(b + c*x)) - (Sqrt[2]*Sqrt[c]*(4*c*d + (-2*b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(4*c*d - (2*b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(-b^2 + 4*a*c)","A",1
2299,1,366,428,1.2562578,"\int \frac{1}{\sqrt{d+e x} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^2),x]","\frac{\frac{\sqrt{c} \left(\frac{\left(2 c e \left(d \sqrt{b^2-4 a c}+6 a e-4 b d\right)-b e^2 \left(\sqrt{b^2-4 a c}+b\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(-2 c e \left(d \sqrt{b^2-4 a c}-6 a e+4 b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{b^2-4 a c}}+\frac{\sqrt{d+e x} \left(-2 c (a e+c d x)+b^2 e+b c (e x-d)\right)}{a+x (b+c x)}}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{\sqrt{c} \left(-2 c e \left(-d \sqrt{b^2-4 a c}-6 a e+4 b d\right)-b e^2 \left(\sqrt{b^2-4 a c}+b\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{c} \left(-2 c e \left(d \sqrt{b^2-4 a c}-6 a e+4 b d\right)-b e^2 \left(b-\sqrt{b^2-4 a c}\right)+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{d+e x} \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}",1,"((Sqrt[d + e*x]*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x)) + (Sqrt[c]*(((8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c]))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2300,1,552,604,2.4698131,"\int \frac{1}{(d+e x)^{3/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^2),x]","\frac{\frac{e \left(2 c e (5 a e+b d)-3 b^2 e^2-2 c^2 d^2\right)}{\sqrt{d+e x} \left(e (a e-b d)+c d^2\right)}-\frac{\sqrt{c} \left(\frac{\left(2 c^2 d e \left(d \sqrt{b^2-4 a c}+16 a e-6 b d\right)-2 c e^2 \left(b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right)+3 b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(-2 c^2 d e \left(d \sqrt{b^2-4 a c}-16 a e+6 b d\right)+2 c e^2 \left(b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}-8 a b e+b^2 (-d)\right)+3 b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \left(e (b d-a e)-c d^2\right)}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{\sqrt{d+e x} (a+x (b+c x))}}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","-\frac{e \left(-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right)}{\left(b^2-4 a c\right) \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}+\frac{\sqrt{c} \left(-2 c^2 d e \left(-d \sqrt{b^2-4 a c}-16 a e+6 b d\right)-2 c e^2 \left(b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right)+3 b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)^2}-\frac{\sqrt{c} \left(-2 c^2 d e \left(d \sqrt{b^2-4 a c}-16 a e+6 b d\right)-2 c e^2 \left(-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right)+3 b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+8 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left(b^2-4 a c\right) \sqrt{d+e x} \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}",1,"((e*(-2*c^2*d^2 - 3*b^2*e^2 + 2*c*e*(b*d + 5*a*e)))/((c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x]) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/(Sqrt[d + e*x]*(a + x*(b + c*x))) - (Sqrt[c]*(((8*c^3*d^3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 16*a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((8*c^3*d^3 + 3*b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 16*a*e) + 2*c*e^2*(-(b^2*d) + b*Sqrt[b^2 - 4*a*c]*d - 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2301,1,337,360,0.903359,"\int \frac{1}{x^{5/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*x + c*x^2)^2),x]","\frac{\frac{b \left(5 b^2-19 a c\right)}{a^2 \sqrt{x}}-\frac{\sqrt{c} \left(\frac{\left(28 a^2 c^2-29 a b^2 c+19 a b c \sqrt{b^2-4 a c}-5 b^3 \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a^2 \sqrt{b^2-4 a c}}+\frac{-2 a c+b^2+b c x}{x^{3/2} (a+x (b+c x))}+\frac{14 a c-5 b^2}{3 a x^{3/2}}}{a \left(b^2-4 a c\right)}","\frac{b \left(5 b^2-19 a c\right)}{a^3 \sqrt{x} \left(b^2-4 a c\right)}-\frac{5 b^2-14 a c}{3 a^2 x^{3/2} \left(b^2-4 a c\right)}+\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c+b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c-b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^3 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x}{a x^{3/2} \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"((-5*b^2 + 14*a*c)/(3*a*x^(3/2)) + (b*(5*b^2 - 19*a*c))/(a^2*Sqrt[x]) + (b^2 - 2*a*c + b*c*x)/(x^(3/2)*(a + x*(b + c*x))) - (Sqrt[c]*(-(((5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - 5*b^3*Sqrt[b^2 - 4*a*c] + 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]))/(a*(b^2 - 4*a*c))","A",1
2302,1,17950,751,6.6319953,"\int \frac{(d+e x)^{7/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]","\text{Result too large to show}","\frac{\sqrt{d+e x} \left(x (2 c d-b e) \left(-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right)-\left(b^2 \left(a e^3+11 c d^2 e\right)\right)+12 b c d \left(3 a e^2+c d^2\right)-4 a c e \left(5 a e^2+7 c d^2\right)\right)}{4 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{\left(-8 c^3 d^2 e \left(-3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right)+2 c^2 e^2 \left(-2 b d \left(9 d \sqrt{b^2-4 a c}+38 a e\right)+4 a e \left(4 d \sqrt{b^2-4 a c}+5 a e\right)+53 b^2 d^2\right)-2 b c e^3 \left(-5 b d \sqrt{b^2-4 a c}+8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right)-b^3 e^4 \left(b-\sqrt{b^2-4 a c}\right)+96 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{4 \sqrt{2} c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\left(-8 c^3 d^2 e \left(3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right)+2 c^2 e^2 \left(2 b d \left(9 d \sqrt{b^2-4 a c}-38 a e\right)-4 a e \left(4 d \sqrt{b^2-4 a c}-5 a e\right)+53 b^2 d^2\right)-2 b c e^3 \left(5 b d \sqrt{b^2-4 a c}-8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right)-b^3 e^4 \left(\sqrt{b^2-4 a c}+b\right)+96 c^4 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{4 \sqrt{2} c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}",1,"Result too large to show","B",1
2303,1,506,577,5.0111486,"\int \frac{(d+e x)^{5/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^3,x]","\frac{\sqrt{d+e x} \left(4 b \left(3 a^2 e^2+a c \left(5 d^2-9 d e x+4 e^2 x^2\right)+3 c^2 d x^2 (3 d-2 e x)\right)+4 c \left(-a^2 e (7 d+e x)+a c x \left(10 d^2+d e x+3 e^2 x^2\right)+6 c^2 d^2 x^3\right)+b^2 \left(a e (19 e x-5 d)+c x \left(8 d^2-37 d e x+3 e^2 x^2\right)\right)+b^3 \left(-2 d^2-9 d e x+5 e^2 x^2\right)\right)}{4 \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}-\frac{3 \sqrt{2 e \left(\sqrt{b^2-4 a c}-b\right)+4 c d} \left(4 c e \left(-d \sqrt{b^2-4 a c}+a e-4 b d\right)+b e^2 \left(2 \sqrt{b^2-4 a c}+3 b\right)+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{8 \sqrt{c} \left(b^2-4 a c\right)^{5/2}}+\frac{3 \sqrt{4 c d-2 e \left(\sqrt{b^2-4 a c}+b\right)} \left(4 c e \left(d \sqrt{b^2-4 a c}+a e-4 b d\right)+b e^2 \left(3 b-2 \sqrt{b^2-4 a c}\right)+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{8 \sqrt{c} \left(b^2-4 a c\right)^{5/2}}","-\frac{3 \sqrt{d+e x} \left(-x \left(-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right)-4 b \left(a e^2+c d^2\right)+4 a c d e+3 b^2 d e\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 \left(-8 c^2 d e \left(-d \sqrt{b^2-4 a c}-3 a e+6 b d\right)+2 c e^2 \left(-4 b d \sqrt{b^2-4 a c}+2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right)-b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+32 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{4 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{3 \left(-8 c^2 d e \left(d \sqrt{b^2-4 a c}-3 a e+6 b d\right)+2 c e^2 \left(4 b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+32 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{4 \sqrt{2} \sqrt{c} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(Sqrt[d + e*x]*(b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2) + b^2*(a*e*(-5*d + 19*e*x) + c*x*(8*d^2 - 37*d*e*x + 3*e^2*x^2)) + 4*c*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)) + 4*b*(3*a^2*e^2 + 3*c^2*d*x^2*(3*d - 2*e*x) + a*c*(5*d^2 - 9*d*e*x + 4*e^2*x^2))))/(4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(5/2)) + (3*Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(8*Sqrt[c]*(b^2 - 4*a*c)^(5/2))","A",1
2304,1,4707,441,6.2603931,"\int \frac{(d+e x)^{3/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^3,x]","\text{Result too large to show}","-\frac{3 \sqrt{c} \left(-4 c e \left(-d \sqrt{b^2-4 a c}-a e+4 b d\right)+b e^2 \left(3 b-2 \sqrt{b^2-4 a c}\right)+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{2 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{3 \sqrt{c} \left(-4 c e \left(d \sqrt{b^2-4 a c}-a e+4 b d\right)+b e^2 \left(2 \sqrt{b^2-4 a c}+3 b\right)+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{2 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{\sqrt{d+e x} \left(4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}",1,"-1/2*((d + e*x)^(5/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(5/2)*(-1/2*(a*c*e*(2*c*d - b*e)^2) + ((b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 + b^2*e^2 - c*e*(11*b*d - 6*a*e)))/2 + c*(-1/2*(c*e*(b*d - 2*a*e)*(2*c*d - b*e)) + ((2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - c*e*(11*b*d - 6*a*e)))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (-1/2*(e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*(d + e*x)^(3/2)) + (2*((3*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4)*Sqrt[d + e*x])/c + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*(((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)/2 - (-1/2*((-2*c*d + b*e)*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)) + 2*c*((e*((3*c*d*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*a*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 - (d*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*(((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)/2 + (-1/2*((-2*c*d + b*e)*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)) + 2*c*((e*((3*c*d*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*a*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 - (d*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))","B",1
2305,1,580,634,4.0232885,"\int \frac{\sqrt{d+e x}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]","-\frac{\sqrt{c} \left(\frac{\left(8 c^2 d e \left(3 d \sqrt{b^2-4 a c}+13 a e-18 b d\right)+2 c e^2 \left(-2 b \left(6 d \sqrt{b^2-4 a c}+13 a e\right)+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+96 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\left(-8 c^2 d e \left(3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right)+2 c e^2 \left(12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right)+b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+96 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \left(e (a e-b d)+c d^2\right)}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left(b^2-4 a c\right) (a+x (b+c x))^2}-\frac{\sqrt{d+e x} \left(4 b c \left(2 a e^2+3 c d (d-2 e x)\right)+4 c^2 \left(a e (d+5 e x)+6 c d^2 x\right)+b^3 e^2+b^2 c e (e x-13 d)\right)}{4 \left(b^2-4 a c\right)^2 (a+x (b+c x)) \left(e (b d-a e)-c d^2\right)}","-\frac{\sqrt{c} \left(-8 c^2 d e \left(-3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right)+2 c e^2 \left(-2 b \left(6 d \sqrt{b^2-4 a c}+13 a e\right)+10 a e \sqrt{b^2-4 a c}+23 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+96 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{c} \left(-8 c^2 d e \left(3 d \sqrt{b^2-4 a c}-13 a e+18 b d\right)+2 c e^2 \left(12 b d \sqrt{b^2-4 a c}-10 a e \sqrt{b^2-4 a c}-26 a b e+23 b^2 d\right)+b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+96 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \left(a e^2-b d e+c d^2\right)}-\frac{(b+2 c x) \sqrt{d+e x}}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{\sqrt{d+e x} \left(-c x \left(-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right)-4 b c \left(2 a e^2+3 c d^2\right)-4 a c^2 d e+b^3 \left(-e^2\right)+13 b^2 c d e\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}",1,"-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (Sqrt[d + e*x]*(b^3*e^2 + b^2*c*e*(-13*d + e*x) + 4*b*c*(2*a*e^2 + 3*c*d*(d - 2*e*x)) + 4*c^2*(6*c*d^2*x + a*e*(d + 5*e*x))))/(4*(b^2 - 4*a*c)^2*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - (Sqrt[c]*(((96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2306,1,799,835,6.1186921,"\int \frac{1}{\sqrt{d+e x} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^3),x]","\frac{\frac{\sqrt{d+e x} \left(e b^2+c (e x-d) b-2 c (a e+c d x)\right)}{(a+x (b+c x))^2}-\frac{\sqrt{d+e x} \left(3 e^3 b^4+c e^2 (4 d+3 e x) b^3+c e \left(c d (6 e x-19 d)-25 a e^2\right) b^2+4 c^2 \left(3 c (d-3 e x) d^2+a e^2 (5 d-6 e x)\right) b+4 c^2 \left(6 c^2 x d^3+a c e (d+12 e x) d+7 a^2 e^3\right)\right)}{2 \left(b^2-4 a c\right) \left(e (b d-a e)-c d^2\right) (a+x (b+c x))}+\frac{3 \sqrt{c} \left(\frac{\left(32 c^4 d^4-8 c^3 e \left(8 b d+\sqrt{b^2-4 a c} d-9 a e\right) d^2+b^3 \left(b-\sqrt{b^2-4 a c}\right) e^4+2 b c e^3 \left(d b^2-\left(\sqrt{b^2-4 a c} d+5 a e\right) b+4 a \sqrt{b^2-4 a c} e\right)+2 c^2 e^2 \left(15 b^2 d^2+6 b \left(\sqrt{b^2-4 a c} d-6 a e\right) d+4 a e \left(7 a e-2 \sqrt{b^2-4 a c} d\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}\right)}{\sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}-\frac{\left(32 c^4 d^4+8 c^3 e \left(-8 b d+\sqrt{b^2-4 a c} d+9 a e\right) d^2+b^3 \left(b+\sqrt{b^2-4 a c}\right) e^4+2 b c e^3 \left(d b^2+\sqrt{b^2-4 a c} d b-5 a e b-4 a \sqrt{b^2-4 a c} e\right)+2 c^2 e^2 \left(15 b^2 d^2-6 b \left(\sqrt{b^2-4 a c} d+6 a e\right) d+4 a e \left(2 \sqrt{b^2-4 a c} d+7 a e\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right)}{\sqrt{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}}\right)}{2 \sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(c d^2+e (a e-b d)\right)}}{2 \left(b^2-4 a c\right) \left(c d^2+e (a e-b d)\right)}","-\frac{\sqrt{d+e x} \left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right)}{2 \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \left(c x^2+b x+a\right)^2}-\frac{3 \sqrt{c} \left(32 c^4 d^4-8 c^3 e \left(8 b d-\sqrt{b^2-4 a c} d-9 a e\right) d^2+b^3 \left(b+\sqrt{b^2-4 a c}\right) e^4+2 b c e^3 \left(d b^2+\sqrt{b^2-4 a c} d b-5 a e b-4 a \sqrt{b^2-4 a c} e\right)+2 c^2 e^2 \left(15 b^2 d^2-6 b \left(\sqrt{b^2-4 a c} d+6 a e\right) d+4 a e \left(2 \sqrt{b^2-4 a c} d+7 a e\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e} \left(c d^2-b e d+a e^2\right)^2}+\frac{3 \sqrt{c} \left(32 c^4 d^4-8 c^3 e \left(8 b d+\sqrt{b^2-4 a c} d-9 a e\right) d^2+b^3 \left(b-\sqrt{b^2-4 a c}\right) e^4+2 c^2 e^2 \left(15 b^2 d^2+6 b \left(\sqrt{b^2-4 a c} d-6 a e\right) d-4 a e \left(2 \sqrt{b^2-4 a c} d-7 a e\right)\right)+2 b c e^3 \left(d b^2-\left(\sqrt{b^2-4 a c} d+5 a e\right) b+4 a \sqrt{b^2-4 a c} e\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}\right)}{4 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e} \left(c d^2-e (b d-a e)\right)^2}-\frac{\sqrt{d+e x} \left(5 a c e (2 c d-b e)^2-3 c \left(4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(12 c^2 d^2-3 b^2 e^2-7 c e (b d-2 a e)\right)\right)}{4 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 \left(c x^2+b x+a\right)}",1,"((Sqrt[d + e*x]*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/(a + x*(b + c*x))^2 - (Sqrt[d + e*x]*(3*b^4*e^3 + b^3*c*e^2*(4*d + 3*e*x) + 4*b*c^2*(a*e^2*(5*d - 6*e*x) + 3*c*d^2*(d - 3*e*x)) + b^2*c*e*(-25*a*e^2 + c*d*(-19*d + 6*e*x)) + 4*c^2*(7*a^2*e^3 + 6*c^2*d^3*x + a*c*d*e*(d + 12*e*x))))/(2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) + (3*Sqrt[c]*(-(((32*c^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 + 8*c^3*d^2*e*(-8*b*d + Sqrt[b^2 - 4*a*c]*d + 9*a*e) + 2*b*c*e^3*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 5*a*b*e - 4*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(15*b^2*d^2 - 6*b*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e) + 4*a*e*(2*Sqrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((32*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d - 9*a*e) + 2*b*c*e^3*(b^2*d + 4*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + 2*c^2*e^2*(15*b^2*d^2 + 6*b*d*(Sqrt[b^2 - 4*a*c]*d - 6*a*e) + 4*a*e*(-2*Sqrt[b^2 - 4*a*c]*d + 7*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(2*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))))/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2307,1,197,629,0.4979858,"\int \frac{\sqrt{d+e x}}{a+i b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]/(a + I*b*x + c*x^2),x]","\frac{\sqrt{2} \left(\sqrt{2 c d-e \left(\sqrt{-4 a c-b^2}+i b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{-4 a c-b^2}+i b\right)}}\right)-\sqrt{e \sqrt{-4 a c-b^2}-i b e+2 c d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{-4 a c-b^2}-i b e+2 c d}}\right)\right)}{\sqrt{c} \sqrt{-4 a c-b^2}}","\frac{e \log \left(-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right)}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \log \left(\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right)}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left(\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right)}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left(\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right)}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}",1,"(Sqrt[2]*(-(Sqrt[2*c*d - I*b*e + Sqrt[-b^2 - 4*a*c]*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - I*b*e + Sqrt[-b^2 - 4*a*c]*e]]) + Sqrt[2*c*d - (I*b + Sqrt[-b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (I*b + Sqrt[-b^2 - 4*a*c])*e]]))/(Sqrt[c]*Sqrt[-b^2 - 4*a*c])","A",1
2308,1,198,705,0.6165277,"\int \frac{1}{\sqrt{d+e x} \left(a+i b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + I*b*x + c*x^2)),x]","\frac{2 \sqrt{2} \sqrt{c} \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{-4 a c-b^2}+i b\right)}}\right)}{\sqrt{2 c d-e \left(\sqrt{-4 a c-b^2}+i b\right)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{-4 a c-b^2}-i b e+2 c d}}\right)}{\sqrt{2 c d+e \left(\sqrt{-4 a c-b^2}-i b\right)}}\right)}{\sqrt{-4 a c-b^2}}","-\frac{e \log \left(-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right)}{2 \sqrt{c d^2-e (-a e+i b d)} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \log \left(\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right)}{2 \sqrt{c d^2-e (-a e+i b d)} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left(\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right)}{\sqrt{c d^2-e (-a e+i b d)} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left(\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right)}{\sqrt{c d^2-e (-a e+i b d)} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}",1,"(2*Sqrt[2]*Sqrt[c]*(-(ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - I*b*e + Sqrt[-b^2 - 4*a*c]*e]]/Sqrt[2*c*d + ((-I)*b + Sqrt[-b^2 - 4*a*c])*e]) + ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (I*b + Sqrt[-b^2 - 4*a*c])*e]]/Sqrt[2*c*d - (I*b + Sqrt[-b^2 - 4*a*c])*e]))/Sqrt[-b^2 - 4*a*c]","A",1
2309,1,134,279,0.3138641,"\int \frac{(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx","Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]","\frac{2 \left(620 \sqrt{2 x+1} \left(30 x^2+50 x-11\right)+3 \sqrt{10-5 i \sqrt{31}} \left(589+178 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+3 \sqrt{10+5 i \sqrt{31}} \left(589-178 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{58125}","\frac{4}{25} (2 x+1)^{5/2}+\frac{16}{75} (2 x+1)^{3/2}-\frac{76}{125} \sqrt{2 x+1}-\frac{1}{125} \sqrt{\frac{1}{310} \left(168698+42875 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{125} \sqrt{\frac{1}{310} \left(168698+42875 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{125} \sqrt{\frac{2}{155} \left(42875 \sqrt{35}-168698\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{125} \sqrt{\frac{2}{155} \left(42875 \sqrt{35}-168698\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(620*Sqrt[1 + 2*x]*(-11 + 50*x + 30*x^2) + 3*Sqrt[10 - (5*I)*Sqrt[31]]*(589 + (178*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 3*Sqrt[10 + (5*I)*Sqrt[31]]*(589 - (178*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/58125","C",1
2310,1,133,266,0.2604615,"\int \frac{(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx","Integrate[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2),x]","\frac{2 \left(310 \sqrt{2 x+1} (10 x+17)+3 i \sqrt{10-5 i \sqrt{31}} \left(27 \sqrt{31}+124 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)-3 i \sqrt{10+5 i \sqrt{31}} \left(27 \sqrt{31}-124 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{11625}","\frac{4}{15} (2 x+1)^{3/2}+\frac{16}{25} \sqrt{2 x+1}+\frac{1}{25} \sqrt{\frac{1}{310} \left(1225 \sqrt{35}-7162\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{25} \sqrt{\frac{1}{310} \left(1225 \sqrt{35}-7162\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{25} \sqrt{\frac{2}{155} \left(7162+1225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{25} \sqrt{\frac{2}{155} \left(7162+1225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(310*Sqrt[1 + 2*x]*(17 + 10*x) + (3*I)*Sqrt[10 - (5*I)*Sqrt[31]]*(124*I + 27*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] - (3*I)*Sqrt[10 + (5*I)*Sqrt[31]]*(-124*I + 27*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/11625","C",1
2311,1,128,253,0.2563812,"\int \frac{(1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx","Integrate[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2),x]","\frac{2}{775} \left(310 \sqrt{2 x+1}-i \sqrt{10-5 i \sqrt{31}} \left(2 \sqrt{31}-31 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+i \sqrt{10+5 i \sqrt{31}} \left(2 \sqrt{31}+31 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)","\frac{4}{5} \sqrt{2 x+1}+\frac{1}{5} \sqrt{\frac{1}{310} \left(178+35 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{5} \sqrt{\frac{1}{310} \left(178+35 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{5} \sqrt{\frac{2}{155} \left(35 \sqrt{35}-178\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{5} \sqrt{\frac{2}{155} \left(35 \sqrt{35}-178\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(310*Sqrt[1 + 2*x] - I*Sqrt[10 - (5*I)*Sqrt[31]]*(-31*I + 2*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + I*Sqrt[10 + (5*I)*Sqrt[31]]*(31*I + 2*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/775","C",1
2312,1,112,222,0.1987787,"\int \frac{\sqrt{1+2 x}}{2+3 x+5 x^2} \, dx","Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2),x]","\frac{2 \left(\sqrt{-2+i \sqrt{31}} \left(\sqrt{31}-2 i\right) \tan ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right)+\sqrt{-2-i \sqrt{31}} \left(\sqrt{31}+2 i\right) \tan ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right)\right)}{5 \sqrt{217}}","\frac{\log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{\sqrt{10 \left(2+\sqrt{35}\right)}}-\frac{\log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{\sqrt{10 \left(2+\sqrt{35}\right)}}-\sqrt{\frac{2}{5 \left(\sqrt{35}-2\right)}} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\sqrt{\frac{2}{5 \left(\sqrt{35}-2\right)}} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(Sqrt[-2 + I*Sqrt[31]]*(-2*I + Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]] + Sqrt[-2 - I*Sqrt[31]]*(2*I + Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]]))/(5*Sqrt[217])","C",1
2313,1,112,218,0.2351789,"\int \frac{1}{\sqrt{1+2 x} \left(2+3 x+5 x^2\right)} \, dx","Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)),x]","\frac{2 \left(\sqrt{2-i \sqrt{31}} \left(\sqrt{31}-2 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{2+i \sqrt{31}} \left(\sqrt{31}+2 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{7 \sqrt{155}}","-\frac{\log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{\sqrt{14 \left(2+\sqrt{35}\right)}}+\frac{\log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{\sqrt{14 \left(2+\sqrt{35}\right)}}-\sqrt{\frac{2}{217} \left(2+\sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\sqrt{\frac{2}{217} \left(2+\sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(Sqrt[2 - I*Sqrt[31]]*(-2*I + Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(2*I + Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(7*Sqrt[155])","C",1
2314,1,122,253,0.4291432,"\int \frac{1}{(1+2 x)^{3/2} \left(2+3 x+5 x^2\right)} \, dx","Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)),x]","\frac{2 \left(-\frac{2170}{\sqrt{2 x+1}}+\sqrt{10-5 i \sqrt{31}} \left(124+27 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{10+5 i \sqrt{31}} \left(124-27 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{7595}","-\frac{4}{7 \sqrt{2 x+1}}-\frac{1}{7} \sqrt{\frac{1}{434} \left(178+35 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{7} \sqrt{\frac{1}{434} \left(178+35 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{7} \sqrt{\frac{2}{217} \left(35 \sqrt{35}-178\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{7} \sqrt{\frac{2}{217} \left(35 \sqrt{35}-178\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*(-2170/Sqrt[1 + 2*x] + Sqrt[10 - (5*I)*Sqrt[31]]*(124 + (27*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[10 + (5*I)*Sqrt[31]]*(124 - (27*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/7595","C",1
2315,1,133,266,0.5143294,"\int \frac{1}{(1+2 x)^{5/2} \left(2+3 x+5 x^2\right)} \, dx","Integrate[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)),x]","\frac{2 \left(-\frac{2170 (24 x+19)}{(2 x+1)^{3/2}}+3 i \sqrt{10-5 i \sqrt{31}} \left(178 \sqrt{31}+589 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)-3 i \sqrt{10+5 i \sqrt{31}} \left(178 \sqrt{31}-589 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{159495}","-\frac{16}{49 \sqrt{2 x+1}}-\frac{4}{21 (2 x+1)^{3/2}}-\frac{1}{49} \sqrt{\frac{1}{434} \left(1225 \sqrt{35}-7162\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{49} \sqrt{\frac{1}{434} \left(1225 \sqrt{35}-7162\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{49} \sqrt{\frac{2}{217} \left(7162+1225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{49} \sqrt{\frac{2}{217} \left(7162+1225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(2*((-2170*(19 + 24*x))/(1 + 2*x)^(3/2) + (3*I)*Sqrt[10 - (5*I)*Sqrt[31]]*(589*I + 178*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] - (3*I)*Sqrt[10 + (5*I)*Sqrt[31]]*(-589*I + 178*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/159495","C",1
2316,1,199,296,0.4404134,"\int \frac{(1+2 x)^{7/2}}{\left(2+3 x+5 x^2\right)^2} \, dx","Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]","\frac{1}{217} \left(\frac{(20 x+37) (2 x+1)^{9/2}}{5 x^2+3 x+2}-8 (2 x+1)^{7/2}-28 (2 x+1)^{5/2}-\frac{56}{5} (2 x+1)^{3/2}+\frac{4228}{25} \sqrt{2 x+1}-\frac{14 i \left(\sqrt{2-i \sqrt{31}} \left(512 \sqrt{31}-4681 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)-\sqrt{2+i \sqrt{31}} \left(512 \sqrt{31}+4681 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{775 \sqrt{5}}\right)","-\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left(5 x^2+3 x+2\right)}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left(5682718+968975 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{775} \sqrt{\frac{1}{310} \left(5682718+968975 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{775} \sqrt{\frac{2}{155} \left(968975 \sqrt{35}-5682718\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)-\frac{1}{775} \sqrt{\frac{2}{155} \left(968975 \sqrt{35}-5682718\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"((4228*Sqrt[1 + 2*x])/25 - (56*(1 + 2*x)^(3/2))/5 - 28*(1 + 2*x)^(5/2) - 8*(1 + 2*x)^(7/2) + ((1 + 2*x)^(9/2)*(37 + 20*x))/(2 + 3*x + 5*x^2) - (((14*I)/775)*(Sqrt[2 - I*Sqrt[31]]*(-4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] - Sqrt[2 + I*Sqrt[31]]*(4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/Sqrt[5])/217","C",1
2317,1,141,283,0.6659925,"\int \frac{(1+2 x)^{5/2}}{\left(2+3 x+5 x^2\right)^2} \, dx","Integrate[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^2,x]","\frac{-\frac{155 \sqrt{2 x+1} (54 x+41)}{5 x^2+3 x+2}+2 \sqrt{10-5 i \sqrt{31}} \left(62-101 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+2 \sqrt{10+5 i \sqrt{31}} \left(62+101 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{24025}","-\frac{(5-4 x) (2 x+1)^{3/2}}{31 \left(5 x^2+3 x+2\right)}-\frac{8}{155} \sqrt{2 x+1}+\frac{1}{155} \sqrt{\frac{1}{310} \left(10325 \sqrt{35}-32678\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{155} \sqrt{\frac{1}{310} \left(10325 \sqrt{35}-32678\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{155} \sqrt{\frac{2}{155} \left(32678+10325 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{1}{155} \sqrt{\frac{2}{155} \left(32678+10325 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"((-155*Sqrt[1 + 2*x]*(41 + 54*x))/(2 + 3*x + 5*x^2) + 2*Sqrt[10 - (5*I)*Sqrt[31]]*(62 - (101*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(62 + (101*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/24025","C",1
2318,1,141,270,0.7124386,"\int \frac{(1+2 x)^{3/2}}{\left(2+3 x+5 x^2\right)^2} \, dx","Integrate[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^2,x]","\frac{\frac{155 \sqrt{2 x+1} (4 x-5)}{5 x^2+3 x+2}+2 \left(31-4 i \sqrt{31}\right) \sqrt{10-5 i \sqrt{31}} \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+2 \left(31+4 i \sqrt{31}\right) \sqrt{10+5 i \sqrt{31}} \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{4805}","-\frac{\sqrt{2 x+1} (5-4 x)}{31 \left(5 x^2+3 x+2\right)}-\frac{1}{31} \sqrt{\frac{1}{310} \left(47 \sqrt{35}-218\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{31} \sqrt{\frac{1}{310} \left(47 \sqrt{35}-218\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{31} \sqrt{\frac{2}{155} \left(218+47 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{1}{31} \sqrt{\frac{2}{155} \left(218+47 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"((155*Sqrt[1 + 2*x]*(-5 + 4*x))/(2 + 3*x + 5*x^2) + 2*(31 - (4*I)*Sqrt[31])*Sqrt[10 - (5*I)*Sqrt[31]]*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*(31 + (4*I)*Sqrt[31])*Sqrt[10 + (5*I)*Sqrt[31]]*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/4805","C",1
2319,1,145,270,0.3555878,"\int \frac{\sqrt{1+2 x}}{\left(2+3 x+5 x^2\right)^2} \, dx","Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^2,x]","\frac{1}{31} \left(\frac{\sqrt{2 x+1} (10 x+3)}{5 x^2+3 x+2}+\frac{2 \sqrt{10-5 i \sqrt{31}} \left(62-39 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+2 \sqrt{10+5 i \sqrt{31}} \left(62+39 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{1085}\right)","\frac{\sqrt{2 x+1} (10 x+3)}{31 \left(5 x^2+3 x+2\right)}+\frac{1}{31} \sqrt{\frac{1}{434} \left(47 \sqrt{35}-218\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{31} \sqrt{\frac{1}{434} \left(47 \sqrt{35}-218\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{31} \sqrt{\frac{2}{217} \left(218+47 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{1}{31} \sqrt{\frac{2}{217} \left(218+47 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"((Sqrt[1 + 2*x]*(3 + 10*x))/(2 + 3*x + 5*x^2) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(62 - (39*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(62 + (39*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/1085)/31","C",1
2320,1,166,270,0.3189514,"\int \frac{1}{\sqrt{1+2 x} \left(2+3 x+5 x^2\right)^2} \, dx","Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2),x]","\frac{1}{217} \left(\frac{\sqrt{2 x+1} (20 x+37)}{5 x^2+3 x+2}+\frac{2 \sqrt{10-5 i \sqrt{31}} \left(101 \sqrt{31}-62 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)}{31 \left(\sqrt{31}+2 i\right)}+\frac{2 \sqrt{10+5 i \sqrt{31}} \left(101 \sqrt{31}+62 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{31 \left(\sqrt{31}-2 i\right)}\right)","\frac{\sqrt{2 x+1} (20 x+37)}{217 \left(5 x^2+3 x+2\right)}-\frac{1}{217} \sqrt{\frac{1}{434} \left(10325 \sqrt{35}-32678\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)+\frac{1}{217} \sqrt{\frac{1}{434} \left(10325 \sqrt{35}-32678\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)-\frac{1}{217} \sqrt{\frac{2}{217} \left(32678+10325 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{1}{217} \sqrt{\frac{2}{217} \left(32678+10325 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"((Sqrt[1 + 2*x]*(37 + 20*x))/(2 + 3*x + 5*x^2) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(-62*I + 101*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]])/(31*(2*I + Sqrt[31])) + (2*Sqrt[10 + (5*I)*Sqrt[31]]*(62*I + 101*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/(31*(-2*I + Sqrt[31])))/217","C",1
2321,1,158,283,0.3878824,"\int \frac{1}{(1+2 x)^{3/2} \left(2+3 x+5 x^2\right)^2} \, dx","Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^2),x]","\frac{1}{217} \left(\frac{20 x+37}{\sqrt{2 x+1} \left(5 x^2+3 x+2\right)}-\frac{604}{7 \sqrt{2 x+1}}+\frac{2 \sqrt{10-5 i \sqrt{31}} \left(25234+3657 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+2 \sqrt{10+5 i \sqrt{31}} \left(25234-3657 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{7595}\right)","\frac{20 x+37}{217 \sqrt{2 x+1} \left(5 x^2+3 x+2\right)}-\frac{604}{1519 \sqrt{2 x+1}}-\frac{\sqrt{\frac{1}{434} \left(5682718+968975 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1519}+\frac{\sqrt{\frac{1}{434} \left(5682718+968975 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1519}+\frac{\sqrt{\frac{2}{217} \left(968975 \sqrt{35}-5682718\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{1519}-\frac{\sqrt{\frac{2}{217} \left(968975 \sqrt{35}-5682718\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{1519}",1,"(-604/(7*Sqrt[1 + 2*x]) + (37 + 20*x)/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(25234 + (3657*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(25234 - (3657*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/7595)/217","C",1
2322,1,176,296,0.5182065,"\int \frac{1}{(1+2 x)^{5/2} \left(2+3 x+5 x^2\right)^2} \, dx","Integrate[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)^2),x]","\frac{1}{217} \left(\frac{20 x+37}{(2 x+1)^{3/2} \left(5 x^2+3 x+2\right)}-\frac{4680}{49 \sqrt{2 x+1}}-\frac{820}{21 (2 x+1)^{3/2}}+\frac{2 i \sqrt{5} \left(\sqrt{2-i \sqrt{31}} \left(9188 \sqrt{31}+15469 i\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\left(-9188 \sqrt{31}+15469 i\right) \sqrt{2+i \sqrt{31}} \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{10633}\right)","\frac{20 x+37}{217 (2 x+1)^{3/2} \left(5 x^2+3 x+2\right)}-\frac{4680}{10633 \sqrt{2 x+1}}-\frac{820}{4557 (2 x+1)^{3/2}}-\frac{5 \sqrt{\frac{1}{434} \left(2632525 \sqrt{35}-12504542\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{10633}+\frac{5 \sqrt{\frac{1}{434} \left(2632525 \sqrt{35}-12504542\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{10633}+\frac{5 \sqrt{\frac{2}{217} \left(12504542+2632525 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{10633}-\frac{5 \sqrt{\frac{2}{217} \left(12504542+2632525 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{10633}",1,"(-820/(21*(1 + 2*x)^(3/2)) - 4680/(49*Sqrt[1 + 2*x]) + (37 + 20*x)/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)) + ((2*I)/10633)*Sqrt[5]*(Sqrt[2 - I*Sqrt[31]]*(15469*I + 9188*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + (15469*I - 9188*Sqrt[31])*Sqrt[2 + I*Sqrt[31]]*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/217","C",1
2323,1,236,313,0.7381968,"\int \frac{(1+2 x)^{9/2}}{\left(2+3 x+5 x^2\right)^3} \, dx","Integrate[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]","\frac{-\frac{7 (640 x+409) (2 x+1)^{11/2}}{5 x^2+3 x+2}+\frac{217 (20 x+37) (2 x+1)^{11/2}}{\left(5 x^2+3 x+2\right)^2}+1792 (2 x+1)^{9/2}+1932 (2 x+1)^{7/2}-2352 (2 x+1)^{5/2}-\frac{47236}{5} (2 x+1)^{3/2}-\frac{155232}{25} \sqrt{2 x+1}+\frac{294 \left(\sqrt{2-i \sqrt{31}} \left(8184-7907 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{2+i \sqrt{31}} \left(8184+7907 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{775 \sqrt{5}}}{94178}","-\frac{(5-4 x) (2 x+1)^{7/2}}{62 \left(5 x^2+3 x+2\right)^2}-\frac{(1143-1088 x) (2 x+1)^{3/2}}{9610 \left(5 x^2+3 x+2\right)}-\frac{1584 \sqrt{2 x+1}}{24025}+\frac{3 \sqrt{\frac{1}{310} \left(64681225 \sqrt{35}-250141922\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{48050}-\frac{3 \sqrt{\frac{1}{310} \left(64681225 \sqrt{35}-250141922\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{48050}-\frac{3 \sqrt{\frac{1}{310} \left(250141922+64681225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{24025}+\frac{3 \sqrt{\frac{1}{310} \left(250141922+64681225 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{24025}",1,"((-155232*Sqrt[1 + 2*x])/25 - (47236*(1 + 2*x)^(3/2))/5 - 2352*(1 + 2*x)^(5/2) + 1932*(1 + 2*x)^(7/2) + 1792*(1 + 2*x)^(9/2) + (217*(1 + 2*x)^(11/2)*(37 + 20*x))/(2 + 3*x + 5*x^2)^2 - (7*(1 + 2*x)^(11/2)*(409 + 640*x))/(2 + 3*x + 5*x^2) + (294*(Sqrt[2 - I*Sqrt[31]]*(8184 - (7907*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(8184 + (7907*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(775*Sqrt[5]))/94178","C",1
2324,1,223,300,0.6705097,"\int \frac{(1+2 x)^{7/2}}{\left(2+3 x+5 x^2\right)^3} \, dx","Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^3,x]","\frac{-\frac{5 (400 x+89) (2 x+1)^{9/2}}{5 x^2+3 x+2}+\frac{217 (20 x+37) (2 x+1)^{9/2}}{\left(5 x^2+3 x+2\right)^2}+800 (2 x+1)^{7/2}+196 (2 x+1)^{5/2}-2352 (2 x+1)^{3/2}-\frac{35084}{5} \sqrt{2 x+1}+\frac{98 \left(\sqrt{2-i \sqrt{31}} \left(5549-902 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{2+i \sqrt{31}} \left(5549+902 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{155 \sqrt{5}}}{94178}","-\frac{(5-4 x) (2 x+1)^{5/2}}{62 \left(5 x^2+3 x+2\right)^2}-\frac{(957-592 x) \sqrt{2 x+1}}{9610 \left(5 x^2+3 x+2\right)}-\frac{\sqrt{\frac{1}{310} \left(1806875 \sqrt{35}-9651062\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{9610}+\frac{\sqrt{\frac{1}{310} \left(1806875 \sqrt{35}-9651062\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{9610}-\frac{\sqrt{\frac{1}{310} \left(9651062+1806875 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{4805}+\frac{\sqrt{\frac{1}{310} \left(9651062+1806875 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{4805}",1,"((-35084*Sqrt[1 + 2*x])/5 - 2352*(1 + 2*x)^(3/2) + 196*(1 + 2*x)^(5/2) + 800*(1 + 2*x)^(7/2) + (217*(1 + 2*x)^(9/2)*(37 + 20*x))/(2 + 3*x + 5*x^2)^2 - (5*(1 + 2*x)^(9/2)*(89 + 400*x))/(2 + 3*x + 5*x^2) + (98*(Sqrt[2 - I*Sqrt[31]]*(5549 - (902*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(5549 + (902*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(155*Sqrt[5]))/94178","C",1
2325,1,209,300,0.5808086,"\int \frac{(1+2 x)^{5/2}}{\left(2+3 x+5 x^2\right)^3} \, dx","Integrate[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^3,x]","\frac{\frac{(480 x+1973) (2 x+1)^{7/2}}{5 x^2+3 x+2}+\frac{217 (20 x+37) (2 x+1)^{7/2}}{\left(5 x^2+3 x+2\right)^2}-192 (2 x+1)^{5/2}-1540 (2 x+1)^{3/2}-2352 \sqrt{2 x+1}+\frac{294 \left(\sqrt{2-i \sqrt{31}} \left(124-47 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{2+i \sqrt{31}} \left(124+47 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{31 \sqrt{5}}}{94178}","-\frac{(5-4 x) (2 x+1)^{3/2}}{62 \left(5 x^2+3 x+2\right)^2}+\frac{3 (78 x+11) \sqrt{2 x+1}}{1922 \left(5 x^2+3 x+2\right)}+\frac{3 \sqrt{\frac{1}{310} \left(2705 \sqrt{35}-15082\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1922}-\frac{3 \sqrt{\frac{1}{310} \left(2705 \sqrt{35}-15082\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1922}-\frac{3}{961} \sqrt{\frac{1}{310} \left(15082+2705 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{3}{961} \sqrt{\frac{1}{310} \left(15082+2705 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(-2352*Sqrt[1 + 2*x] - 1540*(1 + 2*x)^(3/2) - 192*(1 + 2*x)^(5/2) + (217*(1 + 2*x)^(7/2)*(37 + 20*x))/(2 + 3*x + 5*x^2)^2 + ((1 + 2*x)^(7/2)*(1973 + 480*x))/(2 + 3*x + 5*x^2) + (294*(Sqrt[2 - I*Sqrt[31]]*(124 - (47*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(124 + (47*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(31*Sqrt[5]))/94178","C",1
2326,1,198,300,0.4961811,"\int \frac{(1+2 x)^{3/2}}{\left(2+3 x+5 x^2\right)^3} \, dx","Integrate[(1 + 2*x)^(3/2)/(2 + 3*x + 5*x^2)^3,x]","\frac{\frac{(2960 x+4391) (2 x+1)^{5/2}}{5 x^2+3 x+2}+\frac{217 (20 x+37) (2 x+1)^{5/2}}{\left(5 x^2+3 x+2\right)^2}-1184 (2 x+1)^{3/2}-3276 \sqrt{2 x+1}+\frac{42 \left(\sqrt{2-i \sqrt{31}} \left(1209-218 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+\sqrt{2+i \sqrt{31}} \left(1209+218 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)\right)}{31 \sqrt{5}}}{94178}","-\frac{\sqrt{2 x+1} (5-4 x)}{62 \left(5 x^2+3 x+2\right)^2}+\frac{\sqrt{2 x+1} (120 x+67)}{1922 \left(5 x^2+3 x+2\right)}-\frac{3 \sqrt{\frac{1}{434} \left(2705 \sqrt{35}-15082\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1922}+\frac{3 \sqrt{\frac{1}{434} \left(2705 \sqrt{35}-15082\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{1922}-\frac{3}{961} \sqrt{\frac{1}{434} \left(15082+2705 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)+\frac{3}{961} \sqrt{\frac{1}{434} \left(15082+2705 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)",1,"(-3276*Sqrt[1 + 2*x] - 1184*(1 + 2*x)^(3/2) + (217*(1 + 2*x)^(5/2)*(37 + 20*x))/(2 + 3*x + 5*x^2)^2 + ((1 + 2*x)^(5/2)*(4391 + 2960*x))/(2 + 3*x + 5*x^2) + (42*(Sqrt[2 - I*Sqrt[31]]*(1209 - (218*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(1209 + (218*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(31*Sqrt[5]))/94178","C",1
2327,1,151,300,0.6431298,"\int \frac{\sqrt{1+2 x}}{\left(2+3 x+5 x^2\right)^3} \, dx","Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^3,x]","\frac{\frac{1085 \sqrt{2 x+1} \left(8950 x^3+8365 x^2+7547 x+1849\right)}{\left(5 x^2+3 x+2\right)^2}+2 \sqrt{10-5 i \sqrt{31}} \left(16864-7353 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+2 \sqrt{10+5 i \sqrt{31}} \left(16864+7353 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{14597590}","\frac{\sqrt{2 x+1} (10 x+3)}{62 \left(5 x^2+3 x+2\right)^2}+\frac{\sqrt{2 x+1} (1790 x+599)}{13454 \left(5 x^2+3 x+2\right)}+\frac{\sqrt{\frac{1}{434} \left(1806875 \sqrt{35}-9651062\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{13454}-\frac{\sqrt{\frac{1}{434} \left(1806875 \sqrt{35}-9651062\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{13454}-\frac{\sqrt{\frac{1}{434} \left(9651062+1806875 \sqrt{35}\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{6727}+\frac{\sqrt{\frac{1}{434} \left(9651062+1806875 \sqrt{35}\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{6727}",1,"((1085*Sqrt[1 + 2*x]*(1849 + 7547*x + 8365*x^2 + 8950*x^3))/(2 + 3*x + 5*x^2)^2 + 2*Sqrt[10 - (5*I)*Sqrt[31]]*(16864 - (7353*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(16864 + (7353*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/14597590","C",1
2328,1,151,314,0.6436511,"\int \frac{1}{\sqrt{1+2 x} \left(2+3 x+5 x^2\right)^3} \, dx","Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]","\frac{\frac{1085 \sqrt{2 x+1} \left(39600 x^3+69895 x^2+47861 x+26483\right)}{\left(5 x^2+3 x+2\right)^2}+6 \sqrt{10-5 i \sqrt{31}} \left(228749-23998 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+6 \sqrt{10+5 i \sqrt{31}} \left(228749+23998 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{102183130}","\frac{\sqrt{2 x+1} (20 x+37)}{434 \left(5 x^2+3 x+2\right)^2}+\frac{\sqrt{2 x+1} (7920 x+9227)}{94178 \left(5 x^2+3 x+2\right)}-\frac{3 \sqrt{\frac{1}{434} \left(64681225 \sqrt{35}-250141922\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{94178}+\frac{3 \sqrt{\frac{1}{434} \left(64681225 \sqrt{35}-250141922\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{94178}-\frac{3 \sqrt{\frac{1}{434} \left(2+\sqrt{35}\right)} \left(7379+264 \sqrt{35}\right) \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{47089}+\frac{3 \sqrt{\frac{1}{434} \left(2+\sqrt{35}\right)} \left(7379+264 \sqrt{35}\right) \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{47089}",1,"((1085*Sqrt[1 + 2*x]*(26483 + 47861*x + 69895*x^2 + 39600*x^3))/(2 + 3*x + 5*x^2)^2 + 6*Sqrt[10 - (5*I)*Sqrt[31]]*(228749 - (23998*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 6*Sqrt[10 + (5*I)*Sqrt[31]]*(228749 + (23998*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/102183130","C",1
2329,1,156,313,0.8573581,"\int \frac{1}{(1+2 x)^{3/2} \left(2+3 x+5 x^2\right)^3} \, dx","Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]","\frac{-\frac{217 \left(4054500 x^4+4501400 x^3+4077245 x^2+1525635 x+429487\right)}{\sqrt{2 x+1} \left(5 x^2+3 x+2\right)^2}+6 \sqrt{10-5 i \sqrt{31}} \left(560852+58421 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right)+6 \sqrt{10+5 i \sqrt{31}} \left(560852-58421 i \sqrt{31}\right) \tanh ^{-1}\left(\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right)}{143056382}","\frac{20 x+37}{434 \sqrt{2 x+1} \left(5 x^2+3 x+2\right)^2}+\frac{5 (2080 x+2329)}{94178 \sqrt{2 x+1} \left(5 x^2+3 x+2\right)}-\frac{81090}{329623 \sqrt{2 x+1}}-\frac{15 \sqrt{\frac{1}{434} \left(2257111762+387427075 \sqrt{35}\right)} \log \left(5 (2 x+1)-\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{659246}+\frac{15 \sqrt{\frac{1}{434} \left(2257111762+387427075 \sqrt{35}\right)} \log \left(5 (2 x+1)+\sqrt{10 \left(2+\sqrt{35}\right)} \sqrt{2 x+1}+\sqrt{35}\right)}{659246}-\frac{15 \sqrt{\frac{1}{434} \left(387427075 \sqrt{35}-2257111762\right)} \tan ^{-1}\left(\frac{\sqrt{10 \left(2+\sqrt{35}\right)}-10 \sqrt{2 x+1}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{329623}+\frac{15 \sqrt{\frac{1}{434} \left(387427075 \sqrt{35}-2257111762\right)} \tan ^{-1}\left(\frac{10 \sqrt{2 x+1}+\sqrt{10 \left(2+\sqrt{35}\right)}}{\sqrt{10 \left(\sqrt{35}-2\right)}}\right)}{329623}",1,"((-217*(429487 + 1525635*x + 4077245*x^2 + 4501400*x^3 + 4054500*x^4))/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + 6*Sqrt[10 - (5*I)*Sqrt[31]]*(560852 + (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 6*Sqrt[10 + (5*I)*Sqrt[31]]*(560852 - (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/143056382","C",1
2330,1,518,389,1.5371453,"\int \frac{x^{9/2}}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[x^(9/2)/(a + b*x + c*x^2)^3,x]","\frac{\frac{x^{11/2} \left(12 a^2 c^2-25 a b^2 c-16 a b c^2 x+7 b^4+7 b^3 c x\right)}{2 a \left(4 a c-b^2\right) (a+x (b+c x))}+\frac{-\frac{6 a^2 b \sqrt{x} \left(b^2-8 a c\right)}{c^2}+\frac{2 a^2 x^{3/2} \left(b^2-28 a c\right)}{c}+\frac{3 \sqrt{2} a^2 \left(\frac{\left(28 a^2 c^2 \sqrt{b^2-4 a c}-44 a^2 b c^2+11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}-b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(28 a^2 c^2 \sqrt{b^2-4 a c}+44 a^2 b c^2-11 a b^3 c-9 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{c^{5/2} \sqrt{b^2-4 a c}}+24 a^2 b x^{5/2}+2 b x^{9/2} \left(7 b^2-16 a c\right)+6 a x^{7/2} \left(4 a c-3 b^2\right)}{4 a \left(b^2-4 a c\right)}+\frac{x^{11/2} \left(-2 a c+b^2+b c x\right)}{(a+x (b+c x))^2}}{2 a \left(b^2-4 a c\right)}","\frac{3 \left(-\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{4 \sqrt{2} c^{5/2} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left(\frac{44 a^2 b c^2-11 a b^3 c+b^5}{\sqrt{b^2-4 a c}}+28 a^2 c^2-9 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 \sqrt{2} c^{5/2} \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{3 b \sqrt{x} \left(b^2-8 a c\right)}{4 c^2 \left(b^2-4 a c\right)^2}+\frac{x^{7/2} (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{x^{3/2} \left(b x \left(b^2-16 a c\right)+a \left(b^2-28 a c\right)\right)}{4 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}",1,"((x^(11/2)*(b^2 - 2*a*c + b*c*x))/(a + x*(b + c*x))^2 + (x^(11/2)*(7*b^4 - 25*a*b^2*c + 12*a^2*c^2 + 7*b^3*c*x - 16*a*b*c^2*x))/(2*a*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + ((-6*a^2*b*(b^2 - 8*a*c)*Sqrt[x])/c^2 + (2*a^2*(b^2 - 28*a*c)*x^(3/2))/c + 24*a^2*b*x^(5/2) + 6*a*(-3*b^2 + 4*a*c)*x^(7/2) + 2*b*(7*b^2 - 16*a*c)*x^(9/2) + (3*Sqrt[2]*a^2*(((-b^5 + 11*a*b^3*c - 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 9*a*b^2*c*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((b^5 - 11*a*b^3*c + 44*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 9*a*b^2*c*Sqrt[b^2 - 4*a*c] + 28*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(c^(5/2)*Sqrt[b^2 - 4*a*c]))/(4*a*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a*c))","A",1
2331,1,440,458,1.2102128,"\int \frac{1}{x^{3/2} \left(a+b x+c x^2\right)^3} \, dx","Integrate[1/(x^(3/2)*(a + b*x + c*x^2)^3),x]","\frac{\frac{-36 a^2 c^2+35 a b^2 c+32 a b c^2 x-5 b^4-5 b^3 c x}{2 a \sqrt{x} \left(4 a c-b^2\right) (a+x (b+c x))}-\frac{\frac{3 \sqrt{c} \left(\frac{b \left(124 a^2 c^2-47 a b^2 c+5 b^4\right)}{\sqrt{b^2-4 a c}}+\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}-5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{3 \left(5 b^2-12 a c\right) \left(b^2-5 a c\right)}{\sqrt{x}}}{2 a^2 \left(b^2-4 a c\right)}+\frac{-2 a c+b^2+b c x}{\sqrt{x} (a+x (b+c x))^2}}{2 a \left(b^2-4 a c\right)}","-\frac{3 \left(5 b^2-12 a c\right) \left(b^2-5 a c\right)}{4 a^3 \sqrt{x} \left(b^2-4 a c\right)^2}+\frac{36 a^2 c^2+b c x \left(5 b^2-32 a c\right)-35 a b^2 c+5 b^4}{4 a^2 \sqrt{x} \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 \sqrt{c} \left(\sqrt{b^2-4 a c} \left(60 a^2 c^2-37 a b^2 c+5 b^4\right)+124 a^2 b c^2-47 a b^3 c+5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{4 \sqrt{2} a^3 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left(-60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c+37 a b^2 c \sqrt{b^2-4 a c}-5 b^4 \sqrt{b^2-4 a c}+5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 \sqrt{2} a^3 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c x}{2 a \sqrt{x} \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"((b^2 - 2*a*c + b*c*x)/(Sqrt[x]*(a + x*(b + c*x))^2) + (-5*b^4 + 35*a*b^2*c - 36*a^2*c^2 - 5*b^3*c*x + 32*a*b*c^2*x)/(2*a*(-b^2 + 4*a*c)*Sqrt[x]*(a + x*(b + c*x))) - ((3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/Sqrt[x] + (3*Sqrt[c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a^2*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a*c))","A",1
2332,1,19,28,0.0050654,"\int \frac{3-x+x^2}{\sqrt[3]{x}} \, dx","Integrate[(3 - x + x^2)/x^(1/3),x]","\frac{3}{40} x^{2/3} \left(5 x^2-8 x+60\right)","\frac{3 x^{8/3}}{8}-\frac{3 x^{5/3}}{5}+\frac{9 x^{2/3}}{2}",1,"(3*x^(2/3)*(60 - 8*x + 5*x^2))/40","A",1
2333,1,206,248,0.3246874,"\int (d+e x)^3 \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[a + b*x + c*x^2],x]","\frac{\frac{e (a+x (b+c x))^{3/2} \left(-2 c e (16 a e+75 b d+21 b e x)+35 b^2 e^2+12 c^2 d (16 d+7 e x)\right)}{48 c^2}+\frac{5 (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\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)\right)}{256 c^{7/2}}+e (d+e x)^2 (a+x (b+c x))^{3/2}}{5 c}","-\frac{\left(b^2-4 a c\right) (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{9/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right)}{128 c^4}+\frac{e \left(a+b x+c x^2\right)^{3/2} \left(-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right)}{240 c^3}+\frac{e (d+e x)^2 \left(a+b x+c x^2\right)^{3/2}}{5 c}",1,"(e*(d + e*x)^2*(a + x*(b + c*x))^(3/2) + (e*(a + x*(b + c*x))^(3/2)*(35*b^2*e^2 + 12*c^2*d*(16*d + 7*e*x) - 2*c*e*(75*b*d + 16*a*e + 21*b*e*x)))/(48*c^2) + (5*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(256*c^(7/2)))/(5*c)","A",1
2334,1,162,191,0.1735877,"\int (d+e x)^2 \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[a + b*x + c*x^2],x]","\frac{\frac{\left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\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)\right)}{32 c^{5/2}}+\frac{5 e (a+x (b+c x))^{3/2} (2 c d-b e)}{6 c}+e (d+e x) (a+x (b+c x))^{3/2}}{4 c}","-\frac{\left(b^2-4 a c\right) \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right)}{64 c^3}+\frac{5 e \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x) \left(a+b x+c x^2\right)^{3/2}}{4 c}",1,"((5*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(6*c) + e*(d + e*x)*(a + x*(b + c*x))^(3/2) + ((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(32*c^(5/2)))/(4*c)","A",1
2335,1,114,115,0.105223,"\int (d+e x) \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)*Sqrt[a + b*x + c*x^2],x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left(4 c (2 a e+c x (3 d+2 e x))-3 b^2 e+2 b c (3 d+e x)\right)+3 \left(b^2-4 a c\right) (b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{48 c^{5/2}}","-\frac{\left(b^2-4 a c\right) (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{5/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left(a+b x+c x^2\right)^{3/2}}{3 c}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3*b^2*e + 2*b*c*(3*d + e*x) + 4*c*(2*a*e + c*x*(3*d + 2*e*x))) + 3*(b^2 - 4*a*c)*(-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(48*c^(5/2))","A",1
2336,1,71,75,0.0583502,"\int \sqrt{a+b x+c x^2} \, dx","Integrate[Sqrt[a + b*x + c*x^2],x]","\frac{(b+2 c x) \sqrt{a+x (b+c x)}}{4 c}-\frac{\left(b^2-4 a c\right) \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)}{8 c^{3/2}}","\frac{(b+2 c x) \sqrt{a+b x+c x^2}}{4 c}-\frac{\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)}{8 c^{3/2}}",1,"((b + 2*c*x)*Sqrt[a + x*(b + c*x)])/(4*c) - ((b^2 - 4*a*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(3/2))","A",1
2337,1,145,152,0.2076126,"\int \frac{\sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x),x]","\frac{-2 \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+\frac{(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+2 e \sqrt{a+x (b+c x)}}{2 e^2}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^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} e^2}+\frac{\sqrt{a+b x+c x^2}}{e}",1,"(2*e*Sqrt[a + x*(b + c*x)] + ((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] - 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(2*e^2)","A",1
2338,1,152,160,0.1732096,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^2,x]","\frac{\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{2 \sqrt{e (a e-b d)+c d^2}}-\frac{e \sqrt{a+x (b+c x)}}{d+e x}+\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{e^2}","-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^2 \sqrt{a e^2-b d e+c d^2}}-\frac{\sqrt{a+b x+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{e^2}",1,"(-((e*Sqrt[a + x*(b + c*x)])/(d + e*x)) + Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + ((2*c*d - b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]))/e^2","A",1
2339,1,149,153,0.1540786,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^3,x]","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}","\frac{\sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left(a e^2-b d e+c d^2\right)}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"(Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))","A",1
2340,1,206,215,0.3386989,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^4,x]","\frac{3 (2 c d-b e) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)-\frac{2 e (a+x (b+c x))^{3/2}}{(d+e x)^3}}{6 \left(e (a e-b d)+c d^2\right)}","-\frac{\left(b^2-4 a c\right) (2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{16 \left(a e^2-b d e+c d^2\right)^{5/2}}-\frac{e \left(a+b x+c x^2\right)^{3/2}}{3 (d+e x)^3 \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^2}",1,"((-2*e*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + 3*(2*c*d - b*e)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(6*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2341,1,276,308,0.6178292,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^5,x]","\frac{3 \left(-2 c e (a e+4 b d)+\frac{5 b^2 e^2}{2}+8 c^2 d^2\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)-\frac{6 e (a+x (b+c x))^{3/2} \left(e (a e-b d)+c d^2\right)}{(d+e x)^4}-\frac{5 e (a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x)^3}}{24 \left(e (a e-b d)+c d^2\right)^2}","\frac{\sqrt{a+b x+c x^2} \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^3}-\frac{\left(b^2-4 a c\right) \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{128 \left(a e^2-b d e+c d^2\right)^{7/2}}-\frac{5 e \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left(a e^2-b d e+c d^2\right)^2}-\frac{e \left(a+b x+c x^2\right)^{3/2}}{4 (d+e x)^4 \left(a e^2-b d e+c d^2\right)}",1,"((-6*e*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 - (5*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + 3*(8*c^2*d^2 + (5*b^2*e^2)/2 - 2*c*e*(4*b*d + a*e))*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(24*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
2342,1,367,402,2.6191154,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^6} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^6,x]","-\frac{-\frac{\frac{15}{4} (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)-\frac{e (a+x (b+c x))^{3/2} \left(-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right)}{2 (d+e x)^3}}{24 \left(e (a e-b d)+c d^2\right)^2}+\frac{7 e (a+x (b+c x))^{3/2} (2 c d-b e)}{8 (d+e x)^4 \left(e (a e-b d)+c d^2\right)}+\frac{e (a+x (b+c x))^{3/2}}{(d+e x)^5}}{5 \left(e (a e-b d)+c d^2\right)}","-\frac{e \left(a+b x+c x^2\right)^{3/2} \left(-4 c e (8 a e+27 b d)+35 b^2 e^2+108 c^2 d^2\right)}{240 (d+e x)^3 \left(a e^2-b d e+c d^2\right)^3}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^4}-\frac{\left(b^2-4 a c\right) (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{256 \left(a e^2-b d e+c d^2\right)^{9/2}}-\frac{7 e \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{40 (d+e x)^4 \left(a e^2-b d e+c d^2\right)^2}-\frac{e \left(a+b x+c x^2\right)^{3/2}}{5 (d+e x)^5 \left(a e^2-b d e+c d^2\right)}",1,"-1/5*((e*(a + x*(b + c*x))^(3/2))/(d + e*x)^5 + (7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(8*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^4) - (-1/2*(e*(108*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(27*b*d + 8*a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + (15*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/4)/(24*(c*d^2 + e*(-(b*d) + a*e))^2))/(c*d^2 + e*(-(b*d) + a*e))","A",1
2343,1,231,321,0.4801135,"\int (d+e x)^3 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{e (a+x (b+c x))^{5/2} \left(-2 c e (8 a e+49 b d+15 b e x)+21 b^2 e^2+4 c^2 d (32 d+15 e x)\right)}{40 c^2}+\frac{7 (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2048 c^{9/2}}+e (d+e x)^2 (a+x (b+c x))^{5/2}}{7 c}","\frac{3 \left(b^2-4 a c\right)^2 (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2048 c^{11/2}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right)}{1024 c^5}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right)}{128 c^4}+\frac{e \left(a+b x+c x^2\right)^{5/2} \left(-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right)}{280 c^3}+\frac{e (d+e x)^2 \left(a+b x+c x^2\right)^{5/2}}{7 c}",1,"(e*(d + e*x)^2*(a + x*(b + c*x))^(5/2) + (e*(a + x*(b + c*x))^(5/2)*(21*b^2*e^2 + 4*c^2*d*(32*d + 15*e*x) - 2*c*e*(49*b*d + 8*a*e + 15*b*e*x)))/(40*c^2) + (7*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2048*c^(9/2)))/(7*c)","A",1
2344,1,188,257,0.3015692,"\int (d+e x)^2 \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{\left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{512 c^{7/2}}+\frac{7 e (a+x (b+c x))^{5/2} (2 c d-b e)}{10 c}+e (d+e x) (a+x (b+c x))^{5/2}}{6 c}","\frac{\left(b^2-4 a c\right)^2 \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{1024 c^{9/2}}-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right)}{512 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right)}{192 c^3}+\frac{7 e \left(a+b x+c x^2\right)^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left(a+b x+c x^2\right)^{5/2}}{6 c}",1,"((7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(10*c) + e*(d + e*x)*(a + x*(b + c*x))^(5/2) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(512*c^(7/2)))/(6*c)","A",1
2345,1,144,161,0.2835521,"\int (d+e x) \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(3/2),x]","\frac{5 (2 c d-b e) \left(\frac{3 \left(b^2-4 a c\right) \left(\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} (b+2 c x) \sqrt{a+x (b+c x)}\right)}{128 c^{5/2}}+\frac{(b+2 c x) (a+x (b+c x))^{3/2}}{8 c}\right)+2 e (a+x (b+c x))^{5/2}}{10 c}","\frac{3 \left(b^2-4 a c\right)^2 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{7/2}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left(a+b x+c x^2\right)^{5/2}}{5 c}",1,"(2*e*(a + x*(b + c*x))^(5/2) + 5*(2*c*d - b*e)*(((b + 2*c*x)*(a + x*(b + c*x))^(3/2))/(8*c) + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(128*c^(5/2))))/(10*c)","A",1
2346,1,114,112,0.3380787,"\int \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2),x]","\frac{\sqrt{a+x (b+c x)} \left(2 (b+2 c x) \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)-\frac{3 \left(b^2-4 a c\right)^{3/2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)}{128 c^2}","\frac{3 \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)}{128 c^{5/2}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{8 c}",1,"(Sqrt[a + x*(b + c*x)]*(2*(b + 2*c*x)*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) - (3*(b^2 - 4*a*c)^(3/2)*ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))/(128*c^2)","A",1
2347,1,236,252,0.4070013,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+x (b+c x)} \left(2 c e (16 a e-15 b d+7 b e x)+3 b^2 e^2+4 c^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-24 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)-3 (2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{48 c^{3/2} e^4}","\frac{\sqrt{a+b x+c x^2} \left(-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right)}{8 c e^3}-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} e^4}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^4}+\frac{\left(a+b x+c x^2\right)^{3/2}}{3 e}",1,"(-3*(2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(3*b^2*e^2 + 2*c*e*(-15*b*d + 16*a*e + 7*b*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 24*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(48*c^(3/2)*e^4)","A",1
2348,1,214,227,0.4200704,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^2,x]","\frac{\frac{3 \left(4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+12 (2 c d-b e) \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)-\frac{8 e^3 (a+x (b+c x))^{3/2}}{d+e x}+6 e \sqrt{a+x (b+c x)} (3 b e-4 c d+2 c e x)}{8 e^4}","\frac{3 \left(-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} e^4}-\frac{3 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^4}-\frac{3 \sqrt{a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left(a+b x+c x^2\right)^{3/2}}{e (d+e x)}",1,"(6*e*(-4*c*d + 3*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)] - (8*e^3*(a + x*(b + c*x))^(3/2))/(d + e*x) + (3*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 12*(2*c*d - b*e)*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*e^4)","A",1
2349,1,311,236,0.8901031,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^3,x]","\frac{\frac{3 \left(-\frac{2 e \sqrt{a+x (b+c x)} \left(-c e (2 a e-5 b d+b e x)-b^2 e^2+2 c^2 d (e x-2 d)\right)}{e (a e-b d)+c d^2}-\frac{\left(4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}-4 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{2 e^3}+\frac{3 (a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x) \left(e (a e-b d)+c d^2\right)}-\frac{2 (a+x (b+c x))^{3/2}}{(d+e x)^2}}{4 e}","\frac{3 \left(-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 e^4 \sqrt{a e^2-b d e+c d^2}}-\frac{3 \sqrt{c} (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 e^4}+\frac{3 \sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left(a+b x+c x^2\right)^{3/2}}{2 e (d+e x)^2}",1,"((-2*(a + x*(b + c*x))^(3/2))/(d + e*x)^2 + (3*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*((-2*e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 2*c^2*d*(-2*d + e*x) - c*e*(-5*b*d + 2*a*e + b*e*x)))/(c*d^2 + e*(-(b*d) + a*e)) - 4*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - ((8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)]))/(2*e^3))/(4*e)","A",1
2350,1,421,307,1.5817853,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]","\frac{\frac{3 \left(\frac{2 \sqrt{a+x (b+c x)} \left(2 c^2 e (2 a e (2 e x-3 d)+b d (7 d-2 e x))-b c e^2 (-10 a e+5 b d+b e x)-b^3 e^3+4 c^3 d^2 (e x-2 d)\right)}{e^2}+\frac{2 (a+x (b+c x))^{3/2} \left(4 c e (b d-2 a e)+b^2 e^2-4 c^2 d^2\right)}{d+e x}+\frac{(2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+16 c^{3/2} \left(e (a e-b d)+c d^2\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{e^3}\right)}{8 \left(e (a e-b d)+c d^2\right)^2}+\frac{3 (a+x (b+c x))^{3/2} (2 c d-b e)}{2 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}-\frac{2 (a+x (b+c x))^{3/2}}{(d+e x)^3}}{6 e}","-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{16 e^4 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left(e x \left(-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right)-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right)}{8 e^3 (d+e x)^2 \left(a e^2-b d e+c d^2\right)}+\frac{c^{3/2} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{e^4}-\frac{\left(a+b x+c x^2\right)^{3/2}}{3 e (d+e x)^3}",1,"((-2*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + (3*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (3*((2*(-4*c^2*d^2 + b^2*e^2 + 4*c*e*(b*d - 2*a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x) + (2*Sqrt[a + x*(b + c*x)]*(-(b^3*e^3) + 4*c^3*d^2*(-2*d + e*x) - b*c*e^2*(5*b*d - 10*a*e + b*e*x) + 2*c^2*e*(b*d*(7*d - 2*e*x) + 2*a*e*(-3*d + 2*e*x))))/e^2 + (16*c^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (2*c*d - b*e)*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/e^3))/(8*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*e)","A",1
2351,1,222,225,0.6578703,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]","-\frac{3 \left(b^2-4 a c\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)+\frac{2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}}{16 \left(e (a e-b d)+c d^2\right)}","-\frac{3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^2}+\frac{3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{128 \left(a e^2-b d e+c d^2\right)^{5/2}}+\frac{\left(a+b x+c x^2\right)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left(a e^2-b d e+c d^2\right)}",1,"-1/16*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(c*d^2 + e*(-(b*d) + a*e))","A",1
2352,1,275,296,1.0996631,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^6} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]","-\frac{(2 c d-b e) \left(3 \left(b^2-4 a c\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)+\frac{2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}\right)}{32 \left(e (a e-b d)+c d^2\right)^2}-\frac{e (a+x (b+c x))^{5/2}}{5 (d+e x)^5 \left(e (a e-b d)+c d^2\right)}","-\frac{3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^3}+\frac{3 \left(b^2-4 a c\right)^2 (2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{256 \left(a e^2-b d e+c d^2\right)^{7/2}}-\frac{e \left(a+b x+c x^2\right)^{5/2}}{5 (d+e x)^5 \left(a e^2-b d e+c d^2\right)}+\frac{\left(a+b x+c x^2\right)^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left(a e^2-b d e+c d^2\right)^2}",1,"-1/5*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - ((2*c*d - b*e)*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))))/(32*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
2353,1,349,409,1.6899956,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^7} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x]","-\frac{\left(-2 c e (a e+6 b d)+\frac{7 b^2 e^2}{2}+12 c^2 d^2\right) \left(3 \left(b^2-4 a c\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}\right)+\frac{2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}\right)}{192 \left(e (a e-b d)+c d^2\right)^3}-\frac{7 e (a+x (b+c x))^{5/2} (2 c d-b e)}{60 (d+e x)^5 \left(e (a e-b d)+c d^2\right)^2}-\frac{e (a+x (b+c x))^{5/2}}{6 (d+e x)^6 \left(e (a e-b d)+c d^2\right)}","\frac{\left(a+b x+c x^2\right)^{3/2} \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{192 (d+e x)^4 \left(a e^2-b d e+c d^2\right)^3}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{512 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^4}+\frac{\left(b^2-4 a c\right)^2 \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{1024 \left(a e^2-b d e+c d^2\right)^{9/2}}-\frac{7 e \left(a+b x+c x^2\right)^{5/2} (2 c d-b e)}{60 (d+e x)^5 \left(a e^2-b d e+c d^2\right)^2}-\frac{e \left(a+b x+c x^2\right)^{5/2}}{6 (d+e x)^6 \left(a e^2-b d e+c d^2\right)}",1,"-1/6*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^6) - (7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(60*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^5) - ((12*c^2*d^2 + (7*b^2*e^2)/2 - 2*c*e*(6*b*d + a*e))*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))))/(192*(c*d^2 + e*(-(b*d) + a*e))^3)","A",1
2354,1,687,510,6.0697332,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^8} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^8,x]","-\frac{(a+x (b+c x))^{3/2} \left(-\frac{\frac{\left(a+b x+c x^2\right)^{5/2} \left(-\frac{3}{4} e \left(-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right)-\frac{9}{2} c d e (2 c d-b e)\right)}{5 (d+e x)^5 \left(a e^2-b d e+c d^2\right)}-\frac{\left(b \left(\frac{3}{4} e \left(-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right)-\frac{9}{2} c d e (2 c d-b e)\right)-2 \left(\frac{3}{4} c d \left(-2 c e (8 a e+31 b d)+21 b^2 e^2+56 c^2 d^2\right)-\frac{9}{2} a c e^2 (2 c d-b e)\right)\right) \left(\frac{\left(a+b x+c x^2\right)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left(a e^2-b d e+c d^2\right)}-\frac{3 \left(b^2-4 a c\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-x (2 c d-b e)-b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \sqrt{a e^2-b d e+c d^2} \left(4 a e^2-4 b d e+4 c d^2\right)}+\frac{\sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left(a e^2-b d e+c d^2\right)}\right)}{16 \left(a e^2-b d e+c d^2\right)}\right)}{2 \left(a e^2-b d e+c d^2\right)}}{6 \left(a e^2-b d e+c d^2\right)}-\frac{\left(a+b x+c x^2\right)^{5/2} \left(\frac{1}{2} e (9 b e-14 c d)-2 c d e\right)}{6 (d+e x)^6 \left(a e^2-b d e+c d^2\right)}\right)}{7 \left(a+b x+c x^2\right)^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{e \left(a+b x+c x^2\right) (a+x (b+c x))^{3/2}}{7 (d+e x)^7 \left(a e^2-b d e+c d^2\right)}","-\frac{e \left(a+b x+c x^2\right)^{5/2} \left(-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right)}{280 (d+e x)^5 \left(a e^2-b d e+c d^2\right)^3}+\frac{\left(a+b x+c x^2\right)^{3/2} (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left(a e^2-b d e+c d^2\right)^4}-\frac{3 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^5}+\frac{3 \left(b^2-4 a c\right)^2 (2 c d-b e) \left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2048 \left(a e^2-b d e+c d^2\right)^{11/2}}-\frac{3 e \left(a+b x+c x^2\right)^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left(a e^2-b d e+c d^2\right)^2}-\frac{e \left(a+b x+c x^2\right)^{5/2}}{7 (d+e x)^7 \left(a e^2-b d e+c d^2\right)}",1,"-1/7*(e*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^7) - ((a + x*(b + c*x))^(3/2)*(-1/6*((-2*c*d*e + (e*(-14*c*d + 9*b*e))/2)*(a + b*x + c*x^2)^(5/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) - ((((-9*c*d*e*(2*c*d - b*e))/2 - (3*e*(56*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(31*b*d + 8*a*e)))/4)*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) - ((-2*((-9*a*c*e^2*(2*c*d - b*e))/2 + (3*c*d*(56*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(31*b*d + 8*a*e)))/4) + b*((-9*c*d*e*(2*c*d - b*e))/2 + (3*e*(56*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(31*b*d + 8*a*e)))/4))*(((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (3*(b^2 - 4*a*c)*(((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*(4*c*d^2 - 4*b*d*e + 4*a*e^2))))/(16*(c*d^2 - b*d*e + a*e^2))))/(2*(c*d^2 - b*d*e + a*e^2)))/(6*(c*d^2 - b*d*e + a*e^2))))/(7*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2))","A",1
2355,1,280,400,0.781654,"\int (d+e x)^3 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]","\frac{\frac{3 (2 c d-b e) \left(-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(16 c^2 \left(33 a^2+26 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(11 c x^2-20 a\right)+32 b c^2 x \left(13 a+8 c x^2\right)+15 b^4-40 b^3 c x\right)-15 \left(b^2-4 a c\right)^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{65536 c^{11/2}}+\frac{e (a+x (b+c x))^{7/2} \left(-2 c e (32 a e+243 b d+77 b e x)+99 b^2 e^2+4 c^2 d (160 d+77 e x)\right)}{224 c^2}+e (d+e x)^2 (a+x (b+c x))^{7/2}}{9 c}","-\frac{5 \left(b^2-4 a c\right)^3 (2 c d-b e) \left(-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{65536 c^{13/2}}+\frac{5 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right)}{32768 c^6}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2} (2 c d-b e) \left(-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right)}{12288 c^5}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/2} (2 c d-b e) \left(-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right)}{768 c^4}+\frac{e \left(a+b x+c x^2\right)^{7/2} \left(-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right)}{2016 c^3}+\frac{e (d+e x)^2 \left(a+b x+c x^2\right)^{7/2}}{9 c}",1,"(e*(d + e*x)^2*(a + x*(b + c*x))^(7/2) + (e*(a + x*(b + c*x))^(7/2)*(99*b^2*e^2 + 4*c^2*d*(160*d + 77*e*x) - 2*c*e*(243*b*d + 32*a*e + 77*b*e*x)))/(224*c^2) + (3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) - 15*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(65536*c^(11/2)))/(9*c)","A",1
2356,1,238,323,0.5909273,"\int (d+e x)^2 \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]","\frac{-\frac{\left(2 c e (a e+8 b d)-\frac{9 b^2 e^2}{2}-16 c^2 d^2\right) \left(256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left(b^2-4 a c\right) \left(16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left(b^2-4 a c\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\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)\right)\right)\right)}{6144 c^{9/2}}+\frac{9 e (a+x (b+c x))^{7/2} (2 c d-b e)}{14 c}+e (d+e x) (a+x (b+c x))^{7/2}}{8 c}","-\frac{5 \left(b^2-4 a c\right)^3 \left(-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{32768 c^{11/2}}+\frac{5 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2} \left(-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right)}{16384 c^5}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right)}{6144 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/2} \left(-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right)}{384 c^3}+\frac{9 e \left(a+b x+c x^2\right)^{7/2} (2 c d-b e)}{112 c^2}+\frac{e (d+e x) \left(a+b x+c x^2\right)^{7/2}}{8 c}",1,"((9*e*(2*c*d - b*e)*(a + x*(b + c*x))^(7/2))/(14*c) + e*(d + e*x)*(a + x*(b + c*x))^(7/2) - ((-16*c^2*d^2 - (9*b^2*e^2)/2 + 2*c*e*(8*b*d + a*e))*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/(6144*c^(9/2)))/(8*c)","A",1
2357,1,180,207,0.2707953,"\int (d+e x) \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]","\frac{(2 c d-b e) \left(256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left(b^2-4 a c\right) \left(16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left(b^2-4 a c\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\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)\right)\right)\right)}{6144 c^{9/2}}+\frac{e (a+x (b+c x))^{7/2}}{7 c}","-\frac{5 \left(b^2-4 a c\right)^3 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2048 c^{9/2}}+\frac{5 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left(a+b x+c x^2\right)^{7/2}}{7 c}",1,"(e*(a + x*(b + c*x))^(7/2))/(7*c) + ((2*c*d - b*e)*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/(6144*c^(9/2))","A",1
2358,1,162,149,0.5505251,"\int \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(a + b*x + c*x^2)^(5/2),x]","\frac{\sqrt{a+x (b+c x)} \left(2 (b+2 c x) \left(16 c^2 \left(33 a^2+26 a c x^2+8 c^2 x^4\right)+8 b^2 c \left(11 c x^2-20 a\right)+32 b c^2 x \left(13 a+8 c x^2\right)+15 b^4-40 b^3 c x\right)+\frac{15 \left(b^2-4 a c\right)^{5/2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)}{3072 c^3}","-\frac{5 \left(b^2-4 a c\right)^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{1024 c^{7/2}}+\frac{5 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^3}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2}}{192 c^2}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/2}}{12 c}",1,"(Sqrt[a + x*(b + c*x)]*(2*(b + 2*c*x)*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) + (15*(b^2 - 4*a*c)^(5/2)*ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))/(3072*c^3)","A",1
2359,1,440,459,1.2282843,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x),x]","-\frac{\frac{2 e \sqrt{a+x (b+c x)} \left(-4 c^2 e^2 \left(32 a^2 e^2+2 a b e (7 e x-39 d)+b^2 d (44 d-5 e x)\right)+2 b^2 c e^3 (-14 a e+5 b d+3 b e x)+16 c^3 d e (a e (7 e x-16 d)+6 b d (3 d-e x))+3 b^4 e^4-64 c^4 d^3 (2 d-e x)\right)}{c^2}+\frac{(2 c d-b e) \left(16 c^2 e^2 \left(15 a^2 e^2-20 a b d e+7 b^2 d^2\right)+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{5/2}}+256 \left(e (a e-b d)+c d^2\right)^{5/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{256 e^6}+\frac{(a+x (b+c x))^{3/2} \left(2 c e (8 a e-11 b d+3 b e x)+3 b^2 e^2+4 c^2 d (4 d-3 e x)\right)}{48 c e^3}+\frac{(a+x (b+c x))^{5/2}}{5 e}","\frac{\sqrt{a+b x+c x^2} \left(8 c^2 e^2 \left(16 a^2 e^2-39 a b d e+22 b^2 d^2\right)-2 c e x (2 c d-b e) \left(-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right)-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right)}{128 c^2 e^5}-\frac{(2 c d-b e) \left(16 c^2 e^2 \left(15 a^2 e^2-20 a b d e+7 b^2 d^2\right)+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{5/2} e^6}+\frac{\left(a+b x+c x^2\right)^{3/2} \left(-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right)}{48 c e^3}+\frac{\left(a e^2-b d e+c d^2\right)^{5/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^6}+\frac{\left(a+b x+c x^2\right)^{5/2}}{5 e}",1,"(a + x*(b + c*x))^(5/2)/(5*e) + ((a + x*(b + c*x))^(3/2)*(3*b^2*e^2 + 4*c^2*d*(4*d - 3*e*x) + 2*c*e*(-11*b*d + 8*a*e + 3*b*e*x)))/(48*c*e^3) - ((2*e*Sqrt[a + x*(b + c*x)]*(3*b^4*e^4 - 64*c^4*d^3*(2*d - e*x) + 2*b^2*c*e^3*(5*b*d - 14*a*e + 3*b*e*x) - 4*c^2*e^2*(32*a^2*e^2 + b^2*d*(44*d - 5*e*x) + 2*a*b*e*(-39*d + 7*e*x)) + 16*c^3*d*e*(6*b*d*(3*d - e*x) + a*e*(-16*d + 7*e*x))))/c^2 + ((2*c*d - b*e)*(128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(2*b*d - 5*a*e) - 64*c^3*d^2*e*(4*b*d - 5*a*e) + 16*c^2*e^2*(7*b^2*d^2 - 20*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2) + 256*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(256*e^6)","A",1
2360,1,370,388,1.0804132,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^2,x]","\frac{5 \left(\left(48 c^2 e^2 \left(a^2 e^2-4 a b d e+3 b^2 d^2\right)-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} \left(e \sqrt{a+x (b+c x)} \left(8 c^2 e (a e (3 e x-8 d)+2 b d (7 d-2 e x))+2 b c e^2 (22 a e-24 b d+b e x)+b^3 e^3+32 c^3 d^2 (e x-2 d)\right)+32 c (2 c d-b e) \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)\right)}{128 c^{3/2} e^6}+\frac{5 (a+x (b+c x))^{3/2} (7 b e-8 c d+6 c e x)}{24 e^3}-\frac{(a+x (b+c x))^{5/2}}{e (d+e x)}","\frac{5 \left(48 c^2 e^2 \left(a^2 e^2-4 a b d e+3 b^2 d^2\right)-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{3/2} e^6}-\frac{5 \sqrt{a+b x+c x^2} \left(-2 c e x \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right)-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right)}{64 c e^5}-\frac{5 (2 c d-b e) \left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 e^6}-\frac{5 \left(a+b x+c x^2\right)^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left(a+b x+c x^2\right)^{5/2}}{e (d+e x)}",1,"(5*(-8*c*d + 7*b*e + 6*c*e*x)*(a + x*(b + c*x))^(3/2))/(24*e^3) - (a + x*(b + c*x))^(5/2)/(e*(d + e*x)) + (5*((128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(b^3*e^3 + 32*c^3*d^2*(-2*d + e*x) + 2*b*c*e^2*(-24*b*d + 22*a*e + b*e*x) + 8*c^2*e*(2*b*d*(7*d - 2*e*x) + a*e*(-8*d + 3*e*x))) + 32*c*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])))/(128*c^(3/2)*e^6)","A",1
2361,1,445,331,1.7380323,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]","\frac{\frac{5 \left(\frac{-2 c^2 e \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right) \left(4 c e (a e-5 b d+b e x)+5 b^2 e^2+8 c^2 d (2 d-e x)\right)+2 c^2 \left(4 c e (a e-4 b d)+3 b^2 e^2+16 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+c^{3/2} (2 c d-b e) \left(e (a e-b d)+c d^2\right) \left(4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{4 c^2 e^5}-\frac{(a+x (b+c x))^{3/2} \left(c e (2 a e-11 b d+3 b e x)+3 b^2 e^2+2 c^2 d (4 d-3 e x)\right)}{3 e^2}\right)}{e (b d-a e)-c d^2}+\frac{5 (a+x (b+c x))^{5/2} (2 c d-b e)}{(d+e x) \left(e (a e-b d)+c d^2\right)}-\frac{2 (a+x (b+c x))^{5/2}}{(d+e x)^2}}{4 e}","-\frac{5 (2 c d-b e) \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 \sqrt{c} e^6}+\frac{5 \sqrt{a e^2-b d e+c d^2} \left(-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 e^6}+\frac{5 \sqrt{a+b x+c x^2} \left(-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right)}{8 e^5}+\frac{5 \left(a+b x+c x^2\right)^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left(a+b x+c x^2\right)^{5/2}}{2 e (d+e x)^2}",1,"((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 + (5*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (5*(-1/3*((a + x*(b + c*x))^(3/2)*(3*b^2*e^2 + 2*c^2*d*(4*d - 3*e*x) + c*e*(-11*b*d + 2*a*e + 3*b*e*x)))/e^2 + (-2*c^2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]*(5*b^2*e^2 + 8*c^2*d*(2*d - e*x) + 4*c*e*(-5*b*d + a*e + b*e*x)) + c^(3/2)*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*c^2*(16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*c^2*e^5)))/(-(c*d^2) + e*(b*d - a*e)))/(4*e)","A",1
2362,1,559,337,3.0120961,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^4,x]","\frac{\frac{5 \left(\frac{(a+x (b+c x))^{3/2} \left(2 c^2 e (2 a e (2 e x-3 d)+3 b d (5 d-2 e x))+b c e^2 (10 a e-15 b d+b e x)+b^3 e^3-4 c^3 d^2 (4 d-3 e x)\right)}{2 e^2}-\frac{(a+x (b+c x))^{5/2} \left(4 c e (2 a e-3 b d)+b^2 e^2+12 c^2 d^2\right)}{2 (d+e x)}+\frac{3 \left(2 e \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right) \left(4 c^2 e (a e (e x-3 d)+b d (7 d-2 e x))+b c e^2 (8 a e-13 b d+b e x)+b^3 e^3+8 c^3 d^2 (e x-2 d)\right)+2 \sqrt{c} \left(4 c e (a e-4 b d)+3 b^2 e^2+16 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+(2 c d-b e) \left(4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right) \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)}{4 e^5}\right)}{2 \left(e (a e-b d)+c d^2\right)^2}+\frac{5 (a+x (b+c x))^{5/2} (2 c d-b e)}{2 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}-\frac{2 (a+x (b+c x))^{5/2}}{(d+e x)^3}}{6 e}","\frac{5 \sqrt{c} \left(-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 e^6}-\frac{5 (2 c d-b e) \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{16 e^6 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \sqrt{a+b x+c x^2} \left(-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right)}{8 e^5 (d+e x)}+\frac{5 \left(a+b x+c x^2\right)^{3/2} (-b e+4 c d+2 c e x)}{12 e^3 (d+e x)^2}-\frac{\left(a+b x+c x^2\right)^{5/2}}{3 e (d+e x)^3}",1,"((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^3 + (5*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (5*(-1/2*((12*c^2*d^2 + b^2*e^2 + 4*c*e*(-3*b*d + 2*a*e))*(a + x*(b + c*x))^(5/2))/(d + e*x) + ((a + x*(b + c*x))^(3/2)*(b^3*e^3 - 4*c^3*d^2*(4*d - 3*e*x) + b*c*e^2*(-15*b*d + 10*a*e + b*e*x) + 2*c^2*e*(3*b*d*(5*d - 2*e*x) + 2*a*e*(-3*d + 2*e*x))))/(2*e^2) + (3*(2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]*(b^3*e^3 + 8*c^3*d^2*(-2*d + e*x) + b*c*e^2*(-13*b*d + 8*a*e + b*e*x) + 4*c^2*e*(b*d*(7*d - 2*e*x) + a*e*(-3*d + e*x))) + 2*Sqrt[c]*(16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(4*e^5)))/(2*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*e)","A",1
2363,1,593,492,3.6703753,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^5} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^5,x]","\frac{-\frac{15 \left(48 c^2 e^2 \left(a^2 e^2-4 a b d e+3 b^2 d^2\right)-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left(2 c e^2 \left(-4 a^2 e^2 \left(11 d^2+20 d e x+27 e^2 x^2\right)-2 a b e \left(45 d^3+169 d^2 e x+191 d e^2 x^2+139 e^3 x^3\right)+b^2 d \left(120 d^3+435 d^2 e x+566 d e^2 x^2+323 e^3 x^3\right)\right)+e^3 \left(-48 a^3 e^3+8 a^2 b e^2 (d-17 e x)+2 a b^2 e \left(5 d^2+18 d e x-59 e^2 x^2\right)+b^3 \left(15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right)\right)-8 c^2 e \left(b d \left(150 d^4+530 d^3 e x+665 d^2 e^2 x^2+327 d e^3 x^3+24 e^4 x^4\right)-a e \left(100 d^4+355 d^3 e x+448 d^2 e^2 x^2+235 d e^3 x^3+24 e^4 x^4\right)\right)+16 c^3 d^2 \left(60 d^4+210 d^3 e x+260 d^2 e^2 x^2+125 d e^3 x^3+12 e^4 x^4\right)\right)}{(d+e x)^4 \left(e (a e-b d)+c d^2\right)}-960 c^{3/2} (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{384 e^6}","\frac{5 \left(48 c^2 e^2 \left(a^2 e^2-4 a b d e+3 b^2 d^2\right)-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{128 e^6 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{5 \left(a+b x+c x^2\right)^{3/2} \left(3 e x \left(-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right)-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right)}{96 e^3 (d+e x)^3 \left(a e^2-b d e+c d^2\right)}+\frac{5 \sqrt{a+b x+c x^2} \left(2 c e x \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right)-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right)}{64 e^5 (d+e x) \left(a e^2-b d e+c d^2\right)}-\frac{5 c^{3/2} (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 e^6}-\frac{\left(a+b x+c x^2\right)^{5/2}}{4 e (d+e x)^4}",1,"((2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*d^2*(60*d^4 + 210*d^3*e*x + 260*d^2*e^2*x^2 + 125*d*e^3*x^3 + 12*e^4*x^4) + e^3*(-48*a^3*e^3 + 8*a^2*b*e^2*(d - 17*e*x) + 2*a*b^2*e*(5*d^2 + 18*d*e*x - 59*e^2*x^2) + b^3*(15*d^3 + 55*d^2*e*x + 73*d*e^2*x^2 - 15*e^3*x^3)) + 2*c*e^2*(-4*a^2*e^2*(11*d^2 + 20*d*e*x + 27*e^2*x^2) - 2*a*b*e*(45*d^3 + 169*d^2*e*x + 191*d*e^2*x^2 + 139*e^3*x^3) + b^2*d*(120*d^3 + 435*d^2*e*x + 566*d*e^2*x^2 + 323*e^3*x^3)) - 8*c^2*e*(-(a*e*(100*d^4 + 355*d^3*e*x + 448*d^2*e^2*x^2 + 235*d*e^3*x^3 + 24*e^4*x^4)) + b*d*(150*d^4 + 530*d^3*e*x + 665*d^2*e^2*x^2 + 327*d*e^3*x^3 + 24*e^4*x^4))))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^4) - 960*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - (15*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(384*e^6)","A",1
2364,1,84,88,0.0383998,"\int \frac{\sqrt{-2-3 x+5 x^2}}{x} \, dx","Integrate[Sqrt[-2 - 3*x + 5*x^2]/x,x]","\sqrt{5 x^2-3 x-2}-\sqrt{2} \tan ^{-1}\left(\frac{-3 x-4}{2 \sqrt{10 x^2-6 x-4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{10 x-3}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right)}{2 \sqrt{5}}","\sqrt{5 x^2-3 x-2}+\sqrt{2} \tan ^{-1}\left(\frac{3 x+4}{2 \sqrt{2} \sqrt{5 x^2-3 x-2}}\right)+\frac{3 \tanh ^{-1}\left(\frac{3-10 x}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right)}{2 \sqrt{5}}",1,"Sqrt[-2 - 3*x + 5*x^2] - Sqrt[2]*ArcTan[(-4 - 3*x)/(2*Sqrt[-4 - 6*x + 10*x^2])] - (3*ArcTanh[(-3 + 10*x)/(2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])","A",1
2365,1,68,68,0.0339988,"\int \frac{\sqrt{2-x-x^2}}{x^2} \, dx","Integrate[Sqrt[2 - x - x^2]/x^2,x]","-\frac{\sqrt{-x^2-x+2}}{x}+\frac{\tanh ^{-1}\left(\frac{4-x}{2 \sqrt{2} \sqrt{-x^2-x+2}}\right)}{2 \sqrt{2}}+\sin ^{-1}\left(\frac{1}{3} (-2 x-1)\right)","-\frac{\sqrt{-x^2-x+2}}{x}+\frac{\tanh ^{-1}\left(\frac{4-x}{2 \sqrt{2} \sqrt{-x^2-x+2}}\right)}{2 \sqrt{2}}+\sin ^{-1}\left(\frac{1}{3} (-2 x-1)\right)",1,"-(Sqrt[2 - x - x^2]/x) + ArcSin[(-1 - 2*x)/3] + ArcTanh[(4 - x)/(2*Sqrt[2]*Sqrt[2 - x - x^2])]/(2*Sqrt[2])","A",1
2366,1,26,38,0.009675,"\int (1+x)^3 \sqrt{2+2 x+x^2} \, dx","Integrate[(1 + x)^3*Sqrt[2 + 2*x + x^2],x]","\frac{1}{15} \left(x^2+2 x+2\right)^{3/2} \left(3 x^2+6 x+1\right)","\frac{1}{5} (x+1)^2 \left(x^2+2 x+2\right)^{3/2}-\frac{2}{15} \left(x^2+2 x+2\right)^{3/2}",1,"((2 + 2*x + x^2)^(3/2)*(1 + 6*x + 3*x^2))/15","A",1
2367,1,40,54,0.0213409,"\int (-2+3 x) \sqrt{8+12 x+9 x^2} \, dx","Integrate[(-2 + 3*x)*Sqrt[8 + 12*x + 9*x^2],x]","\frac{1}{9} \left(\left(9 x^2-6 x-4\right) \sqrt{9 x^2+12 x+8}-24 \sinh ^{-1}\left(\frac{3 x}{2}+1\right)\right)","\frac{1}{9} \left(9 x^2+12 x+8\right)^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left(\frac{3 x}{2}+1\right)",1,"((-4 - 6*x + 9*x^2)*Sqrt[8 + 12*x + 9*x^2] - 24*ArcSinh[1 + (3*x)/2])/9","A",1
2368,1,43,56,0.0277688,"\int (7-2 x) \sqrt{9+16 x-4 x^2} \, dx","Integrate[(7 - 2*x)*Sqrt[9 + 16*x - 4*x^2],x]","-\frac{1}{6} \sqrt{-4 x^2+16 x+9} \left(4 x^2-25 x+9\right)-\frac{75}{4} \sin ^{-1}\left(\frac{4}{5}-\frac{2 x}{5}\right)","\frac{1}{6} \left(-4 x^2+16 x+9\right)^{3/2}-\frac{3}{2} (2-x) \sqrt{-4 x^2+16 x+9}-\frac{75}{4} \sin ^{-1}\left(\frac{2 (2-x)}{5}\right)",1,"-1/6*(Sqrt[9 + 16*x - 4*x^2]*(9 - 25*x + 4*x^2)) - (75*ArcSin[4/5 - (2*x)/5])/4","A",1
2369,1,63,61,0.0166777,"\int \frac{\sqrt{-1-x+x^2}}{1+x} \, dx","Integrate[Sqrt[-1 - x + x^2]/(1 + x),x]","\sqrt{x^2-x-1}-\tanh ^{-1}\left(\frac{-3 x-1}{2 \sqrt{x^2-x-1}}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{2 x-1}{2 \sqrt{x^2-x-1}}\right)","\sqrt{x^2-x-1}+\frac{3}{2} \tanh ^{-1}\left(\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right)+\tanh ^{-1}\left(\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right)",1,"Sqrt[-1 - x + x^2] - ArcTanh[(-1 - 3*x)/(2*Sqrt[-1 - x + x^2])] - (3*ArcTanh[(-1 + 2*x)/(2*Sqrt[-1 - x + x^2])])/2","A",1
2370,1,65,65,0.0142306,"\int \frac{\sqrt{-1-x+x^2}}{1-x} \, dx","Integrate[Sqrt[-1 - x + x^2]/(1 - x),x]","-\sqrt{x^2-x-1}-\tan ^{-1}\left(\frac{3-x}{2 \sqrt{x^2-x-1}}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{2 x-1}{2 \sqrt{x^2-x-1}}\right)","-\sqrt{x^2-x-1}-\tan ^{-1}\left(\frac{3-x}{2 \sqrt{x^2-x-1}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right)",1,"-Sqrt[-1 - x + x^2] - ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] - ArcTanh[(-1 + 2*x)/(2*Sqrt[-1 - x + x^2])]/2","A",1
2371,1,263,261,0.3155346,"\int \frac{x^6}{\sqrt{a+b x+c x^2}} \, dx","Integrate[x^6/Sqrt[a + b*x + c*x^2],x]","\frac{\left(-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{1024 c^{13/2}}+\frac{48 a^3 c^2 (50 c x-231 b)+8 a^2 c \left(1785 b^3-2268 b^2 c x-618 b c^2 x^2+100 c^3 x^3\right)+a \left(-3465 b^5+16590 b^4 c x+5376 b^3 c^2 x^2-1728 b^2 c^3 x^3+736 b c^4 x^4-320 c^5 x^5\right)+x \left(-3465 b^6-1155 b^5 c x+462 b^4 c^2 x^2-264 b^3 c^3 x^3+176 b^2 c^4 x^4-128 b c^5 x^5+1280 c^6 x^6\right)}{7680 c^6 \sqrt{a+x (b+c x)}}","-\frac{\left(7 b \left(528 a^2 c^2-680 a b^2 c+165 b^4\right)-2 c x \left(400 a^2 c^2-1176 a b^2 c+385 b^4\right)\right) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left(-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{1024 c^{13/2}}-\frac{b x^2 \left(77 b^2-156 a c\right) \sqrt{a+b x+c x^2}}{320 c^4}+\frac{x^3 \left(99 b^2-100 a c\right) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}",1,"(48*a^3*c^2*(-231*b + 50*c*x) + 8*a^2*c*(1785*b^3 - 2268*b^2*c*x - 618*b*c^2*x^2 + 100*c^3*x^3) + a*(-3465*b^5 + 16590*b^4*c*x + 5376*b^3*c^2*x^2 - 1728*b^2*c^3*x^3 + 736*b*c^4*x^4 - 320*c^5*x^5) + x*(-3465*b^6 - 1155*b^5*c*x + 462*b^4*c^2*x^2 - 264*b^3*c^3*x^3 + 176*b^2*c^4*x^4 - 128*b*c^5*x^5 + 1280*c^6*x^6))/(7680*c^6*Sqrt[a + x*(b + c*x)]) + ((231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(1024*c^(13/2))","A",1
2372,1,213,202,0.2016865,"\int \frac{x^5}{\sqrt{a+b x+c x^2}} \, dx","Integrate[x^5/Sqrt[a + b*x + c*x^2],x]","\frac{1024 a^3 c^2+4 a^2 c \left(-735 b^2+578 b c x+128 c^2 x^2\right)+a \left(945 b^4-3570 b^3 c x-1148 b^2 c^2 x^2+344 b c^3 x^3-128 c^4 x^4\right)+3 x \left(315 b^5+105 b^4 c x-42 b^3 c^2 x^2+24 b^2 c^3 x^3-16 b c^4 x^4+128 c^5 x^5\right)}{1920 c^5 \sqrt{a+x (b+c x)}}-\frac{b \left(240 a^2 c^2-280 a b^2 c+63 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{256 c^{11/2}}","-\frac{b \left(240 a^2 c^2-280 a b^2 c+63 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{256 c^{11/2}}+\frac{\left(1024 a^2 c^2-14 b c x \left(45 b^2-92 a c\right)-2940 a b^2 c+945 b^4\right) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{x^2 \left(63 b^2-64 a c\right) \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}",1,"(1024*a^3*c^2 + 4*a^2*c*(-735*b^2 + 578*b*c*x + 128*c^2*x^2) + a*(945*b^4 - 3570*b^3*c*x - 1148*b^2*c^2*x^2 + 344*b*c^3*x^3 - 128*c^4*x^4) + 3*x*(315*b^5 + 105*b^4*c*x - 42*b^3*c^2*x^2 + 24*b^2*c^3*x^3 - 16*b*c^4*x^4 + 128*c^5*x^5))/(1920*c^5*Sqrt[a + x*(b + c*x)]) - (b*(63*b^4 - 280*a*b^2*c + 240*a^2*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(11/2))","A",1
2373,1,352,296,0.4599981,"\int \frac{(d+e x)^4}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^4/Sqrt[a + b*x + c*x^2],x]","\frac{\left(48 c^2 e^2 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{128 c^{9/2}}+\frac{e \left(-4 a^2 c e^2 (2 c (64 d+9 e x)-55 b e)+a \left(-105 b^3 e^3+10 b^2 c e^2 (48 d+29 e x)+4 b c^2 e \left(-216 d^2-208 d e x+23 e^2 x^2\right)+8 c^3 \left(96 d^3+72 d^2 e x-32 d e^2 x^2-3 e^3 x^3\right)\right)+x (b+c x) \left(-105 b^3 e^3+10 b^2 c e^2 (48 d+7 e x)-8 b c^2 e \left(108 d^2+40 d e x+7 e^2 x^2\right)+16 c^3 \left(48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right)\right)\right)}{192 c^4 \sqrt{a+x (b+c x)}}","\frac{\left(48 c^2 e^2 \left(a^2 e^2+8 a b d e+6 b^2 d^2\right)-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{9/2}}+\frac{e \sqrt{a+b x+c x^2} \left(2 c e x \left(-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right)-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right)}{192 c^4}+\frac{7 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x)^3 \sqrt{a+b x+c x^2}}{4 c}",1,"(e*(-4*a^2*c*e^2*(-55*b*e + 2*c*(64*d + 9*e*x)) + a*(-105*b^3*e^3 + 10*b^2*c*e^2*(48*d + 29*e*x) + 4*b*c^2*e*(-216*d^2 - 208*d*e*x + 23*e^2*x^2) + 8*c^3*(96*d^3 + 72*d^2*e*x - 32*d*e^2*x^2 - 3*e^3*x^3)) + x*(b + c*x)*(-105*b^3*e^3 + 10*b^2*c*e^2*(48*d + 7*e*x) - 8*b*c^2*e*(108*d^2 + 40*d*e*x + 7*e^2*x^2) + 16*c^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3))))/(192*c^4*Sqrt[a + x*(b + c*x)]) + ((128*c^4*d^4 + 35*b^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c*e^3*(4*b*d + 3*a*e) + 48*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(128*c^(9/2))","A",1
2374,1,210,174,0.2124481,"\int \frac{(d+e x)^3}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^3/Sqrt[a + b*x + c*x^2],x]","\frac{e \left(-16 a^2 c e^2+a \left(15 b^2 e^2-2 b c e (27 d+13 e x)+4 c^2 \left(18 d^2+9 d e x-2 e^2 x^2\right)\right)+x (b+c x) \left(15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)\right)\right)}{24 c^3 \sqrt{a+x (b+c x)}}+\frac{(2 c d-b e) \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{16 c^{7/2}}","\frac{(2 c d-b e) \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{7/2}}+\frac{e \sqrt{a+b x+c x^2} \left(-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right)}{24 c^3}+\frac{e (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c}",1,"(e*(-16*a^2*c*e^2 + a*(15*b^2*e^2 - 2*b*c*e*(27*d + 13*e*x) + 4*c^2*(18*d^2 + 9*d*e*x - 2*e^2*x^2)) + x*(b + c*x)*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*e*x) + 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2))))/(24*c^3*Sqrt[a + x*(b + c*x)]) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(16*c^(7/2))","A",1
2375,1,103,127,0.110566,"\int \frac{(d+e x)^2}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[a + b*x + c*x^2],x]","\frac{\left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{8 c^{5/2}}+\frac{e \sqrt{a+x (b+c x)} (-3 b e+8 c d+2 c e x)}{4 c^2}","\frac{\left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{5/2}}+\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c}",1,"(e*(8*c*d - 3*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)])/(4*c^2) + ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(5/2))","A",1
2376,1,66,68,0.0750713,"\int \frac{d+e x}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)/Sqrt[a + b*x + c*x^2],x]","\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 c^{3/2}}+\frac{e \sqrt{a+x (b+c x)}}{c}","\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 c^{3/2}}+\frac{e \sqrt{a+b x+c x^2}}{c}",1,"(e*Sqrt[a + x*(b + c*x)])/c + ((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(3/2))","A",1
2377,1,34,36,0.0210455,"\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx","Integrate[1/Sqrt[a + b*x + c*x^2],x]","\frac{\log \left(2 \sqrt{c} \sqrt{a+b x+c x^2}+b+2 c x\right)}{\sqrt{c}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c}}",1,"Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]]/Sqrt[c]","A",1
2378,1,78,79,0.0186553,"\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}","\frac{\tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\sqrt{a e^2-b d e+c d^2}}",1,"-(ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d^2 + e*(-(b*d) + a*e)])","A",1
2379,1,131,134,0.084937,"\int \frac{1}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + b*x + c*x^2]),x]","-\frac{e \sqrt{a+x (b+c x)}}{(d+e x) \left(e (a e-b d)+c d^2\right)}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{2 \left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{e \sqrt{a+b x+c x^2}}{(d+e x) \left(a e^2-b d e+c d^2\right)}",1,"-((e*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x))) - ((2*c*d - b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(2*(c*d^2 + e*(-(b*d) + a*e))^(3/2))","A",1
2380,1,205,209,0.2605404,"\int \frac{1}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]","-\frac{\left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{5/2}}-\frac{3 e \sqrt{a+x (b+c x)} (2 c d-b e)}{4 (d+e x) \left(e (a e-b d)+c d^2\right)^2}-\frac{e \sqrt{a+x (b+c x)}}{2 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}","\frac{\left(-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 \left(a e^2-b d e+c d^2\right)^{5/2}}-\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left(a e^2-b d e+c d^2\right)^2}-\frac{e \sqrt{a+b x+c x^2}}{2 (d+e x)^2 \left(a e^2-b d e+c d^2\right)}",1,"-1/2*(e*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) - (3*e*(2*c*d - b*e)*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) - ((8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(5/2))","A",1
2381,1,288,293,0.6070723,"\int \frac{1}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]","-\frac{\frac{e \sqrt{a+x (b+c x)} \left(-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right)}{4 (d+e x) \left(e (a e-b d)+c d^2\right)}+\frac{3 (2 c d-b e) \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{8 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}{(d+e x)^3}+\frac{5 e \sqrt{a+x (b+c x)} (2 c d-b e)}{2 (d+e x)^2}}{6 \left(e (a e-b d)+c d^2\right)^2}","-\frac{e \sqrt{a+b x+c x^2} \left(-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right)}{24 (d+e x) \left(a e^2-b d e+c d^2\right)^3}+\frac{(2 c d-b e) \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{16 \left(a e^2-b d e+c d^2\right)^{7/2}}-\frac{5 e \sqrt{a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^2}-\frac{e \sqrt{a+b x+c x^2}}{3 (d+e x)^3 \left(a e^2-b d e+c d^2\right)}",1,"-1/6*((2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + (5*e*(2*c*d - b*e)*Sqrt[a + x*(b + c*x)])/(2*(d + e*x)^2) + (e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(c*d^2 + e*(-(b*d) + a*e))^2","A",1
2382,1,316,286,0.4310862,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{c} \left(4 b c \left(-13 a^2 e^4+a c e^2 \left(12 d^2+40 d e x-5 e^2 x^2\right)+2 c^2 d^3 (d-4 e x)\right)+8 c^2 \left(a^2 e^3 (16 d+3 e x)+a c e \left(-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right)+2 c^2 d^4 x\right)+b^3 e^3 (15 a e+c x (5 e x-48 d))-2 b^2 c e^2 \left(a e (24 d+31 e x)+c x \left(-24 d^2+8 d e x+e^2 x^2\right)\right)+15 b^4 e^4 x\right)-3 e^2 \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{8 c^{7/2} \left(4 a c-b^2\right) \sqrt{a+x (b+c x)}}","\frac{3 e^2 \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{7/2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left(b^2-4 a c\right)}+\frac{e \sqrt{a+b x+c x^2} \left(2 c e x \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right)-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right)}{4 c^3 \left(b^2-4 a c\right)}",1,"(2*Sqrt[c]*(15*b^4*e^4*x + b^3*e^3*(15*a*e + c*x*(-48*d + 5*e*x)) + 4*b*c*(-13*a^2*e^4 + 2*c^2*d^3*(d - 4*e*x) + a*c*e^2*(12*d^2 + 40*d*e*x - 5*e^2*x^2)) - 2*b^2*c*e^2*(a*e*(24*d + 31*e*x) + c*x*(-24*d^2 + 8*d*e*x + e^2*x^2)) + 8*c^2*(2*c^2*d^4*x + a^2*e^3*(16*d + 3*e*x) + a*c*e*(-8*d^3 - 12*d^2*e*x + 8*d*e^2*x^2 + e^3*x^3))) - 3*(b^2 - 4*a*c)*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(7/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])","A",1
2383,1,196,177,0.402891,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(4 c \left(2 a^2 e^3+a c e \left(-3 d^2-3 d e x+e^2 x^2\right)+c^2 d^3 x\right)-b^2 e^2 (3 a e+c x (e x-6 d))+2 b c \left(a e^2 (3 d+5 e x)+c d^2 (d-3 e x)\right)-3 b^3 e^3 x\right)}{\sqrt{a+x (b+c x)}}+3 e^2 \left(b^2-4 a c\right) (b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 c^{5/2} \left(4 a c-b^2\right)}","\frac{e \sqrt{a+b x+c x^2} \left(-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right)}{c^2 \left(b^2-4 a c\right)}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{3 e^2 (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 c^{5/2}}",1,"((2*Sqrt[c]*(-3*b^3*e^3*x - b^2*e^2*(3*a*e + c*x*(-6*d + e*x)) + 2*b*c*(c*d^2*(d - 3*e*x) + a*e^2*(3*d + 5*e*x)) + 4*c*(2*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + e^2*x^2))))/Sqrt[a + x*(b + c*x)] + 3*(b^2 - 4*a*c)*e^2*(-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(5/2)*(-b^2 + 4*a*c))","A",1
2384,1,127,129,0.2342859,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right)}{\sqrt{a+x (b+c x)}}-e^2 \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)}{c^{3/2} \left(4 a c-b^2\right)}","\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left(b^2-4 a c\right)}-\frac{2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{e^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2}}",1,"((2*Sqrt[c]*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*e^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(c^(3/2)*(-b^2 + 4*a*c))","A",1
2385,1,43,45,0.1834991,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(3/2),x]","\frac{4 a e-2 b d+2 b e x-4 c d x}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}",1,"(-2*b*d + 4*a*e - 4*c*d*x + 2*b*e*x)/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])","A",1
2386,1,31,32,0.0171333,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^(-3/2),x]","-\frac{2 (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)}}","-\frac{2 (b+2 c x)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}",1,"(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])","A",1
2387,1,162,155,0.2706894,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{\frac{e^2 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{\sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}}{4 a c-b^2}","\frac{e^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"((-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/((c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]) + ((b^2 - 4*a*c)*e^2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(-b^2 + 4*a*c)","A",1
2388,1,250,255,0.3388055,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 \left(-\frac{e \sqrt{a+x (b+c x)} \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right)}{2 (d+e x) \left(e (a e-b d)+c d^2\right)}+\frac{3 e^2 \left(b^2-4 a c\right) (b e-2 c d) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{4 \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x) \sqrt{a+x (b+c x)}}\right)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","-\frac{e \sqrt{a+b x+c x^2} \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right)}{\left(b^2-4 a c\right) (d+e x) \left(a e^2-b d e+c d^2\right)^2}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) (d+e x) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{5/2}}",1,"(2*(-1/2*(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*Sqrt[a + x*(b + c*x)]) + (3*(b^2 - 4*a*c)*e^2*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2389,1,353,371,1.2596428,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 \left(-\frac{e \sqrt{a+x (b+c x)} \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right)}{4 (d+e x)^2 \left(e (a e-b d)+c d^2\right)}+\frac{1}{16} e \left(-\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left(-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right)}{(d+e x) \left(e (a e-b d)+c d^2\right)^2}-\frac{3 e \left(b^2-4 a c\right) \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{5/2}}\right)+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^2 \sqrt{a+x (b+c x)}}\right)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","-\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right)}{4 \left(b^2-4 a c\right) (d+e x) \left(a e^2-b d e+c d^2\right)^3}-\frac{e \sqrt{a+b x+c x^2} \left(-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right)}{2 \left(b^2-4 a c\right) (d+e x)^2 \left(a e^2-b d e+c d^2\right)^2}+\frac{3 e^2 \left(-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 \left(a e^2-b d e+c d^2\right)^{7/2}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) (d+e x)^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"(2*(-1/4*(e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((-2*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)))/16))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2390,1,486,519,2.2729684,"\int \frac{1}{(d+e x)^4 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^4*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 \left(\frac{e \left(-\frac{2 \sqrt{a+x (b+c x)} \left(4 c^2 e^2 \left(64 a^2 e^2+332 a b d e+119 b^2 d^2\right)-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right)}{(d+e x) \left(e (a e-b d)+c d^2\right)}-\frac{4 \sqrt{a+x (b+c x)} (2 c d-b e) \left(-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right)}{(d+e x)^2}+\frac{15 e \left(b^2-4 a c\right) (b e-2 c d) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}\right)}{96 \left(e (a e-b d)+c d^2\right)^2}-\frac{e \sqrt{a+x (b+c x)} \left(-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right)}{6 (d+e x)^3 \left(e (a e-b d)+c d^2\right)}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^3 \sqrt{a+x (b+c x)}}\right)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","-\frac{e \sqrt{a+b x+c x^2} \left(4 c^2 e^2 \left(64 a^2 e^2+332 a b d e+119 b^2 d^2\right)-20 b^2 c e^3 (23 a e+19 b d)-16 c^3 d^2 e (83 a e+12 b d)+105 b^4 e^4+96 c^4 d^4\right)}{24 \left(b^2-4 a c\right) (d+e x) \left(a e^2-b d e+c d^2\right)^4}-\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (29 a e+6 b d)+35 b^2 e^2+24 c^2 d^2\right)}{12 \left(b^2-4 a c\right) (d+e x)^2 \left(a e^2-b d e+c d^2\right)^3}-\frac{e \sqrt{a+b x+c x^2} \left(-4 c e (4 a e+3 b d)+7 b^2 e^2+12 c^2 d^2\right)}{3 \left(b^2-4 a c\right) (d+e x)^3 \left(a e^2-b d e+c d^2\right)^2}+\frac{5 e^2 (2 c d-b e) \left(-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{16 \left(a e^2-b d e+c d^2\right)^{9/2}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) (d+e x)^3 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"(2*(-1/6*(e*(12*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(3*b*d + 4*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^3) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^3*Sqrt[a + x*(b + c*x)]) + (e*((-4*(2*c*d - b*e)*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(6*b*d + 29*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 - (2*(96*c^4*d^4 + 105*b^4*e^4 - 20*b^2*c*e^3*(19*b*d + 23*a*e) - 16*c^3*d^2*e*(12*b*d + 83*a*e) + 4*c^2*e^2*(119*b^2*d^2 + 332*a*b*d*e + 64*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (15*(b^2 - 4*a*c)*e*(-2*c*d + b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(96*(c*d^2 + e*(-(b*d) + a*e))^2)))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2391,1,566,389,1.8090507,"\int \frac{(d+e x)^5}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^5/(a + b*x + c*x^2)^(5/2),x]","\frac{b^4 e^4 \left(15 a^2 e-30 a c x (2 d+3 e x)+c^2 x^3 (3 e x-40 d)\right)-2 b^3 c \left(15 a^2 e^4 (d+7 e x)+2 a c e^4 x^2 (37 e x-45 d)+c^2 d^2 \left(d^3+15 d^2 e x-30 d e^2 x^2-10 e^3 x^3\right)\right)+4 b^2 c \left(-25 a^3 e^5+3 a^2 c e^4 x (35 d+4 e x)+a c^2 e \left(-5 d^4+60 d^3 e x-30 d^2 e^2 x^2+70 d e^3 x^3-6 e^4 x^4\right)+c^3 d^3 x \left(3 d^2-30 d e x+10 e^2 x^2\right)\right)+8 b c^2 \left(a^3 e^4 (25 d+39 e x)+4 a^2 c e^2 \left(5 d^3-15 d^2 e x+8 e^3 x^3\right)+3 a c^2 d^2 \left(d^3-5 d^2 e x+10 d e^2 x^2-10 e^3 x^3\right)+2 c^3 d^4 x^2 (3 d-5 e x)\right)+16 c^2 \left(8 a^4 e^5+a^3 c e^3 \left(-20 d^2-15 d e x+12 e^2 x^2\right)+a^2 c^2 e \left(-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right)+a c^3 d^3 x \left(3 d^2+10 e^2 x^2\right)+2 c^4 d^5 x^3\right)+10 b^5 e^4 x (3 a e+c x (2 e x-3 d))+15 b^6 e^5 x^2}{3 c^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 c^{7/2}}","-\frac{8 (d+e x)^2 \left(8 a^2 c e^3-x (2 c d-b e) \left(-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right)+b^2 \left(3 c d^2 e-a e^3\right)-2 b c d \left(3 a e^2+c d^2\right)\right)}{3 c \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{e \sqrt{a+b x+c x^2} \left(8 c^2 e^2 \left(-16 a^2 e^2-25 a b d e+b^2 d^2\right)+2 c e x (2 c d-b e) \left(-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right)+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right)}{3 c^3 \left(b^2-4 a c\right)^2}-\frac{2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{5 e^4 (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 c^{7/2}}",1,"(15*b^6*e^5*x^2 + 10*b^5*e^4*x*(3*a*e + c*x*(-3*d + 2*e*x)) + b^4*e^4*(15*a^2*e + c^2*x^3*(-40*d + 3*e*x) - 30*a*c*x*(2*d + 3*e*x)) - 2*b^3*c*(15*a^2*e^4*(d + 7*e*x) + 2*a*c*e^4*x^2*(-45*d + 37*e*x) + c^2*d^2*(d^3 + 15*d^2*e*x - 30*d*e^2*x^2 - 10*e^3*x^3)) + 8*b*c^2*(2*c^3*d^4*x^2*(3*d - 5*e*x) + a^3*e^4*(25*d + 39*e*x) + 3*a*c^2*d^2*(d^3 - 5*d^2*e*x + 10*d*e^2*x^2 - 10*e^3*x^3) + 4*a^2*c*e^2*(5*d^3 - 15*d^2*e*x + 8*e^3*x^3)) + 4*b^2*c*(-25*a^3*e^5 + 3*a^2*c*e^4*x*(35*d + 4*e*x) + c^3*d^3*x*(3*d^2 - 30*d*e*x + 10*e^2*x^2) + a*c^2*e*(-5*d^4 + 60*d^3*e*x - 30*d^2*e^2*x^2 + 70*d*e^3*x^3 - 6*e^4*x^4)) + 16*c^2*(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*x^3 + 3*e^4*x^4)))/(3*c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(7/2))","A",1
2392,1,397,295,0.814145,"\int \frac{(d+e x)^4}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]","\frac{e^4 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{5/2}}-\frac{2 \left(b^3 \left(3 a^2 e^4-18 a c e^4 x^2+c^2 d \left(d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right)\right)-2 b^2 c \left(21 a^2 e^4 x+2 a c e \left(-2 d^3+18 d^2 e x-6 d e^2 x^2+7 e^3 x^3\right)+3 c^2 d^2 x \left(d^2-8 d e x+2 e^2 x^2\right)\right)-4 b c \left(5 a^3 e^4+12 a^2 c d e^2 (d-2 e x)+3 a c^2 d \left(d^3-4 d^2 e x+6 d e^2 x^2-4 e^3 x^3\right)+2 c^3 d^3 x^2 (3 d-4 e x)\right)+8 c^2 \left(a^3 e^3 (8 d+3 e x)+4 a^2 c e \left(d^3+3 d e^2 x^2+e^3 x^3\right)-3 a c^2 d^2 x \left(d^2+2 e^2 x^2\right)-2 c^3 d^4 x^3\right)+2 b^4 e^4 x \left(3 a+2 c x^2\right)+3 b^5 e^4 x^2\right)}{3 c^2 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right)}{3 c^2 \left(b^2-4 a c\right)^2}+\frac{4 (d+e x) \left(x (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right)-\left(b^2 \left(5 c d^2 e-a e^3\right)\right)+4 b c d \left(3 a e^2+c d^2\right)-4 a c e \left(3 a e^2+c d^2\right)\right)}{3 c \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{e^4 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{5/2}}",1,"(-2*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e^4*x^2 + c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4 + 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*a*c^2*d*(d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*e*x) - 3*a*c^2*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2 - 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))))/(3*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (e^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)","A",1
2393,1,190,118,0.7319381,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(5/2),x]","\frac{2 \left(12 b (d-e x) \left(2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right)-8 \left(2 a^3 e^3+3 a^2 c e \left(d^2+e^2 x^2\right)-3 a c^2 d x \left(d^2+e^2 x^2\right)-2 c^3 d^3 x^3\right)-6 b^2 \left(d^2-6 d e x+e^2 x^2\right) (a e-c d x)+b^3 \left(-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right)\right)}{3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{16 \left(a e^2-b d e+c d^2\right) (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(2*(-6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + 12*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d - e*x)^2) - 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2*x^2) - 3*a*c^2*d*x*(d^2 + e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))","A",1
2394,1,167,98,0.6744896,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(5/2),x]","\frac{2 \left(4 b \left(2 a^2 e^2+3 a c (d-e x)^2+2 c^2 d x^2 (3 d-2 e x)\right)+8 c \left(-2 a^2 d e+a c x \left(3 d^2+e^2 x^2\right)+2 c^2 d^2 x^3\right)+b^2 \left(2 c x \left(3 d^2-12 d e x+e^2 x^2\right)-4 a e (d-3 e x)\right)-\left(b^3 \left(d^2+6 d e x-3 e^2 x^2\right)\right)\right)}{3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(2*(-(b^3*(d^2 + 6*d*e*x - 3*e^2*x^2)) + 4*b*(2*a^2*e^2 + 2*c^2*d*x^2*(3*d - 2*e*x) + 3*a*c*(d - e*x)^2) + 8*c*(-2*a^2*d*e + 2*c^2*d^2*x^3 + a*c*x*(3*d^2 + e^2*x^2)) + b^2*(-4*a*e*(d - 3*e*x) + 2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))","A",1
2395,1,110,91,0.1168979,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(5/2),x]","-\frac{2 \left(8 c \left(a^2 e-3 a c d x-2 c^2 d x^3\right)+2 b^2 (a e+3 c x (2 e x-d))+4 b c \left(2 c x^2 (e x-3 d)-3 a (d-e x)\right)+b^3 (d+3 e x)\right)}{3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{8 (b+2 c x) (2 c d-b e)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*(b^3*(d + 3*e*x) + 8*c*(a^2*e - 3*a*c*d*x - 2*c^2*d*x^3) + 4*b*c*(-3*a*(d - e*x) + 2*c*x^2*(-3*d + e*x)) + 2*b^2*(a*e + 3*c*x*(-d + 2*e*x))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))","A",1
2396,1,57,70,0.0277221,"\int \frac{1}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^(-5/2),x]","\frac{2 (b+2 c x) \left(4 c \left(3 a+2 c x^2\right)-b^2+8 b c x\right)}{3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{16 c (b+2 c x)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(2*(b + 2*c*x)*(-b^2 + 8*b*c*x + 4*c*(3*a + 2*c*x^2)))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))","A",1
2397,1,309,310,0.4445281,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]","\frac{2 \left(8 c^2 \left(3 a^2 e^3+5 a c d e^2 x+2 c^2 d^3 x\right)+2 b^2 c e \left(c d (e x-6 d)-11 a e^2\right)+4 b c^2 \left(5 a e^2 (d-e x)+2 c d^2 (d-3 e x)\right)+3 b^4 e^3+b^3 c e^2 (d+3 e x)\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)^2}-\frac{2 \left(2 c (a e+c d x)+b^2 (-e)+b c (d-e x)\right)}{3 \left(b^2-4 a c\right) (a+x (b+c x))^{3/2} \left(e (a e-b d)+c d^2\right)}-\frac{e^4 \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{5/2}}","-\frac{2 \left(-c x (2 c d-b e) \left(-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(-4 c e (b d-3 a e)-3 b^2 e^2+8 c^2 d^2\right)+4 a c e (2 c d-b e)^2\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{e^4 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\left(a e^2-b d e+c d^2\right)^{5/2}}",1,"(-2*(-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x)))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2)) + (2*(3*b^4*e^3 + b^3*c*e^2*(d + 3*e*x) + 8*c^2*(3*a^2*e^3 + 2*c^2*d^3*x + 5*a*c*d*e^2*x) + 4*b*c^2*(2*c*d^2*(d - 3*e*x) + 5*a*e^2*(d - e*x)) + 2*b^2*c*e*(-11*a*e^2 + c*d*(-6*d + e*x))))/(3*(b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) - (e^4*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)","A",1
2398,1,482,473,1.6949516,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]","\frac{2 \left(\frac{e \sqrt{a+x (b+c x)} \left(-4 c^2 e^2 \left(32 a^2 e^2+36 a b d e-3 b^2 d^2\right)+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right)}{2 \left(b^2-4 a c\right) (d+e x) \left(e (a e-b d)+c d^2\right)^2}+\frac{-8 c^2 \left(4 a^2 e^3-a c d e (d-7 e x)+2 c^2 d^3 x\right)+2 b^2 c e \left(16 a e^2+c d (5 d+e x)\right)-4 b c^2 \left(a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right)-5 b^4 e^3+b^3 c e^2 (3 d-5 e x)}{\left(b^2-4 a c\right) (d+e x) \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{15 e^4 \left(b^2-4 a c\right) (b e-2 c d) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{4 \left(e (a e-b d)+c d^2\right)^{5/2}}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x) (a+x (b+c x))^{3/2}}\right)}{3 \left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{e \sqrt{a+b x+c x^2} \left(4 c^2 e^2 \left(-32 a^2 e^2-36 a b d e+3 b^2 d^2\right)+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right)}{3 \left(b^2-4 a c\right)^2 (d+e x) \left(a e^2-b d e+c d^2\right)^3}-\frac{2 \left(-c x (2 c d-b e) \left(-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right)+6 a c e (2 c d-b e)^2\right)}{3 \left(b^2-4 a c\right)^2 (d+e x) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{3 \left(b^2-4 a c\right) (d+e x) \left(a+b x+c x^2\right)^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{2 \left(a e^2-b d e+c d^2\right)^{7/2}}",1,"(2*((e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*Sqrt[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (-5*b^4*e^3 + b^3*c*e^2*(3*d - 5*e*x) - 8*c^2*(4*a^2*e^3 + 2*c^2*d^3*x - a*c*d*e*(d - 7*e*x)) - 4*b*c^2*(a*e^2*(9*d - 7*e*x) + 2*c*d^2*(d - 3*e*x)) + 2*b^2*c*e*(16*a*e^2 + c*d*(5*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)*Sqrt[a + x*(b + c*x)]) + (15*(b^2 - 4*a*c)*e^4*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2399,1,626,621,2.7649051,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]","\frac{2 \left(\frac{e \left(\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left(-16 c^2 e^2 \left(81 a^2 e^2+28 a b d e+b^2 d^2\right)+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right)}{(d+e x) \left(e (a e-b d)+c d^2\right)}-\frac{15 e^3 \left(b^2-4 a c\right)^2 \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{4 \sqrt{a+x (b+c x)} \left(8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right)}{(d+e x)^2}\right)}{16 \left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)^2}+\frac{-8 c^2 \left(5 a^2 e^3+a c d e (9 e x-2 d)+2 c^2 d^3 x\right)+2 b^2 c e \left(21 a e^2+c d (4 d+3 e x)\right)-4 b c^2 \left(a e^2 (13 d-9 e x)+2 c d^2 (d-3 e x)\right)-7 b^4 e^3+7 b^3 c e^2 (d-e x)}{\left(b^2-4 a c\right) (d+e x)^2 \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^2 (a+x (b+c x))^{3/2}}\right)}{3 \left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left(-16 c^2 e^2 \left(81 a^2 e^2+28 a b d e+b^2 d^2\right)+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right)}{12 \left(b^2-4 a c\right)^2 (d+e x) \left(a e^2-b d e+c d^2\right)^4}-\frac{2 \left(-c x (2 c d-b e) \left(-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(20 a c e^2-7 b^2 e^2+8 c^2 d^2\right)+8 a c e (2 c d-b e)^2\right)}{3 \left(b^2-4 a c\right)^2 (d+e x)^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}+\frac{5 e^4 \left(-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{8 \left(a e^2-b d e+c d^2\right)^{9/2}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{3 \left(b^2-4 a c\right) (d+e x)^2 \left(a+b x+c x^2\right)^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{e \sqrt{a+b x+c x^2} \left(8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right)}{6 \left(b^2-4 a c\right)^2 (d+e x)^2 \left(a e^2-b d e+c d^2\right)^3}",1,"(2*((b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*(a + x*(b + c*x))^(3/2)) + (-7*b^4*e^3 + 7*b^3*c*e^2*(d - e*x) - 4*b*c^2*(a*e^2*(13*d - 9*e*x) + 2*c*d^2*(d - 3*e*x)) + 2*b^2*c*e*(21*a*e^2 + c*d*(4*d + 3*e*x)) - 8*c^2*(5*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(-2*d + 9*e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((4*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 + (2*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e + 81*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) - (15*(b^2 - 4*a*c)^2*e^3*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(16*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2)))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2400,1,25,29,0.0080979,"\int \frac{3+x}{\sqrt{5-4 x-x^2}} \, dx","Integrate[(3 + x)/Sqrt[5 - 4*x - x^2],x]","\sin ^{-1}\left(\frac{x+2}{3}\right)-\sqrt{-x^2-4 x+5}","-\sqrt{-x^2-4 x+5}-\sin ^{-1}\left(\frac{1}{3} (-x-2)\right)",1,"-Sqrt[5 - 4*x - x^2] + ArcSin[(2 + x)/3]","A",1
2401,1,25,25,0.009281,"\int \frac{5-4 x}{\sqrt{-8+12 x-4 x^2}} \, dx","Integrate[(5 - 4*x)/Sqrt[-8 + 12*x - 4*x^2],x]","\sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x)","\sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x)",1,"Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2","A",1
2402,1,25,23,0.0064737,"\int \frac{3+2 x}{\sqrt{5+2 x+x^2}} \, dx","Integrate[(3 + 2*x)/Sqrt[5 + 2*x + x^2],x]","2 \sqrt{x^2+2 x+5}+\sinh ^{-1}\left(\frac{1}{4} (2 x+2)\right)","2 \sqrt{x^2+2 x+5}+\sinh ^{-1}\left(\frac{x+1}{2}\right)",1,"2*Sqrt[5 + 2*x + x^2] + ArcSinh[(2 + 2*x)/4]","A",1
2403,1,30,34,0.0196878,"\int \frac{-1+x}{\sqrt{3-4 x+x^2}} \, dx","Integrate[(-1 + x)/Sqrt[3 - 4*x + x^2],x]","\sqrt{x^2-4 x+3}+\tanh ^{-1}\left(\frac{x-2}{\sqrt{x^2-4 x+3}}\right)","\sqrt{x^2-4 x+3}-\tanh ^{-1}\left(\frac{2-x}{\sqrt{x^2-4 x+3}}\right)",1,"Sqrt[3 - 4*x + x^2] + ArcTanh[(-2 + x)/Sqrt[3 - 4*x + x^2]]","A",1
2404,1,22,19,0.0059629,"\int \frac{1}{(1-x) \sqrt{-4+2 x+x^2}} \, dx","Integrate[1/((1 - x)*Sqrt[-4 + 2*x + x^2]),x]","\tan ^{-1}\left(\frac{6-4 x}{2 \sqrt{x^2+2 x-4}}\right)","\tan ^{-1}\left(\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right)",1,"ArcTan[(6 - 4*x)/(2*Sqrt[-4 + 2*x + x^2])]","A",1
2405,1,12,13,0.0036899,"\int \frac{1}{(-2+x) \sqrt{3-4 x+x^2}} \, dx","Integrate[1/((-2 + x)*Sqrt[3 - 4*x + x^2]),x]","\tan ^{-1}\left(\sqrt{(x-2)^2-1}\right)","\tan ^{-1}\left(\sqrt{x^2-4 x+3}\right)",1,"ArcTan[Sqrt[-1 + (-2 + x)^2]]","A",1
2406,1,19,17,0.037659,"\int \frac{1+x}{\left(2+3 x+x^2\right)^{3/2}} \, dx","Integrate[(1 + x)/(2 + 3*x + x^2)^(3/2),x]","\frac{2 \sqrt{x^2+3 x+2}}{x+2}","\frac{2 (x+1)}{\sqrt{x^2+3 x+2}}",1,"(2*Sqrt[2 + 3*x + x^2])/(2 + x)","A",1
2407,1,51,96,0.0314775,"\int \frac{1}{(d+e x) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]","\frac{2 (b+2 c x) (\log (b+2 c x)-\log (d+e x))}{\sqrt{\frac{(b+2 c x)^2}{c}} (2 c d-b e)}","\frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}",1,"(2*(b + 2*c*x)*(Log[b + 2*c*x] - Log[d + e*x]))/((2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/c])","A",1
2408,1,55,56,0.0376342,"\int \frac{1}{\left(\frac{b e}{2 c}+e x\right) \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(((b*e)/(2*c) + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right)}{e \sqrt{b^2-4 a c}}","\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right)}{e \sqrt{b^2-4 a c}}",1,"(2*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*e)","A",1
2409,1,45,48,0.0561976,"\int \frac{1}{(d+e x) \sqrt{\frac{-c d^2+b d e}{e^2}+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x + c*x^2]),x]","-\frac{2 e \sqrt{\frac{(d+e x) (b e-c d+c e x)}{e^2}}}{(d+e x) (b e-2 c d)}","\frac{2 e \sqrt{-\frac{d (c d-b e)}{e^2}+b x+c x^2}}{(d+e x) (2 c d-b e)}",1,"(-2*e*Sqrt[((d + e*x)*(-(c*d) + b*e + c*e*x))/e^2])/((-2*c*d + b*e)*(d + e*x))","A",1
2410,1,21,27,0.0112737,"\int \frac{1}{\left(\frac{b e}{2 c}+e x\right) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx","Integrate[1/(((b*e)/(2*c) + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]","-\frac{2}{e \sqrt{\frac{(b+2 c x)^2}{c}}}","-\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}",1,"-2/(e*Sqrt[(b + 2*c*x)^2/c])","A",1
2411,1,40,40,0.0141897,"\int \frac{x}{\sqrt{2+4 x+3 x^2}} \, dx","Integrate[x/Sqrt[2 + 4*x + 3*x^2],x]","\frac{1}{9} \left(3 \sqrt{3 x^2+4 x+2}-2 \sqrt{3} \sinh ^{-1}\left(\frac{3 x+2}{\sqrt{2}}\right)\right)","\frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left(\frac{3 x+2}{\sqrt{2}}\right)}{3 \sqrt{3}}",1,"(3*Sqrt[2 + 4*x + 3*x^2] - 2*Sqrt[3]*ArcSinh[(2 + 3*x)/Sqrt[2]])/9","A",1
2412,1,40,40,0.016626,"\int \frac{x}{\sqrt{2+4 x-3 x^2}} \, dx","Integrate[x/Sqrt[2 + 4*x - 3*x^2],x]","\frac{1}{9} \left(-3 \sqrt{-3 x^2+4 x+2}-2 \sqrt{3} \sin ^{-1}\left(\frac{2-3 x}{\sqrt{10}}\right)\right)","-\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left(\frac{2-3 x}{\sqrt{10}}\right)}{3 \sqrt{3}}",1,"(-3*Sqrt[2 + 4*x - 3*x^2] - 2*Sqrt[3]*ArcSin[(2 - 3*x)/Sqrt[10]])/9","A",1
2413,1,52,57,0.0190037,"\int \frac{x}{\sqrt{2+5 x+3 x^2}} \, dx","Integrate[x/Sqrt[2 + 5*x + 3*x^2],x]","\frac{1}{18} \left(6 \sqrt{3 x^2+5 x+2}-5 \sqrt{3} \tanh ^{-1}\left(\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right)\right)","\frac{1}{3} \sqrt{3 x^2+5 x+2}-\frac{5 \tanh ^{-1}\left(\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right)}{6 \sqrt{3}}",1,"(6*Sqrt[2 + 5*x + 3*x^2] - 5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/18","A",1
2414,1,38,38,0.0096356,"\int \frac{x}{\sqrt{2+5 x-3 x^2}} \, dx","Integrate[x/Sqrt[2 + 5*x - 3*x^2],x]","-\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left(\frac{1}{7} (5-6 x)\right)}{6 \sqrt{3}}","-\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left(\frac{1}{7} (5-6 x)\right)}{6 \sqrt{3}}",1,"-1/3*Sqrt[2 + 5*x - 3*x^2] - (5*ArcSin[(5 - 6*x)/7])/(6*Sqrt[3])","A",1
2415,1,49,54,0.017638,"\int \frac{x}{\sqrt{-2+4 x+3 x^2}} \, dx","Integrate[x/Sqrt[-2 + 4*x + 3*x^2],x]","\frac{1}{9} \left(3 \sqrt{3 x^2+4 x-2}-2 \sqrt{3} \tanh ^{-1}\left(\frac{3 x+2}{\sqrt{9 x^2+12 x-6}}\right)\right)","\frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left(\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right)}{3 \sqrt{3}}",1,"(3*Sqrt[-2 + 4*x + 3*x^2] - 2*Sqrt[3]*ArcTanh[(2 + 3*x)/Sqrt[-6 + 12*x + 9*x^2]])/9","A",1
2416,1,49,54,0.0197011,"\int \frac{x}{\sqrt{-2+4 x-3 x^2}} \, dx","Integrate[x/Sqrt[-2 + 4*x - 3*x^2],x]","\frac{1}{9} \left(-3 \sqrt{-3 x^2+4 x-2}-2 \sqrt{3} \tan ^{-1}\left(\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right)\right)","-\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left(\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right)}{3 \sqrt{3}}",1,"(-3*Sqrt[-2 + 4*x - 3*x^2] - 2*Sqrt[3]*ArcTan[(2 - 3*x)/Sqrt[-6 + 12*x - 9*x^2]])/9","A",1
2417,1,52,57,0.0189601,"\int \frac{x}{\sqrt{-2+5 x+3 x^2}} \, dx","Integrate[x/Sqrt[-2 + 5*x + 3*x^2],x]","\frac{1}{18} \left(6 \sqrt{3 x^2+5 x-2}-5 \sqrt{3} \tanh ^{-1}\left(\frac{6 x+5}{2 \sqrt{9 x^2+15 x-6}}\right)\right)","\frac{1}{3} \sqrt{3 x^2+5 x-2}-\frac{5 \tanh ^{-1}\left(\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right)}{6 \sqrt{3}}",1,"(6*Sqrt[-2 + 5*x + 3*x^2] - 5*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[-6 + 15*x + 9*x^2])])/18","A",1
2418,1,34,34,0.0083748,"\int \frac{x}{\sqrt{-2+5 x-3 x^2}} \, dx","Integrate[x/Sqrt[-2 + 5*x - 3*x^2],x]","-\frac{1}{3} \sqrt{-3 x^2+5 x-2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}}","-\frac{1}{3} \sqrt{-3 x^2+5 x-2}-\frac{5 \sin ^{-1}(5-6 x)}{6 \sqrt{3}}",1,"-1/3*Sqrt[-2 + 5*x - 3*x^2] - (5*ArcSin[5 - 6*x])/(6*Sqrt[3])","A",1
2419,1,31,27,0.0114448,"\int \frac{1}{x \sqrt{4+12 x+9 x^2}} \, dx","Integrate[1/(x*Sqrt[4 + 12*x + 9*x^2]),x]","\frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{(3 x+2)^2}}","-\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{9 x^2+12 x+4}}",1,"((2 + 3*x)*(Log[x] - Log[2 + 3*x]))/(2*Sqrt[(2 + 3*x)^2])","A",1
2420,1,31,27,0.0157744,"\int \frac{1}{x \sqrt{4-12 x+9 x^2}} \, dx","Integrate[1/(x*Sqrt[4 - 12*x + 9*x^2]),x]","\frac{(3 x-2) (\log (2-3 x)-\log (x))}{2 \sqrt{(2-3 x)^2}}","-\frac{(2-3 x) \tanh ^{-1}(1-3 x)}{\sqrt{9 x^2-12 x+4}}",1,"((-2 + 3*x)*(Log[2 - 3*x] - Log[x]))/(2*Sqrt[(2 - 3*x)^2])","A",1
2421,1,33,27,0.0102356,"\int \frac{1}{x \sqrt{-4+12 x-9 x^2}} \, dx","Integrate[1/(x*Sqrt[-4 + 12*x - 9*x^2]),x]","\frac{(3 x-2) (\log (2-3 x)-\log (x))}{2 \sqrt{-(2-3 x)^2}}","-\frac{(2-3 x) \tanh ^{-1}(1-3 x)}{\sqrt{-9 x^2+12 x-4}}",1,"((-2 + 3*x)*(Log[2 - 3*x] - Log[x]))/(2*Sqrt[-(2 - 3*x)^2])","A",1
2422,1,33,27,0.0080212,"\int \frac{1}{x \sqrt{-4-12 x-9 x^2}} \, dx","Integrate[1/(x*Sqrt[-4 - 12*x - 9*x^2]),x]","\frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{-(3 x+2)^2}}","-\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{-9 x^2-12 x-4}}",1,"((2 + 3*x)*(Log[x] - Log[2 + 3*x]))/(2*Sqrt[-(2 + 3*x)^2])","A",1
2423,1,31,68,0.0132332,"\int \frac{1}{x \sqrt{a^2+2 a b x+b^2 x^2}} \, dx","Integrate[1/(x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]","\frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{(a+b x)^2}}","\frac{\log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}",1,"((a + b*x)*(Log[x] - Log[a + b*x]))/(a*Sqrt[(a + b*x)^2])","A",1
2424,1,34,71,0.0184518,"\int \frac{1}{x \sqrt{a^2-2 a b x+b^2 x^2}} \, dx","Integrate[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]","\frac{(a-b x) (\log (x)-\log (a-b x))}{a \sqrt{(a-b x)^2}}","\frac{\log (x) (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}}",1,"((a - b*x)*(Log[x] - Log[a - b*x]))/(a*Sqrt[(a - b*x)^2])","A",1
2425,1,36,77,0.0133409,"\int \frac{1}{x \sqrt{-a^2+2 a b x-b^2 x^2}} \, dx","Integrate[1/(x*Sqrt[-a^2 + 2*a*b*x - b^2*x^2]),x]","\frac{(a-b x) (\log (x)-\log (a-b x))}{a \sqrt{-(a-b x)^2}}","\frac{\log (x) (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}}",1,"((a - b*x)*(Log[x] - Log[a - b*x]))/(a*Sqrt[-(a - b*x)^2])","A",1
2426,1,33,74,0.0132792,"\int \frac{1}{x \sqrt{-a^2-2 a b x-b^2 x^2}} \, dx","Integrate[1/(x*Sqrt[-a^2 - 2*a*b*x - b^2*x^2]),x]","\frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{-(a+b x)^2}}","\frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}",1,"((a + b*x)*(Log[x] - Log[a + b*x]))/(a*Sqrt[-(a + b*x)^2])","A",1
2427,1,37,52,0.0182101,"\int x \sqrt{3-2 x-x^2} \, dx","Integrate[x*Sqrt[3 - 2*x - x^2],x]","\frac{1}{6} \sqrt{-x^2-2 x+3} \left(2 x^2+x-9\right)-2 \sin ^{-1}\left(\frac{x+1}{2}\right)","-\frac{1}{3} \left(-x^2-2 x+3\right)^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left(\frac{1}{2} (-x-1)\right)",1,"(Sqrt[3 - 2*x - x^2]*(-9 + x + 2*x^2))/6 - 2*ArcSin[(1 + x)/2]","A",1
2428,1,42,56,0.0200614,"\int x \sqrt{8+2 x-x^2} \, dx","Integrate[x*Sqrt[8 + 2*x - x^2],x]","\frac{1}{6} \left(\sqrt{-x^2+2 x+8} \left(2 x^2-x-19\right)-27 \sin ^{-1}\left(\frac{1}{3}-\frac{x}{3}\right)\right)","-\frac{1}{3} \left(-x^2+2 x+8\right)^{3/2}-\frac{1}{2} (1-x) \sqrt{-x^2+2 x+8}-\frac{9}{2} \sin ^{-1}\left(\frac{1-x}{3}\right)",1,"(Sqrt[8 + 2*x - x^2]*(-19 - x + 2*x^2) - 27*ArcSin[1/3 - x/3])/6","A",1
2429,1,38,50,0.0213955,"\int x \sqrt{4+2 x+x^2} \, dx","Integrate[x*Sqrt[4 + 2*x + x^2],x]","\frac{1}{6} \left(\sqrt{x^2+2 x+4} \left(2 x^2+x+5\right)-9 \sinh ^{-1}\left(\frac{x+1}{\sqrt{3}}\right)\right)","\frac{1}{3} \left(x^2+2 x+4\right)^{3/2}-\frac{1}{2} (x+1) \sqrt{x^2+2 x+4}-\frac{3}{2} \sinh ^{-1}\left(\frac{x+1}{\sqrt{3}}\right)",1,"(Sqrt[4 + 2*x + x^2]*(5 + x + 2*x^2) - 9*ArcSinh[(1 + x)/Sqrt[3]])/6","A",1
2430,1,28,31,0.0093777,"\int \frac{1}{x \sqrt{2+4 x+3 x^2}} \, dx","Integrate[1/(x*Sqrt[2 + 4*x + 3*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{x+1}{\sqrt{\frac{3 x^2}{2}+2 x+1}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} (x+1)}{\sqrt{3 x^2+4 x+2}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(1 + x)/Sqrt[1 + 2*x + (3*x^2)/2]]/Sqrt[2])","A",1
2431,1,28,31,0.0087848,"\int \frac{1}{x \sqrt{2+4 x-3 x^2}} \, dx","Integrate[1/(x*Sqrt[2 + 4*x - 3*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{x+1}{\sqrt{-\frac{3 x^2}{2}+2 x+1}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} (x+1)}{\sqrt{-3 x^2+4 x+2}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(1 + x)/Sqrt[1 + 2*x - (3*x^2)/2]]/Sqrt[2])","A",1
2432,1,31,36,0.0102934,"\int \frac{1}{x \sqrt{2+5 x+3 x^2}} \, dx","Integrate[1/(x*Sqrt[2 + 5*x + 3*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{5 x+4}{2 \sqrt{6 x^2+10 x+4}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(4 + 5*x)/(2*Sqrt[4 + 10*x + 6*x^2])]/Sqrt[2])","A",1
2433,1,31,36,0.0102436,"\int \frac{1}{x \sqrt{2+5 x-3 x^2}} \, dx","Integrate[1/(x*Sqrt[2 + 5*x - 3*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{5 x+4}{2 \sqrt{-6 x^2+10 x+4}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{5 x+4}{2 \sqrt{2} \sqrt{-3 x^2+5 x+2}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(4 + 5*x)/(2*Sqrt[4 + 10*x - 6*x^2])]/Sqrt[2])","A",1
2434,1,27,33,0.0083546,"\int \frac{1}{x \sqrt{-2+4 x+3 x^2}} \, dx","Integrate[1/(x*Sqrt[-2 + 4*x + 3*x^2]),x]","\frac{\tan ^{-1}\left(\frac{x-1}{\sqrt{\frac{3 x^2}{2}+2 x-1}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} (1-x)}{\sqrt{3 x^2+4 x-2}}\right)}{\sqrt{2}}",1,"ArcTan[(-1 + x)/Sqrt[-1 + 2*x + (3*x^2)/2]]/Sqrt[2]","A",1
2435,1,27,33,0.0079785,"\int \frac{1}{x \sqrt{-2+4 x-3 x^2}} \, dx","Integrate[1/(x*Sqrt[-2 + 4*x - 3*x^2]),x]","\frac{\tan ^{-1}\left(\frac{x-1}{\sqrt{-\frac{3 x^2}{2}+2 x-1}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} (1-x)}{\sqrt{-3 x^2+4 x-2}}\right)}{\sqrt{2}}",1,"ArcTan[(-1 + x)/Sqrt[-1 + 2*x - (3*x^2)/2]]/Sqrt[2]","A",1
2436,1,30,36,0.0092268,"\int \frac{1}{x \sqrt{-2+5 x+3 x^2}} \, dx","Integrate[1/(x*Sqrt[-2 + 5*x + 3*x^2]),x]","\frac{\tan ^{-1}\left(\frac{5 x-4}{2 \sqrt{6 x^2+10 x-4}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{4-5 x}{2 \sqrt{2} \sqrt{3 x^2+5 x-2}}\right)}{\sqrt{2}}",1,"ArcTan[(-4 + 5*x)/(2*Sqrt[-4 + 10*x + 6*x^2])]/Sqrt[2]","A",1
2437,1,30,36,0.009783,"\int \frac{1}{x \sqrt{-2+5 x-3 x^2}} \, dx","Integrate[1/(x*Sqrt[-2 + 5*x - 3*x^2]),x]","\frac{\tan ^{-1}\left(\frac{5 x-4}{2 \sqrt{-6 x^2+10 x-4}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{4-5 x}{2 \sqrt{2} \sqrt{-3 x^2+5 x-2}}\right)}{\sqrt{2}}",1,"ArcTan[(-4 + 5*x)/(2*Sqrt[-4 + 10*x - 6*x^2])]/Sqrt[2]","A",1
2438,1,43,57,0.0178423,"\int \frac{1}{x^3 \sqrt{1+x+x^2}} \, dx","Integrate[1/(x^3*Sqrt[1 + x + x^2]),x]","\frac{1}{8} \left(\frac{2 \sqrt{x^2+x+1} (3 x-2)}{x^2}+\tanh ^{-1}\left(\frac{x+2}{2 \sqrt{x^2+x+1}}\right)\right)","\frac{3 \sqrt{x^2+x+1}}{4 x}-\frac{\sqrt{x^2+x+1}}{2 x^2}+\frac{1}{8} \tanh ^{-1}\left(\frac{x+2}{2 \sqrt{x^2+x+1}}\right)",1,"((2*(-2 + 3*x)*Sqrt[1 + x + x^2])/x^2 + ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])/8","A",1
2439,1,27,23,0.0546615,"\int \left(\frac{1}{x}-\frac{1}{x \sqrt{1+b x+c x^2}}\right) \, dx","Integrate[x^(-1) - 1/(x*Sqrt[1 + b*x + c*x^2]),x]","\tanh ^{-1}\left(\frac{b x+2}{2 \sqrt{b x+c x^2+1}}\right)+\log (x)","\log \left(-2 \sqrt{b x+c x^2+1}-b x-2\right)",1,"ArcTanh[(2 + b*x)/(2*Sqrt[1 + b*x + c*x^2])] + Log[x]","A",1
2440,1,594,504,3.4614938,"\int (d x)^{5/2} \sqrt{a+b x+c x^2} \, dx","Integrate[(d*x)^(5/2)*Sqrt[a + b*x + c*x^2],x]","\frac{(d x)^{5/2} \left(-\frac{4 \left(21 a^2 c^2-36 a b^2 c+8 b^4\right) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left(21 a^2 c^2-36 a b^2 c+8 b^4\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{4 a}{x \left(\sqrt{b^2-4 a c}+b\right)}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}-\frac{i x \left(21 a^2 c^2 \sqrt{b^2-4 a c}-48 a^2 b c^2+44 a b^3 c-36 a b^2 c \sqrt{b^2-4 a c}+8 b^4 \sqrt{b^2-4 a c}-8 b^5\right) \sqrt{\frac{4 a}{x \left(\sqrt{b^2-4 a c}+b\right)}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+2 c \sqrt{x} \left(b c \left(5 c x^2-27 a\right)+7 c^2 x \left(2 a+5 c x^2\right)+8 b^3-6 b^2 c x\right) (a+x (b+c x))\right)}{315 c^4 x^{5/2} \sqrt{a+x (b+c x)}}","-\frac{4 d^3 x \left(21 a^2 c^2-36 a b^2 c+8 b^4\right) \sqrt{a+b x+c x^2}}{315 c^{7/2} \sqrt{d x} \left(\sqrt{a}+\sqrt{c} x\right)}-\frac{\sqrt[4]{a} d^3 \sqrt{x} \left(42 a^2 c^2-72 a b^2 c+\sqrt{a} b \sqrt{c} \left(8 b^2-27 a c\right)+16 b^4\right) \left(\sqrt{a}+\sqrt{c} x\right) \sqrt{\frac{a+b x+c x^2}{\left(\sqrt{a}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt[4]{a} d^3 \sqrt{x} \left(21 a^2 c^2-36 a b^2 c+8 b^4\right) \left(\sqrt{a}+\sqrt{c} x\right) \sqrt{\frac{a+b x+c x^2}{\left(\sqrt{a}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{2 d^2 \sqrt{d x} \left(3 c x \left(8 b^2-7 a c\right)+b \left(3 a c+8 b^2\right)\right) \sqrt{a+b x+c x^2}}{315 c^3}-\frac{4 b d^2 \sqrt{d x} \left(a+b x+c x^2\right)^{3/2}}{21 c^2}+\frac{2 d (d x)^{3/2} \left(a+b x+c x^2\right)^{3/2}}{9 c}",1,"((d*x)^(5/2)*((-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(a + x*(b + c*x)))/Sqrt[x] + 2*c*Sqrt[x]*(a + x*(b + c*x))*(8*b^3 - 6*b^2*c*x + b*c*(-27*a + 5*c*x^2) + 7*c^2*x*(2*a + 5*c*x^2)) + (I*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] - (I*(-8*b^5 + 44*a*b^3*c - 48*a^2*b*c^2 + 8*b^4*Sqrt[b^2 - 4*a*c] - 36*a*b^2*c*Sqrt[b^2 - 4*a*c] + 21*a^2*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/(315*c^4*x^(5/2)*Sqrt[a + x*(b + c*x)])","C",1
2441,1,5328,581,13.0408616,"\int (d+e x)^{3/2} \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right)}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2}}{7 c}",1,"Result too large to show","C",1
2442,1,697,513,10.494854,"\int \sqrt{d+e x} \sqrt{a+b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2],x]","\frac{-\frac{4 e^2 (a+x (b+c x)) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{d+e x}}+\frac{i (d+e x) \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{4 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+2} \left(\left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(c \left(a e^2 \left(3 \sqrt{e^2 \left(b^2-4 a c\right)}+8 c d\right)-c d^2 \sqrt{e^2 \left(b^2-4 a c\right)}\right)-b^2 e^2 \left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 c d\right)+b c e \left(d \sqrt{e^2 \left(b^2-4 a c\right)}-4 a e^2\right)+b^3 e^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{\sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}+2 c e^2 \sqrt{d+e x} (a+x (b+c x)) (b e+c (d+3 e x))}{15 c^2 e^3 \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^2 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^2 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{5 e}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (2 c d-b e)}{15 c e}",1,"((-4*e^2*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(a + x*(b + c*x)))/Sqrt[d + e*x] + 2*c*e^2*Sqrt[d + e*x]*(a + x*(b + c*x))*(b*e + c*(d + 3*e*x)) + (I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^3*e^3 - b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/(15*c^2*e^3*Sqrt[a + x*(b + c*x)])","C",1
2443,1,936,444,8.1290018,"\int \frac{\sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x],x]","\frac{\sqrt{d+e x} \left(4 (a+x (b+c x)) e^2+\frac{(d+e x) \left(\frac{4 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x)) e^2}{(d+e x)^2}+\frac{i \sqrt{2} (2 c d-b e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(b^2 e^2-4 a c e^2-b \sqrt{\left(b^2-4 a c\right) e^2} e+2 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{c \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}\right)}{6 e^3 \sqrt{a+x (b+c x)}}","\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 e}",1,"(Sqrt[d + e*x]*(4*e^2*(a + x*(b + c*x)) + ((d + e*x)*((4*e^2*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))/(d + e*x)^2 + (I*Sqrt[2]*(2*c*d - b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(b^2*e^2 - 4*a*c*e^2 + 2*c*d*Sqrt[(b^2 - 4*a*c)*e^2] - b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(c*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(6*e^3*Sqrt[a + x*(b + c*x)])","C",1
2444,1,865,419,4.3739079,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^{3/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^(3/2),x]","\frac{4 c x^2 e^2+4 (a+b x) e^2-2 (a+x (b+c x)) e^2-\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)^{3/2} \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}+\frac{i \sqrt{2} \sqrt{\left(b^2-4 a c\right) e^2} (d+e x)^{3/2} \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}}{e^3 \sqrt{d+e x} \sqrt{a+x (b+c x)}}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{a+b x+c x^2}}{e \sqrt{d+e x}}",1,"(4*c*e^2*x^2 + 4*e^2*(a + b*x) - 2*e^2*(a + x*(b + c*x)) - (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])] + (I*Sqrt[2]*Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/(e^3*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)])","C",1
2445,1,962,497,9.797029,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^{5/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^(5/2),x]","\frac{\frac{i \sqrt{2} (2 c d-b e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right) (d+e x)^{5/2}}{\sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}+\frac{i \sqrt{2} \left(-b^2 e^2+4 a c e^2+b \sqrt{\left(b^2-4 a c\right) e^2} e-2 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right) (d+e x)^{5/2}}{\sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}+4 e^2 (b e-2 c d) (a+x (b+c x)) (d+e x)+4 e^2 (a+x (b+c x)) \left(c d (d+2 e x)-e^2 (a+b x)\right)}{6 e^3 \left(c d^2+e (a e-b d)\right) (d+e x)^{3/2} \sqrt{a+x (b+c x)}}","-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 e^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2} (2 c d-b e)}{3 e \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{a+b x+c x^2}}{3 e (d+e x)^{3/2}}",1,"(4*e^2*(-2*c*d + b*e)*(d + e*x)*(a + x*(b + c*x)) + 4*e^2*(a + x*(b + c*x))*(-(e^2*(a + b*x)) + c*d*(d + 2*e*x)) + (I*Sqrt[2]*(2*c*d - b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x)^(5/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])] + (I*Sqrt[2]*(-(b^2*e^2) + 4*a*c*e^2 - 2*c*d*Sqrt[(b^2 - 4*a*c)*e^2] + b*e*Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x)^(5/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/(6*e^3*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)])","C",1
2446,1,3493,617,12.7069747,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^{7/2}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^(7/2),x]","\text{Result too large to show}","\frac{4 \sqrt{a+b x+c x^2} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{15 e \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{2 \sqrt{a+b x+c x^2} (2 c d-b e)}{15 e (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{a+b x+c x^2}}{5 e (d+e x)^{5/2}}",1,"Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]*(-2/(5*e*(d + e*x)^3) - (2*(-2*c*d + b*e))/(15*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (4*(-(c^2*d^2) + b*c*d*e - b^2*e^2 + 3*a*c*e^2))/(15*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x))) - (2*c*Sqrt[a + x*(b + c*x)]*((2*(c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2)*(d + e*x)^(3/2)*(c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x)))/(c*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2]) - ((c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x)]*((I*c^2*d^2*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - (I*b*c*d*e*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + (I*b^2*e^2*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - ((3*I)*a*c*e^2*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + (I*Sqrt[2]*c^2*d*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - (I*b*c*e*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)])))/(c*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])))/(15*e^3*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x + c*x^2])","C",1
2447,1,10848,816,13.8437258,"\int (d+e x)^{3/2} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2),x]","\text{Result too large to show}","\frac{2 e \sqrt{d+e x} \left(c x^2+b x+a\right)^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left(c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right) \left(c x^2+b x+a\right)^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left(8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left(2 b^2 d^2+17 a b e d-10 a^2 e^2\right)-3 c e (2 c d-b e) \left(c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right) x\right) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left(c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right) \left(c^2 d^2+b^2 e^2-c e (b d+3 a e)\right) \sqrt{d+e x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c d^2-b e d+a e^2\right) \left(16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left(b^2 d^2-28 a b e d-20 a^2 e^2\right)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
2448,1,7541,712,13.7282453,"\int \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2),x]","\text{Result too large to show}","-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(3 c^2 e^2 \left(-28 a^2 e^2-20 a b d e+3 b^2 d^2\right)+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-6 c e x \left(-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right)-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right)}{315 c^2 e^3}+\frac{2 (d+e x)^{3/2} \left(a+b x+c x^2\right)^{3/2}}{9 e}-\frac{2 \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2} (2 c d-b e)}{21 c e}",1,"Result too large to show","C",0
2449,1,5338,579,13.2924344,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/Sqrt[d + e*x],x]","\text{Result too large to show}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{35 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{35 c^2 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-c e (11 b d-10 a e)+b^2 e^2-3 c e x (2 c d-b e)+8 c^2 d^2\right)}{35 c e^3}+\frac{2 \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2}}{7 e}",1,"Result too large to show","C",0
2450,1,732,515,10.6502887,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(3/2),x]","\frac{\frac{4 e^2 (a+x (b+c x)) \left(4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right)}{c \sqrt{d+e x}}-\frac{i (d+e x) \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{4 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+2} \left(\left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) \left(4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)-\left(4 c \left(4 c d^2 \sqrt{e^2 \left(b^2-4 a c\right)}+a e^2 \left(3 \sqrt{e^2 \left(b^2-4 a c\right)}-2 c d\right)\right)+b^2 e^2 \left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 c d\right)+4 b \left(a c e^3-4 c d e \sqrt{e^2 \left(b^2-4 a c\right)}\right)-b^3 e^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{c \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}+\frac{4 e^2 (a+x (b+c x)) \left(e (-5 a e+7 b d+2 b e x)+c \left(-8 d^2-2 d e x+e^2 x^2\right)\right)}{\sqrt{d+e x}}}{10 e^5 \sqrt{a+x (b+c x)}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{5 c e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{16 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{5 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left(a+b x+c x^2\right)^{3/2}}{e \sqrt{d+e x}}",1,"((4*e^2*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*(a + x*(b + c*x)))/(c*Sqrt[d + e*x]) + (4*e^2*(a + x*(b + c*x))*(e*(7*b*d - 5*a*e + 2*b*e*x) + c*(-8*d^2 - 2*d*e*x + e^2*x^2)))/Sqrt[d + e*x] - (I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(c*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]))/(10*e^5*Sqrt[a + x*(b + c*x)])","C",1
2451,1,978,499,8.5325352,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 (a+x (b+c x)) \left(c \left(8 d^2+10 e x d+e^2 x^2\right)-e (3 b d+a e+4 b e x)\right)}{e^3 (d+e x)^2}-\frac{(d+e x) \left(-\frac{16 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x)) e^2}{(d+e x)^2}-\frac{4 i \sqrt{2} (2 c d-b e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left(b^2 e^2-4 a c e^2-4 b \sqrt{\left(b^2-4 a c\right) e^2} e+8 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{e^5 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}\right)}{3 \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{a+b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left(a+b x+c x^2\right)^{3/2}}{3 e (d+e x)^{3/2}}",1,"(Sqrt[d + e*x]*((2*(a + x*(b + c*x))*(-(e*(3*b*d + a*e + 4*b*e*x)) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(e^3*(d + e*x)^2) - ((d + e*x)*((-16*e^2*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))/(d + e*x)^2 - ((4*I)*Sqrt[2]*(2*c*d - b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(b^2*e^2 - 4*a*c*e^2 + 8*c*d*Sqrt[(b^2 - 4*a*c)*e^2] - 4*b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(e^5*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(3*Sqrt[a + x*(b + c*x)])","C",1
2452,1,1386,578,12.7125104,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]","\frac{\sqrt{d+e x} (a+x (b+c x))^{3/2} \left(\frac{4 (2 c d-b e)}{5 e^3 (d+e x)^2}-\frac{2 \left(11 c^2 d^2-11 b c e d+b^2 e^2+7 a c e^2\right)}{5 e^3 \left(c d^2-b e d+a e^2\right) (d+e x)}-\frac{2 \left(c d^2-b e d+a e^2\right)}{5 e^3 (d+e x)^3}\right)}{c x^2+b x+a}-\frac{(d+e x)^{3/2} (a+x (b+c x))^{3/2} \left(-4 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(16 d^2 \left(\frac{d}{d+e x}-1\right)^2 c^3+4 e \left(a e \left(\frac{7 d^2}{(d+e x)^2}-\frac{6 d}{d+e x}+3\right)-4 b d \left(\frac{2 d^2}{(d+e x)^2}-\frac{3 d}{d+e x}+1\right)\right) c^2+e^2 \left(\left(\frac{17 d^2}{(d+e x)^2}-\frac{18 d}{d+e x}+1\right) b^2-\frac{4 a e \left(\frac{7 d}{d+e x}-3\right) b}{d+e x}+\frac{12 a^2 e^2}{(d+e x)^2}\right) c+\frac{b^2 e^3 \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(16 c^2 d^2+b^2 e^2+4 c e (3 a e-4 b d)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-b^3 e^3+b^2 \left(2 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+4 b \left(a c e^3-4 c d e \sqrt{\left(b^2-4 a c\right) e^2}\right)+4 c \left(4 c \sqrt{\left(b^2-4 a c\right) e^2} d^2+a e^2 \left(3 \sqrt{\left(b^2-4 a c\right) e^2}-2 c d\right)\right)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{10 e^5 \left(c d^2-b e d+a e^2\right) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(c x^2+b x+a\right)^{3/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{5 e^4 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{a+b x+c x^2} \left(e x \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right)}{5 e^3 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{16 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{5 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \left(a+b x+c x^2\right)^{3/2}}{5 e (d+e x)^{5/2}}",1,"(Sqrt[d + e*x]*(a + x*(b + c*x))^(3/2)*((-2*(c*d^2 - b*d*e + a*e^2))/(5*e^3*(d + e*x)^3) + (4*(2*c*d - b*e))/(5*e^3*(d + e*x)^2) - (2*(11*c^2*d^2 - 11*b*c*d*e + b^2*e^2 + 7*a*c*e^2))/(5*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x))))/(a + b*x + c*x^2) - ((d + e*x)^(3/2)*(a + x*(b + c*x))^(3/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^3*d^2*(-1 + d/(d + e*x))^2 + (b^2*e^3*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x) + 4*c^2*e*(a*e*(3 + (7*d^2)/(d + e*x)^2 - (6*d)/(d + e*x)) - 4*b*d*(1 + (2*d^2)/(d + e*x)^2 - (3*d)/(d + e*x))) + c*e^2*((12*a^2*e^2)/(d + e*x)^2 + b^2*(1 + (17*d^2)/(d + e*x)^2 - (18*d)/(d + e*x)) - (4*a*b*e*(-3 + (7*d)/(d + e*x)))/(d + e*x))) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(10*e^5*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + b*x + c*x^2)^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",0
2453,1,5469,721,13.3849166,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]","\text{Result too large to show}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{35 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{35 e^4 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{4 \sqrt{a+b x+c x^2} (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right)}{35 e^3 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 \sqrt{a+b x+c x^2} \left(e x \left(-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right)-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right)}{35 e^3 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \left(a+b x+c x^2\right)^{3/2}}{7 e (d+e x)^{7/2}}",1,"Result too large to show","C",0
2454,1,708,616,4.3746089,"\int \sqrt{d x} \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[Sqrt[d*x]*(a + b*x + c*x^2)^(5/2),x]","\frac{\sqrt{d x} \left(2 c \sqrt{x} \left(b c^2 \left(708 a^2+3071 a c x^2+1701 c^2 x^4\right)+77 c^3 x \left(31 a^2+28 a c x^2+9 c^2 x^4\right)+b^3 c \left(15 c x^2-241 a\right)+3 b^2 c^2 x \left(54 a+371 c x^2\right)+24 b^5-18 b^4 c x\right) (a+x (b+c x))-\frac{4 \left(-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left(-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right) \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{4 a}{x \left(\sqrt{b^2-4 a c}+b\right)}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+\frac{i x \left(924 a^3 c^3 \sqrt{b^2-4 a c}-1632 a^3 b c^3+1192 a^2 b^3 c^2-951 a^2 b^2 c^2 \sqrt{b^2-4 a c}-292 a b^5 c-24 b^6 \sqrt{b^2-4 a c}+268 a b^4 c \sqrt{b^2-4 a c}+24 b^7\right) \sqrt{\frac{4 a}{x \left(\sqrt{b^2-4 a c}+b\right)}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}\right)}{9009 c^4 \sqrt{x} \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{d x} \left(3 c x \left(308 a^2 c^2-181 a b^2 c+24 b^4\right)+b \left(108 a^2 c^2-151 a b^2 c+24 b^4\right)\right) \sqrt{a+b x+c x^2}}{9009 c^3}-\frac{4 d x \left(-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right) \sqrt{a+b x+c x^2}}{9009 c^{7/2} \sqrt{d x} \left(\sqrt{a}+\sqrt{c} x\right)}-\frac{\sqrt[4]{a} d \sqrt{x} \left(\sqrt{a} b \sqrt{c} \left(708 a^2 c^2-241 a b^2 c+24 b^4\right)+2 \left(-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right)\right) \left(\sqrt{a}+\sqrt{c} x\right) \sqrt{\frac{a+b x+c x^2}{\left(\sqrt{a}+\sqrt{c} x\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt[4]{a} d \sqrt{x} \left(-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right) \left(\sqrt{a}+\sqrt{c} x\right) \sqrt{\frac{a+b x+c x^2}{\left(\sqrt{a}+\sqrt{c} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt{x}}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}-\frac{10 \sqrt{d x} \left(14 c x \left(3 b^2-11 a c\right)+3 b \left(6 b^2-19 a c\right)\right) \left(a+b x+c x^2\right)^{3/2}}{9009 c^2}+\frac{2 (d x)^{3/2} \left(a+b x+c x^2\right)^{5/2}}{13 d}+\frac{10 b \sqrt{d x} \left(a+b x+c x^2\right)^{5/2}}{143 c}",1,"(Sqrt[d*x]*((-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(a + x*(b + c*x)))/Sqrt[x] + 2*c*Sqrt[x]*(a + x*(b + c*x))*(24*b^5 - 18*b^4*c*x + b^3*c*(-241*a + 15*c*x^2) + 3*b^2*c^2*x*(54*a + 371*c*x^2) + 77*c^3*x*(31*a^2 + 28*a*c*x^2 + 9*c^2*x^4) + b*c^2*(708*a^2 + 3071*a*c*x^2 + 1701*c^2*x^4)) + (I*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(24*b^7 - 292*a*b^5*c + 1192*a^2*b^3*c^2 - 1632*a^3*b*c^3 - 24*b^6*Sqrt[b^2 - 4*a*c] + 268*a*b^4*c*Sqrt[b^2 - 4*a*c] - 951*a^2*b^2*c^2*Sqrt[b^2 - 4*a*c] + 924*a^3*c^3*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/(9009*c^4*Sqrt[x]*Sqrt[a + x*(b + c*x)])","C",1
2455,1,10879,847,13.8702155,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/Sqrt[d + e*x],x]","\text{Result too large to show}","\frac{2 \sqrt{d+e x} \left(c x^2+b x+a\right)^{5/2}}{11 e}+\frac{10 \sqrt{d+e x} \left(16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right) \left(c x^2+b x+a\right)^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left(128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left(65 b^2 d^2-124 a b e d+60 a^2 e^2\right)-12 c e (2 c d-b e) \left(4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right) x\right) \sqrt{c x^2+b x+a}}{693 c^2 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left(128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left(33 b^2 d^2-124 a b e d+124 a^2 e^2\right)\right) \sqrt{d+e x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left(c d^2-b e d+a e^2\right) \left(128 c^4 d^4-4 c^3 e (64 b d-69 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left(41 b^2 d^2-92 a b e d+60 a^2 e^2\right)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{693 c^3 e^6 \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
2456,1,7946,716,13.7629108,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]","\text{Result too large to show}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(3 c^2 e^2 \left(28 a^2 e^2-76 a b d e+45 b^2 d^2\right)-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{63 c^2 e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \left(-4 c e (32 b d-33 a e)-b^2 e^2+128 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{63 c^2 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-3 c e x \left(-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right)-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right)}{63 c e^5}-\frac{10 \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left(a+b x+c x^2\right)^{5/2}}{e \sqrt{d+e x}}",1,"Result too large to show","C",0
2457,1,5407,622,13.4367124,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(5/2),x]","\text{Result too large to show}","\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{21 c e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right)}{21 e^5}+\frac{10 \left(a+b x+c x^2\right)^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left(a+b x+c x^2\right)^{5/2}}{3 e (d+e x)^{3/2}}",1,"Result too large to show","C",0
2458,1,8961,603,13.4392097,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{a+b x+c x^2} \left(-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right)}{15 e^5 \sqrt{d+e x}}+\frac{2 \left(a+b x+c x^2\right)^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^3 (d+e x)^{3/2}}-\frac{2 \left(a+b x+c x^2\right)^{5/2}}{5 e (d+e x)^{5/2}}",1,"Result too large to show","C",0
2459,1,5482,731,13.1977989,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^{9/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]","\text{Result too large to show}","\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{21 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{21 e^6 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 c \sqrt{a+b x+c x^2} \left(e x \left(-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right)-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right)}{21 e^5 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}-\frac{2 \left(a+b x+c x^2\right)^{3/2} \left(e x \left(-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right)-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right)}{21 e^3 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \left(a+b x+c x^2\right)^{5/2}}{7 e (d+e x)^{7/2}}",1,"Result too large to show","C",0
2460,1,8108,923,14.1987675,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x)^{11/2}} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]","\text{Result too large to show}","-\frac{2 \left(c x^2+b x+a\right)^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \left(16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left(26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right) x\right) \left(c x^2+b x+a\right)^{3/2}}{63 e^3 \left(c d^2-b e d+a e^2\right) (d+e x)^{7/2}}-\frac{2 \left(128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left(37 b^2 d^2-52 a b e d+12 a^2 e^2\right) d-2 a b^3 e^5-b c e^3 \left(b^2 d^2+9 a b e d-24 a^2 e^2\right)+e \left(160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left(57 b^2 d^2-92 a b e d+28 a^2 e^2\right)\right) x\right) \sqrt{c x^2+b x+a}}{63 e^5 \left(c d^2-b e d+a e^2\right)^2 (d+e x)^{3/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left(45 b^2 d^2-76 a b e d+28 a^2 e^2\right)\right) \sqrt{d+e x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{63 e^6 \left(c d^2-b e d+a e^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left(128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{63 e^6 \left(c d^2-b e d+a e^2\right) \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
2461,1,5340,600,13.3373042,"\int \frac{(d+e x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^(7/2)/Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right)}{105 c^3}+\frac{12 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}",1,"Result too large to show","C",1
2462,1,868,509,9.0751734,"\int \frac{(d+e x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^(5/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 \sqrt{c x^2+b x+a} \left(\left(23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right) \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)-\frac{i \sqrt{1-\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}+1} \left(\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(-30 c^3 d^3-c^2 \left(-34 a e^2-45 b d e+23 d \sqrt{\left(b^2-4 a c\right) e^2}\right) d+8 b^2 e^2 \left(b e-\sqrt{\left(b^2-4 a c\right) e^2}\right)+c e \left(-31 d e b^2-17 a e^2 b+23 d \sqrt{\left(b^2-4 a c\right) e^2} b+9 a e \sqrt{\left(b^2-4 a c\right) e^2}\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{d+e x}}\right) (d+e x)^{3/2}}{15 c^3 e \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}+\frac{\left(\frac{2 e^2 x}{5 c}-\frac{2 e (4 b e-11 c d)}{15 c^2}\right) \left(c x^2+b x+a\right) \sqrt{d+e x}}{\sqrt{a+x (b+c x)}}","-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^3 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{8 e \sqrt{d+e x} \sqrt{a+b x+c x^2} (2 c d-b e)}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{5 c}",1,"(Sqrt[d + e*x]*((-2*e*(-11*c*d + 4*b*e))/(15*c^2) + (2*e^2*x)/(5*c))*(a + b*x + c*x^2))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) - ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (-30*c^3*d^3 + 8*b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) - c^2*d*(-45*b*d*e - 34*a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 23*b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 9*a*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(15*c^3*e*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2463,1,609,439,5.4849796,"\int \frac{(d+e x)^{3/2}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[a + b*x + c*x^2],x]","\frac{2 \sqrt{d+e x} \left(c e^2 (a+x (b+c x))+(d+e x) \left(\frac{2 e^2 (a+x (b+c x)) (2 c d-b e)}{(d+e x)^2}+\frac{i \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+1} \left(\left(c \left(2 d \sqrt{e^2 \left(b^2-4 a c\right)}-a e^2-3 b d e\right)+b e \left(b e-\sqrt{e^2 \left(b^2-4 a c\right)}\right)+3 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+(b e-2 c d) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{\sqrt{2} \sqrt{d+e x} \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}\right)\right)}{3 c^2 e \sqrt{a+x (b+c x)}}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c}",1,"(2*Sqrt[d + e*x]*(c*e^2*(a + x*(b + c*x)) + (d + e*x)*((2*e^2*(2*c*d - b*e)*(a + x*(b + c*x)))/(d + e*x)^2 + (I*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (3*c^2*d^2 + b*e*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-3*b*d*e - a*e^2 + 2*d*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x]))))/(3*c^2*e*Sqrt[a + x*(b + c*x)])","C",1
2464,1,365,188,0.7704466,"\int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[a + b*x + c*x^2],x]","\frac{i \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} \sqrt{1-\frac{2 c (d+e x)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c d}} \sqrt{d+e x}\right)|\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c d}} \sqrt{d+e x}\right)|\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)\right)}{\sqrt{2} c e \sqrt{a+x (b+c x)} \sqrt{\frac{c}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(I*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]))/(Sqrt[2]*c*e*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + x*(b + c*x)])","C",1
2465,1,308,189,0.60799,"\int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{i (d+e x) \sqrt{2-\frac{4 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{e \sqrt{a+x (b+c x)} \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c \sqrt{d+e x} \sqrt{a+b x+c x^2}}",1,"(I*(d + e*x)*Sqrt[2 - (4*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/(e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + x*(b + c*x)])","C",1
2466,1,408,248,0.7531632,"\int \frac{1}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{-\frac{4 e^2 (a+x (b+c x))}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} \sqrt{1-\frac{2 c (d+e x)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c d}} \sqrt{d+e x}\right)|\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) e-2 c d}} \sqrt{d+e x}\right)|\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)\right)}{\sqrt{\frac{c}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}}}{2 e \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 e \sqrt{a+b x+c x^2}}{\sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}",1,"((-4*e^2*(a + x*(b + c*x)))/Sqrt[d + e*x] + (I*Sqrt[2]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]))/Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)])/(2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])","C",1
2467,1,643,523,6.0495893,"\int \frac{1}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \left(\frac{i (d+e x)^{5/2} \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+1} \left(\left(c \left(2 d \sqrt{e^2 \left(b^2-4 a c\right)}-a e^2-3 b d e\right)+b e \left(b e-\sqrt{e^2 \left(b^2-4 a c\right)}\right)+3 c^2 d^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+(b e-2 c d) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{\sqrt{2} \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}+2 e^2 (d+e x) (a+x (b+c x)) (2 c d-b e)+e^2 (a+x (b+c x)) (e (-a e+3 b d+2 b e x)-c d (5 d+4 e x))\right)}{3 e (d+e x)^{3/2} \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)^2}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{4 e \sqrt{a+b x+c x^2} (2 c d-b e)}{3 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 e \sqrt{a+b x+c x^2}}{3 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}",1,"(2*(2*e^2*(2*c*d - b*e)*(d + e*x)*(a + x*(b + c*x)) + e^2*(a + x*(b + c*x))*(-(c*d*(5*d + 4*e*x)) + e*(3*b*d - a*e + 2*b*e*x)) + (I*(d + e*x)^(5/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (3*c^2*d^2 + b*e*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-3*b*d*e - a*e^2 + 2*d*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(3*e*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)])","C",1
2468,1,983,629,8.6704578,"\int \frac{1}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{c x^2+b x+a} \left(\left(23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right) \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)-\frac{i \sqrt{1-\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}+1} \left(\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(-30 c^3 d^3-c^2 \left(-34 a e^2-45 b d e+23 d \sqrt{\left(b^2-4 a c\right) e^2}\right) d+8 b^2 e^2 \left(b e-\sqrt{\left(b^2-4 a c\right) e^2}\right)+c e \left(-31 d e b^2-17 a e^2 b+23 d \sqrt{\left(b^2-4 a c\right) e^2} b+9 a e \sqrt{\left(b^2-4 a c\right) e^2}\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{d+e x}}\right) (d+e x)^{3/2}}{15 e \left(c d^2-b e d+a e^2\right)^3 \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}+\frac{\left(c x^2+b x+a\right) \left(\frac{2 \left(-23 c^2 d^2+23 b c e d-8 b^2 e^2+9 a c e^2\right) e}{15 \left(c d^2-b e d+a e^2\right)^3 (d+e x)}+\frac{8 (b e-2 c d) e}{15 \left(c d^2-b e d+a e^2\right)^2 (d+e x)^2}-\frac{2 e}{5 \left(c d^2-b e d+a e^2\right) (d+e x)^3}\right) \sqrt{d+e x}}{\sqrt{a+x (b+c x)}}","-\frac{2 e \sqrt{a+b x+c x^2} \left(-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right)}{15 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}-\frac{8 e \sqrt{a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 e \sqrt{a+b x+c x^2}}{5 (d+e x)^{5/2} \left(a e^2-b d e+c d^2\right)}",1,"(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e)/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + (8*e*(-2*c*d + b*e))/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + (2*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2 + 9*a*c*e^2))/(15*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) - ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (-30*c^3*d^3 + 8*b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) - c^2*d*(-45*b*d*e - 34*a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 23*b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 9*a*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(15*e*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2469,1,5433,641,12.9193529,"\int \frac{(d+e x)^{7/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^(3/2),x]","\text{Result too large to show}","\frac{4 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right)}{3 c^2 \left(b^2-4 a c\right)}-\frac{4 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left(b^2-4 a c\right)}",1,"Result too large to show","C",1
2470,1,1323,533,12.3551475,"\int \frac{(d+e x)^{5/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]","\frac{(d+e x)^{3/2} \left(-4 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(d^2 \left(\frac{d}{d+e x}-1\right)^2 c^3-e \left(a e \left(\frac{2 d^2}{(d+e x)^2}-\frac{6 d}{d+e x}+3\right)+b d \left(\frac{2 d^2}{(d+e x)^2}-\frac{3 d}{d+e x}+1\right)\right) c^2+e^2 \left(\left(\frac{2 d^2}{(d+e x)^2}-\frac{3 d}{d+e x}+1\right) b^2+\frac{a e \left(\frac{2 d}{d+e x}-3\right) b}{d+e x}-\frac{3 a^2 e^2}{(d+e x)^2}\right) c+\frac{b^2 e^3 \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(c^2 d^2+b^2 e^2-c e (b d+3 a e)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-b^3 e^3+b^2 \left(2 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+b c \left(4 a e^2-d \sqrt{\left(b^2-4 a c\right) e^2}\right) e+c \left(c d^2 \sqrt{\left(b^2-4 a c\right) e^2}-a e^2 \left(8 c d+3 \sqrt{\left(b^2-4 a c\right) e^2}\right)\right)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right) \left(c x^2+b x+a\right)^{3/2}}{c^2 \left(4 a c-b^2\right) e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x))^{3/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}+\frac{2 \sqrt{d+e x} \left(b c d^2+2 c^2 x d^2-4 a c e d-2 b c e x d+a b e^2+b^2 e^2 x-2 a c e^2 x\right) \left(c x^2+b x+a\right)}{c \left(4 a c-b^2\right) (a+x (b+c x))^{3/2}}","-\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c^2 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c^2 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left(b^2-4 a c\right)}",1,"(2*Sqrt[d + e*x]*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + 2*c^2*d^2*x - 2*b*c*d*e*x + b^2*e^2*x - 2*a*c*e^2*x)*(a + b*x + c*x^2))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^(3/2)) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c^3*d^2*(-1 + d/(d + e*x))^2 + (b^2*e^3*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x) - c^2*e*(a*e*(3 + (2*d^2)/(d + e*x)^2 - (6*d)/(d + e*x)) + b*d*(1 + (2*d^2)/(d + e*x)^2 - (3*d)/(d + e*x))) + c*e^2*((-3*a^2*e^2)/(d + e*x)^2 + b^2*(1 + (2*d^2)/(d + e*x)^2 - (3*d)/(d + e*x)) + (a*b*e*(-3 + (2*d)/(d + e*x)))/(d + e*x))) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(4*a*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(c^2*(-b^2 + 4*a*c)*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2471,1,964,457,5.2702222,"\int \frac{(d+e x)^{3/2}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^(3/2),x]","\frac{\sqrt{d+e x} \left(\frac{4 (-b d-2 c x d+2 a e+b e x)}{b^2-4 a c}+\frac{(d+e x) \left(\frac{4 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x)) e^2}{(d+e x)^2}+\frac{i \sqrt{2} (2 c d-b e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(b^2 e^2-4 a c e^2-b \sqrt{\left(b^2-4 a c\right) e^2} e+2 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{c \left(4 a c-b^2\right) e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}\right)}{2 \sqrt{a+x (b+c x)}}","-\frac{4 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(Sqrt[d + e*x]*((4*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(b^2 - 4*a*c) + ((d + e*x)*((4*e^2*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))/(d + e*x)^2 + (I*Sqrt[2]*(2*c*d - b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(b^2*e^2 - 4*a*c*e^2 + 2*c*d*Sqrt[(b^2 - 4*a*c)*e^2] - b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(c*(-b^2 + 4*a*c)*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(2*Sqrt[a + x*(b + c*x)])","C",1
2472,1,996,426,8.4201001,"\int \frac{\sqrt{d+e x}}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^(3/2),x]","\frac{(d+e x)^{3/2} \left(c x^2+b x+a\right)^{3/2} \left(-4 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \sqrt{\left(b^2-4 a c\right) e^2} \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{\left(4 a c-b^2\right) e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x))^{3/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}-\frac{2 (b+2 c x) \sqrt{d+e x} \left(c x^2+b x+a\right)}{\left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}","-\frac{2 (b+2 c x) \sqrt{d+e x}}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(-2*(b + 2*c*x)*Sqrt[d + e*x]*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2)) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*Sqrt[(b^2 - 4*a*c)*e^2]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/((-b^2 + 4*a*c)*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2473,1,976,480,7.1530097,"\int \frac{1}{\sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2)),x]","\frac{\sqrt{d+e x} \left(-4 e b^2+4 c (d-e x) b+8 c (a e+c d x)-\frac{(d+e x) \left(-\frac{4 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x)) e^2}{(d+e x)^2}-\frac{i \sqrt{2} (2 c d-b e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left(b^2 e^2-4 a c e^2-b \sqrt{\left(b^2-4 a c\right) e^2} e+2 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 a e^2+2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (d-e x) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 a e^2-2 c d x e+\sqrt{\left(b^2-4 a c\right) e^2} x e+b (e x-d) e+d \sqrt{\left(b^2-4 a c\right) e^2}}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}\right)}{2 \left(b^2-4 a c\right) \left(e (b d-a e)-c d^2\right) \sqrt{a+x (b+c x)}}","-\frac{2 \sqrt{d+e x} \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{4 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}",1,"(Sqrt[d + e*x]*(-4*b^2*e + 8*c*(a*e + c*d*x) + 4*b*c*(d - e*x) - ((d + e*x)*((-4*e^2*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x)))/(d + e*x)^2 - (I*Sqrt[2]*(2*c*d - b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(b^2*e^2 - 4*a*c*e^2 + 2*c*d*Sqrt[(b^2 - 4*a*c)*e^2] - b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)])","C",1
2474,1,784,607,10.3914168,"\int \frac{1}{(d+e x)^{3/2} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]","\frac{-\frac{4 e (a+x (b+c x)) \left(c e (3 a e+b d)-b^2 e^2-c^2 d^2\right)}{\sqrt{d+e x}}-\frac{2 \left(e^3 \left(b^2-4 a c\right) (a+x (b+c x))+(d+e x) \left(b c \left(c d (d-2 e x)-3 a e^2\right)+2 c^2 \left(a e (2 d-e x)+c d^2 x\right)+b^3 e^2+b^2 c e (e x-2 d)\right)\right)}{\sqrt{d+e x}}-\frac{i (d+e x) \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{4 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+2} \left(\left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(c \left(a e^2 \left(3 \sqrt{e^2 \left(b^2-4 a c\right)}+8 c d\right)-c d^2 \sqrt{e^2 \left(b^2-4 a c\right)}\right)-b^2 e^2 \left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 c d\right)+b c e \left(d \sqrt{e^2 \left(b^2-4 a c\right)}-4 a e^2\right)+b^3 e^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{e \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)^2}","-\frac{4 e \sqrt{a+b x+c x^2} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\left(b^2-4 a c\right) \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"((-4*e*(-(c^2*d^2) - b^2*e^2 + c*e*(b*d + 3*a*e))*(a + x*(b + c*x)))/Sqrt[d + e*x] - (2*((b^2 - 4*a*c)*e^3*(a + x*(b + c*x)) + (d + e*x)*(b^3*e^2 + b^2*c*e*(-2*d + e*x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^2*(c*d^2*x + a*e*(2*d - e*x)))))/Sqrt[d + e*x] - (I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^3*e^3 - b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)])","C",1
2475,1,5565,744,13.0229955,"\int \frac{1}{(d+e x)^{5/2} \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]","\text{Result too large to show}","-\frac{4 e \sqrt{a+b x+c x^2} \left(-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right)}{3 \left(b^2-4 a c\right) (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)^2}-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left(-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right)}{3 \left(b^2-4 a c\right) \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^3}-\frac{4 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) (d+e x)^{3/2} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"Result too large to show","C",1
2476,1,5598,659,14.2963828,"\int \frac{(d+e x)^{7/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^(5/2),x]","\text{Result too large to show}","\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a e^2-b d e+c d^2\right) \left(-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 \left(b^2-4 a c\right)^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 \left(b^2-4 a c\right)^{3/2} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{d+e x} \left(x (2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right)-\left(b^2 \left(9 c d^2 e-a e^3\right)\right)+8 b c d \left(3 a e^2+c d^2\right)-4 a c e \left(5 a e^2+3 c d^2\right)\right)}{3 c \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"Result too large to show","C",1
2477,1,1460,590,12.8642374,"\int \frac{(d+e x)^{5/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(b c d^2+2 c^2 x d^2-4 a c e d-2 b c e x d+a b e^2+b^2 e^2 x-2 a c e^2 x\right)}{3 c \left(4 a c-b^2\right) \left(c x^2+b x+a\right)^2}+\frac{2 \left(e^2 b^3-9 c d e b^2+c e^2 x b^2+8 c^2 d^2 b+4 a c e^2 b-16 c^2 d e x b+4 a c^2 d e+16 c^3 d^2 x+12 a c^2 e^2 x\right)}{3 c \left(4 a c-b^2\right)^2 \left(c x^2+b x+a\right)}\right) \left(c x^2+b x+a\right)^3}{(a+x (b+c x))^{5/2}}+\frac{(d+e x)^{3/2} \left(-4 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(16 d^2 \left(\frac{d}{d+e x}-1\right)^2 c^3+4 e \left(a e \left(\frac{7 d^2}{(d+e x)^2}-\frac{6 d}{d+e x}+3\right)-4 b d \left(\frac{2 d^2}{(d+e x)^2}-\frac{3 d}{d+e x}+1\right)\right) c^2+e^2 \left(\left(\frac{17 d^2}{(d+e x)^2}-\frac{18 d}{d+e x}+1\right) b^2-\frac{4 a e \left(\frac{7 d}{d+e x}-3\right) b}{d+e x}+\frac{12 a^2 e^2}{(d+e x)^2}\right) c+\frac{b^2 e^3 \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(16 c^2 d^2+b^2 e^2+4 c e (3 a e-4 b d)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-b^3 e^3+b^2 \left(2 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+4 b \left(a c e^3-4 c d e \sqrt{\left(b^2-4 a c\right) e^2}\right)+4 c \left(4 c \sqrt{\left(b^2-4 a c\right) e^2} d^2+a e^2 \left(3 \sqrt{\left(b^2-4 a c\right) e^2}-2 c d\right)\right)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right) \left(c x^2+b x+a\right)^{5/2}}{6 c \left(4 a c-b^2\right)^2 e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x))^{5/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}","-\frac{2 \sqrt{d+e x} \left(-x \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right)-8 b \left(a e^2+c d^2\right)+4 a c d e+7 b^2 d e\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c \left(b^2-4 a c\right)^{3/2} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{16 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c \left(b^2-4 a c\right)^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((2*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + 2*c^2*d^2*x - 2*b*c*d*e*x + b^2*e^2*x - 2*a*c*e^2*x))/(3*c*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^2) + (2*(8*b*c^2*d^2 - 9*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 4*a*b*c*e^2 + 16*c^3*d^2*x - 16*b*c^2*d*e*x + b^2*c*e^2*x + 12*a*c^2*e^2*x))/(3*c*(-b^2 + 4*a*c)^2*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^3*d^2*(-1 + d/(d + e*x))^2 + (b^2*e^3*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x) + 4*c^2*e*(a*e*(3 + (7*d^2)/(d + e*x)^2 - (6*d)/(d + e*x)) - 4*b*d*(1 + (2*d^2)/(d + e*x)^2 - (3*d)/(d + e*x))) + c*e^2*((12*a^2*e^2)/(d + e*x)^2 + b^2*(1 + (17*d^2)/(d + e*x)^2 - (18*d)/(d + e*x)) - (4*a*b*e*(-3 + (7*d)/(d + e*x)))/(d + e*x))) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(6*c*(-b^2 + 4*a*c)^2*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(5/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2478,1,1141,542,7.3031442,"\int \frac{(d+e x)^{3/2}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(c x^2+b x+a\right)^3 \left(\frac{2 (-b d-2 c x d+2 a e+b e x)}{3 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^2}-\frac{2 \left(5 e b^2-8 c d b+8 c e x b-4 a c e-16 c^2 d x\right)}{3 \left(b^2-4 a c\right)^2 \left(c x^2+b x+a\right)}\right)}{(a+x (b+c x))^{5/2}}-\frac{(d+e x)^{3/2} \left(c x^2+b x+a\right)^{5/2} \left(-16 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{4 i \sqrt{2} (b e-2 c d) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-b^2 e^2+4 a c e^2+4 b \sqrt{\left(b^2-4 a c\right) e^2} e-8 c d \sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{3 \left(4 a c-b^2\right)^2 e \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x))^{5/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}","\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c \left(b^2-4 a c\right)^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{2 \sqrt{d+e x} \left(4 a c e-5 b^2 e+8 c x (2 c d-b e)+8 b c d\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{8 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (2*(-8*b*c*d + 5*b^2*e - 4*a*c*e - 16*c^2*d*x + 8*b*c*e*x))/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) - ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-16*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + ((4*I)*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^2*e^2) + 4*a*c*e^2 - 8*c*d*Sqrt[(b^2 - 4*a*c)*e^2] + 4*b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(3*(-b^2 + 4*a*c)^2*e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(5/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2479,1,1440,605,11.7956591,"\int \frac{\sqrt{d+e x}}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^(5/2),x]","\frac{\sqrt{d+e x} \left(\frac{2 \left(e^2 b^3-9 c d e b^2+c e^2 x b^2+8 c^2 d^2 b+4 a c e^2 b-16 c^2 d e x b+4 a c^2 d e+16 c^3 d^2 x+12 a c^2 e^2 x\right)}{3 \left(4 a c-b^2\right)^2 \left(c d^2-b e d+a e^2\right) \left(c x^2+b x+a\right)}-\frac{2 (b+2 c x)}{3 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^2}\right) \left(c x^2+b x+a\right)^3}{(a+x (b+c x))^{5/2}}+\frac{(d+e x)^{3/2} \left(-4 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(16 d^2 \left(\frac{d}{d+e x}-1\right)^2 c^3+4 e \left(a e \left(\frac{7 d^2}{(d+e x)^2}-\frac{6 d}{d+e x}+3\right)-4 b d \left(\frac{2 d^2}{(d+e x)^2}-\frac{3 d}{d+e x}+1\right)\right) c^2+e^2 \left(\left(\frac{17 d^2}{(d+e x)^2}-\frac{18 d}{d+e x}+1\right) b^2-\frac{4 a e \left(\frac{7 d}{d+e x}-3\right) b}{d+e x}+\frac{12 a^2 e^2}{(d+e x)^2}\right) c+\frac{b^2 e^3 \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(16 c^2 d^2+b^2 e^2+4 c e (3 a e-4 b d)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-b^3 e^3+b^2 \left(2 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+4 b \left(a c e^3-4 c d e \sqrt{\left(b^2-4 a c\right) e^2}\right)+4 c \left(4 c \sqrt{\left(b^2-4 a c\right) e^2} d^2+a e^2 \left(3 \sqrt{\left(b^2-4 a c\right) e^2}-2 c d\right)\right)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right) \left(c x^2+b x+a\right)^{5/2}}{6 \left(4 a c-b^2\right)^2 e \left(c d^2-b e d+a e^2\right) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (a+x (b+c x))^{5/2} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}","-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 (b+2 c x) \sqrt{d+e x}}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}+\frac{16 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \left(-c x \left(-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right)-4 b c \left(a e^2+2 c d^2\right)-4 a c^2 d e+b^3 \left(-e^2\right)+9 b^2 c d e\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((-2*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (2*(8*b*c^2*d^2 - 9*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 4*a*b*c*e^2 + 16*c^3*d^2*x - 16*b*c^2*d*e*x + b^2*c*e^2*x + 12*a*c^2*e^2*x))/(3*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^3*d^2*(-1 + d/(d + e*x))^2 + (b^2*e^3*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x) + 4*c^2*e*(a*e*(3 + (7*d^2)/(d + e*x)^2 - (6*d)/(d + e*x)) - 4*b*d*(1 + (2*d^2)/(d + e*x)^2 - (3*d)/(d + e*x))) + c*e^2*((12*a^2*e^2)/(d + e*x)^2 + b^2*(1 + (17*d^2)/(d + e*x)^2 - (18*d)/(d + e*x)) - (4*a*b*e*(-3 + (7*d)/(d + e*x)))/(d + e*x))) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(6*(-b^2 + 4*a*c)^2*e*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a + x*(b + c*x))^(5/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
2480,1,5566,725,12.8743732,"\int \frac{1}{\sqrt{d+e x} \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2)),x]","\text{Result too large to show}","-\frac{2 \sqrt{d+e x} \left(-2 c x (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right)-\left(2 a c e+b^2 (-e)+b c d\right) \left(-5 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right)+3 a c e (2 c d-b e)^2\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \left(-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{d+e x} \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2} \left(a e^2-b d e+c d^2\right)}",1,"Result too large to show","C",1
2481,1,7870,918,13.841944,"\int \frac{1}{(d+e x)^{3/2} \left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]","\text{Result too large to show}","-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) \left(16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left(3 b^2 d^2-20 a b e d-28 a^2 e^2\right)\right)}{3 \left(b^2-4 a c\right)^{3/2} \left(c d^2-b e d+a e^2\right)^3 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{2 e \sqrt{c x^2+b x+a} \left(16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left(3 b^2 d^2-20 a b e d-28 a^2 e^2\right)\right)}{3 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^3 \sqrt{d+e x}}+\frac{8 \sqrt{2} (2 c d-b e) \left(2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 \left(b^2-4 a c\right)^{3/2} \left(c d^2-b e d+a e^2\right)^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left(5 a c e (2 c d-b e)^2-4 c \left(2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right) x (2 c d-b e)-\left(-e b^2+c d b+2 a c e\right) \left(8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right)\right)}{3 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right)}{3 \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt{d+e x} \left(c x^2+b x+a\right)^{3/2}}",1,"Result too large to show","C",0
2482,1,86,30,0.0763034,"\int \frac{\sqrt{3+5 x}}{\sqrt{2+5 x-12 x^2}} \, dx","Integrate[Sqrt[3 + 5*x]/Sqrt[2 + 5*x - 12*x^2],x]","\frac{\sqrt{19} \sqrt{-4 x-1} \sqrt{2-3 x} \left(F\left(\sin ^{-1}\left(\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right)|\frac{21}{76}\right)-E\left(\sin ^{-1}\left(\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right)|\frac{21}{76}\right)\right)}{3 \sqrt{-12 x^2+5 x+2}}","-\frac{1}{3} \sqrt{19} E\left(\sin ^{-1}\left(\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right)|\frac{55}{76}\right)",1,"(Sqrt[19]*Sqrt[-1 - 4*x]*Sqrt[2 - 3*x]*(-EllipticE[ArcSin[(2*Sqrt[3 + 5*x])/Sqrt[7]], 21/76] + EllipticF[ArcSin[(2*Sqrt[3 + 5*x])/Sqrt[7]], 21/76]))/(3*Sqrt[2 + 5*x - 12*x^2])","B",1
2483,1,164,638,0.3531559,"\int (d+e x)^2 \left(a+b x+c x^2\right)^{4/3} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(4/3),x]","\frac{3 (a+x (b+c x))^{4/3} \left(\frac{14 \sqrt[3]{2} (b+2 c x) \left(-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{3 c^2 \left(-\frac{c (a+x (b+c x))}{b^2-4 a c}\right)^{4/3}}-\frac{160 e (a+x (b+c x)) (b e-2 c d)}{c}+224 e (d+e x) (a+x (b+c x))\right)}{3808 c}","\frac{\sqrt[3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left(b^2-4 a c\right)^2 \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right) \sqrt{\frac{-2^{2/3} \sqrt[3]{c} \sqrt[3]{b^2-4 a c} \sqrt[3]{a+b x+c x^2}+\left(b^2-4 a c\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} \left(a+b x+c x^2\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}} \left(-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{935 c^{13/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt[3]{a+b x+c x^2} \left(-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right)}{935 c^4}+\frac{3 (b+2 c x) \left(a+b x+c x^2\right)^{4/3} \left(-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right)}{374 c^3}+\frac{15 e \left(a+b x+c x^2\right)^{7/3} (2 c d-b e)}{119 c^2}+\frac{3 e (d+e x) \left(a+b x+c x^2\right)^{7/3}}{17 c}",1,"(3*(a + x*(b + c*x))^(4/3)*((-160*e*(-2*c*d + b*e)*(a + x*(b + c*x)))/c + 224*e*(d + e*x)*(a + x*(b + c*x)) + (14*2^(1/3)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*Hypergeometric2F1[-4/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c^2*(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(4/3))))/(3808*c)","C",1
2484,1,113,539,0.1992793,"\int (d+e x) \left(a+b x+c x^2\right)^{4/3} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(4/3),x]","\frac{(a+x (b+c x))^{4/3} \left(48 c^2 e (a+x (b+c x))-\frac{7 \sqrt[3]{2} c (b+2 c x) (b e-2 c d) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{\left(-\frac{c (a+x (b+c x))}{b^2-4 a c}\right)^{4/3}}\right)}{224 c^3}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \left(b^2-4 a c\right)^2 \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right) \sqrt{\frac{-2^{2/3} \sqrt[3]{c} \sqrt[3]{b^2-4 a c} \sqrt[3]{a+b x+c x^2}+\left(b^2-4 a c\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} \left(a+b x+c x^2\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}} (2 c d-b e) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt[3]{a+b x+c x^2} (2 c d-b e)}{110 c^3}+\frac{3 (b+2 c x) \left(a+b x+c x^2\right)^{4/3} (2 c d-b e)}{44 c^2}+\frac{3 e \left(a+b x+c x^2\right)^{7/3}}{14 c}",1,"((a + x*(b + c*x))^(4/3)*(48*c^2*e*(a + x*(b + c*x)) - (7*2^(1/3)*c*(-2*c*d + b*e)*(b + 2*c*x)*Hypergeometric2F1[-4/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(4/3)))/(224*c^3)","C",1
2485,1,95,490,0.0535874,"\int \left(a+b x+c x^2\right)^{4/3} \, dx","Integrate[(a + b*x + c*x^2)^(4/3),x]","-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt[3]{a+x (b+c x)} \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8\ 2^{2/3} c^2 \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}","\frac{\sqrt[3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left(b^2-4 a c\right)^2 \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right) \sqrt{\frac{-2^{2/3} \sqrt[3]{c} \sqrt[3]{b^2-4 a c} \sqrt[3]{a+b x+c x^2}+\left(b^2-4 a c\right)^{2/3}+2 \sqrt[3]{2} c^{2/3} \left(a+b x+c x^2\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{55 c^{7/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{a+b x+c x^2}\right)^2}}}-\frac{3 \left(b^2-4 a c\right) (b+2 c x) \sqrt[3]{a+b x+c x^2}}{55 c^2}+\frac{3 (b+2 c x) \left(a+b x+c x^2\right)^{4/3}}{22 c}",1,"-1/8*((b^2 - 4*a*c)*(b + 2*c*x)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2^(2/3)*c^2*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))","C",1
2486,0,0,180,1.2168936,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x),x]","\int \frac{\left(a+b x+c x^2\right)^{4/3}}{d+e x} \, dx","\frac{3 \left(a+b x+c x^2\right)^{4/3} F_1\left(-\frac{8}{3};-\frac{4}{3},-\frac{4}{3};-\frac{5}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{\sqrt[3]{2} e \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3}}",1,"Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x), x]","F",-1
2487,0,0,189,0.8062422,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x)^2,x]","\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(d+e x)^2} \, dx","\frac{12\ 2^{2/3} \left(a+b x+c x^2\right)^{4/3} F_1\left(-\frac{5}{3};-\frac{4}{3},-\frac{4}{3};-\frac{2}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{5 e (d+e x) \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3}}",1,"Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x)^2, x]","F",-1
2488,0,0,187,0.8295996,"\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x)^3,x]","\int \frac{\left(a+b x+c x^2\right)^{4/3}}{(d+e x)^3} \, dx","\frac{6\ 2^{2/3} \left(a+b x+c x^2\right)^{4/3} F_1\left(-\frac{2}{3};-\frac{4}{3},-\frac{4}{3};\frac{1}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{e (d+e x)^2 \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{4/3}}",1,"Integrate[(a + b*x + c*x^2)^(4/3)/(d + e*x)^3, x]","F",-1
2489,1,446,1224,1.1736666,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(7/3),x]","\frac{3 c \left(b^2 \left(a^2 e^3+a c e \left(-9 d^2+42 d e x-11 e^2 x^2\right)+c^2 d x \left(8 d^2-45 d e x+6 e^2 x^2\right)\right)+2 b c \left(a^2 e^2 (15 d-19 e x)+a c \left(7 d^3-21 d^2 e x+27 d e^2 x^2-9 e^3 x^3\right)+15 c^2 d^2 x^2 (d-e x)\right)+4 c \left(-6 a^3 e^3+a^2 c e \left(-6 d^2+3 d e x-8 e^2 x^2\right)+a c^2 d x \left(7 d^2+9 e^2 x^2\right)+5 c^3 d^3 x^3\right)+b^3 \left(2 a e^3 x-c \left(d^3+12 d^2 e x-9 d e^2 x^2-2 e^3 x^3\right)\right)+b^4 e^3 x^2\right)-2^{2/3} (b+2 c x) \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right) \left(9 a^2 c e^2+a \left(-b^2 e^2+b c e (9 e x-5 d)+c^2 \left(5 d^2+9 e^2 x^2\right)\right)+x (b+c x) \left(-b^2 e^2-5 b c d e+5 c^2 d^2\right)\right)}{4 c^2 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{4/3}}","-\frac{3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{4 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^{4/3}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (2 c d-b e) \left(5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt[3]{2} c^{5/3} \left(b^2-4 a c\right)^{5/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}}}-\frac{3^{3/4} (2 c d-b e) \left(5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{2^{5/6} c^{5/3} \left(b^2-4 a c\right)^{5/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}}}+\frac{3 \left(-\left(\left(11 c d^2 e-a e^3\right) b^2\right)+10 c d \left(c d^2+3 a e^2\right) b-8 a c e \left(2 c d^2+3 a e^2\right)+(2 c d-b e) \left(10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right) x\right)}{4 c \left(b^2-4 a c\right)^2 \sqrt[3]{c x^2+b x+a}}-\frac{3 (2 c d-b e) \left(5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right) (b+2 c x)}{2 \sqrt[3]{2} c^{5/3} \left(b^2-4 a c\right)^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}",1,"(3*c*(b^4*e^3*x^2 + b^2*(a^2*e^3 + a*c*e*(-9*d^2 + 42*d*e*x - 11*e^2*x^2) + c^2*d*x*(8*d^2 - 45*d*e*x + 6*e^2*x^2)) + 4*c*(-6*a^3*e^3 + 5*c^3*d^3*x^3 + a^2*c*e*(-6*d^2 + 3*d*e*x - 8*e^2*x^2) + a*c^2*d*x*(7*d^2 + 9*e^2*x^2)) + 2*b*c*(a^2*e^2*(15*d - 19*e*x) + 15*c^2*d^2*x^2*(d - e*x) + a*c*(7*d^3 - 21*d^2*e*x + 27*d*e^2*x^2 - 9*e^3*x^3)) + b^3*(2*a*e^3*x - c*(d^3 + 12*d^2*e*x - 9*d*e^2*x^2 - 2*e^3*x^3))) - 2^(2/3)*(2*c*d - b*e)*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*(9*a^2*c*e^2 + (5*c^2*d^2 - 5*b*c*d*e - b^2*e^2)*x*(b + c*x) + a*(-(b^2*e^2) + b*c*e*(-5*d + 9*e*x) + c^2*(5*d^2 + 9*e^2*x^2)))*Hypergeometric2F1[1/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))","C",1
2490,1,342,1153,0.7857661,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(7/3),x]","\frac{3 c \left(2 b \left(5 a^2 e^2+a c \left(7 d^2-14 d e x+9 e^2 x^2\right)+5 c^2 d x^2 (3 d-2 e x)\right)+4 c \left(a^2 e (e x-4 d)+a c x \left(7 d^2+3 e^2 x^2\right)+5 c^2 d^2 x^3\right)+b^2 \left(2 a e (7 e x-3 d)+2 c x \left(4 d^2-15 d e x+e^2 x^2\right)\right)-\left(b^3 \left(d^2+8 d e x-3 e^2 x^2\right)\right)\right)-2^{2/3} (b+2 c x) \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right) \left(6 a^2 c e^2+a \left(b^2 e^2+2 b c e (3 e x-5 d)+2 c^2 \left(5 d^2+3 e^2 x^2\right)\right)+x (b+c x) \left(b^2 e^2-10 b c d e+10 c^2 d^2\right)\right)}{4 c \left(b^2-4 a c\right)^2 (a+x (b+c x))^{4/3}}","-\frac{3 \left(10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right) (b+2 c x)}{2 \sqrt[3]{2} c^{2/3} \left(b^2-4 a c\right)^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}-\frac{3 \left(4 d e b^2-5 \left(c d^2+a e^2\right) b+4 a c d e-\left(10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right) x\right)}{2 \left(b^2-4 a c\right)^2 \sqrt[3]{c x^2+b x+a}}-\frac{3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^{4/3}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt[3]{2} c^{2/3} \left(b^2-4 a c\right)^{5/3} \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} (b+2 c x)}-\frac{3^{3/4} \left(10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{2^{5/6} c^{2/3} \left(b^2-4 a c\right)^{5/3} \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} (b+2 c x)}",1,"(3*c*(-(b^3*(d^2 + 8*d*e*x - 3*e^2*x^2)) + b^2*(2*a*e*(-3*d + 7*e*x) + 2*c*x*(4*d^2 - 15*d*e*x + e^2*x^2)) + 4*c*(5*c^2*d^2*x^3 + a^2*e*(-4*d + e*x) + a*c*x*(7*d^2 + 3*e^2*x^2)) + 2*b*(5*a^2*e^2 + 5*c^2*d*x^2*(3*d - 2*e*x) + a*c*(7*d^2 - 14*d*e*x + 9*e^2*x^2))) - 2^(2/3)*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*(6*a^2*c*e^2 + (10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*x*(b + c*x) + a*(b^2*e^2 + 2*b*c*e*(-5*d + 3*e*x) + 2*c^2*(5*d^2 + 3*e^2*x^2)))*Hypergeometric2F1[1/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))","C",1
2491,1,200,1043,0.5150325,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(7/3),x]","\frac{3 \left(5\ 2^{2/3} \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \sqrt[3]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)+\frac{4 \left(b^2-4 a c\right) (2 a e-b d+b e x-2 c d x)}{a+x (b+c x)}+20 (b+2 c x) (2 c d-b e)\right)}{16 \left(b^2-4 a c\right)^2 \sqrt[3]{a+x (b+c x)}}","\frac{15 (2 c d-b e) (b+2 c x)}{4 \left(b^2-4 a c\right)^2 \sqrt[3]{c x^2+b x+a}}-\frac{15 \sqrt[3]{c} (2 c d-b e) (b+2 c x)}{2 \sqrt[3]{2} \left(b^2-4 a c\right)^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}-\frac{3 (b d-2 a e+(2 c d-b e) x)}{4 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^{4/3}}+\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} (2 c d-b e) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt[3]{2} \left(b^2-4 a c\right)^{5/3} \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} (b+2 c x)}-\frac{5\ 3^{3/4} \sqrt[3]{c} (2 c d-b e) \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right)}{2^{5/6} \left(b^2-4 a c\right)^{5/3} \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} (b+2 c x)}",1,"(3*(20*(2*c*d - b*e)*(b + 2*c*x) + (4*(b^2 - 4*a*c)*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x)) + 5*2^(2/3)*(-2*c*d + b*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(16*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(1/3))","C",1
2492,1,138,993,0.1907589,"\int \frac{1}{\left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[(a + b*x + c*x^2)^(-7/3),x]","\frac{(b+2 c x) \left(3 \sqrt[3]{2} \left(2 c \left(7 a+5 c x^2\right)-b^2+10 b c x\right)-20 c (a+x (b+c x)) \sqrt[3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{4 \sqrt[3]{2} \left(b^2-4 a c\right)^2 (a+x (b+c x))^{4/3}}","\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right) c^{4/3}}{2 \sqrt[3]{2} \left(b^2-4 a c\right)^{5/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}}}-\frac{5 \sqrt[6]{2} 3^{3/4} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right) \sqrt{\frac{\left(b^2-4 a c\right)^{2/3}-2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a} \sqrt[3]{b^2-4 a c}+2 \sqrt[3]{2} c^{2/3} \left(c x^2+b x+a\right)^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}}\right)|-7-4 \sqrt{3}\right) c^{4/3}}{\left(b^2-4 a c\right)^{5/3} (b+2 c x) \sqrt{\frac{\sqrt[3]{b^2-4 a c} \left(\sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)^2}}}-\frac{15 (b+2 c x) c^{4/3}}{\sqrt[3]{2} \left(b^2-4 a c\right)^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{b^2-4 a c}+2^{2/3} \sqrt[3]{c} \sqrt[3]{c x^2+b x+a}\right)}+\frac{15 (b+2 c x) c}{2 \left(b^2-4 a c\right)^2 \sqrt[3]{c x^2+b x+a}}-\frac{3 (b+2 c x)}{4 \left(b^2-4 a c\right) \left(c x^2+b x+a\right)^{4/3}}",1,"((b + 2*c*x)*(3*2^(1/3)*(-b^2 + 10*b*c*x + 2*c*(7*a + 5*c*x^2)) - 20*c*(a + x*(b + c*x))*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(4*2^(1/3)*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))","C",1
2493,1,180,182,0.6137382,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(7/3)),x]","-\frac{3 \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} F_1\left(\frac{14}{3};\frac{7}{3},\frac{7}{3};\frac{17}{3};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{224\ 2^{2/3} e (a+x (b+c x))^{7/3}}","-\frac{3 \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} F_1\left(\frac{14}{3};\frac{7}{3},\frac{7}{3};\frac{17}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{224\ 2^{2/3} e \left(a+b x+c x^2\right)^{7/3}}",1,"(-3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(7/3)*AppellF1[14/3, 7/3, 7/3, 17/3, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(224*2^(2/3)*e*(a + x*(b + c*x))^(7/3))","A",0
2494,1,190,189,0.9708023,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(7/3)),x]","-\frac{3 e^3 \sqrt[3]{\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} \sqrt[3]{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} F_1\left(\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{17\ 2^{2/3} c^2 (d+e x)^5 \sqrt[3]{a+x (b+c x)}}","-\frac{3 \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} F_1\left(\frac{17}{3};\frac{7}{3},\frac{7}{3};\frac{20}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{272\ 2^{2/3} e (d+e x) \left(a+b x+c x^2\right)^{7/3}}",1,"(-3*e^3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*AppellF1[17/3, 7/3, 7/3, 20/3, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(17*2^(2/3)*c^2*(d + e*x)^5*(a + x*(b + c*x))^(1/3))","A",0
2495,1,190,189,1.8523062,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)^{7/3}} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(7/3)),x]","-\frac{3 e^3 \sqrt[3]{\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} \sqrt[3]{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} F_1\left(\frac{20}{3};\frac{7}{3},\frac{7}{3};\frac{23}{3};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{20\ 2^{2/3} c^2 (d+e x)^6 \sqrt[3]{a+x (b+c x)}}","-\frac{3 \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{7/3} F_1\left(\frac{20}{3};\frac{7}{3},\frac{7}{3};\frac{23}{3};\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{320\ 2^{2/3} e (d+e x)^2 \left(a+b x+c x^2\right)^{7/3}}",1,"(-3*e^3*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/3)*AppellF1[20/3, 7/3, 7/3, 23/3, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(20*2^(2/3)*c^2*(d + e*x)^6*(a + x*(b + c*x))^(1/3))","A",0
2496,1,317,242,0.4769044,"\int \frac{1}{(d+e x) \sqrt[3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx","Integrate[1/((d + e*x)*(c^2*d^2 - b*c*d*e + b^2*e^2 + 3*b*c*e^2*x + 3*c^2*e^2*x^2)^(1/3)),x]","-\frac{\sqrt[3]{3} \sqrt[3]{\frac{-\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} \sqrt[3]{\frac{\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{-6 d e c^2+3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)},\frac{6 d e c^2-3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)}\right)}{2\ 2^{2/3} e \sqrt[3]{b^2 e^2+b c e (3 e x-d)+c^2 \left(d^2+3 e^2 x^2\right)}}","-\frac{\tan ^{-1}\left(\frac{2 (-b e+c d-c e x)}{\sqrt{3} \sqrt[3]{2 c d-b e} \sqrt[3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac{1}{\sqrt{3}}\right)}{\sqrt{3} e (2 c d-b e)^{2/3}}+\frac{\log \left(-3 c e^2 \sqrt[3]{2 c d-b e} \sqrt[3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right)}{2 e (2 c d-b e)^{2/3}}-\frac{\log (d+e x)}{2 e (2 c d-b e)^{2/3}}",1,"-1/2*(3^(1/3)*((3*b*c*e^2 - Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*((3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -1/6*(-6*c^2*d*e + 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(c^2*e*(d + e*x)), (6*c^2*d*e - 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(6*c^2*e*(d + e*x))])/(2^(2/3)*e*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3))","C",0
2497,1,59,635,0.0404713,"\int \frac{(2+3 x)^3}{\sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)^3/(52 - 54*x + 27*x^2)^(1/3),x]","\frac{1}{210} \left(3000 \sqrt[3]{5} (x-1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27}{25} (x-1)^2\right)+\left(27 x^2-54 x+52\right)^{2/3} \left(63 x^2+324 x+838\right)\right)","\frac{1}{30} \left(27 x^2-54 x+52\right)^{2/3} (3 x+2)^2+\frac{1}{7} (8 x+27) \left(27 x^2-54 x+52\right)^{2/3}+\frac{9000 \sqrt[3]{5} (1-x)}{7 \left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)}+\frac{50\ 5^{5/6} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} F\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{189\ 3^{3/4} (1-x) \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}}}-\frac{25\ 5^{5/6} \sqrt{2+\sqrt{3}} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} E\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{189 \sqrt{2} \sqrt[4]{3} (1-x) \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}}}",1,"((52 - 54*x + 27*x^2)^(2/3)*(838 + 324*x + 63*x^2) + 3000*5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25])/210","C",1
2498,1,54,628,0.0243559,"\int \frac{(2+3 x)^2}{\sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)^2/(52 - 54*x + 27*x^2)^(1/3),x]","\frac{1}{42} \left(180 \sqrt[3]{5} (x-1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27}{25} (x-1)^2\right)+\left(27 x^2-54 x+52\right)^{2/3} (6 x+29)\right)","\frac{1}{21} (3 x+2) \left(27 x^2-54 x+52\right)^{2/3}+\frac{25}{42} \left(27 x^2-54 x+52\right)^{2/3}+\frac{2700 \sqrt[3]{5} (1-x)}{7 \left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)}+\frac{5\ 5^{5/6} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} F\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{63\ 3^{3/4} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}-\frac{5\ 5^{5/6} \sqrt{2+\sqrt{3}} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} E\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{126 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}",1,"((29 + 6*x)*(52 - 54*x + 27*x^2)^(2/3) + 180*5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25])/42","C",1
2499,1,47,603,0.0113135,"\int \frac{2+3 x}{\sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)/(52 - 54*x + 27*x^2)^(1/3),x]","\sqrt[3]{5} (x-1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{27}{25} (x-1)^2\right)+\frac{1}{12} \left(27 x^2-54 x+52\right)^{2/3}","\frac{1}{12} \left(27 x^2-54 x+52\right)^{2/3}+\frac{90 \sqrt[3]{5} (1-x)}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}+\frac{5^{5/6} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} F\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{54\ 3^{3/4} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}-\frac{5^{5/6} \sqrt{2+\sqrt{3}} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} E\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{108 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}",1,"(52 - 54*x + 27*x^2)^(2/3)/12 + 5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25]","C",1
2500,1,126,108,0.0734556,"\int \frac{1}{(2+3 x) \sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{9 x-5 i \sqrt{3}-9}{3 x+2}} \sqrt[3]{\frac{9 x+5 i \sqrt{3}-9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right)}{2\ 3^{2/3} \sqrt[3]{27 x^2-54 x+52}}","\frac{\log \left(-27 \sqrt[3]{10} \sqrt[3]{27 x^2-54 x+52}-81 x+216\right)}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt[3]{5} \sqrt[3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right)}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}}",1,"-1/2*(((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)])/(3^(2/3)*(52 - 54*x + 27*x^2)^(1/3))","C",0
2501,1,243,719,0.2887111,"\int \frac{1}{(2+3 x)^2 \sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(1/3)),x]","\frac{-100 \sqrt[3]{3} (3 x+2) \sqrt[3]{\frac{9 x-5 i \sqrt{3}-9}{3 x+2}} \sqrt[3]{\frac{9 x+5 i \sqrt{3}-9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right)+\sqrt[3]{3} 10^{2/3} \sqrt[3]{-9 i x+5 \sqrt{3}+9 i} (3 x+2) \left(3 \sqrt{3} x-3 \sqrt{3}-5 i\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right)-20 \left(27 x^2-54 x+52\right)}{6000 (3 x+2) \sqrt[3]{27 x^2-54 x+52}}","-\frac{\left(27 x^2-54 x+52\right)^{2/3}}{300 (3 x+2)}+\frac{\log \left(-27 \sqrt[3]{10} \sqrt[3]{27 x^2-54 x+52}-81 x+216\right)}{60\ 10^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt[3]{5} \sqrt[3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right)}{30 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{10\ 5^{2/3} \left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)}-\frac{\log (3 x+2)}{60\ 10^{2/3}}+\frac{\left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} F\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{5400\ 3^{3/4} \sqrt[6]{5} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} E\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{10800 \sqrt{2} \sqrt[4]{3} \sqrt[6]{5} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}",1,"(-20*(52 - 54*x + 27*x^2) - 100*3^(1/3)*(2 + 3*x)*((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)] + 3^(1/3)*10^(2/3)*(9*I + 5*Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)*(-5*I - 3*Sqrt[3] + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (-9*I + 5*Sqrt[3] + (9*I)*x)/(10*Sqrt[3])])/(6000*(2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3))","C",0
2502,1,233,744,0.291637,"\int \frac{1}{(2+3 x)^3 \sqrt[3]{52-54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(1/3)),x]","\frac{-150 \sqrt[3]{3} \sqrt[3]{\frac{9 x-5 i \sqrt{3}-9}{3 x+2}} \sqrt[3]{\frac{9 x+5 i \sqrt{3}-9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right)+3^{5/6} 10^{2/3} \sqrt[3]{-9 i x+5 \sqrt{3}+9 i} \left(9 x-5 i \sqrt{3}-9\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right)-\frac{90 (2 x+3) \left(27 x^2-54 x+52\right)}{(3 x+2)^2}}{90000 \sqrt[3]{27 x^2-54 x+52}}","-\frac{\left(27 x^2-54 x+52\right)^{2/3}}{1500 (3 x+2)}-\frac{\left(27 x^2-54 x+52\right)^{2/3}}{600 (3 x+2)^2}+\frac{\log \left(-27 \sqrt[3]{10} \sqrt[3]{27 x^2-54 x+52}-81 x+216\right)}{600\ 10^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt[3]{5} \sqrt[3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right)}{300 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{50\ 5^{2/3} \left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)}-\frac{\log (3 x+2)}{600\ 10^{2/3}}+\frac{\left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} F\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{27000\ 3^{3/4} \sqrt[6]{5} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left(30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right) \sqrt{\frac{10^{2/3} \left((54 x-54)^2+2700\right)^{2/3}+30 \sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}+900}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} E\left(\sin ^{-1}\left(\frac{30 \left(1+\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}\right)|-7+4 \sqrt{3}\right)}{54000 \sqrt{2} \sqrt[4]{3} \sqrt[6]{5} \sqrt{-\frac{30-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}}{\left(30 \left(1-\sqrt{3}\right)-\sqrt[3]{10} \sqrt[3]{(54 x-54)^2+2700}\right)^2}} (1-x)}",1,"((-90*(3 + 2*x)*(52 - 54*x + 27*x^2))/(2 + 3*x)^2 - 150*3^(1/3)*((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)] + 3^(5/6)*10^(2/3)*(9*I + 5*Sqrt[3] - (9*I)*x)^(1/3)*(-9 - (5*I)*Sqrt[3] + 9*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (-9*I + 5*Sqrt[3] + (9*I)*x)/(10*Sqrt[3])])/(90000*(52 - 54*x + 27*x^2)^(1/3))","C",0
2503,1,53,589,0.0291519,"\int \frac{(2+3 x)^3}{\sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)^3/(28 + 54*x + 27*x^2)^(1/3),x]","\frac{1}{210} \left(27 x^2+54 x+28\right)^{2/3} \left(63 x^2+36 x+22\right)-\frac{4}{7} (x+1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-27 (x+1)^2\right)","\frac{1}{30} \left(27 x^2+54 x+28\right)^{2/3} (3 x+2)^2-\frac{1}{35} (8 x+1) \left(27 x^2+54 x+28\right)^{2/3}+\frac{4 \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} F\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{63\ 3^{3/4} (x+1) \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}}}-\frac{\sqrt{2 \left(2+\sqrt{3}\right)} \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} E\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{63 \sqrt[4]{3} (x+1) \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}}}+\frac{72 (x+1)}{7 \left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)}",1,"((28 + 54*x + 27*x^2)^(2/3)*(22 + 36*x + 63*x^2))/210 - (4*(1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2])/7","C",1
2504,1,47,585,0.0186569,"\int \frac{(2+3 x)^2}{\sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)^2/(28 + 54*x + 27*x^2)^(1/3),x]","\frac{1}{42} \left(36 (x+1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-27 (x+1)^2\right)+\left(27 x^2+54 x+28\right)^{2/3} (6 x-1)\right)","\frac{1}{21} (3 x+2) \left(27 x^2+54 x+28\right)^{2/3}-\frac{5}{42} \left(27 x^2+54 x+28\right)^{2/3}-\frac{2 \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} F\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{21\ 3^{3/4} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} E\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{21 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}-\frac{108 (x+1)}{7 \left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)}",1,"((-1 + 6*x)*(28 + 54*x + 27*x^2)^(2/3) + 36*(1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2])/42","C",1
2505,1,41,560,0.0072364,"\int \frac{2+3 x}{\sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[(2 + 3*x)/(28 + 54*x + 27*x^2)^(1/3),x]","\frac{1}{12} \left(27 x^2+54 x+28\right)^{2/3}-(x+1) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-27 (x+1)^2\right)","\frac{1}{12} \left(27 x^2+54 x+28\right)^{2/3}+\frac{\left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} F\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{9\ 3^{3/4} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} E\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{18 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}+\frac{18 (x+1)}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}",1,"(28 + 54*x + 27*x^2)^(2/3)/12 - (1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2]","C",1
2506,1,127,103,0.0757225,"\int \frac{1}{(2+3 x) \sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt[3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right)}{2\ 3^{2/3} \sqrt[3]{27 x^2+54 x+28}}","\frac{\log \left(27 \sqrt[3]{2} \sqrt[3]{27 x^2+54 x+28}-81 x-108\right)}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt[3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}}",1,"-1/2*(((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)])/(3^(2/3)*(28 + 54*x + 27*x^2)^(1/3))","C",0
2507,1,240,671,0.2637212,"\int \frac{1}{(2+3 x)^2 \sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(1/3)),x]","\frac{4 \sqrt[3]{3} (3 x+2) \sqrt[3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt[3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right)+2^{2/3} \sqrt[3]{3} \sqrt[3]{-9 i x+\sqrt{3}-9 i} (3 x+2) \left(3 \sqrt{3} x+3 \sqrt{3}-i\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right)-4 \left(27 x^2+54 x+28\right)}{48 (3 x+2) \sqrt[3]{27 x^2+54 x+28}}","-\frac{\left(27 x^2+54 x+28\right)^{2/3}}{12 (3 x+2)}-\frac{\log \left(27 \sqrt[3]{2} \sqrt[3]{27 x^2+54 x+28}-81 x-108\right)}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt[3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right)}{6\ 2^{2/3} \sqrt{3}}-\frac{\left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} F\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}+\frac{\sqrt{2+\sqrt{3}} \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} E\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{72 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}-\frac{9 (x+1)}{2 \left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)}+\frac{\log (3 x+2)}{12\ 2^{2/3}}",1,"(-4*(28 + 54*x + 27*x^2) + 4*3^(1/3)*(2 + 3*x)*((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + 2^(2/3)*3^(1/3)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)*(-I + 3*Sqrt[3] + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(48*(2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3))","C",0
2508,1,233,696,0.2323515,"\int \frac{1}{(2+3 x)^3 \sqrt[3]{28+54 x+27 x^2}} \, dx","Integrate[1/((2 + 3*x)^3*(28 + 54*x + 27*x^2)^(1/3)),x]","\frac{-54 \sqrt[3]{3} \sqrt[3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt[3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right)+9 i 2^{2/3} 3^{5/6} \sqrt[3]{-9 i x+\sqrt{3}-9 i} \left(9 i x+\sqrt{3}+9 i\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right)+\frac{162 (2 x+1) \left(27 x^2+54 x+28\right)}{(3 x+2)^2}}{1296 \sqrt[3]{27 x^2+54 x+28}}","\frac{\left(27 x^2+54 x+28\right)^{2/3}}{12 (3 x+2)}-\frac{\left(27 x^2+54 x+28\right)^{2/3}}{24 (3 x+2)^2}+\frac{\log \left(27 \sqrt[3]{2} \sqrt[3]{27 x^2+54 x+28}-81 x-108\right)}{24\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt[3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right)}{12\ 2^{2/3} \sqrt{3}}+\frac{\left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} F\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left(6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right) \sqrt{\frac{\left(27 x^2+54 x+28\right)^{2/3}+\sqrt[3]{27 x^2+54 x+28}+1}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} E\left(\sin ^{-1}\left(\frac{6 \left(1+\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}\right)|-7+4 \sqrt{3}\right)}{72 \sqrt{2} \sqrt[4]{3} \sqrt{-\frac{6-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}}{\left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)^2}} (x+1)}+\frac{9 (x+1)}{2 \left(6 \left(1-\sqrt{3}\right)-\sqrt[3]{2} \sqrt[3]{(54 x+54)^2+108}\right)}-\frac{\log (3 x+2)}{24\ 2^{2/3}}",1,"((162*(1 + 2*x)*(28 + 54*x + 27*x^2))/(2 + 3*x)^2 - 54*3^(1/3)*((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + (9*I)*2^(2/3)*3^(5/6)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(9*I + Sqrt[3] + (9*I)*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(1296*(28 + 54*x + 27*x^2)^(1/3))","C",0
2509,1,290,564,0.4113598,"\int \frac{1}{(d+e x) \sqrt[3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx","Integrate[1/((d + e*x)*(-(c^2*d^2) + b*c*d*e + 2*b^2*e^2 + 9*b*c*e^2*x + 9*c^2*e^2*x^2)^(1/3)),x]","-\frac{\sqrt[3]{3} \sqrt[3]{\frac{-\sqrt{c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} \sqrt[3]{\frac{\sqrt{c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{-6 d e c^2+3 b e^2 c+\sqrt{c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)},\frac{6 d e c^2-3 b e^2 c+\sqrt{c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)}\right)}{2\ 2^{2/3} e \sqrt[3]{2 b^2 e^2+b c e (d+9 e x)-\left(c^2 \left(d^2-9 e^2 x^2\right)\right)}}","-\frac{\log (d+e x) \sqrt[3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt[3]{c e (b e+c d)+3 c^2 e^2 x}}{2 \sqrt[3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt[3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac{3 \sqrt[3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt[3]{c e (b e+c d)+3 c^2 e^2 x} \log \left(-\frac{\sqrt[3]{\frac{3}{2}} \left(3 c^2 e^2 x-c e (c d-2 b e)\right)^{2/3}}{\sqrt[3]{c} \sqrt[3]{e} \sqrt[3]{2 c d-b e}}-\sqrt[3]{6} \sqrt[3]{c e (b e+c d)+3 c^2 e^2 x}\right)}{4 \sqrt[3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt[3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac{\sqrt{3} \sqrt[3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt[3]{c e (b e+c d)+3 c^2 e^2 x} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{2} \left(3 c^2 e^2 x-c e (c d-2 b e)\right)^{2/3}}{\sqrt{3} \sqrt[3]{c} \sqrt[3]{e} \sqrt[3]{2 c d-b e} \sqrt[3]{c e (b e+c d)+3 c^2 e^2 x}}\right)}{2 \sqrt[3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt[3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}",1,"-1/2*(3^(1/3)*((3*b*c*e^2 - Sqrt[c^2*e^2*(-2*c*d + b*e)^2] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*((3*b*c*e^2 + Sqrt[c^2*e^2*(-2*c*d + b*e)^2] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -1/6*(-6*c^2*d*e + 3*b*c*e^2 + Sqrt[c^2*e^2*(-2*c*d + b*e)^2])/(c^2*e*(d + e*x)), (6*c^2*d*e - 3*b*c*e^2 + Sqrt[c^2*e^2*(-2*c*d + b*e)^2])/(6*c^2*e*(d + e*x))])/(2^(2/3)*e*(2*b^2*e^2 + b*c*e*(d + 9*e*x) - c^2*(d^2 - 9*e^2*x^2))^(1/3))","C",0
2510,1,235,374,0.9001612,"\int (d+e x)^3 \sqrt[4]{a+b x+c x^2} \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(1/4),x]","\frac{15 (2 c d-b e) \left(-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right) \left(2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)+8 c^2 e (a+x (b+c x))^2 \left(-2 c e (56 a e+243 b d+65 b e x)+117 b^2 e^2+4 c^2 d (154 d+65 e x)\right)+1120 c^4 e (d+e x)^2 (a+x (b+c x))^2}{5040 c^5 (a+x (b+c x))^{3/4}}","-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{336 \sqrt{2} c^{17/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e) \left(-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right)}{168 c^4}+\frac{e \left(a+b x+c x^2\right)^{5/4} \left(-2 c e (56 a e+243 b d)+117 b^2 e^2+130 c e x (2 c d-b e)+616 c^2 d^2\right)}{630 c^3}+\frac{2 e (d+e x)^2 \left(a+b x+c x^2\right)^{5/4}}{9 c}",1,"(1120*c^4*e*(d + e*x)^2*(a + x*(b + c*x))^2 + 8*c^2*e*(a + x*(b + c*x))^2*(117*b^2*e^2 + 4*c^2*d*(154*d + 65*e*x) - 2*c*e*(243*b*d + 56*a*e + 65*b*e*x)) + 15*(2*c*d - b*e)*(28*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]))/(5040*c^5*(a + x*(b + c*x))^(3/4))","A",1
2511,1,191,319,0.3999652,"\int (d+e x)^2 \sqrt[4]{a+b x+c x^2} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(1/4),x]","\frac{\frac{5 \left(-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right) \left(2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)}{4 c^2}-54 e (a+x (b+c x))^2 (b e-2 c d)+60 c e (d+e x) (a+x (b+c x))^2}{210 c^2 (a+x (b+c x))^{3/4}}","-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{168 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} \left(-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right)}{84 c^3}+\frac{9 e \left(a+b x+c x^2\right)^{5/4} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x) \left(a+b x+c x^2\right)^{5/4}}{7 c}",1,"(-54*e*(-2*c*d + b*e)*(a + x*(b + c*x))^2 + 60*c*e*(d + e*x)*(a + x*(b + c*x))^2 + (5*(28*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(7*b*d + 2*a*e))*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]))/(4*c^2))/(210*c^2*(a + x*(b + c*x))^(3/4))","A",1
2512,1,140,241,0.2412901,"\int (d+e x) \sqrt[4]{a+b x+c x^2} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(1/4),x]","\frac{5 (2 c d-b e) \left(2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)+24 c^2 e (a+x (b+c x))^2}{60 c^3 (a+x (b+c x))^{3/4}}","-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{12 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{6 c^2}+\frac{2 e \left(a+b x+c x^2\right)^{5/4}}{5 c}",1,"(24*c^2*e*(a + x*(b + c*x))^2 + 5*(2*c*d - b*e)*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]))/(60*c^3*(a + x*(b + c*x))^(3/4))","A",1
2513,1,100,201,0.2393365,"\int \sqrt[4]{a+b x+c x^2} \, dx","Integrate[(a + b*x + c*x^2)^(1/4),x]","\frac{\sqrt[4]{a+x (b+c x)} \left(\frac{\sqrt{2} \sqrt{b^2-4 a c} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{\sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+2 (b+2 c x)\right)}{6 c}","\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2}}{3 c}-\frac{\left(b^2-4 a c\right)^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{6 \sqrt{2} c^{5/4} (b+2 c x)}",1,"((a + x*(b + c*x))^(1/4)*(2*(b + 2*c*x) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)))/(6*c)","A",1
2514,1,613,881,1.8738328,"\int \frac{\sqrt[4]{a+b x+c x^2}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(1/4)/(d + e*x),x]","\frac{-\frac{\left(4 a c-b^2\right)^{3/4} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{\sqrt{2} c (b+2 c x)}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} (b e-2 c d) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{c}+2 e (a+x (b+c x))}{e^2 (a+x (b+c x))^{3/4}}","-\frac{\left(4 a c-b^2\right)^{3/4} \sqrt[4]{c d^2-b e d+a e^2} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{c^{3/4} e^{3/2} \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(4 a c-b^2\right)^{3/4} \sqrt[4]{c d^2-b e d+a e^2} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{c^{3/4} e^{3/2} \left(c x^2+b x+a\right)^{3/4}}-\frac{\sqrt[4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} \sqrt[4]{c} e^2 (b+2 c x)}-\frac{\left(b^2-4 a c\right) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} c e^2 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(b^2-4 a c\right) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} c e^2 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{2 \sqrt[4]{c x^2+b x+a}}{e}",1,"(2*e*(a + x*(b + c*x)) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-2*c*d + b*e)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/c - ((-b^2 + 4*a*c)^(3/4)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(Sqrt[2]*c*(b + 2*c*x)))/(e^2*(a + x*(b + c*x))^(3/4))","A",1
2515,1,641,944,2.0403321,"\int \frac{\sqrt[4]{a+b x+c x^2}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(1/4)/(d + e*x)^2,x]","\frac{-\frac{\left(4 a c-b^2\right)^{3/4} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} (b e-2 c d) \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{2 \sqrt{2} c (b+2 c x) \left(e (a e-b d)+c d^2\right)}+2 \sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)-\frac{2 e (a+x (b+c x))}{d+e x}}{2 e^2 (a+x (b+c x))^{3/4}}","\frac{\left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left(c d^2-b e d+a e^2\right) (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{\left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c e^2 \left(c d^2-b e d+a e^2\right) (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{\left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left(c d^2-b e d+a e^2\right)^{3/4} \left(c x^2+b x+a\right)^{3/4}}+\frac{\left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} e^{3/2} \left(c d^2-b e d+a e^2\right)^{3/4} \left(c x^2+b x+a\right)^{3/4}}+\frac{c^{3/4} \sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} e^2 (b+2 c x)}-\frac{\sqrt[4]{c x^2+b x+a}}{e (d+e x)}",1,"((-2*e*(a + x*(b + c*x)))/(d + e*x) + 2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2] - ((-b^2 + 4*a*c)^(3/4)*(-2*c*d + b*e)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(2*Sqrt[2]*c*(c*d^2 + e*(-(b*d) + a*e))*(b + 2*c*x)))/(2*e^2*(a + x*(b + c*x))^(3/4))","A",1
2516,1,234,703,0.6227976,"\int (d+e x)^3 \left(a+b x+c x^2\right)^{3/4} \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(3/4),x]","\frac{77 (b+2 c x) (2 c d-b e) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right) \left(8 c (a+x (b+c x))-3 \sqrt{2} \left(b^2-4 a c\right) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)+160 c^2 e (a+x (b+c x))^2 \left(-2 c e (24 a e+121 b d+35 b e x)+55 b^2 e^2+4 c^2 d (78 d+35 e x)\right)+13440 c^4 e (d+e x)^2 (a+x (b+c x))^2}{73920 c^5 \sqrt[4]{a+x (b+c x)}}","-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right)}{80 c^{9/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}-\frac{\left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{\left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/4} (2 c d-b e) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right)}{120 c^4}+\frac{e \left(a+b x+c x^2\right)^{7/4} \left(-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right)}{462 c^3}+\frac{2 e (d+e x)^2 \left(a+b x+c x^2\right)^{7/4}}{11 c}",1,"(13440*c^4*e*(d + e*x)^2*(a + x*(b + c*x))^2 + 160*c^2*e*(a + x*(b + c*x))^2*(55*b^2*e^2 + 4*c^2*d*(78*d + 35*e*x) - 2*c*e*(121*b*d + 24*a*e + 35*b*e*x)) + 77*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(73920*c^5*(a + x*(b + c*x))^(1/4))","C",1
2517,1,196,630,0.2873964,"\int (d+e x)^2 \left(a+b x+c x^2\right)^{3/4} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(3/4),x]","\frac{-\frac{7 (b+2 c x) \left(c e (2 a e+9 b d)-\frac{11 b^2 e^2}{4}-9 c^2 d^2\right) \left(8 c (a+x (b+c x))-3 \sqrt{2} \left(b^2-4 a c\right) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{40 c^2 \sqrt[4]{a+x (b+c x)}}+11 e (a+x (b+c x))^{7/4} (2 c d-b e)+14 c e (d+e x) (a+x (b+c x))^{7/4}}{63 c^2}","-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt[4]{a+b x+c x^2} \left(-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right)}{120 c^{7/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}-\frac{\left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{240 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{\left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{120 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/4} \left(-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right)}{180 c^3}+\frac{11 e \left(a+b x+c x^2\right)^{7/4} (2 c d-b e)}{63 c^2}+\frac{2 e (d+e x) \left(a+b x+c x^2\right)^{7/4}}{9 c}",1,"(11*e*(2*c*d - b*e)*(a + x*(b + c*x))^(7/4) + 14*c*e*(d + e*x)*(a + x*(b + c*x))^(7/4) - (7*(-9*c^2*d^2 - (11*b^2*e^2)/4 + c*e*(9*b*d + 2*a*e))*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(40*c^2*(a + x*(b + c*x))^(1/4)))/(63*c^2)","C",1
2518,1,141,510,0.2430965,"\int (d+e x) \left(a+b x+c x^2\right)^{3/4} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(3/4),x]","\frac{(b+2 c x) (2 c d-b e) \left(8 c (a+x (b+c x))-3 \sqrt{2} \left(b^2-4 a c\right) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{80 c^3 \sqrt[4]{a+x (b+c x)}}+\frac{2 e (a+x (b+c x))^{7/4}}{7 c}","-\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{20 c^{5/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}-\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{40 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{20 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/4} (2 c d-b e)}{10 c^2}+\frac{2 e \left(a+b x+c x^2\right)^{7/4}}{7 c}",1,"(2*e*(a + x*(b + c*x))^(7/4))/(7*c) + ((2*c*d - b*e)*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(80*c^3*(a + x*(b + c*x))^(1/4))","C",1
2519,1,119,452,0.0862903,"\int \left(a+b x+c x^2\right)^{3/4} \, dx","Integrate[(a + b*x + c*x^2)^(3/4),x]","\frac{(b+2 c x) (a+x (b+c x))^{3/4} \left(3 \sqrt{2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+8 \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4}\right)}{40 c \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4}}","-\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt[4]{a+b x+c x^2}}{10 c^{3/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}-\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{20 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{3 \left(b^2-4 a c\right)^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{10 \sqrt{2} c^{7/4} (b+2 c x)}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/4}}{5 c}",1,"((b + 2*c*x)*(a + x*(b + c*x))^(3/4)*(8*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4) + 3*Sqrt[2]*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(40*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4))","C",1
2520,1,180,1209,0.3941142,"\int \frac{\left(a+b x+c x^2\right)^{3/4}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(3/4)/(d + e*x),x]","\frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left(-\frac{3}{2};-\frac{3}{4},-\frac{3}{4};-\frac{1}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{3 e \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{3/4} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{3/4}}","-\frac{(2 c d-b e) \sqrt[4]{c x^2+b x+a} (b+2 c x)}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\sqrt[4]{4 a c-b^2} \left(c d^2-b e d+a e^2\right)^{3/4} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{\sqrt[4]{c} e^{5/2} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt[4]{4 a c-b^2} \left(c d^2-b e d+a e^2\right)^{3/4} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{\sqrt[4]{c} e^{5/2} \sqrt[4]{c x^2+b x+a}}+\frac{2 \left(c x^2+b x+a\right)^{3/4}}{3 e}+\frac{\left(b^2-4 a c\right)^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left(b^2-4 a c\right)^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} \sqrt{c} e^3 \sqrt[4]{c x^2+b x+a} (b+2 c x)}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} \sqrt{c} e^3 \sqrt[4]{c x^2+b x+a} (b+2 c x)}",1,"(4*Sqrt[2]*(a + x*(b + c*x))^(3/4)*AppellF1[-3/2, -3/4, -3/4, -1/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(3*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4))","C",0
2521,1,185,1220,0.2765608,"\int \frac{\left(a+b x+c x^2\right)^{3/4}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/4)/(d + e*x)^2,x]","\frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left(-\frac{1}{2};-\frac{3}{4},-\frac{3}{4};\frac{1}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{e (d+e x) \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{3/4} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{3/4}}","\frac{3 \sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e^3 \sqrt{c d^2-b e d+a e^2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}-\frac{3 \sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e^3 \sqrt{c d^2-b e d+a e^2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}-\frac{3 \sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} e^{5/2} \sqrt[4]{c d^2-b e d+a e^2} \sqrt[4]{c x^2+b x+a}}+\frac{3 \sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} e^{5/2} \sqrt[4]{c d^2-b e d+a e^2} \sqrt[4]{c x^2+b x+a}}-\frac{3 \sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} e^2 (b+2 c x)}+\frac{3 \sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} e^2 (b+2 c x)}-\frac{\left(c x^2+b x+a\right)^{3/4}}{e (d+e x)}+\frac{3 \sqrt{c} (b+2 c x) \sqrt[4]{c x^2+b x+a}}{\sqrt{b^2-4 a c} e^2 \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}",1,"(4*Sqrt[2]*(a + x*(b + c*x))^(3/4)*AppellF1[-1/2, -3/4, -3/4, 1/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*(d + e*x))","C",0
2522,1,286,448,1.1220195,"\int (d+e x)^3 \left(a+b x+c x^2\right)^{5/4} \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/4),x]","\frac{39 (2 c d-b e) \left(-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right) \left(2 c (b+2 c x) \left(32 a^2 c+a \left(-5 b^2+44 b c x+44 c^2 x^2\right)+x \left(-5 b^3+7 b^2 c x+24 b c^2 x^2+12 c^3 x^3\right)\right)+5 \sqrt{2} \left(b^2-4 a c\right)^{5/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)+224 c^3 e (a+x (b+c x))^3 \left(-2 c e (88 a e+507 b d+153 b e x)+221 b^2 e^2+12 c^2 d (110 d+51 e x)\right)+88704 c^5 e (d+e x)^2 (a+x (b+c x))^3}{576576 c^6 (a+x (b+c x))^{3/4}}","\frac{5 \left(b^2-4 a c\right)^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{14784 \sqrt{2} c^{21/4} (b+2 c x)}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e) \left(-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right)}{7392 c^5}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/4} (2 c d-b e) \left(-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right)}{616 c^4}+\frac{e \left(a+b x+c x^2\right)^{9/4} \left(-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right)}{2574 c^3}+\frac{2 e (d+e x)^2 \left(a+b x+c x^2\right)^{9/4}}{13 c}",1,"(88704*c^5*e*(d + e*x)^2*(a + x*(b + c*x))^3 + 224*c^3*e*(a + x*(b + c*x))^3*(221*b^2*e^2 + 12*c^2*d*(110*d + 51*e*x) - 2*c*e*(507*b*d + 88*a*e + 153*b*e*x)) + 39*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e))*(2*c*(b + 2*c*x)*(32*a^2*c + a*(-5*b^2 + 44*b*c*x + 44*c^2*x^2) + x*(-5*b^3 + 7*b^2*c*x + 24*b*c^2*x^2 + 12*c^3*x^3)) + 5*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]))/(576576*c^6*(a + x*(b + c*x))^(3/4))","A",1
2523,1,233,384,0.5702654,"\int (d+e x)^2 \left(a+b x+c x^2\right)^{5/4} \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/4),x]","\frac{2 \left(-\frac{\left(c e (2 a e+11 b d)-\frac{13 b^2 e^2}{4}-11 c^2 d^2\right) \left(24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left(b^2-4 a c\right) \left(2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)\right)}{336 c^4 (a+x (b+c x))^{3/4}}+\frac{13 e (a+x (b+c x))^{9/4} (2 c d-b e)}{18 c}+e (d+e x) (a+x (b+c x))^{9/4}\right)}{11 c}","\frac{5 \left(b^2-4 a c\right)^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{7392 \sqrt{2} c^{17/4} (b+2 c x)}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \sqrt[4]{a+b x+c x^2} \left(-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right)}{3696 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/4} \left(-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right)}{308 c^3}+\frac{13 e \left(a+b x+c x^2\right)^{9/4} (2 c d-b e)}{99 c^2}+\frac{2 e (d+e x) \left(a+b x+c x^2\right)^{9/4}}{11 c}",1,"(2*((13*e*(2*c*d - b*e)*(a + x*(b + c*x))^(9/4))/(18*c) + e*(d + e*x)*(a + x*(b + c*x))^(9/4) - ((-11*c^2*d^2 - (13*b^2*e^2)/4 + c*e*(11*b*d + 2*a*e))*(24*c^2*(b + 2*c*x)*(a + x*(b + c*x))^2 - 5*(b^2 - 4*a*c)*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])))/(336*c^4*(a + x*(b + c*x))^(3/4))))/(11*c)","A",1
2524,1,175,285,0.3354978,"\int (d+e x) \left(a+b x+c x^2\right)^{5/4} \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/4),x]","\frac{(2 c d-b e) \left(24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left(b^2-4 a c\right) \left(2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left(b^2-4 a c\right)^{3/2} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)\right)\right)}{336 c^4 (a+x (b+c x))^{3/4}}+\frac{2 e (a+x (b+c x))^{9/4}}{9 c}","\frac{5 \left(b^2-4 a c\right)^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{336 \sqrt{2} c^{13/4} (b+2 c x)}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/4} (2 c d-b e)}{14 c^2}+\frac{2 e \left(a+b x+c x^2\right)^{9/4}}{9 c}",1,"(2*e*(a + x*(b + c*x))^(9/4))/(9*c) + ((2*c*d - b*e)*(24*c^2*(b + 2*c*x)*(a + x*(b + c*x))^2 - 5*(b^2 - 4*a*c)*(2*c*(b + 2*c*x)*(a + x*(b + c*x)) - Sqrt[2]*(b^2 - 4*a*c)^(3/2)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])))/(336*c^4*(a + x*(b + c*x))^(3/4))","A",1
2525,1,125,236,0.3602702,"\int \left(a+b x+c x^2\right)^{5/4} \, dx","Integrate[(a + b*x + c*x^2)^(5/4),x]","\frac{\sqrt[4]{a+x (b+c x)} \left(2 (b+2 c x) \left(4 c \left(8 a+3 c x^2\right)-5 b^2+12 b c x\right)-\frac{5 \sqrt{2} \left(b^2-4 a c\right)^{3/2} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{\sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right)}{168 c^2}","\frac{5 \left(b^2-4 a c\right)^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{168 \sqrt{2} c^{9/4} (b+2 c x)}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \sqrt[4]{a+b x+c x^2}}{84 c^2}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/4}}{7 c}",1,"((a + x*(b + c*x))^(1/4)*(2*(b + 2*c*x)*(-5*b^2 + 12*b*c*x + 4*c*(8*a + 3*c*x^2)) - (5*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)))/(168*c^2)","A",1
2526,1,699,1014,4.3295455,"\int \frac{\left(a+b x+c x^2\right)^{5/4}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(5/4)/(d + e*x),x]","\frac{\left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} \left(-\sqrt{b^2-4 a c} (b e-2 c d) \left(4 c e (3 b d-4 a e)+b^2 e^2-12 c^2 d^2\right) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)-\frac{6 c \left(4 a c-b^2\right)^{3/4} \left(e (a e-b d)+c d^2\right) \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{b+2 c x}\right)}{6 \sqrt{2} c^2 e^4 (a+x (b+c x))^{3/4}}+\frac{\sqrt[4]{a+x (b+c x)} \left(2 c e (6 a e-7 b d+b e x)+b^2 e^2+4 c^2 d (3 d-e x)\right)}{6 c e^3}+\frac{2 (a+x (b+c x))^{5/4}}{5 e}","-\frac{\left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) \left(c d^2-b e d+a e^2\right)^{5/4}}{c^{3/4} e^{7/2} \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) \left(c d^2-b e d+a e^2\right)^{5/4}}{c^{3/4} e^{7/2} \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(b^2-4 a c\right) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) \left(c d^2-b e d+a e^2\right)}{\sqrt{2} c e^4 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(b^2-4 a c\right) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) \left(c d^2-b e d+a e^2\right)}{\sqrt{2} c e^4 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{2 \left(c x^2+b x+a\right)^{5/4}}{5 e}-\frac{\sqrt[4]{b^2-4 a c} (2 c d-b e) \left(12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{12 \sqrt{2} c^{5/4} e^4 (b+2 c x)}+\frac{\left(12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right) \sqrt[4]{c x^2+b x+a}}{6 c e^3}",1,"(2*(a + x*(b + c*x))^(5/4))/(5*e) + ((a + x*(b + c*x))^(1/4)*(b^2*e^2 + 4*c^2*d*(3*d - e*x) + 2*c*e*(-7*b*d + 6*a*e + b*e*x)))/(6*c*e^3) + (((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(-(Sqrt[b^2 - 4*a*c]*(-2*c*d + b*e)*(-12*c^2*d^2 + b^2*e^2 + 4*c*e*(3*b*d - 4*a*e))*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2]) - (6*c*(-b^2 + 4*a*c)^(3/4)*(c*d^2 + e*(-(b*d) + a*e))*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(b + 2*c*x)))/(6*Sqrt[2]*c^2*e^4*(a + x*(b + c*x))^(3/4))","A",1
2527,1,658,975,3.7357161,"\int \frac{\left(a+b x+c x^2\right)^{5/4}}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x]","\frac{5 \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} \left(\sqrt{b^2-4 a c} \left(2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)-\frac{3 \left(4 a c-b^2\right)^{3/4} (b e-2 c d) \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{4 (b+2 c x)}\right)}{3 \sqrt{2} c e^4 (a+x (b+c x))^{3/4}}+\frac{5 \sqrt[4]{a+x (b+c x)} (2 b e-3 c d+c e x)}{3 e^3}-\frac{(a+x (b+c x))^{5/4}}{e (d+e x)}","\frac{5 \left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c e^4 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{5 \left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c e^4 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}+\frac{5 \left(4 a c-b^2\right)^{3/4} \sqrt[4]{c d^2-b e d+a e^2} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} e^{7/2} \left(c x^2+b x+a\right)^{3/4}}+\frac{5 \left(4 a c-b^2\right)^{3/4} \sqrt[4]{c d^2-b e d+a e^2} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} e^{7/2} \left(c x^2+b x+a\right)^{3/4}}-\frac{\left(c x^2+b x+a\right)^{5/4}}{e (d+e x)}+\frac{5 \sqrt[4]{b^2-4 a c} \left(6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{6 \sqrt{2} \sqrt[4]{c} e^4 (b+2 c x)}-\frac{5 (3 c d-2 b e-c e x) \sqrt[4]{c x^2+b x+a}}{3 e^3}",1,"(5*(-3*c*d + 2*b*e + c*e*x)*(a + x*(b + c*x))^(1/4))/(3*e^3) - (a + x*(b + c*x))^(5/4)/(e*(d + e*x)) + (5*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[b^2 - 4*a*c]*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2] - (3*(-b^2 + 4*a*c)^(3/4)*(-2*c*d + b*e)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(4*(b + 2*c*x))))/(3*Sqrt[2]*c*e^4*(a + x*(b + c*x))^(3/4))","A",1
2528,1,211,637,0.2853081,"\int \frac{(d+e x)^3}{\sqrt[4]{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(1/4),x]","\frac{\frac{e (a+x (b+c x)) \left(-2 c e (40 a e+147 b d+33 b e x)+77 b^2 e^2+12 c^2 d (30 d+11 e x)\right)}{2 c^2}+\frac{21 (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \left(-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{8 \sqrt{2} c^3}+30 e (d+e x)^2 (a+x (b+c x))}{105 c \sqrt[4]{a+x (b+c x)}}","\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e) \left(-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right)}{20 c^{7/2} \sqrt{b^2-4 a c} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{40 \sqrt{2} c^{15/4} (b+2 c x)}-\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+5 b d)+11 b^2 e^2+20 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{20 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{e \left(a+b x+c x^2\right)^{3/4} \left(-2 c e (40 a e+147 b d)+77 b^2 e^2+66 c e x (2 c d-b e)+360 c^2 d^2\right)}{210 c^3}+\frac{2 e (d+e x)^2 \left(a+b x+c x^2\right)^{3/4}}{7 c}",1,"(30*e*(d + e*x)^2*(a + x*(b + c*x)) + (e*(a + x*(b + c*x))*(77*b^2*e^2 + 12*c^2*d*(30*d + 11*e*x) - 2*c*e*(147*b*d + 40*a*e + 33*b*e*x)))/(2*c^2) + (21*(2*c*d - b*e)*(20*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(5*b*d + 6*a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*Sqrt[2]*c^3))/(105*c*(a + x*(b + c*x))^(1/4))","C",1
2529,1,165,573,0.2082672,"\int \frac{(d+e x)^2}{\sqrt[4]{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(1/4),x]","\frac{\frac{3 (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{2 \sqrt{2} c^2}-\frac{14 e (a+x (b+c x)) (b e-2 c d)}{c}+12 e (d+e x) (a+x (b+c x))}{30 c \sqrt[4]{a+x (b+c x)}}","\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} \left(-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right)}{10 c^{5/2} \sqrt{b^2-4 a c} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{20 \sqrt{2} c^{11/4} (b+2 c x)}-\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{10 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{7 e \left(a+b x+c x^2\right)^{3/4} (2 c d-b e)}{15 c^2}+\frac{2 e (d+e x) \left(a+b x+c x^2\right)^{3/4}}{5 c}",1,"((-14*e*(-2*c*d + b*e)*(a + x*(b + c*x)))/c + 12*e*(d + e*x)*(a + x*(b + c*x)) + (3*(20*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(5*b*d + 2*a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2*Sqrt[2]*c^2))/(30*c*(a + x*(b + c*x))^(1/4))","C",1
2530,1,112,469,0.116211,"\int \frac{d+e x}{\sqrt[4]{a+b x+c x^2}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(1/4),x]","\frac{3 \sqrt{2} (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)+8 c e (a+x (b+c x))}{12 c^2 \sqrt[4]{a+x (b+c x)}}","\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{c^{3/2} \sqrt{b^2-4 a c} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} c^{7/4} (b+2 c x)}-\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{7/4} (b+2 c x)}+\frac{2 e \left(a+b x+c x^2\right)^{3/4}}{3 c}",1,"(8*c*e*(a + x*(b + c*x)) + 3*Sqrt[2]*(2*c*d - b*e)*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(12*c^2*(a + x*(b + c*x))^(1/4))","C",1
2531,1,84,418,0.0319269,"\int \frac{1}{\sqrt[4]{a+b x+c x^2}} \, dx","Integrate[(a + b*x + c*x^2)^(-1/4),x]","\frac{(b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)}{\sqrt{2} c \sqrt[4]{a+x (b+c x)}}","\frac{\left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{3/4} (b+2 c x)}-\frac{\sqrt{2} \left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{c^{3/4} (b+2 c x)}+\frac{2 (b+2 c x) \sqrt[4]{a+b x+c x^2}}{\sqrt{c} \sqrt{b^2-4 a c} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}",1,"((b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(Sqrt[2]*c*(a + x*(b + c*x))^(1/4))","C",1
2532,1,178,733,0.2852194,"\int \frac{1}{(d+e x) \sqrt[4]{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(1/4)),x]","-\frac{\sqrt{2} \sqrt[4]{\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} \sqrt[4]{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} F_1\left(\frac{1}{2};\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{e \sqrt[4]{a+x (b+c x)}}","\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{a e^2-b d e+c d^2}}\right)}{\sqrt[4]{c} \sqrt{e} \sqrt[4]{a+b x+c x^2} \sqrt[4]{a e^2-b d e+c d^2}}-\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{a e^2-b d e+c d^2}}\right)}{\sqrt[4]{c} \sqrt{e} \sqrt[4]{a+b x+c x^2} \sqrt[4]{a e^2-b d e+c d^2}}-\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt[4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}+\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c d-b e) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} \sqrt{c} e (b+2 c x) \sqrt[4]{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}",1,"-((Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*AppellF1[1/2, 1/4, 1/4, 3/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(a + x*(b + c*x))^(1/4)))","C",0
2533,1,187,1280,0.3273514,"\int \frac{1}{(d+e x)^2 \sqrt[4]{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(1/4)),x]","-\frac{\sqrt{2} \sqrt[4]{\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} \sqrt[4]{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} F_1\left(\frac{3}{2};\frac{1}{4},\frac{1}{4};\frac{5}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{3 e (d+e x) \sqrt[4]{a+x (b+c x)}}","-\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e \left(c d^2-b e d+a e^2\right)^{3/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}+\frac{\sqrt{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} e \left(c d^2-b e d+a e^2\right)^{3/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}+\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} \sqrt{e} \left(c d^2-b e d+a e^2\right)^{5/4} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} \sqrt{e} \left(c d^2-b e d+a e^2\right)^{5/4} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} \left(c d^2-b e d+a e^2\right) (b+2 c x)}+\frac{\sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} \left(c d^2-b e d+a e^2\right) (b+2 c x)}-\frac{e \left(c x^2+b x+a\right)^{3/4}}{\left(c d^2-b e d+a e^2\right) (d+e x)}+\frac{\sqrt{c} (b+2 c x) \sqrt[4]{c x^2+b x+a}}{\sqrt{b^2-4 a c} \left(c d^2-b e d+a e^2\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}",1,"-1/3*(Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*AppellF1[3/2, 1/4, 1/4, 5/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(d + e*x)*(a + x*(b + c*x))^(1/4))","C",0
2534,1,187,1465,0.4851845,"\int \frac{1}{(d+e x)^3 \sqrt[4]{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(1/4)),x]","-\frac{\sqrt{2} \sqrt[4]{\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} \sqrt[4]{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}} F_1\left(\frac{5}{2};\frac{1}{4},\frac{1}{4};\frac{7}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{5 e (d+e x)^2 \sqrt[4]{a+x (b+c x)}}","-\frac{5 (2 c d-b e) \left(c x^2+b x+a\right)^{3/4} e}{8 \left(c d^2-b e d+a e^2\right)^2 (d+e x)}-\frac{\left(c x^2+b x+a\right)^{3/4} e}{2 \left(c d^2-b e d+a e^2\right) (d+e x)^2}-\frac{5 \sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{8 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (b+2 c x)}+\frac{5 \sqrt[4]{c} \left(b^2-4 a c\right)^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{16 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (b+2 c x)}+\frac{5 \sqrt{c} (2 c d-b e) (b+2 c x) \sqrt[4]{c x^2+b x+a}}{8 \sqrt{b^2-4 a c} \left(c d^2-b e d+a e^2\right)^2 \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\sqrt[4]{4 a c-b^2} \left(12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right) \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{32 \sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{9/4} \sqrt[4]{c x^2+b x+a} \sqrt{e}}-\frac{\sqrt[4]{4 a c-b^2} \left(12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right) \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{32 \sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{9/4} \sqrt[4]{c x^2+b x+a} \sqrt{e}}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \left(12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{32 \sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{5/2} (b+2 c x) \sqrt[4]{c x^2+b x+a} e}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \left(12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{32 \sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{5/2} (b+2 c x) \sqrt[4]{c x^2+b x+a} e}",1,"-1/5*(Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*AppellF1[5/2, 1/4, 1/4, 7/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(d + e*x)^2*(a + x*(b + c*x))^(1/4))","C",0
2535,1,244,307,0.3961285,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(3/4),x]","\frac{2 c e \left(-16 a^2 c e^2+a \left(15 b^2 e^2-2 b c e (25 d+11 e x)+4 c^2 \left(15 d^2+5 d e x-3 e^2 x^2\right)\right)+x (b+c x) \left(15 b^2 e^2-2 b c e (25 d+3 e x)+4 c^2 \left(15 d^2+5 d e x+e^2 x^2\right)\right)\right)-5 \sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} (b e-2 c d) \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{20 c^4 (a+x (b+c x))^{3/4}}","\frac{\sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{4 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{e \sqrt[4]{a+b x+c x^2} \left(-2 c e (8 a e+25 b d)+15 b^2 e^2+6 c e x (2 c d-b e)+56 c^2 d^2\right)}{10 c^3}+\frac{2 e (d+e x)^2 \sqrt[4]{a+b x+c x^2}}{5 c}",1,"(2*c*e*(-16*a^2*c*e^2 + a*(15*b^2*e^2 - 2*b*c*e*(25*d + 11*e*x) + 4*c^2*(15*d^2 + 5*d*e*x - 3*e^2*x^2)) + x*(b + c*x)*(15*b^2*e^2 - 2*b*c*e*(25*d + 3*e*x) + 4*c^2*(15*d^2 + 5*d*e*x + e^2*x^2))) - 5*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-2*c*d + b*e)*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/(20*c^4*(a + x*(b + c*x))^(3/4))","A",1
2536,1,150,262,0.2502944,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(3/4),x]","\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)+2 c e (a+x (b+c x)) (2 c (6 d+e x)-5 b e)}{6 c^3 (a+x (b+c x))^{3/4}}","\frac{\sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{6 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{5 e \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{3 c^2}+\frac{2 e (d+e x) \sqrt[4]{a+b x+c x^2}}{3 c}",1,"(2*c*e*(a + x*(b + c*x))*(-5*b*e + 2*c*(6*d + e*x)) + Sqrt[2]*Sqrt[b^2 - 4*a*c]*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/(6*c^3*(a + x*(b + c*x))^(3/4))","A",1
2537,1,111,200,0.1688247,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(3/4),x]","\frac{2 c e (a+x (b+c x))-\sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} (b e-2 c d) F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{c^2 (a+x (b+c x))^{3/4}}","\frac{\sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{5/4} (b+2 c x)}+\frac{2 e \sqrt[4]{a+b x+c x^2}}{c}",1,"(2*c*e*(a + x*(b + c*x)) - Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-2*c*d + b*e)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/(c^2*(a + x*(b + c*x))^(3/4))","A",1
2538,1,88,170,0.0400444,"\int \frac{1}{\left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[(a + b*x + c*x^2)^(-3/4),x]","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)}{c (a+x (b+c x))^{3/4}}","\frac{\sqrt{2} \sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt[4]{c} (b+2 c x)}",1,"(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2])/(c*(a + x*(b + c*x))^(3/4))","A",1
2539,1,529,709,0.9346318,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/4)),x]","-\frac{\sqrt[4]{a+x (b+c x)} \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{\sqrt{2} \sqrt[4]{4 a c-b^2} (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(e (a e-b d)+c d^2\right)}","-\frac{\sqrt{e} \left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{a e^2-b d e+c d^2}}\right)}{c^{3/4} \left(a+b x+c x^2\right)^{3/4} \left(a e^2-b d e+c d^2\right)^{3/4}}-\frac{\sqrt{e} \left(4 a c-b^2\right)^{3/4} \left(-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{a e^2-b d e+c d^2}}\right)}{c^{3/4} \left(a+b x+c x^2\right)^{3/4} \left(a e^2-b d e+c d^2\right)^{3/4}}-\frac{\left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}\right)^{3/4} (2 c d-b e) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} c (b+2 c x) \left(a+b x+c x^2\right)^{3/4} \left(a e^2-b d e+c d^2\right)}-\frac{\left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}\right)^{3/4} (2 c d-b e) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{\sqrt{2} c (b+2 c x) \left(a+b x+c x^2\right)^{3/4} \left(a e^2-b d e+c d^2\right)}",1,"-(((a + x*(b + c*x))^(1/4)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(Sqrt[2]*(-b^2 + 4*a*c)^(1/4)*(c*d^2 + e*(-(b*d) + a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)))","A",1
2540,1,652,970,2.3048394,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/4)),x]","\frac{-\frac{3 \left(4 a c-b^2\right)^{3/4} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} (b e-2 c d) \left(\sqrt{2} \sqrt[4]{c} \sqrt{e} (b+2 c x) \sqrt[4]{e (a e-b d)+c d^2} \left(\tan ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt[4]{4 a c-b^2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt[4]{c} \sqrt[4]{e (a e-b d)+c d^2}}\right)\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)+\sqrt[4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left.\sin ^{-1}\left(\sqrt{2} \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right)\right|-1\right)\right)}{4 \sqrt{2} c (b+2 c x) \left(e (a e-b d)+c d^2\right)}+\sqrt{2} \sqrt{b^2-4 a c} \left(\frac{c (a+x (b+c x))}{4 a c-b^2}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)\right|2\right)+\frac{e (a+x (b+c x))}{d+e x}}{(a+x (b+c x))^{3/4} \left(e (b d-a e)-c d^2\right)}","-\frac{3 \left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c \left(c d^2-b e d+a e^2\right)^2 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}-\frac{3 \left(b^2-4 a c\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} c \left(c d^2-b e d+a e^2\right)^2 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}-\frac{3 \left(4 a c-b^2\right)^{3/4} \sqrt{e} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} \left(c d^2-b e d+a e^2\right)^{7/4} \left(c x^2+b x+a\right)^{3/4}}-\frac{3 \left(4 a c-b^2\right)^{3/4} \sqrt{e} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 c^{3/4} \left(c d^2-b e d+a e^2\right)^{7/4} \left(c x^2+b x+a\right)^{3/4}}-\frac{c^{3/4} \sqrt[4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} \left(c d^2-b e d+a e^2\right) (b+2 c x)}-\frac{e \sqrt[4]{c x^2+b x+a}}{\left(c d^2-b e d+a e^2\right) (d+e x)}",1,"((e*(a + x*(b + c*x)))/(d + e*x) + Sqrt[2]*Sqrt[b^2 - 4*a*c]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*EllipticF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2] - (3*(-b^2 + 4*a*c)^(3/4)*(-2*c*d + b*e)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2*c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(4*Sqrt[2]*c*(c*d^2 + e*(-(b*d) + a*e))*(b + 2*c*x)))/((-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^(3/4))","A",1
2541,1,1690,1134,6.1874935,"\int \frac{1}{(d+e x)^3 \left(a+b x+c x^2\right)^{3/4}} \, dx","Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(3/4)),x]","-\frac{e \left(c x^2+b x+a\right)}{2 \left(c d^2-b e d+a e^2\right) (d+e x)^2 (a+x (b+c x))^{3/4}}-\frac{\left(\frac{\sqrt[4]{c x^2+b x+a} \left(\frac{1}{4} e (7 b e-8 c d)-\frac{3 c d e}{2}\right)}{\left(-c d^2+b e d-a e^2\right) (d+e x)}+\frac{\frac{7 \sqrt{b^2-4 a c} \left(\frac{c^2}{\left(b^2-4 a c\right) \left(\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}\right)}\right)^{3/4} (2 c d-b e) \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} F\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b^2-4 a c} \left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)}{c}\right)\right|2\right)}{2 \sqrt{2} \left(c x^2+b x+a\right)^{3/4}}+\frac{2 \sqrt{2} \left(\frac{c^2}{\left(b^2-4 a c\right) \left(\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}\right)}\right)^{3/4} \left(\frac{7}{8} c d e (2 c d-b e)+\frac{1}{16} e \left(32 c^2 d^2+21 b^2 e^2-2 c e (23 b d+12 a e)\right)\right) \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \left(\left(\frac{b c e}{b^2-4 a c}-\frac{2 c^2 d}{b^2-4 a c}\right) \left(\frac{\sqrt{\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}}\right)\right|-1\right)}{\left(e^2-\frac{\left(\frac{b c e}{b^2-4 a c}-\frac{2 c^2 d}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}\right) \left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)}+\frac{\sqrt{\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}}\right)\right|-1\right)}{\left(e^2-\frac{\left(\frac{b c e}{b^2-4 a c}-\frac{2 c^2 d}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}\right) \left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)}\right)-2 e \left(\frac{\sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{\sqrt{2} \sqrt[4]{4 a c-b^2} \sqrt{e} \left(e^2-\frac{\left(\frac{b c e}{b^2-4 a c}-\frac{2 c^2 d}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}\right)}+\frac{\sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{\left(-\frac{2 x c^2}{b^2-4 a c}-\frac{b c}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right)}{\sqrt{2} \sqrt[4]{4 a c-b^2} \sqrt{e} \left(e^2-\frac{\left(\frac{b c e}{b^2-4 a c}-\frac{2 c^2 d}{b^2-4 a c}\right)^2}{\frac{b^2 c^2}{\left(b^2-4 a c\right)^2}-\frac{4 a c^3}{\left(b^2-4 a c\right)^2}}\right)}\right)\right)}{e \left(c x^2+b x+a\right)^{3/4}}}{-c d^2+b e d-a e^2}\right) \left(c x^2+b x+a\right)^{3/4}}{2 \left(c d^2-b e d+a e^2\right) (a+x (b+c x))^{3/4}}","-\frac{7 (2 c d-b e) \sqrt[4]{c x^2+b x+a} e}{8 \left(c d^2-b e d+a e^2\right)^2 (d+e x)}-\frac{\sqrt[4]{c x^2+b x+a} e}{2 \left(c d^2-b e d+a e^2\right) (d+e x)^2}-\frac{3 \left(4 a c-b^2\right)^{3/4} \left(20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right) \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) \sqrt{e}}{32 c^{3/4} \left(c d^2-b e d+a e^2\right)^{11/4} \left(c x^2+b x+a\right)^{3/4}}-\frac{3 \left(4 a c-b^2\right)^{3/4} \left(20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right) \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) \sqrt{e}}{32 c^{3/4} \left(c d^2-b e d+a e^2\right)^{11/4} \left(c x^2+b x+a\right)^{3/4}}-\frac{7 c^{3/4} \sqrt[4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{8 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (b+2 c x)}-\frac{3 \left(b^2-4 a c\right) (2 c d-b e) \left(20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{32 \sqrt{2} c \left(c d^2-b e d+a e^2\right)^3 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}-\frac{3 \left(b^2-4 a c\right) (2 c d-b e) \left(20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left(-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}\right)^{3/4} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right)}{32 \sqrt{2} c \left(c d^2-b e d+a e^2\right)^3 (b+2 c x) \left(c x^2+b x+a\right)^{3/4}}",1,"-1/2*(e*(a + b*x + c*x^2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + x*(b + c*x))^(3/4)) - ((a + b*x + c*x^2)^(3/4)*((((-3*c*d*e)/2 + (e*(-8*c*d + 7*b*e))/4)*(a + b*x + c*x^2)^(1/4))/((-(c*d^2) + b*d*e - a*e^2)*(d + e*x)) + ((7*Sqrt[b^2 - 4*a*c]*(c^2/((b^2 - 4*a*c)*((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)))^(3/4)*(2*c*d - b*e)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*EllipticF[ArcSin[(Sqrt[b^2 - 4*a*c]*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c)))/c]/2, 2])/(2*Sqrt[2]*(a + b*x + c*x^2)^(3/4)) + (2*Sqrt[2]*(c^2/((b^2 - 4*a*c)*((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)))^(3/4)*((7*c*d*e*(2*c*d - b*e))/8 + (e*(32*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(23*b*d + 12*a*e)))/16)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*(-2*e*((c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(Sqrt[2]*(-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(e^2 - ((-2*c^2*d)/(b^2 - 4*a*c) + (b*c*e)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))) + (c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(Sqrt[2]*(-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(e^2 - ((-2*c^2*d)/(b^2 - 4*a*c) + (b*c*e)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)))) + ((-2*c^2*d)/(b^2 - 4*a*c) + (b*c*e)/(b^2 - 4*a*c))*((Sqrt[(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))^(1/4)], -1])/((e^2 - ((-2*c^2*d)/(b^2 - 4*a*c) + (b*c*e)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))) + (Sqrt[(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))^(1/4)], -1])/((e^2 - ((-2*c^2*d)/(b^2 - 4*a*c) + (b*c*e)/(b^2 - 4*a*c))^2/((b^2*c^2)/(b^2 - 4*a*c)^2 - (4*a*c^3)/(b^2 - 4*a*c)^2))*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))))))/(e*(a + b*x + c*x^2)^(3/4)))/(-(c*d^2) + b*d*e - a*e^2)))/(2*(c*d^2 - b*d*e + a*e^2)*(a + x*(b + c*x))^(3/4))","A",1
2542,1,247,662,0.5569785,"\int \frac{(d+e x)^3}{\left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(5/4),x]","\frac{3 \sqrt{2} (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \left(-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-8 c \left(4 c \left(4 a^2 e^3+a c e \left(-9 d^2-9 d e x+e^2 x^2\right)+3 c^2 d^3 x\right)-b^2 e^2 (7 a e+c x (e x-18 d))+2 b c \left(a e^2 (9 d+11 e x)+3 c d^2 (d-3 e x)\right)-7 b^3 e^3 x\right)}{12 c^3 \left(b^2-4 a c\right) \sqrt[4]{a+x (b+c x)}}","\frac{2 e \left(a+b x+c x^2\right)^{3/4} \left(-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right)}{3 c^2 \left(b^2-4 a c\right)}+\frac{(b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e) \left(-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right)}{c^{5/2} \left(b^2-4 a c\right)^{3/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} c^{11/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}-\frac{\sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) \left(-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{11/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}-\frac{4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt[4]{a+b x+c x^2}}",1,"(-8*c*(-7*b^3*e^3*x - b^2*e^2*(7*a*e + c*x*(-18*d + e*x)) + 2*b*c*(3*c*d^2*(d - 3*e*x) + a*e^2*(9*d + 11*e*x)) + 4*c*(4*a^2*e^3 + 3*c^2*d^3*x + a*c*e*(-9*d^2 - 9*d*e*x + e^2*x^2))) + 3*Sqrt[2]*(2*c*d - b*e)*(4*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(b*d + 6*a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(12*c^3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))","C",1
2543,1,177,594,0.3188253,"\int \frac{(d+e x)^2}{\left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(5/4),x]","\frac{\sqrt{2} (b+2 c x) \sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)-8 c \left(a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right)}{2 c^2 \left(b^2-4 a c\right) \sqrt[4]{a+x (b+c x)}}","\frac{2 (b+2 c x) \sqrt[4]{a+b x+c x^2} \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right)}{c^{3/2} \left(b^2-4 a c\right)^{3/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{\sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} c^{7/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}-\frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) \left(-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{c^{7/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}+\frac{4 e \left(a+b x+c x^2\right)^{3/4} (2 c d-b e)}{c \left(b^2-4 a c\right)}-\frac{4 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt[4]{a+b x+c x^2}}",1,"(-8*c*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)) + Sqrt[2]*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2*c^2*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))","C",1
2544,1,167,490,0.2187215,"\int \frac{d+e x}{\left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[(d + e*x)/(a + b*x + c*x^2)^(5/4),x]","-\frac{2 \left(2^{3/4} \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \sqrt[4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)+6 c (-2 a e+b (d-e x)+2 c d x)\right)}{3 c \left(b^2-4 a c\right) \sqrt[4]{a+x (b+c x)}}","\frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}-\frac{2 \sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) (2 c d-b e) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt[4]{b^2-4 a c} (b+2 c x)}+\frac{4 (b+2 c x) \sqrt[4]{a+b x+c x^2} (2 c d-b e)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}-\frac{4 (-2 a e+x (2 c d-b e)+b d)}{\left(b^2-4 a c\right) \sqrt[4]{a+b x+c x^2}}",1,"(-2*(6*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + 2^(3/4)*(-2*c*d + b*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(3*c*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))","C",1
2545,1,100,451,0.0656583,"\int \frac{1}{\left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[(a + b*x + c*x^2)^(-5/4),x]","-\frac{2 \sqrt{2} (b+2 c x) \left(\sqrt{2}-\sqrt[4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right)\right)}{\left(b^2-4 a c\right) \sqrt[4]{a+x (b+c x)}}","-\frac{4 (b+2 c x)}{\left(b^2-4 a c\right) \sqrt[4]{a+b x+c x^2}}+\frac{8 \sqrt{c} (b+2 c x) \sqrt[4]{a+b x+c x^2}}{\left(b^2-4 a c\right)^{3/2} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)}+\frac{2 \sqrt{2} \sqrt[4]{c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt[4]{b^2-4 a c} (b+2 c x)}-\frac{4 \sqrt{2} \sqrt[4]{c} \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt[4]{b^2-4 a c} (b+2 c x)}",1,"(-2*Sqrt[2]*(b + 2*c*x)*(Sqrt[2] - ((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/((b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))","C",1
2546,1,180,1299,0.4819054,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(5/4)),x]","-\frac{\left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{5/4} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{5/4} F_1\left(\frac{5}{2};\frac{5}{4},\frac{5}{4};\frac{7}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{10 \sqrt{2} e (a+x (b+c x))^{5/4}}","\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) e^{3/2}}{\sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{5/4} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt[4]{4 a c-b^2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) e^{3/2}}{\sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{5/4} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) e}{\sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{3/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) e}{\sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{3/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt[4]{c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt[4]{b^2-4 a c} \left(c d^2-b e d+a e^2\right) (b+2 c x)}+\frac{\sqrt{2} \sqrt[4]{c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt[4]{b^2-4 a c} \left(c d^2-b e d+a e^2\right) (b+2 c x)}-\frac{4 \left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right)}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) \sqrt[4]{c x^2+b x+a}}+\frac{4 \sqrt{c} (2 c d-b e) (b+2 c x) \sqrt[4]{c x^2+b x+a}}{\left(b^2-4 a c\right)^{3/2} \left(c d^2-b e d+a e^2\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}",1,"-1/10*(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*AppellF1[5/2, 5/4, 5/4, 7/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(Sqrt[2]*e*(a + x*(b + c*x))^(5/4))","C",0
2547,1,187,1485,0.8223342,"\int \frac{1}{(d+e x)^2 \left(a+b x+c x^2\right)^{5/4}} \, dx","Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/4)),x]","-\frac{\left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{5/4} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{5/4} F_1\left(\frac{7}{2};\frac{5}{4},\frac{5}{4};\frac{9}{2};\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{14 \sqrt{2} e (d+e x) (a+x (b+c x))^{5/4}}","-\frac{5 \sqrt{4 a c-b^2} e \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{5/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}+\frac{5 \sqrt{4 a c-b^2} e \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \Pi \left(\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left.\sin ^{-1}\left(\sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right)\right|-1\right) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^{5/2} (b+2 c x) \sqrt[4]{c x^2+b x+a}}+\frac{5 \sqrt[4]{4 a c-b^2} e^{3/2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{9/4} \sqrt[4]{c x^2+b x+a}}-\frac{5 \sqrt[4]{4 a c-b^2} e^{3/2} \sqrt[4]{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} \tanh ^{-1}\left(\frac{\sqrt[4]{4 a c-b^2} \sqrt{e} \sqrt[4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c d^2-b e d+a e^2}}\right) (2 c d-b e)}{4 \sqrt[4]{c} \left(c d^2-b e d+a e^2\right)^{9/4} \sqrt[4]{c x^2+b x+a}}-\frac{\sqrt[4]{c} \left(8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{\sqrt{2} \sqrt[4]{b^2-4 a c} \left(c d^2-b e d+a e^2\right)^2 (b+2 c x)}+\frac{\sqrt[4]{c} \left(8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right) \sqrt{\frac{(b+2 c x)^2}{\left(b^2-4 a c\right) \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)^2}} \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} \sqrt[4]{c x^2+b x+a}}{\sqrt[4]{b^2-4 a c}}\right)|\frac{1}{2}\right)}{2 \sqrt{2} \sqrt[4]{b^2-4 a c} \left(c d^2-b e d+a e^2\right)^2 (b+2 c x)}-\frac{e \left(8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right) \left(c x^2+b x+a\right)^{3/4}}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 (d+e x)}-\frac{4 \left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right)}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) (d+e x) \sqrt[4]{c x^2+b x+a}}+\frac{\sqrt{c} \left(8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right) (b+2 c x) \sqrt[4]{c x^2+b x+a}}{\left(b^2-4 a c\right)^{3/2} \left(c d^2-b e d+a e^2\right)^2 \left(\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right)}",1,"-1/14*(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*AppellF1[7/2, 5/4, 5/4, 9/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(Sqrt[2]*e*(d + e*x)*(a + x*(b + c*x))^(5/4))","C",0
2548,1,231,239,0.4013284,"\int \frac{1}{(d+e x)^{3/2} \sqrt[4]{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/4)),x]","-\frac{2 \left(\sqrt{b^2-4 a c}+b+2 c x\right) \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{\sqrt{d+e x} \sqrt[4]{a+x (b+c x)} \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{3/4}}","\frac{2 \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \sqrt[4]{\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{\sqrt{d+e x} \sqrt[4]{a+b x+c x^2} \left(e \sqrt{b^2-4 a c}-b e+2 c d\right)}",1,"(-2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometric2F1[-1/2, 1/4, 1/2, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(3/4)*Sqrt[d + e*x]*(a + x*(b + c*x))^(1/4))","A",1
2549,1,1309,485,3.9983045,"\int (d+e x)^m \left(a+b x+c x^2\right)^4 \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^4,x]","\frac{(d+e x)^{m+1} \left((a+x (b+c x))^4-\frac{4 (d+e x) (b e (m+15)+2 c (e (m+8) x-7 d)) (a+x (b+c x))^3}{e^2 (m+8) (m+9)}-\frac{12 (d+e x) \left(c e^4 (m+2) (m+3) (m+4) (m+5) (a+x (b+c x))^2 \left(-28 d^2 (5 d-e (m+6) x) c^3+2 e \left(7 b d (d (m+21)-2 e (m+6) x)+2 a e \left(d \left(m^2+3 m-33\right)+e \left(m^2+14 m+48\right) x\right)\right) c^2-b e^2 \left(-2 a e (11 m+81)+b d \left(m^2+17 m+156\right)+b e \left(m^2+7 m+6\right) x\right) c-2 b^3 e^3 (m+1)\right)-2 \left((m+2) (m+3) \left(c e (m+5) \left(c e (b d-2 a e) (m+7) \left(-d e (m+15) b^2+14 \left(c d^2+a e^2\right) b+4 a c d e (m+1)\right)-(b d (5 c d-2 b e)+a e (2 c d (m+1)-b e (m+2))) \left(28 c^2 d^2-b^2 e^2 (m+1)+4 c e (a e (m+8)-7 b d)\right)\right)-(3 c d-b e) \left(c e (2 c d-b e) (m+7) \left(-d e (m+15) b^2+14 \left(c d^2+a e^2\right) b+4 a c d e (m+1)\right)-\left(10 c^2 d^2-b^2 e^2 (m+4)+c e (b d (m-3)+2 a e (m+6))\right) \left(28 c^2 d^2-b^2 e^2 (m+1)+4 c e (a e (m+8)-7 b d)\right)\right)+c e (m+4) \left(c e (2 c d-b e) (m+7) \left(-d e (m+15) b^2+14 \left(c d^2+a e^2\right) b+4 a c d e (m+1)\right)-\left(10 c^2 d^2-b^2 e^2 (m+4)+c e (b d (m-3)+2 a e (m+6))\right) \left(28 c^2 d^2-b^2 e^2 (m+1)+4 c e (a e (m+8)-7 b d)\right)\right) x\right) (a+x (b+c x)) e^2+(2 c d-b e) (m+3) \left(840 c^5 d^6-40 c^4 e \left(63 b d+a e \left(2 m^2+6 m-59\right)\right) d^4+4 c^3 e^2 \left(5 b^2 \left(m^2+3 m+128\right) d^2+20 a b e \left(2 m^2+6 m-59\right) d-2 a^2 e^2 \left(2 m^3+34 m^2+88 m-259\right)\right) d^2+b^4 e^5 (b d-a e) \left(m^3+7 m^2+14 m+8\right)-b^2 c e^4 \left(m^2+3 m+2\right) \left(b^2 (m-16) d^2+8 a b e (m+9) d-4 a^2 e^2 (2 m+13)\right)-8 c^2 e^3 \left(5 b^3 \left(m^2+3 m+23\right) d^3-a b^2 e \left(m^3+2 m^2-m+313\right) d^2-a^2 b e^2 \left(2 m^3+34 m^2+88 m-259\right) d+a^3 e^3 \left(2 m^3+24 m^2+58 m-69\right)\right)\right)+(m+2) \left(-1680 c^6 d^6+80 c^5 e \left(63 b d+a e \left(2 m^2-m-66\right)\right) d^4-4 c^4 e^2 \left(5 b^2 \left(2 m^2-m+249\right) d^2+40 a b e \left(2 m^2-m-66\right) d-12 a^2 e^2 \left(m^3+8 m^2-11 m-123\right)\right) d^2+b^6 e^6 \left(m^3+8 m^2+19 m+12\right)-b^4 c e^5 \left(m^2+4 m+3\right) (b d (3 m-8)+a e (9 m+56))+b^2 c^2 e^4 (m+1) \left(3 b^2 \left(m^2-13 m+12\right) d^2+8 a b e \left(3 m^2+11 m-39\right) d+12 a^2 e^2 \left(2 m^2+24 m+69\right)\right)-8 c^3 e^3 \left(-5 b^3 \left(2 m^2-m+39\right) d^3+3 a b^2 e \left(m^3-2 m^2-6 m+207\right) d^2+6 a^2 b e^2 \left(m^3+8 m^2-11 m-123\right) d+2 a^3 e^3 \left(m^3+18 m^2+104 m+192\right)\right)\right) (d+e x)\right)\right)}{c^2 e^8 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (m+9)}\right)}{e (m+1)}","\frac{(d+e x)^{m+5} \left(6 c^2 e^2 \left(a^2 e^2-10 a b d e+15 b^2 d^2\right)-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right)}{e^9 (m+5)}+\frac{2 (d+e x)^{m+3} \left(a e^2-b d e+c d^2\right)^2 \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9 (m+3)}-\frac{4 (2 c d-b e) (d+e x)^{m+4} \left(a e^2-b d e+c d^2\right) \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (m+4)}-\frac{4 c (2 c d-b e) (d+e x)^{m+6} \left(-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right)}{e^9 (m+6)}+\frac{2 c^2 (d+e x)^{m+7} \left(-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right)}{e^9 (m+7)}+\frac{(d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)^4}{e^9 (m+1)}-\frac{4 (2 c d-b e) (d+e x)^{m+2} \left(a e^2-b d e+c d^2\right)^3}{e^9 (m+2)}-\frac{4 c^3 (2 c d-b e) (d+e x)^{m+8}}{e^9 (m+8)}+\frac{c^4 (d+e x)^{m+9}}{e^9 (m+9)}",1,"((d + e*x)^(1 + m)*((a + x*(b + c*x))^4 - (4*(d + e*x)*(a + x*(b + c*x))^3*(b*e*(15 + m) + 2*c*(-7*d + e*(8 + m)*x)))/(e^2*(8 + m)*(9 + m)) - (12*(d + e*x)*(-2*((2*c*d - b*e)*(3 + m)*(840*c^5*d^6 + b^4*e^5*(b*d - a*e)*(8 + 14*m + 7*m^2 + m^3) - b^2*c*e^4*(2 + 3*m + m^2)*(b^2*d^2*(-16 + m) + 8*a*b*d*e*(9 + m) - 4*a^2*e^2*(13 + 2*m)) - 40*c^4*d^4*e*(63*b*d + a*e*(-59 + 6*m + 2*m^2)) + 4*c^3*d^2*e^2*(5*b^2*d^2*(128 + 3*m + m^2) + 20*a*b*d*e*(-59 + 6*m + 2*m^2) - 2*a^2*e^2*(-259 + 88*m + 34*m^2 + 2*m^3)) - 8*c^2*e^3*(5*b^3*d^3*(23 + 3*m + m^2) - a*b^2*d^2*e*(313 - m + 2*m^2 + m^3) + a^3*e^3*(-69 + 58*m + 24*m^2 + 2*m^3) - a^2*b*d*e^2*(-259 + 88*m + 34*m^2 + 2*m^3))) + (2 + m)*(-1680*c^6*d^6 + b^6*e^6*(12 + 19*m + 8*m^2 + m^3) - b^4*c*e^5*(3 + 4*m + m^2)*(b*d*(-8 + 3*m) + a*e*(56 + 9*m)) + 80*c^5*d^4*e*(63*b*d + a*e*(-66 - m + 2*m^2)) + b^2*c^2*e^4*(1 + m)*(3*b^2*d^2*(12 - 13*m + m^2) + 12*a^2*e^2*(69 + 24*m + 2*m^2) + 8*a*b*d*e*(-39 + 11*m + 3*m^2)) - 4*c^4*d^2*e^2*(40*a*b*d*e*(-66 - m + 2*m^2) + 5*b^2*d^2*(249 - m + 2*m^2) - 12*a^2*e^2*(-123 - 11*m + 8*m^2 + m^3)) - 8*c^3*e^3*(-5*b^3*d^3*(39 - m + 2*m^2) + 3*a*b^2*d^2*e*(207 - 6*m - 2*m^2 + m^3) + 6*a^2*b*d*e^2*(-123 - 11*m + 8*m^2 + m^3) + 2*a^3*e^3*(192 + 104*m + 18*m^2 + m^3)))*(d + e*x) + e^2*(2 + m)*(3 + m)*(c*e*(5 + m)*(c*e*(b*d - 2*a*e)*(7 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*(1 + m) - b^2*d*e*(15 + m)) - (b*d*(5*c*d - 2*b*e) + a*e*(2*c*d*(1 + m) - b*e*(2 + m)))*(28*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-7*b*d + a*e*(8 + m)))) - (3*c*d - b*e)*(c*e*(2*c*d - b*e)*(7 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*(1 + m) - b^2*d*e*(15 + m)) - (10*c^2*d^2 - b^2*e^2*(4 + m) + c*e*(b*d*(-3 + m) + 2*a*e*(6 + m)))*(28*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-7*b*d + a*e*(8 + m)))) + c*e*(4 + m)*(c*e*(2*c*d - b*e)*(7 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*(1 + m) - b^2*d*e*(15 + m)) - (10*c^2*d^2 - b^2*e^2*(4 + m) + c*e*(b*d*(-3 + m) + 2*a*e*(6 + m)))*(28*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-7*b*d + a*e*(8 + m))))*x)*(a + x*(b + c*x))) + c*e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + x*(b + c*x))^2*(-2*b^3*e^3*(1 + m) - 28*c^3*d^2*(5*d - e*(6 + m)*x) - b*c*e^2*(-2*a*e*(81 + 11*m) + b*d*(156 + 17*m + m^2) + b*e*(6 + 7*m + m^2)*x) + 2*c^2*e*(7*b*d*(d*(21 + m) - 2*e*(6 + m)*x) + 2*a*e*(d*(-33 + 3*m + m^2) + e*(48 + 14*m + m^2)*x)))))/(c^2*e^8*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(9 + m))))/(e*(1 + m))","B",1
2550,1,476,305,1.6314724,"\int (d+e x)^m \left(a+b x+c x^2\right)^3 \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^3,x]","\frac{(d+e x)^{m+1} \left(\frac{3 (d+e x) \left(-2 (m+2) (d+e x) \left(2 c^2 e^2 \left(4 a^2 e^2 \left(m^2+10 m+24\right)+4 a b d e \left(m^2-2 m-33\right)+b^2 d^2 \left(m^2-2 m+57\right)\right)-2 b^2 c e^3 (m+1) (3 a e (m+5)+b d (m-3))-8 c^3 d^2 e \left(a e \left(m^2-2 m-33\right)+30 b d\right)+b^4 e^4 \left(m^2+4 m+3\right)+120 c^4 d^4\right)-2 e^2 (m+2) (m+3) (a+x (b+c x)) \left(2 c^2 e \left(2 a e \left(d \left(m^2+3 m-13\right)+e \left(m^2+10 m+24\right) x\right)+5 b d (d (m+13)-2 e (m+4) x)\right)-b c e^2 \left(-2 a e (7 m+37)+b d \left(m^2+13 m+72\right)+b e \left(m^2+5 m+4\right) x\right)-b^3 e^3 (m+1)+20 c^3 d^2 (e (m+4) x-3 d)\right)+2 (m+3) (2 c d-b e) \left(e (a e-b d)+c d^2\right) \left(-4 c e \left(a e \left(m^2+3 m-13\right)+15 b d\right)+b^2 e^2 \left(m^2+3 m+2\right)+60 c^2 d^2\right)-c e^4 (m+2) (m+3) (m+4) (m+5) (a+x (b+c x))^2 (b e (m+11)+2 c (e (m+6) x-5 d))\right)}{c e^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+(a+x (b+c x))^3\right)}{e (m+1)}","\frac{3 (d+e x)^{m+3} \left(a e^2-b d e+c d^2\right) \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 (m+3)}-\frac{(2 c d-b e) (d+e x)^{m+4} \left(-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right)}{e^7 (m+4)}+\frac{3 c (d+e x)^{m+5} \left(-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right)}{e^7 (m+5)}+\frac{(d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)^3}{e^7 (m+1)}-\frac{3 (2 c d-b e) (d+e x)^{m+2} \left(a e^2-b d e+c d^2\right)^2}{e^7 (m+2)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*((a + x*(b + c*x))^3 + (3*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(3 + m)*(60*c^2*d^2 + b^2*e^2*(2 + 3*m + m^2) - 4*c*e*(15*b*d + a*e*(-13 + 3*m + m^2))) - 2*(2 + m)*(120*c^4*d^4 + b^4*e^4*(3 + 4*m + m^2) - 2*b^2*c*e^3*(1 + m)*(b*d*(-3 + m) + 3*a*e*(5 + m)) - 8*c^3*d^2*e*(30*b*d + a*e*(-33 - 2*m + m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-33 - 2*m + m^2) + b^2*d^2*(57 - 2*m + m^2) + 4*a^2*e^2*(24 + 10*m + m^2)))*(d + e*x) - c*e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + x*(b + c*x))^2*(b*e*(11 + m) + 2*c*(-5*d + e*(6 + m)*x)) - 2*e^2*(2 + m)*(3 + m)*(a + x*(b + c*x))*(-(b^3*e^3*(1 + m)) + 20*c^3*d^2*(-3*d + e*(4 + m)*x) - b*c*e^2*(-2*a*e*(37 + 7*m) + b*d*(72 + 13*m + m^2) + b*e*(4 + 5*m + m^2)*x) + 2*c^2*e*(5*b*d*(d*(13 + m) - 2*e*(4 + m)*x) + 2*a*e*(d*(-13 + 3*m + m^2) + e*(24 + 10*m + m^2)*x)))))/(c*e^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))))/(e*(1 + m))","A",1
2551,1,178,178,0.3773921,"\int (d+e x)^m \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{2 (d+e x) \left(\frac{6 (2 c d-b e) \left(e (a e-b d)+c d^2\right)}{m+2}-\frac{(d+e x) \left(4 c e (a e (m+4)-3 b d)-b^2 e^2 (m+1)+12 c^2 d^2\right)}{m+3}\right)}{e^4 (m+4) (m+5)}-\frac{2 (d+e x) (a+x (b+c x)) (b e (m+7)-6 c d+2 c e (m+4) x)}{e^2 (m+4) (m+5)}+(a+x (b+c x))^2\right)}{e (m+1)}","\frac{(d+e x)^{m+3} \left(-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right)}{e^5 (m+3)}+\frac{(d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)^2}{e^5 (m+1)}-\frac{2 (2 c d-b e) (d+e x)^{m+2} \left(a e^2-b d e+c d^2\right)}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)}",1,"((d + e*x)^(1 + m)*((-2*(d + e*x)*(-6*c*d + b*e*(7 + m) + 2*c*e*(4 + m)*x)*(a + x*(b + c*x)))/(e^2*(4 + m)*(5 + m)) + (a + x*(b + c*x))^2 + (2*(d + e*x)*((6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(2 + m) - ((12*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-3*b*d + a*e*(4 + m)))*(d + e*x))/(3 + m)))/(e^4*(4 + m)*(5 + m))))/(e*(1 + m))","A",1
2552,1,83,82,0.0730496,"\int (d+e x)^m \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2),x]","\frac{(d+e x)^{m+1} \left(c d^2-e (b d-a e)\right)}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)}","\frac{(d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)}",1,"((c*d^2 - e*(b*d - a*e))*(d + e*x)^(1 + m))/(e^3*(1 + m)) - ((2*c*d - b*e)*(d + e*x)^(2 + m))/(e^3*(2 + m)) + (c*(d + e*x)^(3 + m))/(e^3*(3 + m))","A",1
2553,1,163,191,0.4294436,"\int \frac{(d+e x)^m}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^m/(a + b*x + c*x^2),x]","\frac{2 c (d+e x)^{m+1} \left(\frac{\, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)}{(m+1) \sqrt{b^2-4 a c}}","\frac{2 c (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{(m+1) \sqrt{b^2-4 a c} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{2 c (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{(m+1) \sqrt{b^2-4 a c} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}",1,"(2*c*(d + e*x)^(1 + m)*(-(Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)) + Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)))/(Sqrt[b^2 - 4*a*c]*(1 + m))","A",1
2554,1,339,425,1.1507415,"\int \frac{(d+e x)^m}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(d + e*x)^m/(a + b*x + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(-\frac{c \left(\frac{4 c e (a e (m-1)+b d)-b^2 e^2 m-4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right) \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{(m+1) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}-\frac{c \left(\frac{-4 c e (a e (m-1)+b d)+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right) \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{(m+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{a+x (b+c x)}\right)}{\left(b^2-4 a c\right) \left(e (a e-b d)+c d^2\right)}","\frac{c (d+e x)^{m+1} \left(-2 c e \left(d m \sqrt{b^2-4 a c}-2 a e (1-m)+2 b d\right)+b e^2 m \left(\sqrt{b^2-4 a c}+b\right)+4 c^2 d^2\right) \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{(m+1) \left(b^2-4 a c\right)^{3/2} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a e^2-b d e+c d^2\right)}-\frac{c (d+e x)^{m+1} \left(\frac{-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right) \, _2F_1\left(1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{(m+1) \left(b^2-4 a c\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \left(a e^2-b d e+c d^2\right)}-\frac{(d+e x)^{m+1} \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right)}",1,"((d + e*x)^(1 + m)*((b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/(a + x*(b + c*x)) - (c*(e*(2*c*d - b*e)*m + (-4*c^2*d^2 + 4*c*e*(b*d + a*e*(-1 + m)) - b^2*e^2*m)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(1 + m)) - (c*(e*(2*c*d - b*e)*m + (4*c^2*d^2 - 4*c*e*(b*d + a*e*(-1 + m)) + b^2*e^2*m)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(1 + m))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))","A",1
2555,0,0,189,0.1212607,"\int (d+e x)^m \left(a+b x+c x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(5/2),x]","\int (d+e x)^m \left(a+b x+c x^2\right)^{5/2} \, dx","\frac{\left(a+b x+c x^2\right)^{5/2} (d+e x)^{m+1} F_1\left(m+1;-\frac{5}{2},-\frac{5}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{5/2} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{5/2}}",1,"Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(5/2), x]","F",-1
2556,0,0,189,0.0652694,"\int (d+e x)^m \left(a+b x+c x^2\right)^{3/2} \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(3/2),x]","\int (d+e x)^m \left(a+b x+c x^2\right)^{3/2} \, dx","\frac{\left(a+b x+c x^2\right)^{3/2} (d+e x)^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{3/2} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{3/2}}",1,"Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(3/2), x]","F",-1
2557,1,207,189,0.1983004,"\int (d+e x)^m \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1) \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}}","\frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"((d + e*x)^(1 + m)*Sqrt[a + x*(b + c*x)]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)])","A",0
2558,1,207,189,0.1733014,"\int \frac{(d+e x)^m}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[a + b*x + c*x^2],x]","\frac{(d+e x)^{m+1} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1) \sqrt{a+x (b+c x)}}","\frac{(d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \sqrt{a+b x+c x^2}}",1,"(Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*Sqrt[a + x*(b + c*x)])","A",0
2559,1,267,189,0.3482458,"\int \frac{(d+e x)^m}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^m/(a + b*x + c*x^2)^(3/2),x]","-\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(\sqrt{b^2-4 a c}+b+2 c x\right) (d+e x)^{m+1} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{4 c (m+1) (a+x (b+c x))^{3/2} \left(e (a e-b d)+c d^2\right)}","\frac{(d+e x)^{m+1} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{3/2} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{3/2} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \left(a+b x+c x^2\right)^{3/2}}",1,"-1/4*(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 3/2, 3/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(c*(c*d^2 + e*(-(b*d) + a*e))*(1 + m)*(a + x*(b + c*x))^(3/2))","A",0
2560,1,225,189,0.318422,"\int \frac{(d+e x)^m}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^m/(a + b*x + c*x^2)^(5/2),x]","\frac{e^3 (d+e x)^{m+1} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} F_1\left(m+1;\frac{5}{2},\frac{5}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{(m+1) \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)^2}","\frac{(d+e x)^{m+1} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{5/2} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{5/2} F_1\left(m+1;\frac{5}{2},\frac{5}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \left(a+b x+c x^2\right)^{5/2}}",1,"(e^3*Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 5/2, 5/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/((c*d^2 + e*(-(b*d) + a*e))^2*(1 + m)*Sqrt[a + x*(b + c*x)])","A",0
2561,1,160,137,0.2590505,"\int (d x)^m \left(a+b x+c x^2\right)^p \, dx","Integrate[(d*x)^m*(a + b*x + c*x^2)^p,x]","\frac{x (d x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} (a+x (b+c x))^p F_1\left(m+1;-p,-p;m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)}{m+1}","\frac{(d x)^{m+1} \left(\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x+c x^2\right)^p F_1\left(m+1;-p,-p;m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1)}",1,"(x*(d*x)^m*(a + x*(b + c*x))^p*AppellF1[1 + m, -p, -p, 2 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
2562,1,205,187,0.3703695,"\int (d+e x)^m \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^p,x]","\frac{(d+e x)^{m+1} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1)}","\frac{(d+e x)^{m+1} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1)}",1,"((d + e*x)^(1 + m)*(a + x*(b + c*x))^p*AppellF1[1 + m, -p, -p, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p)","A",0
2563,1,558,327,1.4657431,"\int (d+e x)^3 \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^p,x]","\frac{1}{4} (a+x (b+c x))^p \left(6 d^2 e x^2 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+4 d e^2 x^3 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+e^3 x^4 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(4;-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+\frac{d^3 2^{p+1} \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p} \, _2F_1\left(-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1)}\right)","-\frac{2^{p-1} (2 c d-b e) \left(a+b x+c x^2\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right) \, _2F_1\left(-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{e \left(a+b x+c x^2\right)^{p+1} \left(-2 c (2 p+3) \left(c d^2 (2 p+5)-e (a e+b d (p+2))\right)-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right)}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac{e (d+e x)^2 \left(a+b x+c x^2\right)^{p+1}}{2 c (p+2)}",1,"((a + x*(b + c*x))^p*((6*d^2*e*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (4*d*e^2*x^3*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (e^3*x^4*AppellF1[4, -p, -p, 5, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (2^(1 + p)*d^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^p)))/4","C",0
2564,1,414,248,0.7374105,"\int (d+e x)^2 \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^2*(a + b*x + c*x^2)^p,x]","\frac{1}{6} (a+x (b+c x))^p \left(6 d e x^2 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+2 e^2 x^3 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+\frac{3 d^2 2^p \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p} \, _2F_1\left(-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1)}\right)","-\frac{2^p \left(a+b x+c x^2\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right) \, _2F_1\left(-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{e (p+2) (2 c d-b e) \left(a+b x+c x^2\right)^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac{e (d+e x) \left(a+b x+c x^2\right)^{p+1}}{c (2 p+3)}",1,"((a + x*(b + c*x))^p*((6*d*e*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (2*e^2*x^3*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (3*2^p*d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^p)))/6","C",0
2565,1,268,160,0.346799,"\int (d+e x) \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)*(a + b*x + c*x^2)^p,x]","\frac{1}{2} (a+x (b+c x))^p \left(e x^2 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right)^{-p} F_1\left(2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right)+\frac{d 2^p \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p} \, _2F_1\left(-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1)}\right)","\frac{e \left(a+b x+c x^2\right)^{p+1}}{2 c (p+1)}-\frac{2^p (2 c d-b e) \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(a+b x+c x^2\right)^{p+1} \, _2F_1\left(-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1) \sqrt{b^2-4 a c}}",1,"((a + x*(b + c*x))^p*((e*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (2^p*d*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^p)))/2","C",0
2566,1,126,122,0.0557977,"\int \left(a+b x+c x^2\right)^p \, dx","Integrate[(a + b*x + c*x^2)^p,x]","\frac{2^{p-1} \left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p} (a+x (b+c x))^p \, _2F_1\left(-p,p+1;p+2;\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{c (p+1)}","-\frac{2^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(a+b x+c x^2\right)^{p+1} \, _2F_1\left(-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{(p+1) \sqrt{b^2-4 a c}}",1,"(2^(-1 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*x))^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^p)","A",1
2567,1,182,184,0.2388759,"\int \frac{\left(a+b x+c x^2\right)^p}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^p/(d + e*x),x]","\frac{2^{2 p-1} (a+x (b+c x))^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{e p}","\frac{2^{2 p-1} \left(a+b x+c x^2\right)^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{e p}",1,"(2^(-1 + 2*p)*(a + x*(b + c*x))^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*p*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p)","A",0
2568,1,191,196,0.2491372,"\int \frac{\left(a+b x+c x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(a + b*x + c*x^2)^p/(d + e*x)^2,x]","\frac{4^p (a+x (b+c x))^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{e (2 p-1) (d+e x)}","-\frac{4^p \left(a+b x+c x^2\right)^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(1-2 p;-p,-p;2 (1-p);\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{e (1-2 p) (d+e x)}",1,"(4^p*(a + x*(b + c*x))^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(-1 + 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x))","A",0
2569,1,193,200,0.3575707,"\int \frac{\left(a+b x+c x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)^p/(d + e*x)^3,x]","\frac{2^{2 p-1} (a+x (b+c x))^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right)}{e (p-1) (d+e x)^2}","-\frac{2^{2 p-1} \left(a+b x+c x^2\right)^p \left(\frac{e \left(-\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{c (d+e x)}\right)^{-p} F_1\left(2 (1-p);-p,-p;3-2 p;\frac{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}{2 c (d+e x)},\frac{2 d-\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{c}}{2 (d+e x)}\right)}{e (1-p) (d+e x)^2}",1,"(2^(-1 + 2*p)*(a + x*(b + c*x))^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(-1 + p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x)^2)","A",0
2570,1,239,185,1.3973083,"\int (d+e x)^{3/2} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(3/2)*(a + b*x + c*x^2)^p,x]","\frac{2 (d+e x)^{5/2} (a+x (b+c x))^p \left(\frac{e \left(e \sqrt{\frac{b^2-4 a c}{e^2}}-b-2 c x\right)}{e \left(e \sqrt{\frac{b^2-4 a c}{e^2}}-b\right)+2 c d}\right)^{-p} \left(\frac{e \left(e \sqrt{\frac{b^2-4 a c}{e^2}}+b+2 c x\right)}{e \left(e \sqrt{\frac{b^2-4 a c}{e^2}}+b\right)-2 c d}\right)^{-p} F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-e \left(b+\sqrt{\frac{b^2-4 a c}{e^2}} e\right)},\frac{2 c (d+e x)}{2 c d+e \left(\sqrt{\frac{b^2-4 a c}{e^2}} e-b\right)}\right)}{5 e}","\frac{2 (d+e x)^{5/2} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(\frac{5}{2};-p,-p;\frac{7}{2};\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{5 e}",1,"(2*(d + e*x)^(5/2)*(a + x*(b + c*x))^p*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x))/(2*c*d - e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e)), (2*c*(d + e*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))])/(5*e*((e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e - 2*c*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e)))^p*((e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e + 2*c*x))/(-2*c*d + e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e)))^p)","A",0
2571,1,211,185,0.420424,"\int \sqrt{d+e x} \left(a+b x+c x^2\right)^p \, dx","Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2)^p,x]","\frac{2^{1-2 p} (d+e x)^{3/2} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{4 e \left(\sqrt{b^2-4 a c}-b\right)+8 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{3 e}","\frac{2 (d+e x)^{3/2} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 e}",1,"(2^(1 - 2*p)*(d + e*x)^(3/2)*(a + x*(b + c*x))^p*AppellF1[3/2, -p, -p, 5/2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(3*e*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(8*c*d + 4*(-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p)","A",0
2572,1,208,183,0.3075177,"\int \frac{\left(a+b x+c x^2\right)^p}{\sqrt{d+e x}} \, dx","Integrate[(a + b*x + c*x^2)^p/Sqrt[d + e*x],x]","\frac{2^{1-2 p} \sqrt{d+e x} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{4 e \left(\sqrt{b^2-4 a c}-b\right)+8 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e}","\frac{2 \sqrt{d+e x} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(\frac{1}{2};-p,-p;\frac{3}{2};\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e}",1,"(2^(1 - 2*p)*Sqrt[d + e*x]*(a + x*(b + c*x))^p*AppellF1[1/2, -p, -p, 3/2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(8*c*d + 4*(-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p)","A",0
2573,1,209,183,0.3779577,"\int \frac{\left(a+b x+c x^2\right)^p}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^p/(d + e*x)^(3/2),x]","-\frac{2^{1-2 p} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{4 e \left(\sqrt{b^2-4 a c}-b\right)+8 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(-\frac{1}{2};-p,-p;\frac{1}{2};\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e \sqrt{d+e x}}","-\frac{2 \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(-\frac{1}{2};-p,-p;\frac{1}{2};\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e \sqrt{d+e x}}",1,"-((2^(1 - 2*p)*(a + x*(b + c*x))^p*AppellF1[-1/2, -p, -p, 1/2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(8*c*d + 4*(-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p*Sqrt[d + e*x]))","A",0
2574,1,214,195,0.2906612,"\int (d+e x)^{-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(a + b*x + c*x^2)^p/(d + e*x)^(2*p),x]","-\frac{(d+e x)^{1-2 p} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (2 p-1)}","\frac{(d+e x)^{1-2 p} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (1-2 p)}",1,"-(((d + e*x)^(1 - 2*p)*(a + x*(b + c*x))^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(-1 + 2*p)*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p))","A",0
2575,1,208,190,0.35551,"\int (d+e x)^{-1-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p,x]","-\frac{(d+e x)^{-2 p} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{2 e p}","-\frac{(d+e x)^{-2 p} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 e p}",1,"-1/2*((a + x*(b + c*x))^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*p*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p*(d + e*x)^(2*p))","A",0
2576,1,242,248,0.327593,"\int (d+e x)^{-2-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-2 - 2*p)*(a + b*x + c*x^2)^p,x]","-\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) (d+e x)^{-2 p-1} (a+x (b+c x))^p \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{-p-1} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}","\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (d+e x)^{-2 p-1} \left(a+b x+c x^2\right)^p \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}",1,"-(((b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)^(-1 - 2*p)*(a + x*(b + c*x))^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + 2*p)))","A",1
2577,1,294,332,0.3018212,"\int (d+e x)^{-3-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^p,x]","\frac{(d+e x)^{-2 (p+1)} (a+x (b+c x))^p \left(\frac{(d+e x) \left(\sqrt{b^2-4 a c}+b+2 c x\right) (b e-2 c d) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{-p-1} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}-\frac{e (a+x (b+c x))}{p+1}\right)}{2 \left(e (a e-b d)+c d^2\right)}","\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (2 c d-b e) (d+e x)^{-2 p-1} \left(a+b x+c x^2\right)^p \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 (2 p+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a e^2-b d e+c d^2\right)}-\frac{e (d+e x)^{-2 (p+1)} \left(a+b x+c x^2\right)^{p+1}}{2 (p+1) \left(a e^2-b d e+c d^2\right)}",1,"((a + x*(b + c*x))^p*(-((e*(a + x*(b + c*x)))/(1 + p)) + ((-2*c*d + b*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + 2*p))))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^(2*(1 + p)))","A",1
2578,1,399,442,1.3355927,"\int (d+e x)^{-4-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-4 - 2*p)*(a + b*x + c*x^2)^p,x]","-\frac{(d+e x)^{-2 p-3} (a+x (b+c x))^p \left(\frac{(d+e x)^2 \left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{-p-1} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \left(e (a e-b d)+c d^2\right)}+\frac{e (p+2) (d+e x) (a+x (b+c x)) (2 c d-b e)}{(p+1) \left(e (a e-b d)+c d^2\right)}+2 e (a+x (b+c x))\right)}{2 (2 p+3) \left(e (a e-b d)+c d^2\right)}","\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (d+e x)^{-2 p-1} \left(a+b x+c x^2\right)^p \left(-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 (2 p+1) (2 p+3) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{e (d+e x)^{-2 p-3} \left(a+b x+c x^2\right)^{p+1}}{(2 p+3) \left(a e^2-b d e+c d^2\right)}-\frac{e (p+2) (2 c d-b e) (d+e x)^{-2 (p+1)} \left(a+b x+c x^2\right)^{p+1}}{2 (p+1) (2 p+3) \left(a e^2-b d e+c d^2\right)^2}",1,"-1/2*((d + e*x)^(-3 - 2*p)*(a + x*(b + c*x))^p*(2*e*(a + x*(b + c*x)) + (e*(2*c*d - b*e)*(2 + p)*(d + e*x)*(a + x*(b + c*x)))/((c*d^2 + e*(-(b*d) + a*e))*(1 + p)) + ((b^2*e^2*(2 + p) + 2*c^2*d^2*(3 + 2*p) - 2*c*e*(a*e + b*d*(3 + 2*p)))*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)^2*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(c*d^2 + e*(-(b*d) + a*e))*(1 + 2*p))))/((c*d^2 + e*(-(b*d) + a*e))*(3 + 2*p))","A",1
2579,1,731,577,4.8788978,"\int (d+e x)^{-5-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^p,x]","-\frac{(d+e x)^{-2 (p+2)} (a+x (b+c x))^p \left(-\frac{(p+3) (d+e x) (b e-2 c d) \left(\frac{(d+e x)^2 \left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{-p-1} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \left(e (a e-b d)+c d^2\right)}+\frac{e (p+2) (d+e x) (a+x (b+c x)) (2 c d-b e)}{(p+1) \left(e (a e-b d)+c d^2\right)}+2 e (a+x (b+c x))\right)}{(2 p+3) \left(e (a e-b d)+c d^2\right)}+\frac{2 c (d+e x)^2 \left(\frac{(d+e x) \left(\sqrt{b^2-4 a c}+b+2 c x\right) (b e-2 c d) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right)}{\left(\sqrt{b^2-4 a c}-b-2 c x\right) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}\right)^{-p-1} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(\left(b+\sqrt{b^2-4 a c}\right) e-2 c d\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}\right)}{(2 p+1) \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)}-\frac{e (a+x (b+c x))}{p+1}\right)}{e (a e-b d)+c d^2}+2 e (a+x (b+c x))\right)}{4 (p+2) \left(e (a e-b d)+c d^2\right)}","\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (2 c d-b e) (d+e x)^{-2 p-1} \left(a+b x+c x^2\right)^p \left(-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{-p} \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{4 (2 p+1) (2 p+3) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a e^2-b d e+c d^2\right)^3}-\frac{e (d+e x)^{-2 (p+1)} \left(a+b x+c x^2\right)^{p+1} \left(-2 c e \left(a e (2 p+3)+b d \left(2 p^2+8 p+9\right)\right)+b^2 e^2 \left(p^2+5 p+6\right)+2 c^2 d^2 \left(2 p^2+8 p+9\right)\right)}{4 (p+1) (p+2) (2 p+3) \left(a e^2-b d e+c d^2\right)^3}-\frac{e (p+3) (2 c d-b e) (d+e x)^{-2 p-3} \left(a+b x+c x^2\right)^{p+1}}{2 (p+2) (2 p+3) \left(a e^2-b d e+c d^2\right)^2}-\frac{e (d+e x)^{-2 (p+2)} \left(a+b x+c x^2\right)^{p+1}}{2 (p+2) \left(a e^2-b d e+c d^2\right)}",1,"-1/4*((a + x*(b + c*x))^p*(2*e*(a + x*(b + c*x)) + (2*c*(d + e*x)^2*(-((e*(a + x*(b + c*x)))/(1 + p)) + ((-2*c*d + b*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + 2*p))))/(c*d^2 + e*(-(b*d) + a*e)) - ((-2*c*d + b*e)*(3 + p)*(d + e*x)*(2*e*(a + x*(b + c*x)) + (e*(2*c*d - b*e)*(2 + p)*(d + e*x)*(a + x*(b + c*x)))/((c*d^2 + e*(-(b*d) + a*e))*(1 + p)) + ((b^2*e^2*(2 + p) + 2*c^2*d^2*(3 + 2*p) - 2*c*e*(a*e + b*d*(3 + 2*p)))*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)^2*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(c*d^2 + e*(-(b*d) + a*e))*(1 + 2*p))))/((c*d^2 + e*(-(b*d) + a*e))*(3 + 2*p))))/((c*d^2 + e*(-(b*d) + a*e))*(2 + p)*(d + e*x)^(2*(2 + p)))","A",1
2580,1,1577,809,6.2733828,"\int (d+e x)^{-6-2 p} \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]","\frac{e (d+e x)^{1-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{\left(c d^2-b e d+a e^2\right) (1-2 (p+3))}+\frac{3 c \left(\frac{e (d+e x)^{3-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{\left(c d^2-b e d+a e^2\right) (3-2 (p+3))}+\frac{\frac{(c d e-e (b e (p+2)-c d (2 p+3))) (d+e x)^{4-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{2 \left(c d^2-b e d+a e^2\right) (p+1)}-\frac{\left(b (c d e+(b e (p+2)-c d (2 p+3)) e)-2 \left(a c e^2+c d (b e (p+2)-c d (2 p+3))\right)\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right) \left(\frac{\left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)^{-p} (d+e x)^{5-2 (p+3)} \left(c x^2+b x+a\right)^p \, _2F_1\left(-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 \left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(c d^2-b e d+a e^2\right) (5-2 (p+3))}}{\left(c d^2-b e d+a e^2\right) (3-2 (p+3))}\right)+\frac{(e (b e (p+4)-c d (2 p+5))-3 c d e) \left(\frac{e (d+e x)^{2-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{\left(c d^2-b e d+a e^2\right) (2-2 (p+3))}+\frac{2 c \left(\frac{(2 c d-b e) \left(-b-2 c x+\sqrt{b^2-4 a c}\right) (d+e x)^{5-2 (p+3)} \left(c x^2+b x+a\right)^p \, _2F_1\left(-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right) \left(\frac{\left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)^{-p}}{2 \left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(c d^2-b e d+a e^2\right) (5-2 (p+3))}+\frac{e (d+e x)^{4-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{\left(c d^2-b e d+a e^2\right) (4-2 (p+3))}\right)+\frac{(e (b e (p+3)-2 c d (p+2))-2 c d e) \left(\frac{e (d+e x)^{3-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{\left(c d^2-b e d+a e^2\right) (3-2 (p+3))}+\frac{\frac{(c d e-e (b e (p+2)-c d (2 p+3))) (d+e x)^{4-2 (p+3)} \left(c x^2+b x+a\right)^{p+1}}{2 \left(c d^2-b e d+a e^2\right) (p+1)}-\frac{\left(b (c d e+(b e (p+2)-c d (2 p+3)) e)-2 \left(a c e^2+c d (b e (p+2)-c d (2 p+3))\right)\right) \left(-b-2 c x+\sqrt{b^2-4 a c}\right) \left(\frac{\left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)^{-p} (d+e x)^{5-2 (p+3)} \left(c x^2+b x+a\right)^p \, _2F_1\left(-p,5-2 (p+3);6-2 (p+3);-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 \left(2 c d-b e+\sqrt{b^2-4 a c} e\right) \left(c d^2-b e d+a e^2\right) (5-2 (p+3))}}{\left(c d^2-b e d+a e^2\right) (3-2 (p+3))}\right)}{e}}{\left(c d^2-b e d+a e^2\right) (2-2 (p+3))}\right)}{e}}{\left(c d^2-b e d+a e^2\right) (1-2 (p+3))}","-\frac{e \left(c x^2+b x+a\right)^{p+1} (d+e x)^{-2 p-5}}{\left(c d^2-b e d+a e^2\right) (2 p+5)}-\frac{e \left(2 c^2 \left(2 p^2+11 p+18\right) d^2+b^2 e^2 \left(p^2+7 p+12\right)-2 c e \left(3 a e (p+2)+b d \left(2 p^2+11 p+18\right)\right)\right) \left(c x^2+b x+a\right)^{p+1} (d+e x)^{-2 p-3}}{2 \left(c d^2-b e d+a e^2\right)^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left(4 c^4 \left(4 p^2+16 p+15\right) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left(p^2+7 p+12\right)-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left(b^2 \left(2 p^2+9 p+10\right) d^2+2 a b e (2 p+5) d+a^2 e^2\right)\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right) \left(\frac{\left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(c x^2+b x+a\right)^p \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right) (d+e x)^{-2 p-1}}{4 \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) \left(c d^2-b e d+a e^2\right)^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left(2 c^2 \left(2 p^2+7 p+8\right) d^2+b^2 e^2 \left(p^2+6 p+8\right)-2 c e \left(a e (5 p+8)+b d \left(2 p^2+7 p+8\right)\right)\right) \left(c x^2+b x+a\right)^{p+1} (d+e x)^{-2 (p+1)}}{4 \left(c d^2-b e d+a e^2\right)^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left(c x^2+b x+a\right)^{p+1} (d+e x)^{-2 (p+2)}}{2 \left(c d^2-b e d+a e^2\right)^2 (p+2) (2 p+5)}",1,"(e*(d + e*x)^(1 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(1 - 2*(3 + p))) + (3*c*((e*(d + e*x)^(3 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p))) + (((c*d*e - e*(b*e*(2 + p) - c*d*(3 + 2*p)))*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*(1 + p)) - ((-2*(a*c*e^2 + c*d*(b*e*(2 + p) - c*d*(3 + 2*p))) + b*(c*d*e + e*(b*e*(2 + p) - c*d*(3 + 2*p))))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p)))) + ((-3*c*d*e + e*(b*e*(4 + p) - c*d*(5 + 2*p)))*((e*(d + e*x)^(2 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(2 - 2*(3 + p))) + (2*c*((e*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(4 - 2*(3 + p))) + ((2*c*d - b*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p)) + ((-2*c*d*e + e*(-2*c*d*(2 + p) + b*e*(3 + p)))*((e*(d + e*x)^(3 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p))) + (((c*d*e - e*(b*e*(2 + p) - c*d*(3 + 2*p)))*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*(1 + p)) - ((-2*(a*c*e^2 + c*d*(b*e*(2 + p) - c*d*(3 + 2*p))) + b*(c*d*e + e*(b*e*(2 + p) - c*d*(3 + 2*p))))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p)))))/e)/((c*d^2 - b*d*e + a*e^2)*(2 - 2*(3 + p)))))/e)/((c*d^2 - b*d*e + a*e^2)*(1 - 2*(3 + p)))","A",1
2581,1,492,440,6.2566418,"\int (d+e x)^m \left(a+b x+c x^2\right)^{-2-\frac{m}{2}} \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(-2 - m/2),x]","\frac{\frac{(d+e x)^{m+2} \left(a+b x+c x^2\right)^{-\frac{m}{2}-1} \left(c d e-\frac{1}{2} e (2 c d (m+1)-b e m)\right)}{2 \left(-\frac{m}{2}-1\right) \left(a e^2-b d e+c d^2\right)}-\frac{\left(\sqrt{b^2-4 a c}-b-2 c x\right) (d+e x)^{m+3} \left(a+b x+c x^2\right)^{-\frac{m}{2}-2} \left(b \left(\frac{1}{2} e (2 c d (m+1)-b e m)+c d e\right)-2 \left(a c e^2+\frac{1}{2} c d (2 c d (m+1)-b e m)\right)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(e \sqrt{b^2-4 a c}-b e+2 c d\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(-e \sqrt{b^2-4 a c}-b e+2 c d\right)}\right)^{\frac{m}{2}+2} \, _2F_1\left(\frac{m}{2}+2,m+3;m+4;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-b e-\sqrt{b^2-4 a c} e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 (m+3) \left(e \sqrt{b^2-4 a c}-b e+2 c d\right) \left(a e^2-b d e+c d^2\right)}}{(m+1) \left(a e^2-b d e+c d^2\right)}+\frac{e (d+e x)^{m+1} \left(a+b x+c x^2\right)^{-\frac{m}{2}-1}}{(m+1) \left(a e^2-b d e+c d^2\right)}","-\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (d+e x)^{m+3} \left(a+b x+c x^2\right)^{-\frac{m}{2}-2} \left(4 c e (a e-b d (m+1))+b^2 e^2 m+4 c^2 d^2 (m+1)\right) \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{\frac{m+4}{2}} \, _2F_1\left(m+3,\frac{m+4}{2};m+4;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{4 (m+1) (m+3) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{e (d+e x)^{m+1} \left(a+b x+c x^2\right)^{-\frac{m}{2}-1}}{(m+1) \left(a e^2-b d e+c d^2\right)}+\frac{e m (2 c d-b e) (d+e x)^{m+2} \left(a+b x+c x^2\right)^{-\frac{m}{2}-1}}{2 (m+1) (m+2) \left(a e^2-b d e+c d^2\right)^2}",1,"(e*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(-1 - m/2))/((c*d^2 - b*d*e + a*e^2)*(1 + m)) + (((c*d*e - (e*(-(b*e*m) + 2*c*d*(1 + m)))/2)*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^(-1 - m/2))/(2*(c*d^2 - b*d*e + a*e^2)*(-1 - m/2)) - ((-2*(a*c*e^2 + (c*d*(-(b*e*m) + 2*c*d*(1 + m)))/2) + b*(c*d*e + (e*(-(b*e*m) + 2*c*d*(1 + m)))/2))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^(2 + m/2)*(d + e*x)^(3 + m)*(a + b*x + c*x^2)^(-2 - m/2)*Hypergeometric2F1[2 + m/2, 3 + m, 4 + m, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(3 + m)))/((c*d^2 - b*d*e + a*e^2)*(1 + m))","A",1
2582,1,132,102,0.0686918,"\int \frac{1}{\sqrt[3]{1+x} \sqrt[3]{1-x+x^2}} \, dx","Integrate[1/((1 + x)^(1/3)*(1 - x + x^2)^(1/3)),x]","\frac{3 \sqrt[3]{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt[3]{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} (x+1)^{2/3} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right)}{2 \sqrt[3]{x^2-x+1}}","\frac{\sqrt[3]{x^3+1} \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{x^3+1}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{x+1} \sqrt[3]{x^2-x+1}}-\frac{\sqrt[3]{x^3+1} \log \left(\sqrt[3]{x^3+1}-x\right)}{2 \sqrt[3]{x+1} \sqrt[3]{x^2-x+1}}",1,"(3*((I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3]))^(1/3)*((-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3]))^(1/3)*(1 + x)^(2/3)*AppellF1[2/3, 1/3, 1/3, 5/3, ((2*I)*(1 + x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3])])/(2*(1 - x + x^2)^(1/3))","C",0
2583,1,143,45,0.0730835,"\int \frac{1}{(1+x)^{2/3} \left(1-x+x^2\right)^{2/3}} \, dx","Integrate[1/((1 + x)^(2/3)*(1 - x + x^2)^(2/3)),x]","\frac{3 \left(2 i x+\sqrt{3}-i\right) \sqrt[3]{x+1} \left(-\frac{\left(\sqrt{3}-3 i\right) x+\sqrt{3}+3 i}{\left(\sqrt{3}+3 i\right) x+\sqrt{3}-3 i}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 i \sqrt{3} (x+1)}{\left(3 i+\sqrt{3}\right) \left(2 i x+\sqrt{3}-i\right)}\right)}{\left(\sqrt{3}-3 i\right) \left(x^2-x+1\right)^{2/3}}","\frac{x \left(x^3+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right)}{(x+1)^{2/3} \left(x^2-x+1\right)^{2/3}}",1,"(3*(-I + Sqrt[3] + (2*I)*x)*(1 + x)^(1/3)*(-((3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/(-3*I + Sqrt[3] + (3*I + Sqrt[3])*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, ((4*I)*Sqrt[3]*(1 + x))/((3*I + Sqrt[3])*(-I + Sqrt[3] + (2*I)*x))])/((-3*I + Sqrt[3])*(1 - x + x^2)^(2/3))","C",1
2584,1,132,41,0.1060759,"\int (1+x)^p \left(1-x+x^2\right)^p \, dx","Integrate[(1 + x)^p*(1 - x + x^2)^p,x]","\frac{\left(\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}\right)^{-p} \left(\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}\right)^{-p} (x+1)^{p+1} \left(x^2-x+1\right)^p F_1\left(p+1;-p,-p;p+2;\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right)}{p+1}","x (x+1)^p \left(x^2-x+1\right)^p \left(x^3+1\right)^{-p} \, _2F_1\left(\frac{1}{3},-p;\frac{4}{3};-x^3\right)",1,"((1 + x)^(1 + p)*(1 - x + x^2)^p*AppellF1[1 + p, -p, -p, 2 + p, ((2*I)*(1 + x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3])])/((1 + p)*((I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3]))^p*((-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3]))^p)","C",0
2585,1,132,109,0.0634321,"\int \frac{1}{\sqrt[3]{1-x} \sqrt[3]{1+x+x^2}} \, dx","Integrate[1/((1 - x)^(1/3)*(1 + x + x^2)^(1/3)),x]","-\frac{3 (1-x)^{2/3} \sqrt[3]{\frac{-2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \sqrt[3]{\frac{2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right)}{2 \sqrt[3]{x^2+x+1}}","\frac{\sqrt[3]{1-x^3} \log \left(\sqrt[3]{1-x^3}+x\right)}{2 \sqrt[3]{1-x} \sqrt[3]{x^2+x+1}}-\frac{\sqrt[3]{1-x^3} \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{1-x} \sqrt[3]{x^2+x+1}}",1,"(-3*(1 - x)^(2/3)*((-I + Sqrt[3] - (2*I)*x)/(-3*I + Sqrt[3]))^(1/3)*((I + Sqrt[3] + (2*I)*x)/(3*I + Sqrt[3]))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, ((-2*I)*(-1 + x))/(3*I + Sqrt[3]), ((2*I)*(-1 + x))/(-3*I + Sqrt[3])])/(2*(1 + x + x^2)^(1/3))","C",0
2586,1,145,45,0.0837747,"\int \frac{1}{(1-x)^{2/3} \left(1+x+x^2\right)^{2/3}} \, dx","Integrate[1/((1 - x)^(2/3)*(1 + x + x^2)^(2/3)),x]","-\frac{3 \sqrt[3]{1-x} \left(2 i x+\sqrt{3}+i\right) \left(\frac{\left(\sqrt{3}+3 i\right) x-\sqrt{3}+3 i}{-\left(\left(\sqrt{3}-3 i\right) x\right)+\sqrt{3}+3 i}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 i \sqrt{3} (x-1)}{\left(-3 i+\sqrt{3}\right) \left(2 i x+\sqrt{3}+i\right)}\right)}{\left(\sqrt{3}+3 i\right) \left(x^2+x+1\right)^{2/3}}","\frac{x \left(1-x^3\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right)}{(1-x)^{2/3} \left(x^2+x+1\right)^{2/3}}",1,"(-3*(1 - x)^(1/3)*(I + Sqrt[3] + (2*I)*x)*((3*I - Sqrt[3] + (3*I + Sqrt[3])*x)/(3*I + Sqrt[3] - (-3*I + Sqrt[3])*x))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, ((4*I)*Sqrt[3]*(-1 + x))/((-3*I + Sqrt[3])*(I + Sqrt[3] + (2*I)*x))])/((3*I + Sqrt[3])*(1 + x + x^2)^(2/3))","C",1
2587,1,133,41,0.1153024,"\int (1-x)^p \left(1+x+x^2\right)^p \, dx","Integrate[(1 - x)^p*(1 + x + x^2)^p,x]","\frac{(x-1) (1-x)^p \left(\frac{-2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}\right)^{-p} \left(\frac{2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}\right)^{-p} \left(x^2+x+1\right)^p F_1\left(p+1;-p,-p;p+2;\frac{2 i (x-1)}{-3 i+\sqrt{3}},-\frac{2 i (x-1)}{3 i+\sqrt{3}}\right)}{p+1}","x (1-x)^p \left(x^2+x+1\right)^p \left(1-x^3\right)^{-p} \, _2F_1\left(\frac{1}{3},-p;\frac{4}{3};x^3\right)",1,"((1 - x)^p*(-1 + x)*(1 + x + x^2)^p*AppellF1[1 + p, -p, -p, 2 + p, ((2*I)*(-1 + x))/(-3*I + Sqrt[3]), ((-2*I)*(-1 + x))/(3*I + Sqrt[3])])/((1 + p)*((-I + Sqrt[3] - (2*I)*x)/(-3*I + Sqrt[3]))^p*((I + Sqrt[3] + (2*I)*x)/(3*I + Sqrt[3]))^p)","C",0
2588,1,241,196,0.185847,"\int \frac{1}{\sqrt[3]{b e-c e x} \sqrt[3]{b^2+b c x+c^2 x^2}} \, dx","Integrate[1/((b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/3)),x]","-\frac{3 \sqrt[3]{\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}} \sqrt[3]{\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}} F_1\left(\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right) (e (b-c x))^{2/3}}{2 c e \sqrt[3]{b^2+b c x+c^2 x^2}}","\frac{\sqrt[3]{b^3 e-c^3 e x^3} \log \left(\sqrt[3]{b^3 e-c^3 e x^3}+c \sqrt[3]{e} x\right)}{2 c \sqrt[3]{e} \sqrt[3]{b^2+b c x+c^2 x^2} \sqrt[3]{b e-c e x}}-\frac{\sqrt[3]{b^3 e-c^3 e x^3} \tan ^{-1}\left(\frac{1-\frac{2 c \sqrt[3]{e} x}{\sqrt[3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right)}{\sqrt{3} c \sqrt[3]{e} \sqrt[3]{b^2+b c x+c^2 x^2} \sqrt[3]{b e-c e x}}",1,"(-3*(e*(b - c*x))^(2/3)*((b*c - Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*((b*c + Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (2*c*(b - c*x))/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]), (2*c*(b - c*x))/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)])])/(2*c*e*(b^2 + b*c*x + c^2*x^2)^(1/3))","C",0
2589,1,258,71,0.2078033,"\int \frac{1}{(b e-c e x)^{2/3} \left(b^2+b c x+c^2 x^2\right)^{2/3}} \, dx","Integrate[1/((b*e - c*e*x)^(2/3)*(b^2 + b*c*x + c^2*x^2)^(2/3)),x]","-\frac{3 \left(-\sqrt{3} \sqrt{-b^2}+b+2 c x\right) \left(\frac{-\sqrt{3} \sqrt{-b^2} c x+3 b^2+\sqrt{3} \sqrt{-b^2} b+3 b c x}{\sqrt{3} \sqrt{-b^2} c x+3 b^2-\sqrt{3} \sqrt{-b^2} b+3 b c x}\right)^{2/3} \sqrt[3]{e (b-c x)} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 \sqrt{3} \sqrt{-b^2} (b-c x)}{\left(3 b+\sqrt{3} \sqrt{-b^2}\right) \left(-b-2 c x+\sqrt{3} \sqrt{-b^2}\right)}\right)}{\left(3 b-\sqrt{3} \sqrt{-b^2}\right) c e \left(b^2+b c x+c^2 x^2\right)^{2/3}}","\frac{x \left(1-\frac{c^3 x^3}{b^3}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right)}{\left(b^2+b c x+c^2 x^2\right)^{2/3} (b e-c e x)^{2/3}}",1,"(-3*(e*(b - c*x))^(1/3)*(b - Sqrt[3]*Sqrt[-b^2] + 2*c*x)*((3*b^2 + Sqrt[3]*b*Sqrt[-b^2] + 3*b*c*x - Sqrt[3]*Sqrt[-b^2]*c*x)/(3*b^2 - Sqrt[3]*b*Sqrt[-b^2] + 3*b*c*x + Sqrt[3]*Sqrt[-b^2]*c*x))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, (4*Sqrt[3]*Sqrt[-b^2]*(b - c*x))/((3*b + Sqrt[3]*Sqrt[-b^2])*(-b + Sqrt[3]*Sqrt[-b^2] - 2*c*x))])/((3*b - Sqrt[3]*Sqrt[-b^2])*c*e*(b^2 + b*c*x + c^2*x^2)^(2/3))","B",1
2590,1,243,67,0.2985651,"\int (b e-c e x)^p \left(b^2+b c x+c^2 x^2\right)^p \, dx","Integrate[(b*e - c*e*x)^p*(b^2 + b*c*x + c^2*x^2)^p,x]","\frac{(c x-b) \left(\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right)^{-p} \left(\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}\right)^{-p} \left(b^2+b c x+c^2 x^2\right)^p F_1\left(p+1;-p,-p;p+2;\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right) (e (b-c x))^p}{c (p+1)}","x \left(b^2+b c x+c^2 x^2\right)^p \left(1-\frac{c^3 x^3}{b^3}\right)^{-p} (b e-c e x)^p \, _2F_1\left(\frac{1}{3},-p;\frac{4}{3};\frac{c^3 x^3}{b^3}\right)",1,"((e*(b - c*x))^p*(-b + c*x)*(b^2 + b*c*x + c^2*x^2)^p*AppellF1[1 + p, -p, -p, 2 + p, (2*c*(b - c*x))/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]), (2*c*(b - c*x))/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)])])/(c*(1 + p)*((b*c - Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)]))^p*((b*c + Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]))^p)","C",0