1,1,164,163,0.047183,"\int \left(d+e x^3\right)^5 \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]","\frac{5}{13} d e^2 x^{13} \left(a e^2+2 b d e+2 c d^2\right)+\frac{1}{2} d^2 e x^{10} \left(2 a e^2+2 b d e+c d^2\right)+\frac{1}{16} e^3 x^{16} \left(a e^2+5 b d e+10 c d^2\right)+\frac{1}{7} d^3 x^7 \left(10 a e^2+5 b d e+c d^2\right)+\frac{1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac{1}{19} e^4 x^{19} (b e+5 c d)+\frac{1}{22} c e^5 x^{22}","\frac{1}{16} e^3 x^{16} \left(e (a e+5 b d)+10 c d^2\right)+\frac{5}{13} d e^2 x^{13} \left(e (a e+2 b d)+2 c d^2\right)+\frac{1}{2} d^2 e x^{10} \left(2 e (a e+b d)+c d^2\right)+\frac{1}{7} d^3 x^7 \left(5 e (2 a e+b d)+c d^2\right)+\frac{1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac{1}{19} e^4 x^{19} (b e+5 c d)+\frac{1}{22} c e^5 x^{22}",1,"a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*b*d*e + 10*a*e^2)*x^7)/7 + (d^2*e*(c*d^2 + 2*b*d*e + 2*a*e^2)*x^10)/2 + (5*d*e^2*(2*c*d^2 + 2*b*d*e + a*e^2)*x^13)/13 + (e^3*(10*c*d^2 + 5*b*d*e + a*e^2)*x^16)/16 + (e^4*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22","A",1
2,1,135,135,0.0366774,"\int \left(d+e x^3\right)^4 \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^4*(a + b*x^3 + c*x^6),x]","\frac{1}{13} e^2 x^{13} \left(a e^2+4 b d e+6 c d^2\right)+\frac{1}{5} d e x^{10} \left(2 a e^2+3 b d e+2 c d^2\right)+\frac{1}{7} d^2 x^7 \left(6 a e^2+4 b d e+c d^2\right)+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19}","\frac{1}{13} e^2 x^{13} \left(e (a e+4 b d)+6 c d^2\right)+\frac{1}{7} d^2 x^7 \left(6 a e^2+4 b d e+c d^2\right)+\frac{1}{5} d e x^{10} \left(e (2 a e+3 b d)+2 c d^2\right)+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19}",1,"a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7)/7 + (d*e*(2*c*d^2 + 3*b*d*e + 2*a*e^2)*x^10)/5 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19","A",1
3,1,104,103,0.0286687,"\int \left(d+e x^3\right)^3 \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^3*(a + b*x^3 + c*x^6),x]","\frac{1}{10} e x^{10} \left(a e^2+3 b d e+3 c d^2\right)+\frac{1}{7} d x^7 \left(3 a e^2+3 b d e+c d^2\right)+\frac{1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac{1}{13} e^2 x^{13} (b e+3 c d)+\frac{1}{16} c e^3 x^{16}","\frac{1}{10} e x^{10} \left(e (a e+3 b d)+3 c d^2\right)+\frac{1}{7} d x^7 \left(3 e (a e+b d)+c d^2\right)+\frac{1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac{1}{13} e^2 x^{13} (b e+3 c d)+\frac{1}{16} c e^3 x^{16}",1,"a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^7)/7 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x^10)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16","A",1
4,1,73,73,0.0218201,"\int \left(d+e x^3\right)^2 \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]","\frac{1}{7} x^7 \left(a e^2+2 b d e+c d^2\right)+\frac{1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{13} c e^2 x^{13}","\frac{1}{7} x^7 \left(e (a e+2 b d)+c d^2\right)+\frac{1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{13} c e^2 x^{13}",1,"a*d^2*x + (d*(b*d + 2*a*e)*x^4)/4 + ((c*d^2 + 2*b*d*e + a*e^2)*x^7)/7 + (e*(2*c*d + b*e)*x^10)/10 + (c*e^2*x^13)/13","A",1
5,1,42,42,0.0086942,"\int \left(d+e x^3\right) \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)*(a + b*x^3 + c*x^6),x]","\frac{1}{4} x^4 (a e+b d)+a d x+\frac{1}{7} x^7 (b e+c d)+\frac{1}{10} c e x^{10}","\frac{1}{4} x^4 (a e+b d)+a d x+\frac{1}{7} x^7 (b e+c d)+\frac{1}{10} c e x^{10}",1,"a*d*x + ((b*d + a*e)*x^4)/4 + ((c*d + b*e)*x^7)/7 + (c*e*x^10)/10","A",1
6,1,176,188,0.1545234,"\int \frac{a+b x^3+c x^6}{d+e x^3} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3),x]","\frac{-\frac{2 \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(e (a e-b d)+c d^2\right)}{d^{2/3}}+\frac{4 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(e (a e-b d)+c d^2\right)}{d^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(e (a e-b d)+c d^2\right)}{d^{2/3}}+12 \sqrt[3]{e} x (b e-c d)+3 c e^{4/3} x^4}{12 e^{7/3}}","-\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(a e^2-b d e+c d^2\right)}{6 d^{2/3} e^{7/3}}+\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(a e^2-b d e+c d^2\right)}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(a e^2-b d e+c d^2\right)}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x (c d-b e)}{e^2}+\frac{c x^4}{4 e}",1,"(12*e^(1/3)*(-(c*d) + b*e)*x + 3*c*e^(4/3)*x^4 - (4*Sqrt[3]*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (4*(c*d^2 + e*(-(b*d) + a*e))*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (2*(c*d^2 + e*(-(b*d) + a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3))/(12*e^(7/3))","A",1
7,1,199,213,0.1915679,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^2} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^2,x]","\frac{\frac{6 \sqrt[3]{e} x \left(e (a e-b d)+c d^2\right)}{d \left(d+e x^3\right)}+\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(4 c d^2-e (2 a e+b d)\right)}{d^{5/3}}-\frac{2 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(4 c d^2-e (2 a e+b d)\right)}{d^{5/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(4 c d^2-e (2 a e+b d)\right)}{d^{5/3}}+18 c \sqrt[3]{e} x}{18 e^{7/3}}","\frac{x \left(a e^2-b d e+c d^2\right)}{3 d e^2 \left(d+e x^3\right)}+\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(4 c d^2-e (2 a e+b d)\right)}{18 d^{5/3} e^{7/3}}-\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(4 c d^2-e (2 a e+b d)\right)}{9 d^{5/3} e^{7/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(4 c d^2-e (2 a e+b d)\right)}{3 \sqrt{3} d^{5/3} e^{7/3}}+\frac{c x}{e^2}",1,"(18*c*e^(1/3)*x + (6*e^(1/3)*(c*d^2 + e*(-(b*d) + a*e))*x)/(d*(d + e*x^3)) + (2*Sqrt[3]*(4*c*d^2 - e*(b*d + 2*a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(5/3) - (2*(4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(1/3) + e^(1/3)*x])/d^(5/3) + ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(5/3))/(18*e^(7/3))","A",1
8,1,209,242,0.2669086,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^3} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^3,x]","\frac{2 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(e (5 a e+b d)+2 c d^2\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(e (5 a e+b d)+2 c d^2\right)-\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(e (5 a e+b d)+2 c d^2\right)-\frac{3 d^{2/3} \sqrt[3]{e} x \left(c d^2 \left(4 d+7 e x^3\right)-e \left(a e \left(8 d+5 e x^3\right)+b d \left(e x^3-2 d\right)\right)\right)}{\left(d+e x^3\right)^2}}{54 d^{8/3} e^{7/3}}","-\frac{x \left(7 c d^2-e (5 a e+b d)\right)}{18 d^2 e^2 \left(d+e x^3\right)}+\frac{x \left(a e^2-b d e+c d^2\right)}{6 d e^2 \left(d+e x^3\right)^2}-\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(e (5 a e+b d)+2 c d^2\right)}{54 d^{8/3} e^{7/3}}+\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(e (5 a e+b d)+2 c d^2\right)}{27 d^{8/3} e^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(e (5 a e+b d)+2 c d^2\right)}{9 \sqrt{3} d^{8/3} e^{7/3}}",1,"((-3*d^(2/3)*e^(1/3)*x*(c*d^2*(4*d + 7*e*x^3) - e*(b*d*(-2*d + e*x^3) + a*e*(8*d + 5*e*x^3))))/(d + e*x^3)^2 - 2*Sqrt[3]*(2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(1/3) + e^(1/3)*x] - (2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(54*d^(8/3)*e^(7/3))","A",1
9,1,126,132,0.0682091,"\int \frac{x^8 \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{\left(-a c e+b^2 e-b c d\right) \log \left(a+b x^3+c x^6\right)+\frac{2 \left(3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right) \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+2 c x^3 (c d-b e)+c^2 e x^6}{6 c^3}","-\frac{\left(a c e+b^2 (-e)+b c d\right) \log \left(a+b x^3+c x^6\right)}{6 c^3}-\frac{\left(3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 c^3 \sqrt{b^2-4 a c}}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^6}{6 c}",1,"(2*c*(c*d - b*e)*x^3 + c^2*e*x^6 + (2*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-(b*c*d) + b^2*e - a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)","A",1
10,1,93,97,0.0700685,"\int \frac{x^5 \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^5*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{\frac{2 \left(-2 a c e+b^2 e-b c d\right) \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+(c d-b e) \log \left(a+b x^3+c x^6\right)+2 c e x^3}{6 c^2}","\frac{\left(2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left(a+b x^3+c x^6\right)}{6 c^2}+\frac{e x^3}{3 c}",1,"(2*c*e*x^3 + (2*(-(b*c*d) + b^2*e - 2*a*c*e)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c*d - b*e)*Log[a + b*x^3 + c*x^6])/(6*c^2)","A",1
11,1,71,72,0.05109,"\int \frac{x^2 \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^2*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{e \log \left(a+b x^3+c x^6\right)-\frac{2 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{6 c}","\frac{e \log \left(a+b x^3+c x^6\right)}{6 c}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 c \sqrt{b^2-4 a c}}",1,"((-2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*Log[a + b*x^3 + c*x^6])/(6*c)","A",1
12,1,80,78,0.0352059,"\int \frac{d+e x^3}{x \left(a+b x^3+c x^6\right)} \, dx","Integrate[(d + e*x^3)/(x*(a + b*x^3 + c*x^6)),x]","\frac{d \log (x)}{a}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]}{3 a}","\frac{(b d-2 a e) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 a \sqrt{b^2-4 a c}}-\frac{d \log \left(a+b x^3+c x^6\right)}{6 a}+\frac{d \log (x)}{a}",1,"(d*Log[x])/a - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^3)/(b + 2*c*#1^3) & ]/(3*a)","C",1
13,1,130,112,0.0499395,"\int \frac{d+e x^3}{x^4 \left(a+b x^3+c x^6\right)} \, dx","Integrate[(d + e*x^3)/(x^4*(a + b*x^3 + c*x^6)),x]","\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{-\text{$\#$1}^3 a c e \log (x-\text{$\#$1})+\text{$\#$1}^3 b c d \log (x-\text{$\#$1})-a b e \log (x-\text{$\#$1})-a c d \log (x-\text{$\#$1})+b^2 d \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]}{3 a^2}+\frac{\log (x) (a e-b d)}{a^2}-\frac{d}{3 a x^3}","-\frac{\left(-a b e-2 a c d+b^2 d\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left(a+b x^3+c x^6\right)}{6 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{3 a x^3}",1,"-1/3*d/(a*x^3) + ((-(b*d) + a*e)*Log[x])/a^2 + RootSum[a + b*#1^3 + c*#1^6 & , (b^2*d*Log[x - #1] - a*c*d*Log[x - #1] - a*b*e*Log[x - #1] + b*c*d*Log[x - #1]*#1^3 - a*c*e*Log[x - #1]*#1^3)/(b + 2*c*#1^3) & ]/(3*a^2)","C",1
14,1,88,723,0.0477408,"\int \frac{x^4 \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^4*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{3 e x^2-2 \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b e \log (x-\text{$\#$1})+\text{$\#$1}^3 (-c) d \log (x-\text{$\#$1})+a e \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]}{6 c}","\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{e x^2}{2 c}",1,"(3*e*x^2 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (a*e*Log[x - #1] - c*d*Log[x - #1]*#1^3 + b*e*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ])/(6*c)","C",1
15,1,88,718,0.0502066,"\int \frac{x^3 \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^3*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{e x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b e \log (x-\text{$\#$1})+\text{$\#$1}^3 (-c) d \log (x-\text{$\#$1})+a e \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 c}","-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{e x}{c}",1,"(e*x)/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*e*Log[x - #1] - c*d*Log[x - #1]*#1^3 + b*e*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*c)","C",1
16,1,59,634,0.0296719,"\int \frac{x \left(d+e x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x*(d + e*x^3))/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]","\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , (d*Log[x - #1] + e*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/3","C",1
17,1,61,634,0.0296789,"\int \frac{d+e x^3}{a+b x^3+c x^6} \, dx","Integrate[(d + e*x^3)/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]","-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , (d*Log[x - #1] + e*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/3","C",1
18,1,85,653,0.0465666,"\int \frac{d+e x^3}{x^2 \left(a+b x^3+c x^6\right)} \, dx","Integrate[(d + e*x^3)/(x^2*(a + b*x^3 + c*x^6)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]}{3 a}-\frac{d}{a x}","-\frac{\sqrt[3]{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt[3]{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt[3]{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt[3]{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{d}{a x}",1,"-(d/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/(3*a)","C",1
19,1,89,655,0.0470883,"\int \frac{d+e x^3}{x^3 \left(a+b x^3+c x^6\right)} \, dx","Integrate[(d + e*x^3)/(x^3*(a + b*x^3 + c*x^6)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 a}-\frac{d}{2 a x^2}","\frac{c^{2/3} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{c^{2/3} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{c^{2/3} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{c^{2/3} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{d}{2 a x^2}",1,"-1/2*d/(a*x^2) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*a)","C",1
20,1,46,46,0.016197,"\int \frac{x^8 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]","-\frac{x^6}{6}+\frac{\tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{6} \log \left(x^6-x^3+1\right)","-\frac{x^6}{6}-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{6} \log \left(x^6-x^3+1\right)",1,"-1/6*x^6 + ArcTan[(-1 + 2*x^3)/Sqrt[3]]/(3*Sqrt[3]) + Log[1 - x^3 + x^6]/6","A",1
21,1,31,31,0.0080728,"\int \frac{x^5 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^5*(1 - x^3))/(1 - x^3 + x^6),x]","\frac{2 \tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{x^3}{3}","-\frac{x^3}{3}-\frac{2 \tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"-1/3*x^3 + (2*ArcTan[(-1 + 2*x^3)/Sqrt[3]])/(3*Sqrt[3])","A",1
22,1,39,39,0.0090086,"\int \frac{x^2 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^2*(1 - x^3))/(1 - x^3 + x^6),x]","\frac{\tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)","-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)",1,"ArcTan[(-1 + 2*x^3)/Sqrt[3]]/(3*Sqrt[3]) - Log[1 - x^3 + x^6]/6","A",1
23,1,44,41,0.0131791,"\int \frac{1-x^3}{x \left(1-x^3+x^6\right)} \, dx","Integrate[(1 - x^3)/(x*(1 - x^3 + x^6)),x]","\log (x)-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^3) & ]/3","C",1
24,1,45,31,0.0125757,"\int \frac{1-x^3}{x^4 \left(1-x^3+x^6\right)} \, dx","Integrate[(1 - x^3)/(x^4*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{1}{3 x^3}","\frac{2 \tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{3 x^3}",1,"-1/3*1/x^3 - RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-1 + 2*#1^3) & ]/3","C",1
25,1,47,418,0.0114779,"\int \frac{x^6 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^6*(1 - x^3))/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{x^4}{4}","-\frac{x^4}{4}-\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}+\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"-1/4*x^4 + RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1)/(-1 + 2*#1^3) & ]/3","C",1
26,1,48,382,0.0132492,"\int \frac{x^4 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^4*(1 - x^3))/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right]-\frac{x^2}{2}","-\frac{x^2}{2}-\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{3\ 2^{2/3} \sqrt{3} \sqrt[3]{1-i \sqrt{3}}}+\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{3\ 2^{2/3} \sqrt{3} \sqrt[3]{1+i \sqrt{3}}}+\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{3 \sqrt{3} \sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}-\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{3 \sqrt{3} \sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}+\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}-\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}",1,"-1/2*x^2 + RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-#1 + 2*#1^4) & ]/3","C",1
27,1,46,378,0.0126473,"\int \frac{x^3 \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^3*(1 - x^3))/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right]-x","-\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{3 \sqrt[3]{2} \sqrt{3} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{3 \sqrt[3]{2} \sqrt{3} \left(1+i \sqrt{3}\right)^{2/3}}-x+\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{3 \sqrt{3} \left(\frac{1}{2} \left(1-i \sqrt{3}\right)\right)^{2/3}}-\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{3 \sqrt{3} \left(\frac{1}{2} \left(1+i \sqrt{3}\right)\right)^{2/3}}-\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \left(\frac{1}{2} \left(1-i \sqrt{3}\right)\right)^{2/3}}+\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \left(\frac{1}{2} \left(1+i \sqrt{3}\right)\right)^{2/3}}",1,"-x + RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-#1^2 + 2*#1^5) & ]/3","C",1
28,1,55,411,0.0125399,"\int \frac{x \left(1-x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x*(1 - x^3))/(1 - x^3 + x^6),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right]","\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}",1,"-1/3*RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1 + 2*#1^4) & ]","C",1
29,1,57,411,0.0115961,"\int \frac{1-x^3}{1-x^3+x^6} \, dx","Integrate[(1 - x^3)/(1 - x^3 + x^6),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right]","\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"-1/3*RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1^2 + 2*#1^5) & ]","C",1
30,1,47,416,0.0137628,"\int \frac{1-x^3}{x^2 \left(1-x^3+x^6\right)} \, dx","Integrate[(1 - x^3)/(x^2*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{1}{x}","\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}",1,"-x^(-1) - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^3) & ]/3","C",1
31,1,47,418,0.0124913,"\int \frac{1-x^3}{x^3 \left(1-x^3+x^6\right)} \, dx","Integrate[(1 - x^3)/(x^3*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{1}{2 x^2}","-\frac{1}{2 x^2}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"-1/2*1/x^2 - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1)/(-1 + 2*#1^3) & ]/3","C",1
32,1,37,36,0.0104091,"\int \frac{x^2 \left(-2+x^3\right)}{1-x^3+x^6} \, dx","Integrate[(x^2*(-2 + x^3))/(1 - x^3 + x^6),x]","\frac{1}{6} \log \left(x^6-x^3+1\right)-\frac{\tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{6} \log \left(x^6-x^3+1\right)",1,"-(ArcTan[(-1 + 2*x^3)/Sqrt[3]]/Sqrt[3]) + Log[1 - x^3 + x^6]/6","A",1
33,1,55,39,0.0135703,"\int \frac{1+x^3}{x \left(1-x^3+x^6\right)} \, dx","Integrate[(1 + x^3)/(x*(1 - x^3 + x^6)),x]","\log (x)-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^3 + #1^6 & , (-2*Log[x - #1] + Log[x - #1]*#1^3)/(-1 + 2*#1^3) & ]/3","C",1
34,1,55,39,0.0100219,"\int \frac{1+x^3}{x-x^4+x^7} \, dx","Integrate[(1 + x^3)/(x - x^4 + x^7),x]","\log (x)-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^3 + #1^6 & , (-2*Log[x - #1] + Log[x - #1]*#1^3)/(-1 + 2*#1^3) & ]/3","C",1
35,1,103,396,0.1760979,"\int \left(d+e x^3\right)^{5/2} \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^(5/2)*(a + b*x^3 + c*x^6),x]","\frac{x \sqrt{d+e x^3} \left(\frac{\, _2F_1\left(-\frac{5}{2},\frac{1}{3};\frac{4}{3};-\frac{e x^3}{d}\right) \left(29 d^2 e (23 a e-2 b d)+16 c d^4\right)}{\sqrt{\frac{e x^3}{d}+1}}-2 \left(d+e x^3\right)^3 \left(-29 b e+8 c d-23 c e x^3\right)\right)}{667 e^2}","\frac{2 x \left(d+e x^3\right)^{5/2} \left(667 a e^2-58 b d e+16 c d^2\right)}{11339 e^2}+\frac{30 d x \left(d+e x^3\right)^{3/2} \left(667 a e^2-58 b d e+16 c d^2\right)}{124729 e^2}+\frac{54 d^2 x \sqrt{d+e x^3} \left(667 a e^2-58 b d e+16 c d^2\right)}{124729 e^2}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(667 a e^2-58 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{124729 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}-\frac{2 x \left(d+e x^3\right)^{7/2} (8 c d-29 b e)}{667 e^2}+\frac{2 c x^4 \left(d+e x^3\right)^{7/2}}{29 e}",1,"(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)^3*(8*c*d - 29*b*e - 23*c*e*x^3) + ((16*c*d^4 + 29*d^2*e*(-2*b*d + 23*a*e))*Hypergeometric2F1[-5/2, 1/3, 4/3, -((e*x^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(667*e^2)","C",1
36,1,101,356,0.149762,"\int \left(d+e x^3\right)^{3/2} \left(a+b x^3+c x^6\right) \, dx","Integrate[(d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6),x]","\frac{x \sqrt{d+e x^3} \left(\frac{\, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{e x^3}{d}\right) \left(23 d e (17 a e-2 b d)+16 c d^3\right)}{\sqrt{\frac{e x^3}{d}+1}}-2 \left(d+e x^3\right)^2 \left(-23 b e+8 c d-17 c e x^3\right)\right)}{391 e^2}","\frac{2 x \left(d+e x^3\right)^{3/2} \left(391 a e^2-46 b d e+16 c d^2\right)}{4301 e^2}+\frac{18 d x \sqrt{d+e x^3} \left(391 a e^2-46 b d e+16 c d^2\right)}{21505 e^2}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(391 a e^2-46 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{21505 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}-\frac{2 x \left(d+e x^3\right)^{5/2} (8 c d-23 b e)}{391 e^2}+\frac{2 c x^4 \left(d+e x^3\right)^{5/2}}{23 e}",1,"(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)^2*(8*c*d - 23*b*e - 17*c*e*x^3) + ((16*c*d^3 + 23*d*e*(-2*b*d + 17*a*e))*Hypergeometric2F1[-3/2, 1/3, 4/3, -((e*x^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(391*e^2)","C",1
37,1,98,316,0.1276578,"\int \sqrt{d+e x^3} \left(a+b x^3+c x^6\right) \, dx","Integrate[Sqrt[d + e*x^3]*(a + b*x^3 + c*x^6),x]","\frac{x \sqrt{d+e x^3} \left(\frac{\, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{e x^3}{d}\right) \left(17 e (11 a e-2 b d)+16 c d^2\right)}{\sqrt{\frac{e x^3}{d}+1}}-2 \left(d+e x^3\right) \left(-17 b e+8 c d-11 c e x^3\right)\right)}{187 e^2}","\frac{2 x \sqrt{d+e x^3} \left(187 a e^2-34 b d e+16 c d^2\right)}{935 e^2}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} d \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(187 a e^2-34 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{935 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}-\frac{2 x \left(d+e x^3\right)^{3/2} (8 c d-17 b e)}{187 e^2}+\frac{2 c x^4 \left(d+e x^3\right)^{3/2}}{17 e}",1,"(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)*(8*c*d - 17*b*e - 11*c*e*x^3) + ((16*c*d^2 + 17*e*(-2*b*d + 11*a*e))*Hypergeometric2F1[-1/2, 1/3, 4/3, -((e*x^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(187*e^2)","C",1
38,1,98,278,0.0902939,"\int \frac{a+b x^3+c x^6}{\sqrt{d+e x^3}} \, dx","Integrate[(a + b*x^3 + c*x^6)/Sqrt[d + e*x^3],x]","\frac{x \left(\sqrt{\frac{e x^3}{d}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right) \left(11 e (5 a e-2 b d)+16 c d^2\right)-2 \left(d+e x^3\right) \left(-11 b e+8 c d-5 c e x^3\right)\right)}{55 e^2 \sqrt{d+e x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(55 a e^2-22 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{55 \sqrt[4]{3} e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}-\frac{2 x \sqrt{d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e}",1,"(x*(-2*(d + e*x^3)*(8*c*d - 11*b*e - 5*c*e*x^3) + (16*c*d^2 + 11*e*(-2*b*d + 5*a*e))*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)]))/(55*e^2*Sqrt[d + e*x^3])","C",1
39,1,102,289,0.104712,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^{3/2}} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(3/2),x]","\frac{x \left(\sqrt{\frac{e x^3}{d}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right) \left(5 e (a e+2 b d)-16 c d^2\right)+2 \left(5 e (a e-b d)+c d \left(8 d+3 e x^3\right)\right)\right)}{15 d e^2 \sqrt{d+e x^3}}","\frac{2 x \left(a e^2-b d e+c d^2\right)}{3 d e^2 \sqrt{d+e x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(16 c d^2-5 e (a e+2 b d)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{15 \sqrt[4]{3} d e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}+\frac{2 c x \sqrt{d+e x^3}}{5 e^2}",1,"(x*(2*(5*e*(-(b*d) + a*e) + c*d*(8*d + 3*e*x^3)) + (-16*c*d^2 + 5*e*(2*b*d + a*e))*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)]))/(15*d*e^2*Sqrt[d + e*x^3])","C",1
40,1,129,309,0.1381771,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^{5/2}} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x]","\frac{x \left(d+e x^3\right) \sqrt{\frac{e x^3}{d}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right) \left(e (7 a e+2 b d)+16 c d^2\right)-2 x \left(e \left(b d \left(d-2 e x^3\right)-a e \left(10 d+7 e x^3\right)\right)+c d^2 \left(8 d+11 e x^3\right)\right)}{27 d^2 e^2 \left(d+e x^3\right)^{3/2}}","-\frac{2 x \left(-7 a e^2-2 b d e+11 c d^2\right)}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 x \left(a e^2-b d e+c d^2\right)}{9 d e^2 \left(d+e x^3\right)^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(e (7 a e+2 b d)+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} d^2 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}",1,"(-2*x*(c*d^2*(8*d + 11*e*x^3) + e*(b*d*(d - 2*e*x^3) - a*e*(10*d + 7*e*x^3))) + (16*c*d^2 + e*(2*b*d + 7*a*e))*x*(d + e*x^3)*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(27*d^2*e^2*(d + e*x^3)^(3/2))","C",1
41,1,166,349,0.1882479,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^{7/2}} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x]","\frac{2 x \left(e \left(a e \left(157 d^2+221 d e x^3+91 e^2 x^6\right)+b d \left(-7 d^2+34 d e x^3+14 e^2 x^6\right)\right)+c d^2 \left(-8 d^2-19 d e x^3+16 e^2 x^6\right)\right)+x \sqrt{\frac{e x^3}{d}+1} \left(d+e x^3\right)^2 \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right) \left(7 e (13 a e+2 b d)+16 c d^2\right)}{405 d^3 e^2 \left(d+e x^3\right)^{5/2}}","-\frac{2 x \left(-13 a e^2-2 b d e+17 c d^2\right)}{135 d^2 e^2 \left(d+e x^3\right)^{3/2}}+\frac{2 x \left(a e^2-b d e+c d^2\right)}{15 d e^2 \left(d+e x^3\right)^{5/2}}+\frac{2 x \left(91 a e^2+14 b d e+16 c d^2\right)}{405 d^3 e^2 \sqrt{d+e x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(91 a e^2+14 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{405 \sqrt[4]{3} d^3 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}",1,"(2*x*(c*d^2*(-8*d^2 - 19*d*e*x^3 + 16*e^2*x^6) + e*(b*d*(-7*d^2 + 34*d*e*x^3 + 14*e^2*x^6) + a*e*(157*d^2 + 221*d*e*x^3 + 91*e^2*x^6))) + (16*c*d^2 + 7*e*(2*b*d + 13*a*e))*x*(d + e*x^3)^2*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(405*d^3*e^2*(d + e*x^3)^(5/2))","C",1
42,1,200,389,0.2382048,"\int \frac{a+b x^3+c x^6}{\left(d+e x^3\right)^{9/2}} \, dx","Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]","\frac{7 x \sqrt{\frac{e x^3}{d}+1} \left(d+e x^3\right)^3 \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right) \left(13 e (19 a e+2 b d)+16 c d^2\right)+2 x \left(e \left(a e \left(3388 d^3+7182 d^2 e x^3+5928 d e^2 x^6+1729 e^3 x^9\right)+b d \left(-91 d^3+756 d^2 e x^3+624 d e^2 x^6+182 e^3 x^9\right)\right)+c d^2 \left(-56 d^3-189 d^2 e x^3+384 d e^2 x^6+112 e^3 x^9\right)\right)}{8505 d^4 e^2 \left(d+e x^3\right)^{7/2}}","-\frac{2 x \left(-19 a e^2-2 b d e+23 c d^2\right)}{315 d^2 e^2 \left(d+e x^3\right)^{5/2}}+\frac{2 x \left(a e^2-b d e+c d^2\right)}{21 d e^2 \left(d+e x^3\right)^{7/2}}+\frac{2 x \left(247 a e^2+26 b d e+16 c d^2\right)}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 x \left(247 a e^2+26 b d e+16 c d^2\right)}{2835 d^3 e^2 \left(d+e x^3\right)^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \sqrt{\frac{d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \left(247 a e^2+26 b d e+16 c d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{e} x+\left(1-\sqrt{3}\right) \sqrt[3]{d}}{\sqrt[3]{e} x+\left(1+\sqrt{3}\right) \sqrt[3]{d}}\right)|-7-4 \sqrt{3}\right)}{1215 \sqrt[4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt[3]{d} \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{d}+\sqrt[3]{e} x\right)^2}} \sqrt{d+e x^3}}",1,"(2*x*(c*d^2*(-56*d^3 - 189*d^2*e*x^3 + 384*d*e^2*x^6 + 112*e^3*x^9) + e*(b*d*(-91*d^3 + 756*d^2*e*x^3 + 624*d*e^2*x^6 + 182*e^3*x^9) + a*e*(3388*d^3 + 7182*d^2*e*x^3 + 5928*d*e^2*x^6 + 1729*e^3*x^9))) + 7*(16*c*d^2 + 13*e*(2*b*d + 19*a*e))*x*(d + e*x^3)^3*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(8505*d^4*e^2*(d + e*x^3)^(7/2))","C",1
43,1,88,433,0.0753849,"\int \frac{x^4 \left(d+e x^4\right)}{a+b x^4+c x^8} \, dx","Integrate[(x^4*(d + e*x^4))/(a + b*x^4 + c*x^8),x]","\frac{e x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b e \log (x-\text{$\#$1})+\text{$\#$1}^4 (-c) d \log (x-\text{$\#$1})+a e \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 c}","-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{e x}{c}",1,"(e*x)/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*e*Log[x - #1] - c*d*Log[x - #1]*#1^4 + b*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)","C",1
44,1,71,72,0.0569052,"\int \frac{x^3 \left(d+e x^4\right)}{a+b x^4+c x^8} \, dx","Integrate[(x^3*(d + e*x^4))/(a + b*x^4 + c*x^8),x]","\frac{e \log \left(a+b x^4+c x^8\right)-\frac{2 (b e-2 c d) \tan ^{-1}\left(\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{8 c}","\frac{e \log \left(a+b x^4+c x^8\right)}{8 c}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 c \sqrt{b^2-4 a c}}",1,"((-2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + e*Log[a + b*x^4 + c*x^8])/(8*c)","A",1
45,1,59,375,0.045731,"\int \frac{x^2 \left(d+e x^4\right)}{a+b x^4+c x^8} \, dx","Integrate[(x^2*(d + e*x^4))/(a + b*x^4 + c*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]","\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/4","C",1
46,1,179,184,0.1522152,"\int \frac{x \left(d+e x^4\right)}{a+b x^4+c x^8} \, dx","Integrate[(x*(d + e*x^4))/(a + b*x^4 + c*x^8),x]","\frac{\frac{\left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}","\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]])/(2*Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
47,1,61,375,0.0471409,"\int \frac{d+e x^4}{a+b x^4+c x^8} \, dx","Integrate[(d + e*x^4)/(a + b*x^4 + c*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]","-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/4","C",1
48,1,80,78,0.0331838,"\int \frac{d+e x^4}{x \left(a+b x^4+c x^8\right)} \, dx","Integrate[(d + e*x^4)/(x*(a + b*x^4 + c*x^8)),x]","\frac{d \log (x)}{a}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right]}{4 a}","\frac{(b d-2 a e) \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 a \sqrt{b^2-4 a c}}-\frac{d \log \left(a+b x^4+c x^8\right)}{8 a}+\frac{d \log (x)}{a}",1,"(d*Log[x])/a - RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a)","C",1
49,1,85,392,0.0637975,"\int \frac{d+e x^4}{x^2 \left(a+b x^4+c x^8\right)} \, dx","Integrate[(d + e*x^4)/(x^2*(a + b*x^4 + c*x^8)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{4 a}-\frac{d}{a x}","-\frac{\sqrt[4]{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{d}{a x}",1,"-(d/(a*x)) - RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(4*a)","C",1
50,1,89,199,0.0454711,"\int \frac{d+e x^4}{x^3 \left(a+b x^4+c x^8\right)} \, dx","Integrate[(d + e*x^4)/(x^3*(a + b*x^4 + c*x^8)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^6 c+\text{$\#$1}^2 b}\&\right]}{4 a}-\frac{d}{2 a x^2}","-\frac{\sqrt{c} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{d}{2 a x^2}",1,"-1/2*d/(a*x^2) - RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^4)/(b*#1^2 + 2*c*#1^6) & ]/(4*a)","C",1
51,1,86,394,0.0708395,"\int \frac{d+e x^4}{x^4 \left(a+b x^4+c x^8\right)} \, dx","Integrate[(d + e*x^4)/(x^4*(a + b*x^4 + c*x^8)),x]","-\frac{3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]+\frac{4 d}{x^3}}{12 a}","\frac{c^{3/4} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{d}{3 a x^3}",1,"-1/12*((4*d)/x^3 + 3*RootSum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/a","C",1
52,1,46,278,0.0157423,"\int \frac{x^4 \left(1-x^4\right)}{1-x^4+x^8} \, dx","Integrate[(x^4*(1 - x^4))/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]-x","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-x-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}",1,"-x + RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1^3 + 2*#1^7) & ]/4","C",1
53,1,39,39,0.0135931,"\int \frac{x^3 \left(1-x^4\right)}{1-x^4+x^8} \, dx","Integrate[(x^3*(1 - x^4))/(1 - x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{2 x^4-1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8-x^4+1\right)","-\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8-x^4+1\right)",1,"ArcTan[(-1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 - x^4 + x^8]/8","A",1
54,1,55,355,0.0158953,"\int \frac{x^2 \left(1-x^4\right)}{1-x^4+x^8} \, dx","Integrate[(x^2*(1 - x^4))/(1 - x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\&\right]","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"-1/4*RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1 + 2*#1^5) & ]","C",1
55,1,44,50,0.0161403,"\int \frac{x \left(1-x^4\right)}{1-x^4+x^8} \, dx","Integrate[(x*(1 - x^4))/(1 - x^4 + x^8),x]","\frac{\log \left(x^4+\sqrt{3} x^2+1\right)-\log \left(-x^4+\sqrt{3} x^2-1\right)}{4 \sqrt{3}}","\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{4 \sqrt{3}}-\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{4 \sqrt{3}}",1,"(-Log[-1 + Sqrt[3]*x^2 - x^4] + Log[1 + Sqrt[3]*x^2 + x^4])/(4*Sqrt[3])","A",1
56,1,57,355,0.0138661,"\int \frac{1-x^4}{1-x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"-1/4*RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]","C",1
57,1,44,41,0.0132694,"\int \frac{1-x^4}{x \left(1-x^4+x^8\right)} \, dx","Integrate[(1 - x^4)/(x*(1 - x^4 + x^8)),x]","\log (x)-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]","\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8-x^4+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4","C",1
58,1,47,280,0.0153478,"\int \frac{1-x^4}{x^2 \left(1-x^4+x^8\right)} \, dx","Integrate[(1 - x^4)/(x^2*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{x}","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}",1,"-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^4) & ]/4","C",1
59,1,49,89,0.0160389,"\int \frac{1-x^4}{x^3 \left(1-x^4+x^8\right)} \, dx","Integrate[(1 - x^4)/(x^3*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{2 x^2}","-\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x^2\right)-\frac{1}{4} \tan ^{-1}\left(2 x^2+\sqrt{3}\right)-\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{8 \sqrt{3}}",1,"-1/2*1/x^2 - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^4) & ]/4","C",1
60,1,47,370,0.0149841,"\int \frac{1-x^4}{x^4 \left(1-x^4+x^8\right)} \, dx","Integrate[(1 - x^4)/(x^4*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{3 x^3}","-\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"-1/3*1/x^3 - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4","C",1
61,1,283,280,0.2287466,"\int \frac{x^3}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)} \, dx","Integrate[x^3/((a + c/x^2 + b/x)*(d + e*x)),x]","-\frac{x^2 (a d+b e)}{2 a^2 e^2}+\frac{\left(a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right) \log \left(a x^2+b x+c\right)}{2 a^4 \left(a d^2-b d e+c e^2\right)}+\frac{\left(5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a^4 \sqrt{4 a c-b^2} \left(-a d^2+b d e-c e^2\right)}+\frac{x \left(a^2 d^2+a b d e-a c e^2+b^2 e^2\right)}{a^3 e^3}-\frac{d^5 \log (d+e x)}{e^4 \left(a d^2-b d e+c e^2\right)}+\frac{x^3}{3 a e}","-\frac{x^2 (a d+b e)}{2 a^2 e^2}+\frac{\left(a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right) \log \left(a x^2+b x+c\right)}{2 a^4 \left(a d^2-e (b d-c e)\right)}+\frac{\left(5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a^4 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{x \left(a^2 d^2+a e (b d-c e)+b^2 e^2\right)}{a^3 e^3}-\frac{d^5 \log (d+e x)}{e^4 \left(a d^2-e (b d-c e)\right)}+\frac{x^3}{3 a e}",1,"((a^2*d^2 + a*b*d*e + b^2*e^2 - a*c*e^2)*x)/(a^3*e^3) - ((a*d + b*e)*x^2)/(2*a^2*e^2) + x^3/(3*a*e) + ((b^5*d - 5*a*b^3*c*d + 5*a^2*b*c^2*d - b^4*c*e + 4*a*b^2*c^2*e - 2*a^2*c^3*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^4*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + b*d*e - c*e^2)) - (d^5*Log[d + e*x])/(e^4*(a*d^2 - b*d*e + c*e^2)) + ((b^4*d - 3*a*b^2*c*d + a^2*c^2*d - b^3*c*e + 2*a*b*c^2*e)*Log[c + b*x + a*x^2])/(2*a^4*(a*d^2 - b*d*e + c*e^2))","A",1
62,1,218,218,0.1721859,"\int \frac{x^2}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)} \, dx","Integrate[x^2/((a + c/x^2 + b/x)*(d + e*x)),x]","\frac{\left(2 a b c d-a c^2 e+b^3 (-d)+b^2 c e\right) \log (x (a x+b)+c)}{2 a^3 \left(a d^2+e (c e-b d)\right)}-\frac{x (a d+b e)}{a^2 e^2}+\frac{\left(2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a^3 \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)}+\frac{d^4 \log (d+e x)}{e^3 \left(a d^2+e (c e-b d)\right)}+\frac{x^2}{2 a e}","-\frac{\left(-2 a b c d+a c^2 e+b^3 d-b^2 c e\right) \log \left(a x^2+b x+c\right)}{2 a^3 \left(a d^2-e (b d-c e)\right)}-\frac{x (a d+b e)}{a^2 e^2}-\frac{\left(2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a^3 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{d^4 \log (d+e x)}{e^3 \left(a d^2-e (b d-c e)\right)}+\frac{x^2}{2 a e}",1,"-(((a*d + b*e)*x)/(a^2*e^2)) + x^2/(2*a*e) + ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - b^3*c*e + 3*a*b*c^2*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))) + (d^4*Log[d + e*x])/(e^3*(a*d^2 + e*(-(b*d) + c*e))) + ((-(b^3*d) + 2*a*b*c*d + b^2*c*e - a*c^2*e)*Log[c + x*(b + a*x)])/(2*a^3*(a*d^2 + e*(-(b*d) + c*e)))","A",1
63,1,178,176,0.1850941,"\int \frac{x}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)} \, dx","Integrate[x/((a + c/x^2 + b/x)*(d + e*x)),x]","\frac{\left(-a c d+b^2 d-b c e\right) \log \left(a x^2+b x+c\right)}{2 a^2 \left(a d^2-b d e+c e^2\right)}+\frac{\left(-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a^2 \sqrt{4 a c-b^2} \left(-a d^2+b d e-c e^2\right)}-\frac{d^3 \log (d+e x)}{e^2 \left(a d^2-b d e+c e^2\right)}+\frac{x}{a e}","\frac{\left(-a c d+b^2 d-b c e\right) \log \left(a x^2+b x+c\right)}{2 a^2 \left(a d^2-e (b d-c e)\right)}+\frac{\left(-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}-\frac{d^3 \log (d+e x)}{e^2 \left(a d^2-e (b d-c e)\right)}+\frac{x}{a e}",1,"x/(a*e) + ((b^3*d - 3*a*b*c*d - b^2*c*e + 2*a*c^2*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + b*d*e - c*e^2)) - (d^3*Log[d + e*x])/(e^2*(a*d^2 - b*d*e + c*e^2)) + ((b^2*d - a*c*d - b*c*e)*Log[c + b*x + a*x^2])/(2*a^2*(a*d^2 - b*d*e + c*e^2))","A",1
64,1,132,149,0.1192668,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*(d + e*x)),x]","-\frac{\sqrt{4 a c-b^2} \left(e (b d-c e) \log (x (a x+b)+c)-2 a d^2 \log (d+e x)\right)+2 e \left(2 a c d+b^2 (-d)+b c e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{2 a e \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)}","-\frac{\left(-2 a c d+b^2 d-b c e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{d^2 \log (d+e x)}{e \left(a d^2-b d e+c e^2\right)}-\frac{(b d-c e) \log \left(a x^2+b x+c\right)}{2 a \left(a d^2-e (b d-c e)\right)}",1,"-1/2*(2*e*(-(b^2*d) + 2*a*c*d + b*c*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(-2*a*d^2*Log[d + e*x] + e*(b*d - c*e)*Log[c + x*(b + a*x)]))/(a*Sqrt[-b^2 + 4*a*c]*e*(a*d^2 + e*(-(b*d) + c*e)))","A",1
65,1,107,124,0.0735044,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*x*(d + e*x)),x]","\frac{d \sqrt{4 a c-b^2} (2 \log (d+e x)-\log (x (a x+b)+c))+2 (b d-2 c e) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{2 \sqrt{4 a c-b^2} \left(e (b d-c e)-a d^2\right)}","\frac{(b d-2 c e) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{d \log \left(a x^2+b x+c\right)}{2 \left(a d^2-e (b d-c e)\right)}-\frac{d \log (d+e x)}{a d^2-e (b d-c e)}",1,"(2*(b*d - 2*c*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*d*(2*Log[d + e*x] - Log[c + x*(b + a*x)]))/(2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e)))","A",1
66,1,105,123,0.0733561,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^2 (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^2*(d + e*x)),x]","\frac{e \sqrt{4 a c-b^2} (\log (x (a x+b)+c)-2 \log (d+e x))+(2 b e-4 a d) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{2 \sqrt{4 a c-b^2} \left(e (b d-c e)-a d^2\right)}","-\frac{(2 a d-b e) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}-\frac{e \log \left(a x^2+b x+c\right)}{2 \left(a d^2-b d e+c e^2\right)}+\frac{e \log (d+e x)}{a d^2-b d e+c e^2}",1,"((-4*a*d + 2*b*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*e*(-2*Log[d + e*x] + Log[c + x*(b + a*x)]))/(2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e)))","A",1
67,1,152,158,0.1858392,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^3 (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^3*(d + e*x)),x]","-\frac{\sqrt{4 a c-b^2} \left(-2 \log (x) \left(a d^2+e (c e-b d)\right)+d (a d-b e) \log (x (a x+b)+c)+2 c e^2 \log (d+e x)\right)+2 d \left(a b d+2 a c e+b^2 (-e)\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{2 c d \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)}","\frac{\left(a b d+2 a c e+b^2 (-e)\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}-\frac{e^2 \log (d+e x)}{d \left(a d^2-b d e+c e^2\right)}-\frac{(a d-b e) \log \left(a x^2+b x+c\right)}{2 c \left(a d^2-e (b d-c e)\right)}+\frac{\log (x)}{c d}",1,"-1/2*(2*d*(a*b*d - b^2*e + 2*a*c*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(-2*(a*d^2 + e*(-(b*d) + c*e))*Log[x] + 2*c*e^2*Log[d + e*x] + d*(a*d - b*e)*Log[c + x*(b + a*x)]))/(c*Sqrt[-b^2 + 4*a*c]*d*(a*d^2 + e*(-(b*d) + c*e)))","A",1
68,1,194,193,0.1736817,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^4 (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^4*(d + e*x)),x]","\frac{\left(2 a^2 c d-a b (b d+3 c e)+b^3 e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{c^2 \sqrt{4 a c-b^2} \left(e (b d-c e)-a d^2\right)}+\frac{\left(a b d+a c e+b^2 (-e)\right) \log (x (a x+b)+c)}{2 c^2 \left(a d^2+e (c e-b d)\right)}+\frac{e^3 \log (d+e x)}{a d^4+d^2 e (c e-b d)}-\frac{\log (x) (b d+c e)}{c^2 d^2}-\frac{1}{c d x}","\frac{\left(2 a^2 c d-a b (b d+3 c e)+b^3 e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{\left(a b d+a c e+b^2 (-e)\right) \log \left(a x^2+b x+c\right)}{2 c^2 \left(a d^2-e (b d-c e)\right)}+\frac{e^3 \log (d+e x)}{d^2 \left(a d^2-e (b d-c e)\right)}-\frac{\log (x) (b d+c e)}{c^2 d^2}-\frac{1}{c d x}",1,"-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e))) - ((b*d + c*e)*Log[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(a*d^4 + d^2*e*(-(b*d) + c*e)) + ((a*b*d - b^2*e + a*c*e)*Log[c + x*(b + a*x)])/(2*c^2*(a*d^2 + e*(-(b*d) + c*e)))","A",1
69,1,252,252,0.2197445,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^5 (d+e x)} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^5*(d + e*x)),x]","\frac{\left(a^2 c d-a b (b d+2 c e)+b^3 e\right) \log (x (a x+b)+c)}{2 c^3 \left(a d^2+e (c e-b d)\right)}-\frac{\left(a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{c^3 \sqrt{4 a c-b^2} \left(e (b d-c e)-a d^2\right)}+\frac{\log (x) \left(c \left(c e^2-a d^2\right)+b^2 d^2+b c d e\right)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{a d^5+d^3 e (c e-b d)}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2}","\frac{\left(a^2 c d-a b (b d+2 c e)+b^3 e\right) \log \left(a x^2+b x+c\right)}{2 c^3 \left(a d^2-e (b d-c e)\right)}-\frac{\left(a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c^3 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)}+\frac{\log (x) \left(-c \left(a d^2-c e^2\right)+b^2 d^2+b c d e\right)}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left(a d^2-e (b d-c e)\right)}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2}",1,"-1/2*1/(c*d*x^2) + (b*d + c*e)/(c^2*d^2*x) - ((b^4*e + a^2*c*(3*b*d + 2*c*e) - a*b^2*(b*d + 4*c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e))) + ((b^2*d^2 + b*c*d*e + c*(-(a*d^2) + c*e^2))*Log[x])/(c^3*d^3) - (e^4*Log[d + e*x])/(a*d^5 + d^3*e*(-(b*d) + c*e)) + ((a^2*c*d + b^3*e - a*b*(b*d + 2*c*e))*Log[c + x*(b + a*x)])/(2*c^3*(a*d^2 + e*(-(b*d) + c*e)))","A",1
70,1,338,343,0.3557117,"\int \frac{x^3}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)^2} \, dx","Integrate[x^3/((a + c/x^2 + b/x)*(d + e*x)^2),x]","\frac{\left(b^2 c \left(c e^2-3 a d^2\right)+4 a b c^2 d e+a c^2 \left(a d^2-c e^2\right)+b^4 d^2-2 b^3 c d e\right) \log (x (a x+b)+c)}{2 a^3 \left(a d^2+e (c e-b d)\right)^2}-\frac{x (2 a d+b e)}{a^2 e^3}-\frac{\left(-4 a^2 c^3 d e+b^3 c \left(c e^2-5 a d^2\right)+8 a b^2 c^2 d e+a b c^2 \left(5 a d^2-3 c e^2\right)+b^5 d^2-2 b^4 c d e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a^3 \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)^2}+\frac{d^5}{e^4 (d+e x) \left(a d^2+e (c e-b d)\right)}+\frac{\log (d+e x) \left(3 a d^6+d^4 e (5 c e-4 b d)\right)}{e^4 \left(a d^2+e (c e-b d)\right)^2}+\frac{x^2}{2 a e^2}","\frac{\left(-b^2 c \left(3 a d^2-c e^2\right)+4 a b c^2 d e+a c^2 \left(a d^2-c e^2\right)+b^4 d^2-2 b^3 c d e\right) \log \left(a x^2+b x+c\right)}{2 a^3 \left(a d^2-e (b d-c e)\right)^2}-\frac{x (2 a d+b e)}{a^2 e^3}+\frac{\left(-4 a^2 c^3 d e-b^3 c \left(5 a d^2-c e^2\right)+8 a b^2 c^2 d e+a b c^2 \left(5 a d^2-3 c e^2\right)+b^5 d^2-2 b^4 c d e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a^3 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{d^5}{e^4 (d+e x) \left(a d^2-e (b d-c e)\right)}+\frac{d^4 \log (d+e x) \left(3 a d^2-e (4 b d-5 c e)\right)}{e^4 \left(a d^2-e (b d-c e)\right)^2}+\frac{x^2}{2 a e^2}",1,"-(((2*a*d + b*e)*x)/(a^2*e^3)) + x^2/(2*a*e^2) + d^5/(e^4*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) - ((b^5*d^2 - 2*b^4*c*d*e + 8*a*b^2*c^2*d*e - 4*a^2*c^3*d*e + a*b*c^2*(5*a*d^2 - 3*c*e^2) + b^3*c*(-5*a*d^2 + c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + ((3*a*d^6 + d^4*e*(-4*b*d + 5*c*e))*Log[d + e*x])/(e^4*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b^4*d^2 - 2*b^3*c*d*e + 4*a*b*c^2*d*e + a*c^2*(a*d^2 - c*e^2) + b^2*c*(-3*a*d^2 + c*e^2))*Log[c + x*(b + a*x)])/(2*a^3*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
71,1,269,274,0.2871235,"\int \frac{x^2}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)^2} \, dx","Integrate[x^2/((a + c/x^2 + b/x)*(d + e*x)^2),x]","\frac{(b d-c e) \left(2 a c d+b^2 (-d)+b c e\right) \log (x (a x+b)+c)}{2 a^2 \left(a d^2+e (c e-b d)\right)^2}+\frac{\left(b^2 c \left(c e^2-4 a d^2\right)+6 a b c^2 d e+2 a c^2 \left(a d^2-c e^2\right)+b^4 d^2-2 b^3 c d e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a^2 \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)^2}-\frac{d^4}{e^3 (d+e x) \left(a d^2+e (c e-b d)\right)}-\frac{\log (d+e x) \left(2 a d^5+d^3 e (4 c e-3 b d)\right)}{e^3 \left(a d^2+e (c e-b d)\right)^2}+\frac{x}{a e^2}","-\frac{(b d-c e) \left(-2 a c d+b^2 d-b c e\right) \log \left(a x^2+b x+c\right)}{2 a^2 \left(a d^2-e (b d-c e)\right)^2}-\frac{\left(-b^2 c \left(4 a d^2-c e^2\right)+6 a b c^2 d e+2 a c^2 \left(a d^2-c e^2\right)+b^4 d^2-2 b^3 c d e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}-\frac{d^4}{e^3 (d+e x) \left(a d^2-e (b d-c e)\right)}-\frac{d^3 \log (d+e x) \left(2 a d^2-e (3 b d-4 c e)\right)}{e^3 \left(a d^2-e (b d-c e)\right)^2}+\frac{x}{a e^2}",1,"x/(a*e^2) - d^4/(e^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((b^4*d^2 - 2*b^3*c*d*e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) + b^2*c*(-4*a*d^2 + c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^2*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) - ((2*a*d^5 + d^3*e*(-3*b*d + 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b*d - c*e)*(-(b^2*d) + 2*a*c*d + b*c*e)*Log[c + x*(b + a*x)])/(2*a^2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
72,1,207,246,0.2266245,"\int \frac{x}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)^2} \, dx","Integrate[x/((a + c/x^2 + b/x)*(d + e*x)^2),x]","\frac{\frac{\left(c \left(c e^2-a d^2\right)+b^2 d^2-2 b c d e\right) \log (x (a x+b)+c)}{a}-\frac{2 \left(b c \left(c e^2-3 a d^2\right)+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{a \sqrt{4 a c-b^2}}+\frac{2 \log (d+e x) \left(a d^4+d^2 e (3 c e-2 b d)\right)}{e^2}+\frac{2 d^3 \left(a d^2+e (c e-b d)\right)}{e^2 (d+e x)}}{2 \left(a d^2+e (c e-b d)\right)^2}","\frac{\left(-c \left(a d^2-c e^2\right)+b^2 d^2-2 b c d e\right) \log \left(a x^2+b x+c\right)}{2 a \left(a d^2-e (b d-c e)\right)^2}+\frac{\left(-b c \left(3 a d^2-c e^2\right)+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{a \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{d^2 \log (d+e x) \left(a d^2-e (2 b d-3 c e)\right)}{e^2 \left(a d^2-e (b d-c e)\right)^2}+\frac{d^3}{e^2 (d+e x) \left(a d^2-e (b d-c e)\right)}",1,"((2*d^3*(a*d^2 + e*(-(b*d) + c*e)))/(e^2*(d + e*x)) - (2*(b^3*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e + b*c*(-3*a*d^2 + c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a*Sqrt[-b^2 + 4*a*c]) + (2*(a*d^4 + d^2*e*(-2*b*d + 3*c*e))*Log[d + e*x])/e^2 + ((b^2*d^2 - 2*b*c*d*e + c*(-(a*d^2) + c*e^2))*Log[c + x*(b + a*x)])/a)/(2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
73,1,159,194,0.2253104,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*(d + e*x)^2),x]","\frac{\frac{2 \left(2 c \left(c e^2-a d^2\right)+b^2 d^2-2 b c d e\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{2 d^2 \left(a d^2+e (c e-b d)\right)}{e (d+e x)}-d (b d-2 c e) \log (x (a x+b)+c)+2 d (b d-2 c e) \log (d+e x)}{2 \left(a d^2+e (c e-b d)\right)^2}","-\frac{\left(-2 c \left(a d^2-c e^2\right)+b^2 d^2-2 b c d e\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}-\frac{d^2}{e (d+e x) \left(a d^2-b d e+c e^2\right)}-\frac{d (b d-2 c e) \log \left(a x^2+b x+c\right)}{2 \left(a d^2-e (b d-c e)\right)^2}+\frac{d (b d-2 c e) \log (d+e x)}{\left(a d^2-e (b d-c e)\right)^2}",1,"((-2*d^2*(a*d^2 + e*(-(b*d) + c*e)))/(e*(d + e*x)) + (2*(b^2*d^2 - 2*b*c*d*e + 2*c*(-(a*d^2) + c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*d*(b*d - 2*c*e)*Log[d + e*x] - d*(b*d - 2*c*e)*Log[c + x*(b + a*x)])/(2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
74,1,148,183,0.2442519,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*x*(d + e*x)^2),x]","\frac{-\frac{2 \left(a d (b d-4 c e)+b c e^2\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\left(a d^2-c e^2\right) \log (x (a x+b)+c)+\frac{2 d \left(a d^2+e (c e-b d)\right)}{d+e x}+\left(2 c e^2-2 a d^2\right) \log (d+e x)}{2 \left(a d^2+e (c e-b d)\right)^2}","\frac{\left(a d (b d-4 c e)+b c e^2\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{\left(a d^2-c e^2\right) \log \left(a x^2+b x+c\right)}{2 \left(a d^2-e (b d-c e)\right)^2}+\frac{d}{(d+e x) \left(a d^2-b d e+c e^2\right)}-\frac{\left(a d^2-c e^2\right) \log (d+e x)}{\left(a d^2-e (b d-c e)\right)^2}",1,"((2*d*(a*d^2 + e*(-(b*d) + c*e)))/(d + e*x) - (2*(b*c*e^2 + a*d*(b*d - 4*c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-2*a*d^2 + 2*c*e^2)*Log[d + e*x] + (a*d^2 - c*e^2)*Log[c + x*(b + a*x)])/(2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
75,1,151,189,0.2112305,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^2 (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^2*(d + e*x)^2),x]","\frac{\frac{2 \left(2 a^2 d^2-2 a e (b d+c e)+b^2 e^2\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{2 e \left(a d^2+e (c e-b d)\right)}{d+e x}+e (b e-2 a d) \log (x (a x+b)+c)-2 e (b e-2 a d) \log (d+e x)}{2 \left(a d^2+e (c e-b d)\right)^2}","-\frac{\left(2 a^2 d^2-2 a e (b d+c e)+b^2 e^2\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}-\frac{e}{(d+e x) \left(a d^2-b d e+c e^2\right)}-\frac{e (2 a d-b e) \log \left(a x^2+b x+c\right)}{2 \left(a d^2-e (b d-c e)\right)^2}+\frac{e (2 a d-b e) \log (d+e x)}{\left(a d^2-e (b d-c e)\right)^2}",1,"((-2*e*(a*d^2 + e*(-(b*d) + c*e)))/(d + e*x) + (2*(2*a^2*d^2 + b^2*e^2 - 2*a*e*(b*d + c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*e*(-2*a*d + b*e)*Log[d + e*x] + e*(-2*a*d + b*e)*Log[c + x*(b + a*x)])/(2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
76,1,246,248,0.252518,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^3 (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^3*(d + e*x)^2),x]","\frac{\left(-a^2 d^2+a e (2 b d+c e)-b^2 e^2\right) \log (x (a x+b)+c)}{2 c \left(a d^2+e (c e-b d)\right)^2}-\frac{\left(a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{c \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)^2}+\frac{e^2}{d (d+e x) \left(a d^2+e (c e-b d)\right)}-\frac{e^2 \log (d+e x) \left(3 a d^2+e (c e-2 b d)\right)}{\left(a d^3+d e (c e-b d)\right)^2}+\frac{\log (x)}{c d^2}","-\frac{\left(a^2 d^2-a e (2 b d+c e)+b^2 e^2\right) \log \left(a x^2+b x+c\right)}{2 c \left(a d^2-e (b d-c e)\right)^2}+\frac{\left(a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{e^2}{d (d+e x) \left(a d^2-b d e+c e^2\right)}-\frac{e^2 \log (d+e x) \left(3 a d^2-e (2 b d-c e)\right)}{d^2 \left(a d^2-e (b d-c e)\right)^2}+\frac{\log (x)}{c d^2}",1,"e^2/(d*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) - ((b^3*e^2 - a*b*e*(2*b*d + 3*c*e) + a^2*d*(b*d + 4*c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + Log[x]/(c*d^2) - (e^2*(3*a*d^2 + e*(-2*b*d + c*e))*Log[d + e*x])/(a*d^3 + d*e*(-(b*d) + c*e))^2 + ((-(a^2*d^2) - b^2*e^2 + a*e*(2*b*d + c*e))*Log[c + x*(b + a*x)])/(2*c*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
77,1,287,291,0.3402308,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^4 (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^4*(d + e*x)^2),x]","\frac{\left(-2 a^3 c d^2+a^2 \left(b^2 d^2+6 b c d e+2 c^2 e^2\right)-2 a b^2 e (b d+2 c e)+b^4 e^2\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{c^2 \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)^2}+\frac{(a d-b e) \left(a b d+2 a c e+b^2 (-e)\right) \log (x (a x+b)+c)}{2 c^2 \left(a d^2+e (c e-b d)\right)^2}-\frac{e^3}{d^2 (d+e x) \left(a d^2+e (c e-b d)\right)}+\frac{e^3 \log (d+e x) \left(4 a d^2+e (2 c e-3 b d)\right)}{d^3 \left(a d^2+e (c e-b d)\right)^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x}","\frac{\left(2 a^3 c d^2-a^2 \left(b^2 d^2+6 b c d e+2 c^2 e^2\right)+2 a b^2 e (b d+2 c e)+b^4 \left(-e^2\right)\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{(a d-b e) \left(a b d+2 a c e+b^2 (-e)\right) \log \left(a x^2+b x+c\right)}{2 c^2 \left(a d^2-e (b d-c e)\right)^2}-\frac{e^3}{d^2 (d+e x) \left(a d^2-e (b d-c e)\right)}+\frac{e^3 \log (d+e x) \left(4 a d^2-e (3 b d-2 c e)\right)}{d^3 \left(a d^2-e (b d-c e)\right)^2}-\frac{\log (x) (b d+2 c e)}{c^2 d^3}-\frac{1}{c d^2 x}",1,"-(1/(c*d^2*x)) - e^3/(d^2*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((-2*a^3*c*d^2 + b^4*e^2 - 2*a*b^2*e*(b*d + 2*c*e) + a^2*(b^2*d^2 + 6*b*c*d*e + 2*c^2*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) - ((b*d + 2*c*e)*Log[x])/(c^2*d^3) + (e^3*(4*a*d^2 + e*(-3*b*d + 2*c*e))*Log[d + e*x])/(d^3*(a*d^2 + e*(-(b*d) + c*e))^2) + ((a*d - b*e)*(a*b*d - b^2*e + 2*a*c*e)*Log[c + x*(b + a*x)])/(2*c^2*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
78,1,370,372,0.4250511,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right) x^5 (d+e x)^2} \, dx","Integrate[1/((a + c/x^2 + b/x)*x^5*(d + e*x)^2),x]","-\frac{\left(-a^3 c d^2+a^2 \left(b^2 d^2+4 b c d e+c^2 e^2\right)-a b^2 e (2 b d+3 c e)+b^4 e^2\right) \log (x (a x+b)+c)}{2 c^3 \left(a d^2+e (c e-b d)\right)^2}+\frac{\left(a^3 c d (3 b d+4 c e)-a^2 b \left(b^2 d^2+8 b c d e+5 c^2 e^2\right)+a b^3 e (2 b d+5 c e)+b^5 \left(-e^2\right)\right) \tan ^{-1}\left(\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right)}{c^3 \sqrt{4 a c-b^2} \left(a d^2+e (c e-b d)\right)^2}+\frac{\log (x) \left(c \left(3 c e^2-a d^2\right)+b^2 d^2+2 b c d e\right)}{c^3 d^4}-\frac{e^4 \log (d+e x) \left(5 a d^2+e (3 c e-4 b d)\right)}{d^4 \left(a d^2+e (c e-b d)\right)^2}+\frac{e^4}{d^3 (d+e x) \left(a d^2+e (c e-b d)\right)}+\frac{b d+2 c e}{c^2 d^3 x}-\frac{1}{2 c d^2 x^2}","\frac{\left(a^3 c d^2-a^2 \left(b^2 d^2+4 b c d e+c^2 e^2\right)+a b^2 e (2 b d+3 c e)+b^4 \left(-e^2\right)\right) \log \left(a x^2+b x+c\right)}{2 c^3 \left(a d^2-e (b d-c e)\right)^2}+\frac{\left(-a^3 c d (3 b d+4 c e)+a^2 b \left(b^2 d^2+8 b c d e+5 c^2 e^2\right)-a b^3 e (2 b d+5 c e)+b^5 e^2\right) \tanh ^{-1}\left(\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right)}{c^3 \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right)^2}+\frac{\log (x) \left(-c \left(a d^2-3 c e^2\right)+b^2 d^2+2 b c d e\right)}{c^3 d^4}-\frac{e^4 \log (d+e x) \left(5 a d^2-e (4 b d-3 c e)\right)}{d^4 \left(a d^2-e (b d-c e)\right)^2}+\frac{e^4}{d^3 (d+e x) \left(a d^2-e (b d-c e)\right)}+\frac{b d+2 c e}{c^2 d^3 x}-\frac{1}{2 c d^2 x^2}",1,"-1/2*1/(c*d^2*x^2) + (b*d + 2*c*e)/(c^2*d^3*x) + e^4/(d^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((-(b^5*e^2) + a^3*c*d*(3*b*d + 4*c*e) + a*b^3*e*(2*b*d + 5*c*e) - a^2*b*(b^2*d^2 + 8*b*c*d*e + 5*c^2*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b^2*d^2 + 2*b*c*d*e + c*(-(a*d^2) + 3*c*e^2))*Log[x])/(c^3*d^4) - (e^4*(5*a*d^2 + e*(-4*b*d + 3*c*e))*Log[d + e*x])/(d^4*(a*d^2 + e*(-(b*d) + c*e))^2) - ((-(a^3*c*d^2) + b^4*e^2 - a*b^2*e*(2*b*d + 3*c*e) + a^2*(b^2*d^2 + 4*b*c*d*e + c^2*e^2))*Log[c + x*(b + a*x)])/(2*c^3*(a*d^2 + e*(-(b*d) + c*e))^2)","A",1
79,1,10904,981,14.1986168,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x],x]","\text{Result too large to show}","\frac{2}{11} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x^5+\frac{2 (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac{2 \left(29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac{2 \left(233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left(37 b^2 d^2-135 b c e d+156 c^2 e^2\right) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left(37 b^2 d^2-258 b c e d-771 c^2 e^2\right)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) x}{3465 a^5 e^5 \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \left(a x^2+b x+c\right)}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(a d^2-e (b d-c e)\right) \left(128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left(3 b^2 d^2-29 b c e d+50 c^2 e^2\right)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) x}{3465 a^5 e^5 \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{2 \left(187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left(3 b^2 d^2-29 b c e d+50 c^2 e^2\right)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x}{3465 a^4 e^4}",1,"Result too large to show","C",0
80,1,7531,778,13.7353784,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^3 \sqrt{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]*x^3*Sqrt[d + e*x],x]","\text{Result too large to show}","-\frac{4 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left(8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right)}{315 a^2 e^3}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left(19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right)}{315 a^3 e^3}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} \left(8 a^4 d^4-a^3 d^2 e (4 b d-9 c e)-3 a^2 e^2 \left(b^2 d^2-5 b c d e-7 c^2 e^2\right)-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 a^4 e^4 \left(a x^2+b x+c\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} \left(16 a^3 d^3+6 a^2 c d e^2-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right) \left(a d^2-e (b d-c e)\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 a^4 e^4 \sqrt{d+e x} \left(a x^2+b x+c\right)}+\frac{2 x (d+e x)^{5/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (a d+b e)}{63 a e^3}+\frac{2}{9} x^4 \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}",1,"Result too large to show","C",1
81,1,5350,636,13.0513936,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^2 \sqrt{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]*x^2*Sqrt[d + e*x],x]","\text{Result too large to show}","-\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left(4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right)}{105 a^2 e^2}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} \left(8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right) \left(a d^2-e (b d-c e)\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 a^3 e^3 \sqrt{d+e x} \left(a x^2+b x+c\right)}+\frac{\sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} \left(8 a^3 d^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 a^3 e^3 \left(a x^2+b x+c\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left(a x^2+b x+c\right)}{7 a}",1,"Result too large to show","C",1
82,1,693,550,11.7710943,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x],x]","\frac{x \sqrt{a+\frac{b x+c}{x^2}} \left(\frac{4 e^2 \left(-a^2 d^2+a e (b d+3 c e)-b^2 e^2\right)}{\sqrt{d+e x}}+\frac{i (d+e x) \sqrt{1-\frac{2 \left(a d^2+e (c e-b d)\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 a d-b e\right)}} \sqrt{\frac{4 \left(a d^2+e (c e-b d)\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-2 a d+b e\right)}+2} \left(\left(a^2 d \left(8 c e^2-d \sqrt{e^2 \left(b^2-4 a c\right)}\right)+a e \left(b d \sqrt{e^2 \left(b^2-4 a c\right)}+3 c e \sqrt{e^2 \left(b^2-4 a c\right)}-2 b^2 d e-4 b c e^2\right)+b^2 e^2 \left(b e-\sqrt{e^2 \left(b^2-4 a c\right)}\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 a d-b e\right) \left(a^2 d^2-a e (b d+3 c e)+b^2 e^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{(x (a x+b)+c) \sqrt{\frac{a d^2+e (c e-b d)}{\sqrt{e^2 \left(b^2-4 a c\right)}-2 a d+b e}}}+2 a e^2 \sqrt{d+e x} (a (d+3 e x)+b e)\right)}{15 a^2 e^3}","\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} (2 a d-b e) \left(a d^2-e (b d-c e)\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 a^2 e^2 \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} \left(a^2 d^2-a e (b d+3 c e)+b^2 e^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 a^2 e^2 \left(a x^2+b x+c\right) \sqrt{\frac{a (d+e x)}{2 a d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{5 e}-\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (2 a d-b e)}{15 a e}",1,"(x*Sqrt[a + (c + b*x)/x^2]*((4*e^2*(-(a^2*d^2) - b^2*e^2 + a*e*(b*d + 3*c*e)))/Sqrt[d + e*x] + 2*a*e^2*Sqrt[d + e*x]*(b*e + a*(d + 3*e*x)) + (I*(d + e*x)*Sqrt[1 - (2*(a*d^2 + e*(-(b*d) + c*e)))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(a*d^2 + e*(-(b*d) + c*e)))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(a^2*d^2 + b^2*e^2 - a*e*(b*d + 3*c*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) + a^2*d*(8*c*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e*(-2*b^2*d*e - 4*b*c*e^2 + b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 3*c*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))))/(15*a^2*e^3)","C",1
83,1,1258,955,10.5273854,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x],x]","\frac{x \sqrt{a+\frac{c+b x}{x^2}} \left(\frac{4 (a d+b e) \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (c+x (b+a x)) e^2}{(d+e x)^2}+\frac{6 i \sqrt{2} a c \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \Pi \left(\frac{d \left(2 a d-b e-\sqrt{\left(b^2-4 a c\right) e^2}\right)}{2 \left(a d^2+e (c e-b d)\right)};i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right) e^2}{\sqrt{d+e x}}-\frac{i \sqrt{2} (a d+b e) \left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(-2 a 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{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left(b e \left(\sqrt{\left(b^2-4 a c\right) e^2}-b e\right)+a \left(-2 c e^2+3 b d e+d \sqrt{\left(b^2-4 a c\right) e^2}\right)\right) \sqrt{\frac{-2 c e^2+2 a d x e+b (d-e x) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 c e^2-2 a d x e+b (e x-d) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\left(-2 a 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{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right) (d+e x)^{3/2}}{6 a e^2 \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (c+x (b+a x))}+\frac{2}{3} x \sqrt{a+\frac{c+b x}{x^2}} \sqrt{d+e x}","\frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \left(a x^2+b x+c\right)}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) x}{3 a e \sqrt{d+e x} \left(a x^2+b x+c\right)}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) x}{3 a \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{\sqrt{2} c \sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \Pi \left(\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right) x}{\sqrt{a} \left(a x^2+b x+c\right)}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x",1,"(2*x*Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2])/3 + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*((4*e^2*(a*d + b*e)*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))/(d + e*x)^2 - (I*Sqrt[2]*(a*d + b*e)*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(b*e*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2]) + a*(3*b*d*e - 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + ((6*I)*Sqrt[2]*a*c*e^2*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticPi[(d*(2*a*d - b*e - Sqrt[(b^2 - 4*a*c)*e^2]))/(2*(a*d^2 + e*(-(b*d) + c*e))), I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(6*a*e^2*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))","C",1
84,1,1372,929,11.4130459,"\int \frac{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x}}{x} \, dx","Integrate[(Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/x,x]","\frac{x (d+e x)^{3/2} \sqrt{a+\frac{c+b x}{x^2}} \left(12 d \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(a \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{c e}{d+e x}\right)}{d+e x}\right)-\frac{3 i \sqrt{2} d \left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-\frac{2 c e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 a d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 c e^2}{d+e x}+2 a 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 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}+\frac{i \sqrt{2} \left(4 a d^2-b e d+3 \sqrt{\left(b^2-4 a c\right) e^2} d-2 c e^2\right) \sqrt{\frac{-\frac{2 c e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 a d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 c e^2}{d+e x}+2 a 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 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}+\frac{2 i \sqrt{2} e (b d+c e) \sqrt{\frac{-\frac{2 c e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 a d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 c e^2}{d+e x}+2 a 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 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \Pi \left(\frac{d \left(2 a d-b e-\sqrt{\left(b^2-4 a c\right) e^2}\right)}{2 \left(a d^2+e (c e-b d)\right)};i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right)}{4 d e \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{a x^2+b x+c} \sqrt{\frac{(d+e x)^2 \left(a \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{c e}{d+e x}\right)}{d+e x}\right)}{e^2}}}-\sqrt{d+e x} \sqrt{a+\frac{c+b x}{x^2}}","\frac{3 \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{d+e x} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{2} \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \left(a x^2+b x+c\right)}-\frac{3 \sqrt{2} \sqrt{b^2-4 a c} d \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{d+e x} \left(a x^2+b x+c\right)}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{a \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{(b d+c e) \sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \Pi \left(\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right)}{\sqrt{2} \sqrt{a} d \left(a x^2+b x+c\right)}-\sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x}",1,"-(Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2]) + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*(12*d*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (c*e)/(d + e*x)))/(d + e*x)) - ((3*I)*Sqrt[2]*d*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(4*a*d^2 - b*d*e - 2*c*e^2 + 3*d*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + ((2*I)*Sqrt[2]*e*(b*d + c*e)*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*c*e^2)/(d + e*x) - 2*a*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*c*e^2)/(d + e*x) + 2*a*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticPi[(d*(2*a*d - b*e - Sqrt[(b^2 - 4*a*c)*e^2]))/(2*(a*d^2 + e*(-(b*d) + c*e))), I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(4*d*e*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[c + b*x + a*x^2]*Sqrt[((d + e*x)^2*(a*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (c*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
85,1,811,1287,12.5318817,"\int \frac{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \sqrt{d+e x}}{x^2} \, dx","Integrate[(Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/x^2,x]","\frac{x \sqrt{a+\frac{c+b x}{x^2}} \left(-\frac{8 c d^3}{x^2}-\frac{8 c e d^2}{x}-\frac{4 (b d+c e) d^2}{x}-\frac{i (d+e x)^{3/2} \sqrt{1-\frac{2 \left(a d^2+e (c e-b d)\right)}{\left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{4 \left(a d^2+e (c e-b d)\right)}{\left(-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}+2} \left(d (b d+c e) \left(2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)-\left(b^2 e d^2+b \left(d \sqrt{\left(b^2-4 a c\right) e^2}-5 c e^2\right) d+c e \left(4 a d^2+\sqrt{\left(b^2-4 a c\right) e^2} d+2 c e^2\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+2 e \left(b^2 d^2-2 b c e d+c \left(c e^2-4 a d^2\right)\right) \Pi \left(\frac{d \left(2 a d-b e-\sqrt{\left(b^2-4 a c\right) e^2}\right)}{2 \left(a d^2+e (c e-b d)\right)};i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 a d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{e \sqrt{\frac{a d^2+e (c e-b d)}{-2 a d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} (c+x (b+a x))}\right)}{16 c d^2 \sqrt{d+e x}}","\frac{\sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \Pi \left(\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right) (b d+c e)^2}{4 \sqrt{2} \sqrt{a} c d^2 \left(a x^2+b x+c\right)}+\frac{\sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{d+e x} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) (b d+c e)}{4 \sqrt{2} c d \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \left(a x^2+b x+c\right)}-\frac{\sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) (b d+c e)}{2 \sqrt{2} c \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{\sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} (b d+c e)}{4 c d}+\frac{3 \sqrt{b^2-4 a c} e \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{\frac{a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{-\frac{a \left(a x^2+b x+c\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 a 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 a d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{\sqrt{2} \sqrt{d+e x} \left(a x^2+b x+c\right)}-\frac{(a d+b e) \sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} x \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left(b+\sqrt{b^2-4 a c}\right) e}} \Pi \left(\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left(b-\sqrt{b^2-4 a c}\right) e}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right)}{\sqrt{2} \sqrt{a} d \left(a x^2+b x+c\right)}-\frac{\sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x}}{2 x}",1,"(x*Sqrt[a + (c + b*x)/x^2]*((-8*c*d^3)/x^2 - (8*c*d^2*e)/x - (4*d^2*(b*d + c*e))/x - (I*(d + e*x)^(3/2)*Sqrt[1 - (2*(a*d^2 + e*(-(b*d) + c*e)))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(a*d^2 + e*(-(b*d) + c*e)))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(d*(b*d + c*e)*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (b^2*d^2*e + b*d*(-5*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(4*a*d^2 + 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + 2*e*(b^2*d^2 - 2*b*c*d*e + c*(-4*a*d^2 + c*e^2))*EllipticPi[(d*(2*a*d - b*e - Sqrt[(b^2 - 4*a*c)*e^2]))/(2*(a*d^2 + e*(-(b*d) + c*e))), I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(e*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))))/(16*c*d^2*Sqrt[d + e*x])","C",1
86,0,0,29,0.1951715,"\int (f x)^m \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p,x]","\int (f x)^m \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","\text{Int}\left((f x)^m \left(a+c x^{2 n}\right)^p \left(d+e x^n\right)^q,x\right)",0,"Integrate[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x]","A",-1
87,1,249,358,0.3099786,"\int (f x)^m \left(d+e x^n\right)^3 \left(a+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)^3*(a + c*x^(2*n))^p,x]","x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(\frac{d^3 \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)}{m+1}+e x^n \left(\frac{3 d^2 \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+n+1}+e x^n \left(\frac{3 d \, _2F_1\left(\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+2 n+1}+\frac{e x^n \, _2F_1\left(\frac{m+3 n+1}{2 n},-p;\frac{m+5 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+3 n+1}\right)\right)\right)","\frac{d^3 (f x)^{m+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)}{f (m+1)}+\frac{3 d^2 e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+n+1}+\frac{3 d e^2 x^{2 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+2 n+1}+\frac{e^3 x^{3 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+3 n+1}{2 n},-p;\frac{m+5 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+3 n+1}",1,"(x*(f*x)^m*(a + c*x^(2*n))^p*((d^3*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)])/(1 + m) + e*x^n*((3*d^2*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + n) + e*x^n*((3*d*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + 2*n) + (e*x^n*Hypergeometric2F1[(1 + m + 3*n)/(2*n), -p, (1 + m + 5*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + 3*n)))))/(1 + (c*x^(2*n))/a)^p","A",1
88,1,189,262,0.1585342,"\int (f x)^m \left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)^2*(a + c*x^(2*n))^p,x]","x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(\frac{d^2 \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)}{m+1}+e x^n \left(\frac{2 d \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+n+1}+\frac{e x^n \, _2F_1\left(\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+2 n+1}\right)\right)","\frac{d^2 (f x)^{m+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+2 n+1}",1,"(x*(f*x)^m*(a + c*x^(2*n))^p*((d^2*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)])/(1 + m) + e*x^n*((2*d*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + n) + (e*x^n*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + 2*n))))/(1 + (c*x^(2*n))/a)^p","A",1
89,1,136,166,0.0769312,"\int (f x)^m \left(d+e x^n\right) \left(a+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)*(a + c*x^(2*n))^p,x]","\frac{x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(d (m+n+1) \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)+e (m+1) x^n \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)\right)}{(m+1) (m+n+1)}","\frac{d (f x)^{m+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right)}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right)}{m+n+1}",1,"(x*(f*x)^m*(a + c*x^(2*n))^p*(d*(1 + m + n)*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)] + e*(1 + m)*x^n*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)]))/((1 + m)*(1 + m + n)*(1 + (c*x^(2*n))/a)^p)","A",1
90,0,0,194,0.1098719,"\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{d+e x^n} \, dx","Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n),x]","\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{d+e x^n} \, dx","\frac{x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2 n};-p,1;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d (m+1)}-\frac{e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+n+1}{2 n};-p,1;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (m+n+1)}",1,"Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n), x]","F",-1
91,0,0,302,0.1677097,"\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^2,x]","\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","\frac{x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2 n};-p,2;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (m+1)}+\frac{e^2 x^{2 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+2 n+1}{2 n};-p,2;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (m+2 n+1)}-\frac{2 e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+n+1}{2 n};-p,2;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3 (m+n+1)}",1,"Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^2, x]","F",-1
92,0,0,412,0.7064359,"\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^3,x]","\int \frac{(f x)^m \left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","-\frac{e^3 x^{3 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+3 n+1}{2 n};-p,3;\frac{m+5 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^6 (m+3 n+1)}+\frac{3 e^2 x^{2 n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+2 n+1}{2 n};-p,3;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^5 (m+2 n+1)}-\frac{3 e x^{n+1} (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+n+1}{2 n};-p,3;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (m+n+1)}+\frac{x (f x)^m \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2 n};-p,3;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3 (m+1)}",1,"Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^3, x]","F",-1
93,1,201,16,0.1756953,"\int (b+2 c x) \left(a+b x+c x^2\right)^{13} \, dx","Integrate[(b + 2*c*x)*(a + b*x + c*x^2)^13,x]","\frac{1}{14} x (b+c x) \left(14 a^{13}+91 a^{12} x (b+c x)+364 a^{11} x^2 (b+c x)^2+1001 a^{10} x^3 (b+c x)^3+2002 a^9 x^4 (b+c x)^4+3003 a^8 x^5 (b+c x)^5+3432 a^7 x^6 (b+c x)^6+3003 a^6 x^7 (b+c x)^7+2002 a^5 x^8 (b+c x)^8+1001 a^4 x^9 (b+c x)^9+364 a^3 x^{10} (b+c x)^{10}+91 a^2 x^{11} (b+c x)^{11}+14 a x^{12} (b+c x)^{12}+x^{13} (b+c x)^{13}\right)","\frac{1}{14} \left(a+b x+c x^2\right)^{14}",1,"(x*(b + c*x)*(14*a^13 + 91*a^12*x*(b + c*x) + 364*a^11*x^2*(b + c*x)^2 + 1001*a^10*x^3*(b + c*x)^3 + 2002*a^9*x^4*(b + c*x)^4 + 3003*a^8*x^5*(b + c*x)^5 + 3432*a^7*x^6*(b + c*x)^6 + 3003*a^6*x^7*(b + c*x)^7 + 2002*a^5*x^8*(b + c*x)^8 + 1001*a^4*x^9*(b + c*x)^9 + 364*a^3*x^10*(b + c*x)^10 + 91*a^2*x^11*(b + c*x)^11 + 14*a*x^12*(b + c*x)^12 + x^13*(b + c*x)^13))/14","B",1
94,1,233,18,0.1773674,"\int x \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{13} \, dx","Integrate[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x]","\frac{1}{28} x^2 \left(b+c x^2\right) \left(14 a^{13}+91 a^{12} x^2 \left(b+c x^2\right)+364 a^{11} x^4 \left(b+c x^2\right)^2+1001 a^{10} x^6 \left(b+c x^2\right)^3+2002 a^9 x^8 \left(b+c x^2\right)^4+3003 a^8 x^{10} \left(b+c x^2\right)^5+3432 a^7 x^{12} \left(b+c x^2\right)^6+3003 a^6 x^{14} \left(b+c x^2\right)^7+2002 a^5 x^{16} \left(b+c x^2\right)^8+1001 a^4 x^{18} \left(b+c x^2\right)^9+364 a^3 x^{20} \left(b+c x^2\right)^{10}+91 a^2 x^{22} \left(b+c x^2\right)^{11}+14 a x^{24} \left(b+c x^2\right)^{12}+x^{26} \left(b+c x^2\right)^{13}\right)","\frac{1}{28} \left(a+b x^2+c x^4\right)^{14}",1,"(x^2*(b + c*x^2)*(14*a^13 + 91*a^12*x^2*(b + c*x^2) + 364*a^11*x^4*(b + c*x^2)^2 + 1001*a^10*x^6*(b + c*x^2)^3 + 2002*a^9*x^8*(b + c*x^2)^4 + 3003*a^8*x^10*(b + c*x^2)^5 + 3432*a^7*x^12*(b + c*x^2)^6 + 3003*a^6*x^14*(b + c*x^2)^7 + 2002*a^5*x^16*(b + c*x^2)^8 + 1001*a^4*x^18*(b + c*x^2)^9 + 364*a^3*x^20*(b + c*x^2)^10 + 91*a^2*x^22*(b + c*x^2)^11 + 14*a*x^24*(b + c*x^2)^12 + x^26*(b + c*x^2)^13))/28","B",1
95,1,233,18,0.1817309,"\int x^2 \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{13} \, dx","Integrate[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x]","\frac{1}{42} x^3 \left(b+c x^3\right) \left(14 a^{13}+91 a^{12} x^3 \left(b+c x^3\right)+364 a^{11} x^6 \left(b+c x^3\right)^2+1001 a^{10} x^9 \left(b+c x^3\right)^3+2002 a^9 x^{12} \left(b+c x^3\right)^4+3003 a^8 x^{15} \left(b+c x^3\right)^5+3432 a^7 x^{18} \left(b+c x^3\right)^6+3003 a^6 x^{21} \left(b+c x^3\right)^7+2002 a^5 x^{24} \left(b+c x^3\right)^8+1001 a^4 x^{27} \left(b+c x^3\right)^9+364 a^3 x^{30} \left(b+c x^3\right)^{10}+91 a^2 x^{33} \left(b+c x^3\right)^{11}+14 a x^{36} \left(b+c x^3\right)^{12}+x^{39} \left(b+c x^3\right)^{13}\right)","\frac{1}{42} \left(a+b x^3+c x^6\right)^{14}",1,"(x^3*(b + c*x^3)*(14*a^13 + 91*a^12*x^3*(b + c*x^3) + 364*a^11*x^6*(b + c*x^3)^2 + 1001*a^10*x^9*(b + c*x^3)^3 + 2002*a^9*x^12*(b + c*x^3)^4 + 3003*a^8*x^15*(b + c*x^3)^5 + 3432*a^7*x^18*(b + c*x^3)^6 + 3003*a^6*x^21*(b + c*x^3)^7 + 2002*a^5*x^24*(b + c*x^3)^8 + 1001*a^4*x^27*(b + c*x^3)^9 + 364*a^3*x^30*(b + c*x^3)^10 + 91*a^2*x^33*(b + c*x^3)^11 + 14*a*x^36*(b + c*x^3)^12 + x^39*(b + c*x^3)^13))/42","B",1
96,1,22,23,0.0676116,"\int x^{-1+n} \left(b+2 c x^n\right) \left(a+b x^n+c x^{2 n}\right)^{13} \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^13,x]","\frac{\left(a+x^n \left(b+c x^n\right)\right)^{14}}{14 n}","\frac{\left(a+b x^n+c x^{2 n}\right)^{14}}{14 n}",1,"(a + x^n*(b + c*x^n))^14/(14*n)","A",1
97,1,201,18,0.1754336,"\int (b+2 c x) \left(-a+b x+c x^2\right)^{13} \, dx","Integrate[(b + 2*c*x)*(-a + b*x + c*x^2)^13,x]","\frac{1}{14} x (b+c x) \left(-14 a^{13}+91 a^{12} x (b+c x)-364 a^{11} x^2 (b+c x)^2+1001 a^{10} x^3 (b+c x)^3-2002 a^9 x^4 (b+c x)^4+3003 a^8 x^5 (b+c x)^5-3432 a^7 x^6 (b+c x)^6+3003 a^6 x^7 (b+c x)^7-2002 a^5 x^8 (b+c x)^8+1001 a^4 x^9 (b+c x)^9-364 a^3 x^{10} (b+c x)^{10}+91 a^2 x^{11} (b+c x)^{11}-14 a x^{12} (b+c x)^{12}+x^{13} (b+c x)^{13}\right)","\frac{1}{14} \left(a-b x-c x^2\right)^{14}",1,"(x*(b + c*x)*(-14*a^13 + 91*a^12*x*(b + c*x) - 364*a^11*x^2*(b + c*x)^2 + 1001*a^10*x^3*(b + c*x)^3 - 2002*a^9*x^4*(b + c*x)^4 + 3003*a^8*x^5*(b + c*x)^5 - 3432*a^7*x^6*(b + c*x)^6 + 3003*a^6*x^7*(b + c*x)^7 - 2002*a^5*x^8*(b + c*x)^8 + 1001*a^4*x^9*(b + c*x)^9 - 364*a^3*x^10*(b + c*x)^10 + 91*a^2*x^11*(b + c*x)^11 - 14*a*x^12*(b + c*x)^12 + x^13*(b + c*x)^13))/14","B",1
98,1,233,20,0.1687964,"\int x \left(b+2 c x^2\right) \left(-a+b x^2+c x^4\right)^{13} \, dx","Integrate[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^13,x]","\frac{1}{28} x^2 \left(b+c x^2\right) \left(-14 a^{13}+91 a^{12} x^2 \left(b+c x^2\right)-364 a^{11} x^4 \left(b+c x^2\right)^2+1001 a^{10} x^6 \left(b+c x^2\right)^3-2002 a^9 x^8 \left(b+c x^2\right)^4+3003 a^8 x^{10} \left(b+c x^2\right)^5-3432 a^7 x^{12} \left(b+c x^2\right)^6+3003 a^6 x^{14} \left(b+c x^2\right)^7-2002 a^5 x^{16} \left(b+c x^2\right)^8+1001 a^4 x^{18} \left(b+c x^2\right)^9-364 a^3 x^{20} \left(b+c x^2\right)^{10}+91 a^2 x^{22} \left(b+c x^2\right)^{11}-14 a x^{24} \left(b+c x^2\right)^{12}+x^{26} \left(b+c x^2\right)^{13}\right)","\frac{1}{28} \left(a-b x^2-c x^4\right)^{14}",1,"(x^2*(b + c*x^2)*(-14*a^13 + 91*a^12*x^2*(b + c*x^2) - 364*a^11*x^4*(b + c*x^2)^2 + 1001*a^10*x^6*(b + c*x^2)^3 - 2002*a^9*x^8*(b + c*x^2)^4 + 3003*a^8*x^10*(b + c*x^2)^5 - 3432*a^7*x^12*(b + c*x^2)^6 + 3003*a^6*x^14*(b + c*x^2)^7 - 2002*a^5*x^16*(b + c*x^2)^8 + 1001*a^4*x^18*(b + c*x^2)^9 - 364*a^3*x^20*(b + c*x^2)^10 + 91*a^2*x^22*(b + c*x^2)^11 - 14*a*x^24*(b + c*x^2)^12 + x^26*(b + c*x^2)^13))/28","B",1
99,1,233,20,0.1670923,"\int x^2 \left(b+2 c x^3\right) \left(-a+b x^3+c x^6\right)^{13} \, dx","Integrate[x^2*(b + 2*c*x^3)*(-a + b*x^3 + c*x^6)^13,x]","\frac{1}{42} x^3 \left(b+c x^3\right) \left(-14 a^{13}+91 a^{12} x^3 \left(b+c x^3\right)-364 a^{11} x^6 \left(b+c x^3\right)^2+1001 a^{10} x^9 \left(b+c x^3\right)^3-2002 a^9 x^{12} \left(b+c x^3\right)^4+3003 a^8 x^{15} \left(b+c x^3\right)^5-3432 a^7 x^{18} \left(b+c x^3\right)^6+3003 a^6 x^{21} \left(b+c x^3\right)^7-2002 a^5 x^{24} \left(b+c x^3\right)^8+1001 a^4 x^{27} \left(b+c x^3\right)^9-364 a^3 x^{30} \left(b+c x^3\right)^{10}+91 a^2 x^{33} \left(b+c x^3\right)^{11}-14 a x^{36} \left(b+c x^3\right)^{12}+x^{39} \left(b+c x^3\right)^{13}\right)","\frac{1}{42} \left(a-b x^3-c x^6\right)^{14}",1,"(x^3*(b + c*x^3)*(-14*a^13 + 91*a^12*x^3*(b + c*x^3) - 364*a^11*x^6*(b + c*x^3)^2 + 1001*a^10*x^9*(b + c*x^3)^3 - 2002*a^9*x^12*(b + c*x^3)^4 + 3003*a^8*x^15*(b + c*x^3)^5 - 3432*a^7*x^18*(b + c*x^3)^6 + 3003*a^6*x^21*(b + c*x^3)^7 - 2002*a^5*x^24*(b + c*x^3)^8 + 1001*a^4*x^27*(b + c*x^3)^9 - 364*a^3*x^30*(b + c*x^3)^10 + 91*a^2*x^33*(b + c*x^3)^11 - 14*a*x^36*(b + c*x^3)^12 + x^39*(b + c*x^3)^13))/42","B",1
100,1,24,25,0.0537133,"\int x^{-1+n} \left(b+2 c x^n\right) \left(-a+b x^n+c x^{2 n}\right)^{13} \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^13,x]","\frac{\left(x^n \left(b+c x^n\right)-a\right)^{14}}{14 n}","\frac{\left(a-b x^n-c x^{2 n}\right)^{14}}{14 n}",1,"(-a + x^n*(b + c*x^n))^14/(14*n)","A",1
101,1,172,15,0.0051405,"\int (b+2 c x) \left(b x+c x^2\right)^{13} \, dx","Integrate[(b + 2*c*x)*(b*x + c*x^2)^13,x]","\frac{b^{14} x^{14}}{14}+b^{13} c x^{15}+\frac{13}{2} b^{12} c^2 x^{16}+26 b^{11} c^3 x^{17}+\frac{143}{2} b^{10} c^4 x^{18}+143 b^9 c^5 x^{19}+\frac{429}{2} b^8 c^6 x^{20}+\frac{1716}{7} b^7 c^7 x^{21}+\frac{429}{2} b^6 c^8 x^{22}+143 b^5 c^9 x^{23}+\frac{143}{2} b^4 c^{10} x^{24}+26 b^3 c^{11} x^{25}+\frac{13}{2} b^2 c^{12} x^{26}+b c^{13} x^{27}+\frac{c^{14} x^{28}}{14}","\frac{1}{14} \left(b x+c x^2\right)^{14}",1,"(b^14*x^14)/14 + b^13*c*x^15 + (13*b^12*c^2*x^16)/2 + 26*b^11*c^3*x^17 + (143*b^10*c^4*x^18)/2 + 143*b^9*c^5*x^19 + (429*b^8*c^6*x^20)/2 + (1716*b^7*c^7*x^21)/7 + (429*b^6*c^8*x^22)/2 + 143*b^5*c^9*x^23 + (143*b^4*c^10*x^24)/2 + 26*b^3*c^11*x^25 + (13*b^2*c^12*x^26)/2 + b*c^13*x^27 + (c^14*x^28)/14","B",1
102,1,182,16,0.0061764,"\int x \left(b+2 c x^2\right) \left(b x^2+c x^4\right)^{13} \, dx","Integrate[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x]","\frac{b^{14} x^{28}}{28}+\frac{1}{2} b^{13} c x^{30}+\frac{13}{4} b^{12} c^2 x^{32}+13 b^{11} c^3 x^{34}+\frac{143}{4} b^{10} c^4 x^{36}+\frac{143}{2} b^9 c^5 x^{38}+\frac{429}{4} b^8 c^6 x^{40}+\frac{858}{7} b^7 c^7 x^{42}+\frac{429}{4} b^6 c^8 x^{44}+\frac{143}{2} b^5 c^9 x^{46}+\frac{143}{4} b^4 c^{10} x^{48}+13 b^3 c^{11} x^{50}+\frac{13}{4} b^2 c^{12} x^{52}+\frac{1}{2} b c^{13} x^{54}+\frac{c^{14} x^{56}}{28}","\frac{1}{28} x^{28} \left(b+c x^2\right)^{14}",1,"(b^14*x^28)/28 + (b^13*c*x^30)/2 + (13*b^12*c^2*x^32)/4 + 13*b^11*c^3*x^34 + (143*b^10*c^4*x^36)/4 + (143*b^9*c^5*x^38)/2 + (429*b^8*c^6*x^40)/4 + (858*b^7*c^7*x^42)/7 + (429*b^6*c^8*x^44)/4 + (143*b^5*c^9*x^46)/2 + (143*b^4*c^10*x^48)/4 + 13*b^3*c^11*x^50 + (13*b^2*c^12*x^52)/4 + (b*c^13*x^54)/2 + (c^14*x^56)/28","B",1
103,1,186,16,0.0061884,"\int x^2 \left(b+2 c x^3\right) \left(b x^3+c x^6\right)^{13} \, dx","Integrate[x^2*(b + 2*c*x^3)*(b*x^3 + c*x^6)^13,x]","\frac{b^{14} x^{42}}{42}+\frac{1}{3} b^{13} c x^{45}+\frac{13}{6} b^{12} c^2 x^{48}+\frac{26}{3} b^{11} c^3 x^{51}+\frac{143}{6} b^{10} c^4 x^{54}+\frac{143}{3} b^9 c^5 x^{57}+\frac{143}{2} b^8 c^6 x^{60}+\frac{572}{7} b^7 c^7 x^{63}+\frac{143}{2} b^6 c^8 x^{66}+\frac{143}{3} b^5 c^9 x^{69}+\frac{143}{6} b^4 c^{10} x^{72}+\frac{26}{3} b^3 c^{11} x^{75}+\frac{13}{6} b^2 c^{12} x^{78}+\frac{1}{3} b c^{13} x^{81}+\frac{c^{14} x^{84}}{42}","\frac{1}{42} x^{42} \left(b+c x^3\right)^{14}",1,"(b^14*x^42)/42 + (b^13*c*x^45)/3 + (13*b^12*c^2*x^48)/6 + (26*b^11*c^3*x^51)/3 + (143*b^10*c^4*x^54)/6 + (143*b^9*c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^8*x^66)/2 + (143*b^5*c^9*x^69)/3 + (143*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75)/3 + (13*b^2*c^12*x^78)/6 + (b*c^13*x^81)/3 + (c^14*x^84)/42","B",1
104,1,21,21,0.1167325,"\int x^{-1+n} \left(b+2 c x^n\right) \left(b x^n+c x^{2 n}\right)^{13} \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(b*x^n + c*x^(2*n))^13,x]","\frac{x^{14 n} \left(b+c x^n\right)^{14}}{14 n}","\frac{x^{14 n} \left(b+c x^n\right)^{14}}{14 n}",1,"(x^(14*n)*(b + c*x^n)^14)/(14*n)","A",1
105,1,10,11,0.0029177,"\int \frac{b+2 c x}{a+b x+c x^2} \, dx","Integrate[(b + 2*c*x)/(a + b*x + c*x^2),x]","\log (a+x (b+c x))","\log \left(a+b x+c x^2\right)",1,"Log[a + x*(b + c*x)]","A",1
106,1,17,17,0.005742,"\int \frac{x \left(b+2 c x^2\right)}{a+b x^2+c x^4} \, dx","Integrate[(x*(b + 2*c*x^2))/(a + b*x^2 + c*x^4),x]","\frac{1}{2} \log \left(a+b x^2+c x^4\right)","\frac{1}{2} \log \left(a+b x^2+c x^4\right)",1,"Log[a + b*x^2 + c*x^4]/2","A",1
107,1,17,17,0.006298,"\int \frac{x^2 \left(b+2 c x^3\right)}{a+b x^3+c x^6} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \log \left(a+b x^3+c x^6\right)","\frac{1}{3} \log \left(a+b x^3+c x^6\right)",1,"Log[a + b*x^3 + c*x^6]/3","A",1
108,1,19,19,0.1049093,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{a+b x^n+c x^{2 n}} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n)),x]","\frac{\log \left(a+b x^n+c x^{2 n}\right)}{n}","\frac{\log \left(a+b x^n+c x^{2 n}\right)}{n}",1,"Log[a + b*x^n + c*x^(2*n)]/n","A",1
109,1,15,16,0.0118068,"\int \frac{b+2 c x}{\left(a+b x+c x^2\right)^8} \, dx","Integrate[(b + 2*c*x)/(a + b*x + c*x^2)^8,x]","-\frac{1}{7 (a+x (b+c x))^7}","-\frac{1}{7 \left(a+b x+c x^2\right)^7}",1,"-1/7*1/(a + x*(b + c*x))^7","A",1
110,1,18,18,0.0124285,"\int \frac{x \left(b+2 c x^2\right)}{\left(a+b x^2+c x^4\right)^8} \, dx","Integrate[(x*(b + 2*c*x^2))/(a + b*x^2 + c*x^4)^8,x]","-\frac{1}{14 \left(a+b x^2+c x^4\right)^7}","-\frac{1}{14 \left(a+b x^2+c x^4\right)^7}",1,"-1/14*1/(a + b*x^2 + c*x^4)^7","A",1
111,1,18,18,0.0129815,"\int \frac{x^2 \left(b+2 c x^3\right)}{\left(a+b x^3+c x^6\right)^8} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6)^8,x]","-\frac{1}{21 \left(a+b x^3+c x^6\right)^7}","-\frac{1}{21 \left(a+b x^3+c x^6\right)^7}",1,"-1/21*1/(a + b*x^3 + c*x^6)^7","A",1
112,1,22,23,0.0601993,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{\left(a+b x^n+c x^{2 n}\right)^8} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(a + b*x^n + c*x^(2*n))^8,x]","-\frac{1}{7 n \left(a+x^n \left(b+c x^n\right)\right)^7}","-\frac{1}{7 n \left(a+b x^n+c x^{2 n}\right)^7}",1,"-1/7*1/(n*(a + x^n*(b + c*x^n))^7)","A",1
113,1,12,13,0.0046258,"\int \frac{b+2 c x}{-a+b x+c x^2} \, dx","Integrate[(b + 2*c*x)/(-a + b*x + c*x^2),x]","\log (x (b+c x)-a)","\log \left(a-b x-c x^2\right)",1,"Log[-a + x*(b + c*x)]","A",1
114,1,19,19,0.0067003,"\int \frac{x \left(b+2 c x^2\right)}{-a+b x^2+c x^4} \, dx","Integrate[(x*(b + 2*c*x^2))/(-a + b*x^2 + c*x^4),x]","\frac{1}{2} \log \left(-a+b x^2+c x^4\right)","\frac{1}{2} \log \left(a-b x^2-c x^4\right)",1,"Log[-a + b*x^2 + c*x^4]/2","A",1
115,1,19,19,0.0068964,"\int \frac{x^2 \left(b+2 c x^3\right)}{-a+b x^3+c x^6} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(-a + b*x^3 + c*x^6),x]","\frac{1}{3} \log \left(-a+b x^3+c x^6\right)","\frac{1}{3} \log \left(a-b x^3-c x^6\right)",1,"Log[-a + b*x^3 + c*x^6]/3","A",1
116,1,21,21,0.1172527,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{-a+b x^n+c x^{2 n}} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n)),x]","\frac{\log \left(a-b x^n-c x^{2 n}\right)}{n}","\frac{\log \left(a-b x^n-c x^{2 n}\right)}{n}",1,"Log[a - b*x^n - c*x^(2*n)]/n","A",1
117,1,16,18,0.0133724,"\int \frac{b+2 c x}{\left(-a+b x+c x^2\right)^8} \, dx","Integrate[(b + 2*c*x)/(-a + b*x + c*x^2)^8,x]","\frac{1}{7 (a-x (b+c x))^7}","\frac{1}{7 \left(a-b x-c x^2\right)^7}",1,"1/(7*(a - x*(b + c*x))^7)","A",1
118,1,20,20,0.0169498,"\int \frac{x \left(b+2 c x^2\right)}{\left(-a+b x^2+c x^4\right)^8} \, dx","Integrate[(x*(b + 2*c*x^2))/(-a + b*x^2 + c*x^4)^8,x]","-\frac{1}{14 \left(-a+b x^2+c x^4\right)^7}","\frac{1}{14 \left(a-b x^2-c x^4\right)^7}",1,"-1/14*1/(-a + b*x^2 + c*x^4)^7","A",1
119,1,20,20,0.0165211,"\int \frac{x^2 \left(b+2 c x^3\right)}{\left(-a+b x^3+c x^6\right)^8} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(-a + b*x^3 + c*x^6)^8,x]","-\frac{1}{21 \left(-a+b x^3+c x^6\right)^7}","\frac{1}{21 \left(a-b x^3-c x^6\right)^7}",1,"-1/21*1/(-a + b*x^3 + c*x^6)^7","A",1
120,1,23,25,0.0661379,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{\left(-a+b x^n+c x^{2 n}\right)^8} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n))^8,x]","\frac{1}{7 n \left(a-x^n \left(b+c x^n\right)\right)^7}","\frac{1}{7 n \left(a-b x^n-c x^{2 n}\right)^7}",1,"1/(7*n*(a - x^n*(b + c*x^n))^7)","A",1
121,1,9,10,0.004041,"\int \frac{b+2 c x}{b x+c x^2} \, dx","Integrate[(b + 2*c*x)/(b*x + c*x^2),x]","\log (b+c x)+\log (x)","\log \left(b x+c x^2\right)",1,"Log[x] + Log[b + c*x]","A",1
122,1,15,16,0.0063659,"\int \frac{x \left(b+2 c x^2\right)}{b x^2+c x^4} \, dx","Integrate[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4),x]","\frac{1}{2} \log \left(b+c x^2\right)+\log (x)","\frac{1}{2} \log \left(b x^2+c x^4\right)",1,"Log[x] + Log[b + c*x^2]/2","A",1
123,1,15,16,0.0070938,"\int \frac{x^2 \left(b+2 c x^3\right)}{b x^3+c x^6} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(b*x^3 + c*x^6),x]","\frac{1}{3} \log \left(b+c x^3\right)+\log (x)","\frac{1}{3} \log \left(b x^3+c x^6\right)",1,"Log[x] + Log[b + c*x^3]/3","A",1
124,1,15,15,0.0119826,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{b x^n+c x^{2 n}} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n)),x]","\frac{\log \left(b+c x^n\right)}{n}+\log (x)","\frac{\log \left(b+c x^n\right)}{n}+\log (x)",1,"Log[x] + Log[b + c*x^n]/n","A",1
125,1,14,15,0.020356,"\int \frac{b+2 c x}{\left(b x+c x^2\right)^8} \, dx","Integrate[(b + 2*c*x)/(b*x + c*x^2)^8,x]","-\frac{1}{7 x^7 (b+c x)^7}","-\frac{1}{7 \left(b x+c x^2\right)^7}",1,"-1/7*1/(x^7*(b + c*x)^7)","A",1
126,1,16,16,0.0293706,"\int \frac{x \left(b+2 c x^2\right)}{\left(b x^2+c x^4\right)^8} \, dx","Integrate[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4)^8,x]","-\frac{1}{14 x^{14} \left(b+c x^2\right)^7}","-\frac{1}{14 x^{14} \left(b+c x^2\right)^7}",1,"-1/14*1/(x^14*(b + c*x^2)^7)","A",1
127,1,16,16,0.0366014,"\int \frac{x^2 \left(b+2 c x^3\right)}{\left(b x^3+c x^6\right)^8} \, dx","Integrate[(x^2*(b + 2*c*x^3))/(b*x^3 + c*x^6)^8,x]","-\frac{1}{21 x^{21} \left(b+c x^3\right)^7}","-\frac{1}{21 x^{21} \left(b+c x^3\right)^7}",1,"-1/21*1/(x^21*(b + c*x^3)^7)","A",1
128,1,21,21,0.180939,"\int \frac{x^{-1+n} \left(b+2 c x^n\right)}{\left(b x^n+c x^{2 n}\right)^8} \, dx","Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(b*x^n + c*x^(2*n))^8,x]","-\frac{x^{-7 n}}{7 n \left(b+c x^n\right)^7}","-\frac{x^{-7 n}}{7 n \left(b+c x^n\right)^7}",1,"-1/7*1/(n*x^(7*n)*(b + c*x^n)^7)","A",1
129,1,19,20,0.00933,"\int (b+2 c x) \left(a+b x+c x^2\right)^p \, dx","Integrate[(b + 2*c*x)*(a + b*x + c*x^2)^p,x]","\frac{(a+x (b+c x))^{p+1}}{p+1}","\frac{\left(a+b x+c x^2\right)^{p+1}}{p+1}",1,"(a + x*(b + c*x))^(1 + p)/(1 + p)","A",1
130,1,25,25,0.0105376,"\int x \left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^p \, dx","Integrate[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^p,x]","\frac{\left(a+b x^2+c x^4\right)^{p+1}}{2 (p+1)}","\frac{\left(a+b x^2+c x^4\right)^{p+1}}{2 (p+1)}",1,"(a + b*x^2 + c*x^4)^(1 + p)/(2*(1 + p))","A",1
131,1,25,25,0.0115392,"\int x^2 \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^p,x]","\frac{\left(a+b x^3+c x^6\right)^{p+1}}{3 (p+1)}","\frac{\left(a+b x^3+c x^6\right)^{p+1}}{3 (p+1)}",1,"(a + b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))","A",1
132,1,26,27,0.0286849,"\int x^{-1+n} \left(b+2 c x^n\right) \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^p,x]","\frac{\left(a+x^n \left(b+c x^n\right)\right)^{p+1}}{n (p+1)}","\frac{\left(a+b x^n+c x^{2 n}\right)^{p+1}}{n (p+1)}",1,"(a + x^n*(b + c*x^n))^(1 + p)/(n*(1 + p))","A",1
133,1,21,22,0.0107622,"\int (b+2 c x) \left(-a+b x+c x^2\right)^p \, dx","Integrate[(b + 2*c*x)*(-a + b*x + c*x^2)^p,x]","\frac{(x (b+c x)-a)^{p+1}}{p+1}","\frac{\left(-a+b x+c x^2\right)^{p+1}}{p+1}",1,"(-a + x*(b + c*x))^(1 + p)/(1 + p)","A",1
134,1,27,27,0.0145524,"\int x \left(b+2 c x^2\right) \left(-a+b x^2+c x^4\right)^p \, dx","Integrate[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^p,x]","\frac{\left(-a+b x^2+c x^4\right)^{p+1}}{2 (p+1)}","\frac{\left(-a+b x^2+c x^4\right)^{p+1}}{2 (p+1)}",1,"(-a + b*x^2 + c*x^4)^(1 + p)/(2*(1 + p))","A",1
135,1,27,27,0.0157223,"\int x^2 \left(b+2 c x^3\right) \left(-a+b x^3+c x^6\right)^p \, dx","Integrate[x^2*(b + 2*c*x^3)*(-a + b*x^3 + c*x^6)^p,x]","\frac{\left(-a+b x^3+c x^6\right)^{p+1}}{3 (p+1)}","\frac{\left(-a+b x^3+c x^6\right)^{p+1}}{3 (p+1)}",1,"(-a + b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))","A",1
136,1,28,29,0.0341542,"\int x^{-1+n} \left(b+2 c x^n\right) \left(-a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^p,x]","\frac{\left(x^n \left(b+c x^n\right)-a\right)^{p+1}}{n (p+1)}","\frac{\left(-a+b x^n+c x^{2 n}\right)^{p+1}}{n (p+1)}",1,"(-a + x^n*(b + c*x^n))^(1 + p)/(n*(1 + p))","A",1
137,1,17,19,0.0096539,"\int (b+2 c x) \left(b x+c x^2\right)^p \, dx","Integrate[(b + 2*c*x)*(b*x + c*x^2)^p,x]","\frac{(x (b+c x))^{p+1}}{p+1}","\frac{\left(b x+c x^2\right)^{p+1}}{p+1}",1,"(x*(b + c*x))^(1 + p)/(1 + p)","A",1
138,1,97,24,0.0727212,"\int x \left(b+2 c x^2\right) \left(b x^2+c x^4\right)^p \, dx","Integrate[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^p,x]","\frac{x^2 \left(x^2 \left(b+c x^2\right)\right)^p \left(\frac{c x^2}{b}+1\right)^{-p} \left(2 c (p+1) x^2 \, _2F_1\left(-p,p+2;p+3;-\frac{c x^2}{b}\right)+b (p+2) \, _2F_1\left(-p,p+1;p+2;-\frac{c x^2}{b}\right)\right)}{2 (p+1) (p+2)}","\frac{\left(b x^2+c x^4\right)^{p+1}}{2 (p+1)}",1,"(x^2*(x^2*(b + c*x^2))^p*(b*(2 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^2)/b)] + 2*c*(1 + p)*x^2*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^2)/b)]))/(2*(1 + p)*(2 + p)*(1 + (c*x^2)/b)^p)","C",1
139,1,97,24,0.0739955,"\int x^2 \left(b+2 c x^3\right) \left(b x^3+c x^6\right)^p \, dx","Integrate[x^2*(b + 2*c*x^3)*(b*x^3 + c*x^6)^p,x]","\frac{x^3 \left(x^3 \left(b+c x^3\right)\right)^p \left(\frac{c x^3}{b}+1\right)^{-p} \left(2 c (p+1) x^3 \, _2F_1\left(-p,p+2;p+3;-\frac{c x^3}{b}\right)+b (p+2) \, _2F_1\left(-p,p+1;p+2;-\frac{c x^3}{b}\right)\right)}{3 (p+1) (p+2)}","\frac{\left(b x^3+c x^6\right)^{p+1}}{3 (p+1)}",1,"(x^3*(x^3*(b + c*x^3))^p*(b*(2 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^3)/b)] + 2*c*(1 + p)*x^3*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^3)/b)]))/(3*(1 + p)*(2 + p)*(1 + (c*x^3)/b)^p)","C",1
140,1,111,26,0.1304508,"\int x^{-1+n} \left(b+2 c x^n\right) \left(b x^n+c x^{2 n}\right)^p \, dx","Integrate[x^(-1 + n)*(b + 2*c*x^n)*(b*x^n + c*x^(2*n))^p,x]","\frac{x^{-n p} \left(x^n \left(b+c x^n\right)\right)^p \left(\frac{c x^n}{b}+1\right)^{-p} \left(b (p+2) x^{n (p+1)} \, _2F_1\left(-p,p+1;p+2;-\frac{c x^n}{b}\right)+2 c (p+1) x^{n (p+2)} \, _2F_1\left(-p,p+2;p+3;-\frac{c x^n}{b}\right)\right)}{n (p+1) (p+2)}","\frac{\left(b x^n+c x^{2 n}\right)^{p+1}}{n (p+1)}",1,"((x^n*(b + c*x^n))^p*(b*(2 + p)*x^(n*(1 + p))*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^n)/b)] + 2*c*(1 + p)*x^(n*(2 + p))*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^n)/b)]))/(n*(1 + p)*(2 + p)*x^(n*p)*(1 + (c*x^n)/b)^p)","C",1
141,1,158,196,0.3050247,"\int \frac{(f x)^m \left(d+e x^n\right)}{a+b x^n+c x^{2 n}} \, dx","Integrate[((f*x)^m*(d + e*x^n))/(a + b*x^n + c*x^(2*n)),x]","\frac{x (f x)^m \left(\left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+\left(d \sqrt{b^2-4 a c}+2 a e-b d\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)\right)}{2 a (m+1) \sqrt{b^2-4 a c}}","\frac{(f x)^{m+1} \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{f (m+1) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{(f x)^{m+1} \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{f (m+1) \left(\sqrt{b^2-4 a c}+b\right)}",1,"(x*(f*x)^m*((b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]))/(2*a*Sqrt[b^2 - 4*a*c]*(1 + m))","A",1
142,1,5363,374,6.5191748,"\int \frac{(f x)^m \left(d+e x^n\right)}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[((f*x)^m*(d + e*x^n))/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","-\frac{c (f x)^{m+1} \left((m-n+1) (b d-2 a e)-\frac{2 a b e n+4 a c d (m-2 n+1)+b^2 (-d) (m-n+1)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a f (m+1) n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (f x)^{m+1} \left(\frac{2 a b e n+4 a c d (m-2 n+1)+b^2 (-d) (m-n+1)}{\sqrt{b^2-4 a c}}+(m-n+1) (b d-2 a e)\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a f (m+1) n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(f x)^{m+1} \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{a f n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"Result too large to show","B",1
143,1,20515,816,7.5372684,"\int \frac{(f x)^m \left(d+e x^n\right)}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[((f*x)^m*(d + e*x^n))/(a + b*x^n + c*x^(2*n))^3,x]","\text{Result too large to show}","-\frac{c \left(\left(-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right) (m-n+1)+\frac{-d \left(m^2+(2-3 n) m+2 n^2-3 n+1\right) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left(m^2+(2-4 n) m+3 n^2-4 n+1\right) b^2-4 a^2 c e \left(m^2+(2-n) m-3 n^2-n+1\right) b-8 a^2 c^2 d \left(m^2+(2-6 n) m+8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) (f x)^{m+1}}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) f (m+1) n^2}-\frac{c \left(\left(-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right) (m-n+1)-\frac{-d \left(m^2+(2-3 n) m+2 n^2-3 n+1\right) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left(m^2+(2-4 n) m+3 n^2-4 n+1\right) b^2-4 a^2 c e \left(m^2+(2-n) m-3 n^2-n+1\right) b-8 a^2 c^2 d \left(m^2+(2-6 n) m+8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) (f x)^{m+1}}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) f (m+1) n^2}+\frac{\left(c \left(-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right) x^n+\left(b^2-2 a c\right) \left(-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right)+a b c (b d-2 a e) (m-3 n+1)\right) (f x)^{m+1}}{2 a^2 \left(b^2-4 a c\right)^2 f n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{\left(c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right) (f x)^{m+1}}{2 a \left(b^2-4 a c\right) f n \left(b x^n+c x^{2 n}+a\right)^2}",1,"Result too large to show","B",0
144,1,47,47,0.0229675,"\int \frac{\sqrt[3]{c}-2 \sqrt[3]{d} \sqrt[3]{x}}{c \sqrt[3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt[3]{c} d x^{4/3}} \, dx","Integrate[(c^(1/3) - 2*d^(1/3)*x^(1/3))/(c*d^(1/3)*x^(2/3) - c^(2/3)*d^(2/3)*x + c^(1/3)*d*x^(4/3)),x]","-\frac{3 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{x}+d^{2/3} x^{2/3}\right)}{\sqrt[3]{c} d^{2/3}}","-\frac{3 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} \sqrt[3]{x}+d^{2/3} x^{2/3}\right)}{\sqrt[3]{c} d^{2/3}}",1,"(-3*Log[c^(2/3) - c^(1/3)*d^(1/3)*x^(1/3) + d^(2/3)*x^(2/3)])/(c^(1/3)*d^(2/3))","A",1
145,0,0,245,0.1605058,"\int \frac{(f x)^m \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","Integrate[((f*x)^m*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)),x]","\int \frac{(f x)^m \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","\frac{2 c (f x)^{m+1} \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{f (m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c (f x)^{m+1} \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{f (m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"Integrate[((f*x)^m*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)), x]","F",-1
146,0,0,210,0.1605256,"\int \frac{x^2 \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","Integrate[(x^2*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)),x]","\int \frac{x^2 \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","-\frac{2 c x^3 \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c x^3 \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"Integrate[(x^2*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)), x]","F",-1
147,0,0,206,0.1322057,"\int \frac{x \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","Integrate[(x*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)),x]","\int \frac{x \left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","-\frac{c x^2 \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{2}{n};1,-q;\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c x^2 \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{2}{n};1,-q;\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"Integrate[(x*(d + e*x^n)^q)/(a + b*x^n + c*x^(2*n)), x]","F",-1
148,0,0,194,0.0830967,"\int \frac{\left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)),x]","\int \frac{\left(d+e x^n\right)^q}{a+b x^n+c x^{2 n}} \, dx","-\frac{2 c x \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{1}{n};1,-q;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(\frac{1}{n};1,-q;1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"Integrate[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x]","F",-1
149,1,218,263,0.674683,"\int \frac{\left(d+e x^n\right)^q}{x \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[(d + e*x^n)^q/(x*(a + b*x^n + c*x^(2*n))),x]","\frac{\left(d+e x^n\right)^{q+1} \left(\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^n+d\right)}{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}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^n+d\right)}{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,q+1;q+2;\frac{e x^n}{d}+1\right)}{d}\right)}{a n (q+1)}","\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \left(d+e x^n\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^n+d\right)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e}\right)}{a n (q+1) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \left(d+e x^n\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{2 c \left(e x^n+d\right)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{a n (q+1) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{\left(d+e x^n\right)^{q+1} \, _2F_1\left(1,q+1;q+2;\frac{e x^n}{d}+1\right)}{a d n (q+1)}",1,"((d + e*x^n)^(1 + q)*((c*(1 + b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^n))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + (c*(1 - b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + q, 2 + q, (2*c*(d + e*x^n))/(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 + q, 2 + q, 1 + (e*x^n)/d]/d))/(a*n*(1 + q))","A",1
150,0,0,212,0.1324157,"\int \frac{\left(d+e x^n\right)^q}{x^2 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[(d + e*x^n)^q/(x^2*(a + b*x^n + c*x^(2*n))),x]","\int \frac{\left(d+e x^n\right)^q}{x^2 \left(a+b x^n+c x^{2 n}\right)} \, dx","\frac{2 c \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{x \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{2 c \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{x \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"Integrate[(d + e*x^n)^q/(x^2*(a + b*x^n + c*x^(2*n))), x]","F",-1
151,0,0,210,0.1251675,"\int \frac{\left(d+e x^n\right)^q}{x^3 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[(d + e*x^n)^q/(x^3*(a + b*x^n + c*x^(2*n))),x]","\int \frac{\left(d+e x^n\right)^q}{x^3 \left(a+b x^n+c x^{2 n}\right)} \, dx","\frac{c \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{x^2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{c \left(d+e x^n\right)^q \left(\frac{e x^n}{d}+1\right)^{-q} F_1\left(-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right)}{x^2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"Integrate[(d + e*x^n)^q/(x^3*(a + b*x^n + c*x^(2*n))), x]","F",-1
152,1,391,498,1.1358427,"\int (f x)^m \left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x (f x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p \left(d^2 \left(m^2+m (3 n+2)+2 n^2+3 n+1\right) F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e (m+1) x^n \left(2 d (m+2 n+1) F_1\left(\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e (m+n+1) x^n F_1\left(\frac{m+2 n+1}{n};-p,-p;\frac{m+3 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)\right)}{(m+1) (m+n+1) (m+2 n+1)}","\frac{d^2 (f x)^{m+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+2 n+1}{n};-p,-p;\frac{m+3 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{m+2 n+1}",1,"(x*(f*x)^m*(a + x^n*(b + c*x^n))^p*(d^2*(1 + m^2 + 3*n + 2*n^2 + m*(2 + 3*n))*AppellF1[(1 + m)/n, -p, -p, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + e*(1 + m)*x^n*(2*d*(1 + m + 2*n)*AppellF1[(1 + m + n)/n, -p, -p, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + e*(1 + m + n)*x^n*AppellF1[(1 + m + 2*n)/n, -p, -p, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/((1 + m)*(1 + m + n)*(1 + m + 2*n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
153,1,273,323,0.6002823,"\int (f x)^m \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x (f x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p \left(d (m+n+1) F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e (m+1) x^n F_1\left(\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{(m+1) (m+n+1)}","\frac{d (f x)^{m+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{m+n+1}",1,"(x*(f*x)^m*(a + x^n*(b + c*x^n))^p*(d*(1 + m + n)*AppellF1[(1 + m)/n, -p, -p, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + e*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, -p, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/((1 + m)*(1 + m + n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
154,1,181,158,0.3018737,"\int (f x)^m \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(f*x)^m*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x (f x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{m+1}","\frac{(f x)^{m+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{f (m+1)}",1,"(x*(f*x)^m*(a + x^n*(b + c*x^n))^p*AppellF1[(1 + m)/n, -p, -p, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
155,0,0,34,0.1895344,"\int \frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","Integrate[((f*x)^m*(a + b*x^n + c*x^(2*n))^p)/(d + e*x^n),x]","\int \frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n},x\right)",0,"Integrate[((f*x)^m*(a + b*x^n + c*x^(2*n))^p)/(d + e*x^n), x]","A",-1
156,0,0,34,0.268742,"\int \frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Integrate[((f*x)^m*(a + b*x^n + c*x^(2*n))^p)/(d + e*x^n)^2,x]","\int \frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*x^n + c*x^(2*n))^p)/(d + e*x^n)^2, x]","A",-1