1,1,50,50,0.0019302,"\int (d+e x) \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x)*(a + b*x^2 + c*x^4),x]","a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6","a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6",1,"a*d*x + (a*e*x^2)/2 + (b*d*x^3)/3 + (b*e*x^4)/4 + (c*d*x^5)/5 + (c*e*x^6)/6","A",1
2,1,69,69,0.0205098,"\int \left(d+e x+f x^2\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x + f*x^2)*(a + b*x^2 + c*x^4),x]","\frac{1}{3} x^3 (a f+b d)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{4} b e x^4+\frac{1}{6} c e x^6+\frac{1}{7} c f x^7","\frac{1}{3} x^3 (a f+b d)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{4} b e x^4+\frac{1}{6} c e x^6+\frac{1}{7} c f x^7",1,"a*d*x + (a*e*x^2)/2 + ((b*d + a*f)*x^3)/3 + (b*e*x^4)/4 + ((c*d + b*f)*x^5)/5 + (c*e*x^6)/6 + (c*f*x^7)/7","A",1
3,1,88,88,0.0189669,"\int \left(d+e x+f x^2+g x^3\right) \left(a+b x^2+c x^4\right) \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)*(a + b*x^2 + c*x^4),x]","\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8","\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8",1,"a*d*x + (a*e*x^2)/2 + ((b*d + a*f)*x^3)/3 + ((b*e + a*g)*x^4)/4 + ((c*d + b*f)*x^5)/5 + ((c*e + b*g)*x^6)/6 + (c*f*x^7)/7 + (c*g*x^8)/8","A",1
4,1,105,105,0.0336822,"\int \left(a+b x^2+c x^4\right) \left(d+e x+f x^2+g x^3+h x^4\right) \, dx","Integrate[(a + b*x^2 + c*x^4)*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]","\frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} c g x^8+\frac{1}{9} c h x^9","\frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} c g x^8+\frac{1}{9} c h x^9",1,"a*d*x + (a*e*x^2)/2 + ((b*d + a*f)*x^3)/3 + ((b*e + a*g)*x^4)/4 + ((c*d + b*f + a*h)*x^5)/5 + ((c*e + b*g)*x^6)/6 + ((c*f + b*h)*x^7)/7 + (c*g*x^8)/8 + (c*h*x^9)/9","A",1
5,1,122,122,0.0380028,"\int \left(a+b x^2+c x^4\right) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right) \, dx","Integrate[(a + b*x^2 + c*x^4)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5),x]","\frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{6} x^6 (a i+b g+c e)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} x^8 (b i+c g)+\frac{1}{9} c h x^9+\frac{1}{10} c i x^{10}","\frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{6} x^6 (a i+b g+c e)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} x^8 (b i+c g)+\frac{1}{9} c h x^9+\frac{1}{10} c i x^{10}",1,"a*d*x + (a*e*x^2)/2 + ((b*d + a*f)*x^3)/3 + ((b*e + a*g)*x^4)/4 + ((c*d + b*f + a*h)*x^5)/5 + ((c*e + b*g + a*i)*x^6)/6 + ((c*f + b*h)*x^7)/7 + ((c*g + b*i)*x^8)/8 + (c*h*x^9)/9 + (c*i*x^10)/10","A",1
6,1,97,112,0.0490514,"\int (d+e x) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x)*(a + b*x^2 + c*x^4)^2,x]","\frac{630 a^2 x (2 d+e x)+42 a \left(5 b x^3 (4 d+3 e x)+2 c x^5 (6 d+5 e x)\right)+42 b^2 x^5 (6 d+5 e x)+45 b c x^7 (8 d+7 e x)+14 c^2 x^9 (10 d+9 e x)}{1260}","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{5} d x^5 \left(2 a c+b^2\right)+\frac{1}{6} e x^6 \left(2 a c+b^2\right)+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{2}{7} b c d x^7+\frac{1}{4} b c e x^8+\frac{1}{9} c^2 d x^9+\frac{1}{10} c^2 e x^{10}",1,"(630*a^2*x*(2*d + e*x) + 42*b^2*x^5*(6*d + 5*e*x) + 45*b*c*x^7*(8*d + 7*e*x) + 14*c^2*x^9*(10*d + 9*e*x) + 42*a*(5*b*x^3*(4*d + 3*e*x) + 2*c*x^5*(6*d + 5*e*x)))/1260","A",1
7,1,154,154,0.0451831,"\int \left(d+e x+f x^2\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x + f*x^2)*(a + b*x^2 + c*x^4)^2,x]","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{6} e x^6 \left(2 a c+b^2\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{2} a b e x^4+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{4} b c e x^8+\frac{1}{10} c^2 e x^{10}+\frac{1}{11} c^2 f x^{11}","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{6} e x^6 \left(2 a c+b^2\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{2} a b e x^4+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{4} b c e x^8+\frac{1}{10} c^2 e x^{10}+\frac{1}{11} c^2 f x^{11}",1,"a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*b*e*x^4)/2 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2 + 2*a*c)*e*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + (b*c*e*x^8)/4 + (c*(c*d + 2*b*f)*x^9)/9 + (c^2*e*x^10)/10 + (c^2*f*x^11)/11","A",1
8,1,196,196,0.0568423,"\int \left(d+e x+f x^2+g x^3\right) \left(a+b x^2+c x^4\right)^2 \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)*(a + b*x^2 + c*x^4)^2,x]","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{8} x^8 \left(2 a c g+b^2 g+2 b c e\right)+\frac{1}{6} x^6 \left(2 a b g+2 a c e+b^2 e\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12}","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{8} x^8 \left(2 a c g+b^2 g+2 b c e\right)+\frac{1}{6} x^6 \left(2 a b g+2 a c e+b^2 e\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12}",1,"a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)*x^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11)/11 + (c^2*g*x^12)/12","A",1
9,1,234,234,0.0828115,"\int \left(a+b x^2+c x^4\right)^2 \left(d+e x+f x^2+g x^3+h x^4\right) \, dx","Integrate[(a + b*x^2 + c*x^4)^2*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]","\frac{1}{5} x^5 \left(a^2 h+2 a b f+2 a c d+b^2 d\right)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{9} x^9 \left(2 a c h+b^2 h+2 b c f+c^2 d\right)+\frac{1}{7} x^7 \left(2 a b h+2 a c f+b^2 f+2 b c d\right)+\frac{1}{8} x^8 \left(2 a c g+b^2 g+2 b c e\right)+\frac{1}{6} x^6 \left(2 a b g+2 a c e+b^2 e\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c x^{11} (2 b h+c f)+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13}","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{9} x^9 \left(2 c (a h+b f)+b^2 h+c^2 d\right)+\frac{1}{7} x^7 \left(2 b (a h+c d)+2 a c f+b^2 f\right)+\frac{1}{5} x^5 \left(2 a b f+a (a h+2 c d)+b^2 d\right)+\frac{1}{8} x^8 \left(2 a c g+b^2 g+2 b c e\right)+\frac{1}{6} x^6 \left(2 a b g+2 a c e+b^2 e\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c x^{11} (2 b h+c f)+\frac{1}{12} c^2 g x^{12}+\frac{1}{13} c^2 h x^{13}",1,"a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/4 + ((b^2*d + 2*a*c*d + 2*a*b*f + a^2*h)*x^5)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f + 2*a*b*h)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)*x^8)/8 + ((c^2*d + 2*b*c*f + b^2*h + 2*a*c*h)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c*(c*f + 2*b*h)*x^11)/11 + (c^2*g*x^12)/12 + (c^2*h*x^13)/13","A",1
10,1,50,45,0.0179952,"\int \frac{d+e x}{4-5 x^2+x^4} \, dx","Integrate[(d + e*x)/(4 - 5*x^2 + x^4),x]","\frac{1}{12} (-2 (d+e) \log (1-x)+(d+2 e) \log (2-x)+2 (d-e) \log (x+1)-(d-2 e) \log (x+2))","-\frac{1}{6} d \tanh ^{-1}\left(\frac{x}{2}\right)+\frac{1}{3} d \tanh ^{-1}(x)-\frac{1}{6} e \log \left(1-x^2\right)+\frac{1}{6} e \log \left(4-x^2\right)",1,"(-2*(d + e)*Log[1 - x] + (d + 2*e)*Log[2 - x] + 2*(d - e)*Log[1 + x] - (d - 2*e)*Log[2 + x])/12","A",1
11,1,58,51,0.0260869,"\int \frac{d+e x+f x^2}{4-5 x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4),x]","\frac{1}{12} (-2 \log (1-x) (d+e+f)+\log (2-x) (d+2 e+4 f)+2 \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f))","-\frac{1}{6} (d+4 f) \tanh ^{-1}\left(\frac{x}{2}\right)+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} e \log \left(1-x^2\right)+\frac{1}{6} e \log \left(4-x^2\right)",1,"(-2*(d + e + f)*Log[1 - x] + (d + 2*e + 4*f)*Log[2 - x] + 2*(d - e + f)*Log[1 + x] - (d - 2*e + 4*f)*Log[2 + x])/12","A",1
12,1,68,57,0.0324921,"\int \frac{d+e x+f x^2+g x^3}{4-5 x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(4 - 5*x^2 + x^4),x]","\frac{1}{12} (-2 \log (1-x) (d+e+f+g)+\log (2-x) (d+2 e+4 f+8 g)+2 \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g))","-\frac{1}{6} (d+4 f) \tanh ^{-1}\left(\frac{x}{2}\right)+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} (e+g) \log \left(1-x^2\right)+\frac{1}{6} (e+4 g) \log \left(4-x^2\right)",1,"(-2*(d + e + f + g)*Log[1 - x] + (d + 2*e + 4*f + 8*g)*Log[2 - x] + 2*(d - e + f - g)*Log[1 + x] - (d - 2*e + 4*f - 8*g)*Log[2 + x])/12","A",1
13,1,81,64,0.0451099,"\int \frac{d+e x+f x^2+g x^3+h x^4}{4-5 x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(4 - 5*x^2 + x^4),x]","\frac{1}{12} (-2 \log (1-x) (d+e+f+g+h)+\log (2-x) (d+2 (e+2 f+4 g+8 h))+2 \log (x+1) (d-e+f-g+h)-\log (x+2) (d-2 e+4 f-8 g+16 h)+12 h x)","-\frac{1}{6} \tanh ^{-1}\left(\frac{x}{2}\right) (d+4 f+16 h)+\frac{1}{3} \tanh ^{-1}(x) (d+f+h)-\frac{1}{6} (e+g) \log \left(1-x^2\right)+\frac{1}{6} (e+4 g) \log \left(4-x^2\right)+h x",1,"(12*h*x - 2*(d + e + f + g + h)*Log[1 - x] + (d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] + 2*(d - e + f - g + h)*Log[1 + x] - (d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x])/12","A",1
14,1,98,76,0.0638945,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{4-5 x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4),x]","\frac{1}{12} \left(-2 \log (1-x) (d+e+f+g+h+i)+\log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+2 \log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 (e-2 f+4 g-8 h+16 i))+12 h x+6 i x^2\right)","-\frac{1}{6} \tanh ^{-1}\left(\frac{x}{2}\right) (d+4 f+16 h)+\frac{1}{3} \tanh ^{-1}(x) (d+f+h)-\frac{1}{6} \log \left(1-x^2\right) (e+g+i)+\frac{1}{6} \log \left(4-x^2\right) (e+4 g+16 i)+h x+\frac{i x^2}{2}",1,"(12*h*x + 6*i*x^2 - 2*(d + e + f + g + h + i)*Log[1 - x] + (d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + 2*(d - e + f - g + h - i)*Log[1 + x] - (d - 2*(e - 2*f + 4*g - 8*h + 16*i))*Log[2 + x])/12","A",1
15,1,98,92,0.1779328,"\int \frac{d+e x}{1+x^2+x^4} \, dx","Integrate[(d + e*x)/(1 + x^2 + x^4),x]","\frac{1}{6} i \left(\sqrt{6-6 i \sqrt{3}} d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)-\sqrt{6+6 i \sqrt{3}} d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)+2 i \sqrt{3} e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{4} d \log \left(x^2-x+1\right)+\frac{1}{4} d \log \left(x^2+x+1\right)-\frac{d \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{d \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(I/6)*(Sqrt[6 - (6*I)*Sqrt[3]]*d*ArcTan[((-I + Sqrt[3])*x)/2] - Sqrt[6 + (6*I)*Sqrt[3]]*d*ArcTan[((I + Sqrt[3])*x)/2] + (2*I)*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])","C",0
16,1,121,104,0.1382299,"\int \frac{d+e x+f x^2}{1+x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4),x]","\frac{\left(2 i d+\left(\sqrt{3}-i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{6+6 i \sqrt{3}}}+\frac{\left(\left(\sqrt{3}+i\right) f-2 i d\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{6-6 i \sqrt{3}}}-\frac{e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)}{\sqrt{3}}","-\frac{1}{4} (d-f) \log \left(x^2-x+1\right)+\frac{1}{4} (d-f) \log \left(x^2+x+1\right)-\frac{(d+f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{(d+f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(((2*I)*d + (-I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[6 + (6*I)*Sqrt[3]] + (((-2*I)*d + (I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[6 - (6*I)*Sqrt[3]] - (e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/Sqrt[3]","C",0
17,1,150,127,0.4813997,"\int \frac{d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(1 + x^2 + x^4),x]","\frac{2 \left(\sqrt{2+2 i \sqrt{3}} \left(\left(\sqrt{3}+i\right) f-2 i d\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)+(2 g-4 e) \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)+\sqrt{3} g \log \left(x^4+x^2+1\right)\right)+2 \sqrt{2-2 i \sqrt{3}} \left(2 i d+\left(\sqrt{3}-i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{8 \sqrt{3}}","-\frac{1}{4} (d-f) \log \left(x^2-x+1\right)+\frac{1}{4} (d-f) \log \left(x^2+x+1\right)-\frac{(d+f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{(d+f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{4} g \log \left(x^4+x^2+1\right)",1,"(2*Sqrt[2 - (2*I)*Sqrt[3]]*((2*I)*d + (-I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2] + 2*(Sqrt[2 + (2*I)*Sqrt[3]]*((-2*I)*d + (I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2] + (-4*e + 2*g)*ArcTan[Sqrt[3]/(1 + 2*x^2)] + Sqrt[3]*g*Log[1 + x^2 + x^4]))/(8*Sqrt[3])","C",0
18,1,165,136,0.6031294,"\int \frac{d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(1 + x^2 + x^4),x]","\frac{1}{24} \left(4 \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(\left(\sqrt{3}+3 i\right) d+\left(\sqrt{3}-3 i\right) f-2 \sqrt{3} h\right)+4 \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(\left(\sqrt{3}-3 i\right) d+\left(\sqrt{3}+3 i\right) f-2 \sqrt{3} h\right)-8 \sqrt{3} e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)+4 \sqrt{3} g \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)+6 g \log \left(x^4+x^2+1\right)+24 h x\right)","-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (d+f-2 h)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (d+f-2 h)}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left(x^2-x+1\right)+\frac{1}{4} (d-f) \log \left(x^2+x+1\right)+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{4} g \log \left(x^4+x^2+1\right)+h x",1,"(24*h*x + 4*((3*I + Sqrt[3])*d + (-3*I + Sqrt[3])*f - 2*Sqrt[3]*h)*ArcTan[((-I + Sqrt[3])*x)/2] + 4*((-3*I + Sqrt[3])*d + (3*I + Sqrt[3])*f - 2*Sqrt[3]*h)*ArcTan[((I + Sqrt[3])*x)/2] - 8*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)] + 4*Sqrt[3]*g*ArcTan[Sqrt[3]/(1 + 2*x^2)] + 6*g*Log[1 + x^2 + x^4])/24","C",1
19,1,187,151,0.5820955,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{1+x^2+x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4),x]","\frac{1}{12} \left(\left(1+i \sqrt{3}\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(2 \sqrt{3} d-\left(\sqrt{3}+3 i\right) f-\left(\sqrt{3}-3 i\right) h\right)+\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(-2 i \sqrt{3} d+\left(3+i \sqrt{3}\right) f+i \left(\sqrt{3}+3 i\right) h\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right) (2 e-g-i)+3 (g-i) \log \left(x^4+x^2+1\right)+6 x (2 h+i x)\right)","-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (d+f-2 h)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (d+f-2 h)}{2 \sqrt{3}}-\frac{1}{4} (d-f) \log \left(x^2-x+1\right)+\frac{1}{4} (d-f) \log \left(x^2+x+1\right)+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right) (2 e-g-i)}{2 \sqrt{3}}+\frac{1}{4} (g-i) \log \left(x^4+x^2+1\right)+h x+\frac{i x^2}{2}",1,"(6*x*(2*h + i*x) + (1 + I*Sqrt[3])*(2*Sqrt[3]*d - (3*I + Sqrt[3])*f - (-3*I + Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2] + (I + Sqrt[3])*((-2*I)*Sqrt[3]*d + (3 + I*Sqrt[3])*f + I*(3*I + Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2] - 2*Sqrt[3]*(2*e - g - i)*ArcTan[Sqrt[3]/(1 + 2*x^2)] + 3*(g - i)*Log[1 + x^2 + x^4])/12","C",1
20,1,194,189,0.2512557,"\int \frac{d+e x}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x)/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}+e \left(\log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)-\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)\right)}{2 \sqrt{b^2-4 a c}}","\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"((2*Sqrt[2]*Sqrt[c]*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] - (2*Sqrt[2]*Sqrt[c]*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]] + e*(Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2]))/(2*Sqrt[b^2 - 4*a*c])","A",1
21,1,234,211,0.2193627,"\int \frac{d+e x+f x^2}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4),x]","\frac{\frac{\sqrt{2} \left(f \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(f \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}+e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)-e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{2 \sqrt{b^2-4 a c}}","\frac{\left(\frac{2 c d-b f}{\sqrt{b^2-4 a c}}+f\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(f-\frac{2 c d-b f}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"((Sqrt[2]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*Sqrt[b^2 - 4*a*c])","A",1
22,1,280,245,0.288844,"\int \frac{d+e x+f x^2+g x^3}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \sqrt{2} \sqrt{c} \left(f \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{2 \sqrt{2} \sqrt{c} \left(f \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}+\left(g \left(\sqrt{b^2-4 a c}-b\right)+2 c e\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)+\left(g \left(\sqrt{b^2-4 a c}+b\right)-2 c e\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{4 c \sqrt{b^2-4 a c}}","\frac{\left(\frac{2 c d-b f}{\sqrt{b^2-4 a c}}+f\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(f-\frac{2 c d-b f}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{(2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c}}+\frac{g \log \left(a+b x^2+c x^4\right)}{4 c}",1,"((2*Sqrt[2]*Sqrt[c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + (2*Sqrt[2]*Sqrt[c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]] + (2*c*e + (-b + Sqrt[b^2 - 4*a*c])*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] + (-2*c*e + (b + Sqrt[b^2 - 4*a*c])*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c*Sqrt[b^2 - 4*a*c])","A",1
23,1,383,290,0.5002203,"\int \frac{d+e x+f x^2+g x^3+h x^4}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(c \left(f \sqrt{b^2-4 a c}-2 a h-b f\right)+b h \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-c \left(f \sqrt{b^2-4 a c}+2 a h+b f\right)+b h \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left(g \left(\sqrt{b^2-4 a c}-b\right)+2 c e\right) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\sqrt{b^2-4 a c}}+\frac{\sqrt{c} \left(g \left(\sqrt{b^2-4 a c}+b\right)-2 c e\right) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\sqrt{b^2-4 a c}}+4 \sqrt{c} h x}{4 c^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-c (2 a h+b f)+b^2 h+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-2 a c h+b^2 h-b c f+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{(2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{2 c \sqrt{b^2-4 a c}}+\frac{g \log \left(a+b x^2+c x^4\right)}{4 c}+\frac{h x}{c}",1,"(4*Sqrt[c]*h*x + (2*Sqrt[2]*(2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*h + c*(-(b*f) + Sqrt[b^2 - 4*a*c]*f - 2*a*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*(2*c^2*d + b*(b + Sqrt[b^2 - 4*a*c])*h - c*(b*f + Sqrt[b^2 - 4*a*c]*f + 2*a*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (Sqrt[c]*(-2*c*e + (b + Sqrt[b^2 - 4*a*c])*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*c^(3/2))","A",1
24,1,441,321,0.6470449,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4),x]","\frac{\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(c \left(f \sqrt{b^2-4 a c}-2 a h-b f\right)+b h \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-c \left(f \sqrt{b^2-4 a c}+2 a h+b f\right)+b h \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right) \left(c \left(g \sqrt{b^2-4 a c}-2 a i-b g\right)+b i \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 e\right)}{\sqrt{b^2-4 a c}}-\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-c \left(g \sqrt{b^2-4 a c}+2 a i+b g\right)+b i \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 e\right)}{\sqrt{b^2-4 a c}}+4 c h x+2 c i x^2}{4 c^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-2 a c i+b^2 i-b c g+2 c^2 e\right)}{2 c^2 \sqrt{b^2-4 a c}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-c (2 a h+b f)+b^2 h+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-2 a c h+b^2 h-b c f+2 c^2 d}{\sqrt{b^2-4 a c}}-b h+c f\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{(c g-b i) \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{h x}{c}+\frac{i x^2}{2 c}",1,"(4*c*h*x + 2*c*i*x^2 + (2*Sqrt[2]*Sqrt[c]*(2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*h + c*(-(b*f) + Sqrt[b^2 - 4*a*c]*f - 2*a*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*Sqrt[c]*(2*c^2*d + b*(b + Sqrt[b^2 - 4*a*c])*h - c*(b*f + Sqrt[b^2 - 4*a*c]*f + 2*a*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((2*c^2*e + b*(b - Sqrt[b^2 - 4*a*c])*i + c*(-(b*g) + Sqrt[b^2 - 4*a*c]*g - 2*a*i))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] - ((2*c^2*e + b*(b + Sqrt[b^2 - 4*a*c])*i - c*(b*g + Sqrt[b^2 - 4*a*c]*g + 2*a*i))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*c^2)","A",1
25,1,816,545,1.2929357,"\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4),x]","\frac{m x^5}{5 c}+\frac{l x^4}{4 c}+\frac{(c k-b m) x^3}{3 c^2}+\frac{(c j-b l) x^2}{2 c^2}+\frac{\left(m b^2+c^2 h-c (b k+a m)\right) x}{c^3}+\frac{\left(2 d c^4+\left(-b f+\sqrt{b^2-4 a c} f-2 a h\right) c^3+\left(2 m a^2+3 b k a-\sqrt{b^2-4 a c} k a+b^2 h-b \sqrt{b^2-4 a c} h\right) c^2+b \left(-k b^2+\sqrt{b^2-4 a c} k b-4 a m b+2 a \sqrt{b^2-4 a c} m\right) c+b^3 \left(b-\sqrt{b^2-4 a c}\right) m\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(2 d c^4-\left(b f+\sqrt{b^2-4 a c} f+2 a h\right) c^3+\left(2 m a^2+3 b k a+\sqrt{b^2-4 a c} k a+b^2 h+b \sqrt{b^2-4 a c} h\right) c^2-b \left(k b^2+\sqrt{b^2-4 a c} k b+4 a m b+2 a \sqrt{b^2-4 a c} m\right) c+b^3 \left(b+\sqrt{b^2-4 a c}\right) m\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left(2 e c^3+\left(-b g+\sqrt{b^2-4 a c} g-2 a j\right) c^2+\left(j b^2-\sqrt{b^2-4 a c} j b+3 a l b-a \sqrt{b^2-4 a c} l\right) c+b^2 \left(\sqrt{b^2-4 a c}-b\right) l\right) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{4 c^3 \sqrt{b^2-4 a c}}+\frac{\left(-2 e c^3+\left(b g+\sqrt{b^2-4 a c} g+2 a j\right) c^2-\left(j b^2+\sqrt{b^2-4 a c} j b+3 a l b+a \sqrt{b^2-4 a c} l\right) c+b^2 \left(b+\sqrt{b^2-4 a c}\right) l\right) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{4 c^3 \sqrt{b^2-4 a c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{c^2 \left(2 a^2 m+3 a b k+b^2 h\right)-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right)}{\sqrt{2} c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{c^2 \left(2 a^2 m+3 a b k+b^2 h\right)-b^2 c (4 a m+b k)-c^3 (2 a h+b f)+b^4 m+2 c^4 d}{\sqrt{b^2-4 a c}}-c^2 (a k+b h)+b c (2 a m+b k)+b^3 (-m)+c^3 f\right)}{\sqrt{2} c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\log \left(a+b x^2+c x^4\right) \left(-c (a l+b j)+b^2 l+c^2 g\right)}{4 c^3}+\frac{x \left(-c (a m+b k)+b^2 m+c^2 h\right)}{c^3}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right)}{2 c^3 \sqrt{b^2-4 a c}}+\frac{x^2 (c j-b l)}{2 c^2}+\frac{x^3 (c k-b m)}{3 c^2}+\frac{l x^4}{4 c}+\frac{m x^5}{5 c}",1,"((c^2*h + b^2*m - c*(b*k + a*m))*x)/c^3 + ((c*j - b*l)*x^2)/(2*c^2) + ((c*k - b*m)*x^3)/(3*c^2) + (l*x^4)/(4*c) + (m*x^5)/(5*c) + ((2*c^4*d + c^3*(-(b*f) + Sqrt[b^2 - 4*a*c]*f - 2*a*h) + b^3*(b - Sqrt[b^2 - 4*a*c])*m + c^2*(b^2*h - b*Sqrt[b^2 - 4*a*c]*h + 3*a*b*k - a*Sqrt[b^2 - 4*a*c]*k + 2*a^2*m) + b*c*(-(b^2*k) + b*Sqrt[b^2 - 4*a*c]*k - 4*a*b*m + 2*a*Sqrt[b^2 - 4*a*c]*m))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((2*c^4*d - c^3*(b*f + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b^3*(b + Sqrt[b^2 - 4*a*c])*m + c^2*(b^2*h + b*Sqrt[b^2 - 4*a*c]*h + 3*a*b*k + a*Sqrt[b^2 - 4*a*c]*k + 2*a^2*m) - b*c*(b^2*k + b*Sqrt[b^2 - 4*a*c]*k + 4*a*b*m + 2*a*Sqrt[b^2 - 4*a*c]*m))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((2*c^3*e + c^2*(-(b*g) + Sqrt[b^2 - 4*a*c]*g - 2*a*j) + b^2*(-b + Sqrt[b^2 - 4*a*c])*l + c*(b^2*j - b*Sqrt[b^2 - 4*a*c]*j + 3*a*b*l - a*Sqrt[b^2 - 4*a*c]*l))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(4*c^3*Sqrt[b^2 - 4*a*c]) + ((-2*c^3*e + c^2*(b*g + Sqrt[b^2 - 4*a*c]*g + 2*a*j) + b^2*(b + Sqrt[b^2 - 4*a*c])*l - c*(b^2*j + b*Sqrt[b^2 - 4*a*c]*j + 3*a*b*l + a*Sqrt[b^2 - 4*a*c]*l))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c^3*Sqrt[b^2 - 4*a*c])","A",1
26,1,90,94,0.0535465,"\int \frac{d+e x}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(d + e*x)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{864} \left(\frac{12 \left(d x \left(17-5 x^2\right)+e \left(20-8 x^2\right)\right)}{x^4-5 x^2+4}+8 (d+4 e) \log (1-x)-(19 d+32 e) \log (2-x)-8 (d-4 e) \log (x+1)+(19 d-32 e) \log (x+2)\right)","\frac{d x \left(17-5 x^2\right)}{72 \left(x^4-5 x^2+4\right)}+\frac{19}{432} d \tanh ^{-1}\left(\frac{x}{2}\right)-\frac{1}{54} d \tanh ^{-1}(x)+\frac{1}{27} e \log \left(1-x^2\right)-\frac{1}{27} e \log \left(4-x^2\right)+\frac{e \left(5-2 x^2\right)}{18 \left(x^4-5 x^2+4\right)}",1,"((12*(e*(20 - 8*x^2) + d*x*(17 - 5*x^2)))/(4 - 5*x^2 + x^4) + 8*(d + 4*e)*Log[1 - x] - (19*d + 32*e)*Log[2 - x] - 8*(d - 4*e)*Log[1 + x] + (19*d - 32*e)*Log[2 + x])/864","A",1
27,1,112,115,0.0803746,"\int \frac{d+e x+f x^2}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{864} \left(\frac{12 \left(-5 d x^3+17 d x+e \left(20-8 x^2\right)-8 f x^3+20 f x\right)}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f)-\log (2-x) (19 d+32 e+52 f)-8 \log (x+1) (d-4 e+7 f)+\log (x+2) (19 d-32 e+52 f)\right)","\frac{x \left(-\left(x^2 (5 d+8 f)\right)+17 d+20 f\right)}{72 \left(x^4-5 x^2+4\right)}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left(\frac{x}{2}\right)-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{27} e \log \left(1-x^2\right)-\frac{1}{27} e \log \left(4-x^2\right)+\frac{e \left(5-2 x^2\right)}{18 \left(x^4-5 x^2+4\right)}",1,"((12*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2)))/(4 - 5*x^2 + x^4) + 8*(d + 4*e + 7*f)*Log[1 - x] - (19*d + 32*e + 52*f)*Log[2 - x] - 8*(d - 4*e + 7*f)*Log[1 + x] + (19*d - 32*e + 52*f)*Log[2 + x])/864","A",1
28,1,134,138,0.0541863,"\int \frac{d+e x+f x^2+g x^3}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{864} \left(\frac{12 \left(-5 d x^3+17 d x+e \left(20-8 x^2\right)-8 f x^3+20 f x-4 g \left(5 x^2-8\right)\right)}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f+10 g)-\log (2-x) (19 d+32 e+52 f+80 g)-8 \log (x+1) (d-4 e+7 f-10 g)+\log (x+2) (19 d-32 e+52 f-80 g)\right)","\frac{x \left(-\left(x^2 (5 d+8 f)\right)+17 d+20 f\right)}{72 \left(x^4-5 x^2+4\right)}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left(\frac{x}{2}\right)-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{54} (2 e+5 g) \log \left(1-x^2\right)-\frac{1}{54} (2 e+5 g) \log \left(4-x^2\right)+\frac{-\left(x^2 (2 e+5 g)\right)+5 e+8 g}{18 \left(x^4-5 x^2+4\right)}",1,"((12*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2) - 4*g*(-8 + 5*x^2)))/(4 - 5*x^2 + x^4) + 8*(d + 4*e + 7*f + 10*g)*Log[1 - x] - (19*d + 32*e + 52*f + 80*g)*Log[2 - x] - 8*(d - 4*e + 7*f - 10*g)*Log[1 + x] + (19*d - 32*e + 52*f - 80*g)*Log[2 + x])/864","A",1
29,1,159,150,0.0717051,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{864} \left(-\frac{12 \left(x \left(d \left(5 x^2-17\right)+4 f \left(2 x^2-5\right)+4 h \left(5 x^2-8\right)\right)+4 e \left(2 x^2-5\right)+4 g \left(5 x^2-8\right)\right)}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f+10 g+13 h)-\log (2-x) (19 d+32 e+52 f+80 g+112 h)-8 \log (x+1) (d-4 e+7 f-10 g+13 h)+\log (x+2) (19 d-32 e+52 f-80 g+112 h)\right)","\frac{x \left(-\left(x^2 (5 d+8 f+20 h)\right)+17 d+20 f+32 h\right)}{72 \left(x^4-5 x^2+4\right)}+\frac{1}{432} \tanh ^{-1}\left(\frac{x}{2}\right) (19 d+52 f+112 h)-\frac{1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac{1}{54} (2 e+5 g) \log \left(1-x^2\right)-\frac{1}{54} (2 e+5 g) \log \left(4-x^2\right)+\frac{-\left(x^2 (2 e+5 g)\right)+5 e+8 g}{18 \left(x^4-5 x^2+4\right)}",1,"((-12*(4*e*(-5 + 2*x^2) + 4*g*(-8 + 5*x^2) + x*(4*f*(-5 + 2*x^2) + d*(-17 + 5*x^2) + 4*h*(-8 + 5*x^2))))/(4 - 5*x^2 + x^4) + 8*(d + 4*e + 7*f + 10*g + 13*h)*Log[1 - x] - (19*d + 32*e + 52*f + 80*g + 112*h)*Log[2 - x] - 8*(d - 4*e + 7*f - 10*g + 13*h)*Log[1 + x] + (19*d - 32*e + 52*f - 80*g + 112*h)*Log[2 + x])/864","A",1
30,1,185,162,0.0861908,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4)^2,x]","\frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x-68 i x^2+80 i}{72 \left(x^4-5 x^2+4\right)}+\frac{1}{108} \log (1-x) (d+4 e+7 f+10 g+13 h+16 i)+\frac{1}{864} \log (2-x) (-19 d-32 e-52 f-80 g-112 h-128 i)+\frac{1}{108} \log (x+1) (-d+4 e-7 f+10 g-13 h+16 i)+\frac{1}{864} \log (x+2) (19 d-32 e+52 f-80 g+112 h-128 i)","\frac{x \left(-\left(x^2 (5 d+8 f+20 h)\right)+17 d+20 f+32 h\right)}{72 \left(x^4-5 x^2+4\right)}+\frac{1}{432} \tanh ^{-1}\left(\frac{x}{2}\right) (19 d+52 f+112 h)-\frac{1}{54} \tanh ^{-1}(x) (d+7 f+13 h)+\frac{1}{54} \log \left(1-x^2\right) (2 e+5 g+8 i)-\frac{1}{54} \log \left(4-x^2\right) (2 e+5 g+8 i)+\frac{-\left(x^2 (2 e+5 g+17 i)\right)+5 e+8 g+20 i}{18 \left(x^4-5 x^2+4\right)}",1,"(20*e + 32*g + 80*i + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 68*i*x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(72*(4 - 5*x^2 + x^4)) + ((d + 4*e + 7*f + 10*g + 13*h + 16*i)*Log[1 - x])/108 + ((-19*d - 32*e - 52*f - 80*g - 112*h - 128*i)*Log[2 - x])/864 + ((-d + 4*e - 7*f + 10*g - 13*h + 16*i)*Log[1 + x])/108 + ((19*d - 32*e + 52*f - 80*g + 112*h - 128*i)*Log[2 + x])/864","A",1
31,1,146,140,0.4890284,"\int \frac{d+e x}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(d + e*x)/(1 + x^2 + x^4)^2,x]","\frac{d \left(x-x^3\right)+2 e x^2+e}{6 \left(x^4+x^2+1\right)}-\frac{\left(\sqrt{3}-11 i\right) d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{6 \sqrt{6+6 i \sqrt{3}}}-\frac{\left(\sqrt{3}+11 i\right) d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{6 \sqrt{6-6 i \sqrt{3}}}-\frac{2 e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)}{3 \sqrt{3}}","-\frac{1}{4} d \log \left(x^2-x+1\right)+\frac{1}{4} d \log \left(x^2+x+1\right)+\frac{d x \left(1-x^2\right)}{6 \left(x^4+x^2+1\right)}-\frac{d \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{d \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{2 e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{e \left(2 x^2+1\right)}{6 \left(x^4+x^2+1\right)}",1,"(e + 2*e*x^2 + d*(x - x^3))/(6*(1 + x^2 + x^4)) - ((-11*I + Sqrt[3])*d*ArcTan[((-I + Sqrt[3])*x)/2])/(6*Sqrt[6 + (6*I)*Sqrt[3]]) - ((11*I + Sqrt[3])*d*ArcTan[((I + Sqrt[3])*x)/2])/(6*Sqrt[6 - (6*I)*Sqrt[3]]) - (2*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/(3*Sqrt[3])","C",0
32,1,186,165,0.4194352,"\int \frac{d+e x+f x^2}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^2,x]","\frac{1}{36} \left(\frac{6 \left(x \left(-d x^2+d+2 f x^2+f\right)+2 e x^2+e\right)}{x^4+x^2+1}-\frac{\left(\left(\sqrt{3}-11 i\right) d-2 \left(\sqrt{3}-2 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\left(\left(\sqrt{3}+11 i\right) d-2 \left(\sqrt{3}+2 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-8 \sqrt{3} e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{8} (2 d-f) \log \left(x^2-x+1\right)+\frac{1}{8} (2 d-f) \log \left(x^2+x+1\right)+\frac{x \left(-\left(x^2 (d-2 f)\right)+d+f\right)}{6 \left(x^4+x^2+1\right)}-\frac{(4 d+f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{12 \sqrt{3}}+\frac{2 e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{e \left(2 x^2+1\right)}{6 \left(x^4+x^2+1\right)}",1,"((6*(e + 2*e*x^2 + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4) - (((-11*I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 8*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36","C",0
33,1,200,179,0.4340935,"\int \frac{d+e x+f x^2+g x^3}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(1 + x^2 + x^4)^2,x]","\frac{1}{36} \left(\frac{6 \left(x \left(-d x^2+d+2 f x^2+f\right)+2 e x^2+e-g \left(x^2+2\right)\right)}{x^4+x^2+1}-\frac{\left(\left(\sqrt{3}-11 i\right) d-2 \left(\sqrt{3}-2 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\left(\left(\sqrt{3}+11 i\right) d-2 \left(\sqrt{3}+2 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-4 \sqrt{3} (2 e-g) \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{8} (2 d-f) \log \left(x^2-x+1\right)+\frac{1}{8} (2 d-f) \log \left(x^2+x+1\right)+\frac{x \left(-\left(x^2 (d-2 f)\right)+d+f\right)}{6 \left(x^4+x^2+1\right)}-\frac{(4 d+f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{12 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{x^2 (2 e-g)+e-2 g}{6 \left(x^4+x^2+1\right)}",1,"((6*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4) - (((-11*I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 4*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36","C",0
34,1,234,187,0.606212,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(1 + x^2 + x^4)^2,x]","\frac{1}{36} \left(-\frac{6 \left(x \left(d \left(x^2-1\right)-f \left(2 x^2+1\right)+h \left(x^2+2\right)\right)-e \left(2 x^2+1\right)+g \left(x^2+2\right)\right)}{x^4+x^2+1}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(\left(\sqrt{3}-11 i\right) d-2 \left(\sqrt{3}-2 i\right) f+\left(\sqrt{3}-5 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(\left(\sqrt{3}+11 i\right) d-2 \left(\sqrt{3}+2 i\right) f+\left(\sqrt{3}+5 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-4 \sqrt{3} (2 e-g) \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{8} \log \left(x^2-x+1\right) (2 d-f+h)+\frac{1}{8} \log \left(x^2+x+1\right) (2 d-f+h)+\frac{x \left(-\left(x^2 (d-2 f+h)\right)+d+f-2 h\right)}{6 \left(x^4+x^2+1\right)}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (4 d+f+h)}{12 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (4 d+f+h)}{12 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{x^2 (2 e-g)+e-2 g}{6 \left(x^4+x^2+1\right)}",1,"((-6*(g*(2 + x^2) - e*(1 + 2*x^2) + x*(d*(-1 + x^2) + h*(2 + x^2) - f*(1 + 2*x^2))))/(1 + x^2 + x^4) - (((-11*I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f + (-5*I + Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f + (5*I + Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 4*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36","C",0
35,1,243,194,0.6585231,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(1+x^2+x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^2,x]","\frac{1}{36} \left(\frac{6 \left(-d x^3+d x+2 e x^2+e+2 f x^3+f x-g \left(x^2+2\right)-h x^3-2 h x-i x^2+i\right)}{x^4+x^2+1}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(\left(\sqrt{3}-11 i\right) d-2 \left(\sqrt{3}-2 i\right) f+\left(\sqrt{3}-5 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(\left(\sqrt{3}+11 i\right) d-2 \left(\sqrt{3}+2 i\right) f+\left(\sqrt{3}+5 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-4 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right) (2 e-g+2 i)\right)","-\frac{1}{8} \log \left(x^2-x+1\right) (2 d-f+h)+\frac{1}{8} \log \left(x^2+x+1\right) (2 d-f+h)+\frac{x \left(-\left(x^2 (d-2 f+h)\right)+d+f-2 h\right)}{6 \left(x^4+x^2+1\right)}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (4 d+f+h)}{12 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (4 d+f+h)}{12 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right) (2 e-g+2 i)}{3 \sqrt{3}}+\frac{x^2 (2 e-g-i)+e-2 g+i}{6 \left(x^4+x^2+1\right)}",1,"((6*(e + i + d*x + f*x - 2*h*x + 2*e*x^2 - i*x^2 - d*x^3 + 2*f*x^3 - h*x^3 - g*(2 + x^2)))/(1 + x^2 + x^4) - (((-11*I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f + (-5*I + Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f + (5*I + Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 4*Sqrt[3]*(2*e - g + 2*i)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36","C",0
36,1,341,330,0.7563454,"\int \frac{d+e x}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{2 a b e+4 a c x (d+e x)-2 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} d \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} d \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 c e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{4 c e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{d x \left(-2 a c+b^2+b c x^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} d \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} d \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 c e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*a*b*e + 4*a*c*x*(d + e*x) - 2*b*d*x*(b + c*x^2))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*c*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*c*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
37,1,398,368,1.1698025,"\int \frac{d+e x+f x^2}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{2 a b (e+f x)+4 a c x (d+x (e+f x))-2 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \left(d \sqrt{b^2-4 a c}+4 a f\right)-2 a \left(f \sqrt{b^2-4 a c}+6 c d\right)+b^2 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-4 a b f+12 a c d+b^2 (-d)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 c e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{4 c e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 c e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*a*b*(e + f*x) - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*f - 2*a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*c*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*c*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
38,1,421,386,1.2963145,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{-4 a^2 g+2 a b (e+x (f-g x))+4 a c x (d+x (e+f x))-2 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \left(d \sqrt{b^2-4 a c}+4 a f\right)-2 a \left(f \sqrt{b^2-4 a c}+6 c d\right)+b^2 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-4 a b f+12 a c d+b^2 (-d)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 (b g-2 c e) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 (b g-2 c e) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{(2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*a^2*g - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)) + 2*a*b*(e + x*(f - g*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*f - 2*a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(-2*c*e + b*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*(-2*c*e + b*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
39,1,489,439,1.882141,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{-4 a^2 (g+h x)+2 a b (e+x (f-x (g+h x)))+4 a c x (d+x (e+f x))-2 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}+4 a c f\right)-2 a c \left(f \sqrt{b^2-4 a c}+2 a h+6 c d\right)+b^2 (c d-a h)\right)}{a \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}-4 a c f\right)+2 a c \left(-f \sqrt{b^2-4 a c}+2 a h+6 c d\right)+b^2 (a h-c d)\right)}{a \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 (b g-2 c e) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{2 (b g-2 c e) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{x \left(x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{(2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((-4*a^2*(g + h*x) - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)) + 2*a*b*(e + x*(f - x*(g + h*x))))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d - a*h) - 2*a*c*(6*c*d + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2*(-(c*d) + a*h) + 2*a*c*(6*c*d - Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(-2*c*e + b*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - (2*(-2*c*e + b*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
40,1,524,468,2.1117091,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2,x]","\frac{1}{4} \left(\frac{2 \left(a^2 (b i-2 c (g+x (h+i x)))+a \left(b^2 i x^2+b c (e+x (f-x (g+h x)))+2 c^2 x (d+x (e+f x))\right)-b c d x \left(b+c x^2\right)\right)}{a c \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}+4 a c f\right)-2 a c \left(f \sqrt{b^2-4 a c}+2 a h+6 c d\right)+b^2 (c d-a h)\right)}{a \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b \left(c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}-4 a c f\right)+2 a c \left(-f \sqrt{b^2-4 a c}+2 a h+6 c d\right)+b^2 (a h-c d)\right)}{a \sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right) (-2 a i+b g-2 c e)}{\left(b^2-4 a c\right)^{3/2}}+\frac{2 \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) (2 a i-b g+2 c e)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{-\left(x^2 \left(-2 a c i+b^2 i-b c g+2 c^2 e\right)\right)-b (a i+c e)+2 a c g}{2 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{x \left(x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) (2 a i-b g+2 c e)}{\left(b^2-4 a c\right)^{3/2}}",1,"((2*(-(b*c*d*x*(b + c*x^2)) + a^2*(b*i - 2*c*(g + x*(h + i*x))) + a*(b^2*i*x^2 + 2*c^2*x*(d + x*(e + f*x)) + b*c*(e + x*(f - x*(g + h*x))))))/(a*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d - a*h) - 2*a*c*(6*c*d + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2*(-(c*d) + a*h) + 2*a*c*(6*c*d - Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(-2*c*e + b*g - 2*a*i)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (2*(2*c*e - b*g + 2*a*i)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
41,1,935,770,5.6968421,"\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^2,x]","\frac{4 \sqrt{c} m x+\frac{2 \sqrt{c} \left(2 c (l+m x) a^3-\left((l+m x) b^2-c (j+x (k+3 x (l+m x))) b+2 c^2 (g+x (h+x (j+k x)))\right) a^2+\left(-x^2 (l+m x) b^3+c x^2 (j+k x) b^2+c^2 (e+x (f-x (g+h x))) b+2 c^3 x (d+x (e+f x))\right) a-b c^2 d x \left(c x^2+b\right)\right)}{a \left(4 a c-b^2\right) \left(c x^4+b x^2+a\right)}-\frac{\sqrt{2} \left(-3 a m b^4+a \left(c k+3 \sqrt{b^2-4 a c} m\right) b^3+c \left(-d c^2+a h c+a \left(19 a m-\sqrt{b^2-4 a c} k\right)\right) b^2-c \left(13 \sqrt{b^2-4 a c} m a^2+c \left(\sqrt{b^2-4 a c} h+8 a k\right) a+c^2 \left(\sqrt{b^2-4 a c} d+4 a f\right)\right) b+2 a c^2 \left(-10 m a^2+2 c h a+3 \sqrt{b^2-4 a c} k a+6 c^2 d+c \sqrt{b^2-4 a c} f\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left(3 a m b^4+a \left(3 \sqrt{b^2-4 a c} m-c k\right) b^3+c \left(d c^2-a h c-a \left(\sqrt{b^2-4 a c} k+19 a m\right)\right) b^2-c \left(13 \sqrt{b^2-4 a c} m a^2+c \left(\sqrt{b^2-4 a c} h-8 a k\right) a+c^2 \left(\sqrt{b^2-4 a c} d-4 a f\right)\right) b+2 a c^2 \left(10 m a^2-2 c h a+3 \sqrt{b^2-4 a c} k a-6 c^2 d+c \sqrt{b^2-4 a c} f\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-4 e c^3+2 (b g-2 a j) c^2+a \left(6 b l-4 \sqrt{b^2-4 a c} l\right) c+b^2 \left(\sqrt{b^2-4 a c}-b\right) l\right) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\sqrt{c} \left(4 e c^3+(4 a j-2 b g) c^2-2 a \left(3 b+2 \sqrt{b^2-4 a c}\right) l c+b^2 \left(b+\sqrt{b^2-4 a c}\right) l\right) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{\left(b^2-4 a c\right)^{3/2}}}{4 c^{5/2}}","-\frac{x \left(-\left(b^2 \left(a^2 m+c^2 d\right)\right)+x^2 \left(-b c \left(-3 a^2 m+a c h+c^2 d\right)-a b^3 m+a b^2 c k+2 a c^2 (c f-a k)\right)+2 a c \left(a^2 m-a c h+c^2 d\right)+a b c (a k+c f)\right)}{2 a c^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-\frac{-b^2 c \left(-19 a^2 m-a c h+c^2 d\right)+4 a c^2 \left(-5 a^2 m+a c h+3 c^2 d\right)-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left(13 a^2 m+a c h+c^2 d\right)-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right)}{2 \sqrt{2} a c^{5/2} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{-b^2 c \left(-19 a^2 m-a c h+c^2 d\right)+4 a c^2 \left(-5 a^2 m+a c h+3 c^2 d\right)-3 a b^4 m+a b^3 c k-4 a b c^2 (2 a k+c f)}{\sqrt{b^2-4 a c}}+b c \left(13 a^2 m+a c h+c^2 d\right)-3 a b^3 m+a b^2 c k-2 a c^2 (3 a k+c f)\right)}{2 \sqrt{2} a c^{5/2} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-c^2 (2 b g-4 a j)-6 a b c l+b^3 l+4 c^3 e\right)}{2 c^2 \left(b^2-4 a c\right)^{3/2}}-\frac{x^2 \left(-c^2 (2 a j+b g)+b c (3 a l+b j)+b^3 (-l)+2 c^3 e\right)-a b^2 l+b c (a j+c e)-2 a c (c g-a l)}{2 c^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{l \log \left(a+b x^2+c x^4\right)}{4 c^2}+\frac{m x}{c^2}",1,"(4*Sqrt[c]*m*x + (2*Sqrt[c]*(2*a^3*c*(l + m*x) - b*c^2*d*x*(b + c*x^2) + a*(b^2*c*x^2*(j + k*x) - b^3*x^2*(l + m*x) + 2*c^3*x*(d + x*(e + f*x)) + b*c^2*(e + x*(f - x*(g + h*x)))) - a^2*(b^2*(l + m*x) + 2*c^2*(g + x*(h + x*(j + k*x))) - b*c*(j + x*(k + 3*x*(l + m*x))))))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*a*b^4*m + 2*a*c^2*(6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k - 10*a^2*m) + a*b^3*(c*k + 3*Sqrt[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) + a*c*(Sqrt[b^2 - 4*a*c]*h + 8*a*k) + 13*a^2*Sqrt[b^2 - 4*a*c]*m) + b^2*c*(-(c^2*d) + a*c*h + a*(-(Sqrt[b^2 - 4*a*c]*k) + 19*a*m)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*a*b^4*m + 2*a*c^2*(-6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f - 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*k + 10*a^2*m) + a*b^3*(-(c*k) + 3*Sqrt[b^2 - 4*a*c]*m) - b*c*(c^2*(Sqrt[b^2 - 4*a*c]*d - 4*a*f) + a*c*(Sqrt[b^2 - 4*a*c]*h - 8*a*k) + 13*a^2*Sqrt[b^2 - 4*a*c]*m) + b^2*c*(c^2*d - a*c*h - a*(Sqrt[b^2 - 4*a*c]*k + 19*a*m)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(-4*c^3*e + 2*c^2*(b*g - 2*a*j) + b^2*(-b + Sqrt[b^2 - 4*a*c])*l + a*c*(6*b*l - 4*Sqrt[b^2 - 4*a*c]*l))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (Sqrt[c]*(4*c^3*e + c^2*(-2*b*g + 4*a*j) + b^2*(b + Sqrt[b^2 - 4*a*c])*l - 2*a*c*(3*b + 2*Sqrt[b^2 - 4*a*c])*l)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*c^(5/2))","A",1
42,1,128,143,0.0983431,"\int \frac{d+e x}{\left(4-5 x^2+x^4\right)^3} \, dx","Integrate[(d + e*x)/(4 - 5*x^2 + x^4)^3,x]","\frac{\frac{288 \left(d x \left(17-5 x^2\right)+e \left(20-8 x^2\right)\right)}{\left(x^4-5 x^2+4\right)^2}+\frac{12 \left(d x \left(35 x^2-59\right)+64 e \left(2 x^2-5\right)\right)}{x^4-5 x^2+4}-32 (13 d+16 e) \log (1-x)+(313 d+512 e) \log (2-x)+32 (13 d-16 e) \log (x+1)+(512 e-313 d) \log (x+2)}{41472}","-\frac{d x \left(59-35 x^2\right)}{3456 \left(x^4-5 x^2+4\right)}+\frac{d x \left(17-5 x^2\right)}{144 \left(x^4-5 x^2+4\right)^2}-\frac{313 d \tanh ^{-1}\left(\frac{x}{2}\right)}{20736}+\frac{13}{648} d \tanh ^{-1}(x)-\frac{1}{81} e \log \left(1-x^2\right)+\frac{1}{81} e \log \left(4-x^2\right)-\frac{e \left(5-2 x^2\right)}{54 \left(x^4-5 x^2+4\right)}+\frac{e \left(5-2 x^2\right)}{36 \left(x^4-5 x^2+4\right)^2}",1,"((288*(e*(20 - 8*x^2) + d*x*(17 - 5*x^2)))/(4 - 5*x^2 + x^4)^2 + (12*(64*e*(-5 + 2*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e)*Log[1 - x] + (313*d + 512*e)*Log[2 - x] + 32*(13*d - 16*e)*Log[1 + x] + (-313*d + 512*e)*Log[2 + x])/41472","A",1
43,1,161,175,0.1262875,"\int \frac{d+e x+f x^2}{\left(4-5 x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^3,x]","\frac{\frac{12 \left(d x \left(35 x^2-59\right)+64 e \left(2 x^2-5\right)+20 f x \left(7 x^2-19\right)\right)}{x^4-5 x^2+4}+\frac{288 \left(-5 d x^3+17 d x+e \left(20-8 x^2\right)-8 f x^3+20 f x\right)}{\left(x^4-5 x^2+4\right)^2}-32 \log (1-x) (13 d+16 e+25 f)+\log (2-x) (313 d+512 e+820 f)+32 \log (x+1) (13 d-16 e+25 f)+\log (x+2) (-313 d+512 e-820 f)}{41472}","-\frac{x \left(-35 x^2 (d+4 f)+59 d+380 f\right)}{3456 \left(x^4-5 x^2+4\right)}+\frac{x \left(-\left(x^2 (5 d+8 f)\right)+17 d+20 f\right)}{144 \left(x^4-5 x^2+4\right)^2}-\frac{(313 d+820 f) \tanh ^{-1}\left(\frac{x}{2}\right)}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{81} e \log \left(1-x^2\right)+\frac{1}{81} e \log \left(4-x^2\right)-\frac{e \left(5-2 x^2\right)}{54 \left(x^4-5 x^2+4\right)}+\frac{e \left(5-2 x^2\right)}{36 \left(x^4-5 x^2+4\right)^2}",1,"((288*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2)))/(4 - 5*x^2 + x^4)^2 + (12*(64*e*(-5 + 2*x^2) + 20*f*x*(-19 + 7*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e + 25*f)*Log[1 - x] + (313*d + 512*e + 820*f)*Log[2 - x] + 32*(13*d - 16*e + 25*f)*Log[1 + x] + (-313*d + 512*e - 820*f)*Log[2 + x])/41472","A",1
44,1,193,204,0.0842324,"\int \frac{d+e x+f x^2+g x^3}{\left(4-5 x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(4 - 5*x^2 + x^4)^3,x]","\frac{\frac{12 \left(d x \left(35 x^2-59\right)+64 e \left(2 x^2-5\right)+20 f x \left(7 x^2-19\right)+160 g \left(2 x^2-5\right)\right)}{x^4-5 x^2+4}+\frac{288 \left(-5 d x^3+17 d x+e \left(20-8 x^2\right)-8 f x^3+20 f x-4 g \left(5 x^2-8\right)\right)}{\left(x^4-5 x^2+4\right)^2}-32 \log (1-x) (13 d+16 e+25 f+40 g)+\log (2-x) (313 d+512 e+820 f+1280 g)+32 \log (x+1) (13 d-16 e+25 f-40 g)+\log (x+2) (-313 d+512 e-820 f+1280 g)}{41472}","-\frac{x \left(-35 x^2 (d+4 f)+59 d+380 f\right)}{3456 \left(x^4-5 x^2+4\right)}+\frac{x \left(-\left(x^2 (5 d+8 f)\right)+17 d+20 f\right)}{144 \left(x^4-5 x^2+4\right)^2}-\frac{(313 d+820 f) \tanh ^{-1}\left(\frac{x}{2}\right)}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{162} (2 e+5 g) \log \left(1-x^2\right)+\frac{1}{162} (2 e+5 g) \log \left(4-x^2\right)-\frac{\left(5-2 x^2\right) (2 e+5 g)}{108 \left(x^4-5 x^2+4\right)}+\frac{-\left(x^2 (2 e+5 g)\right)+5 e+8 g}{36 \left(x^4-5 x^2+4\right)^2}",1,"((288*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2) - 4*g*(-8 + 5*x^2)))/(4 - 5*x^2 + x^4)^2 + (12*(64*e*(-5 + 2*x^2) + 160*g*(-5 + 2*x^2) + 20*f*x*(-19 + 7*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e + 25*f + 40*g)*Log[1 - x] + (313*d + 512*e + 820*f + 1280*g)*Log[2 - x] + 32*(13*d - 16*e + 25*f - 40*g)*Log[1 + x] + (-313*d + 512*e - 820*f + 1280*g)*Log[2 + x])/41472","A",1
45,1,231,224,0.1179779,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(4-5 x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(4 - 5*x^2 + x^4)^3,x]","\frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x}{144 \left(x^4-5 x^2+4\right)^2}+\frac{35 d x^3-59 d x+128 e x^2-320 e+140 f x^3-380 f x+320 g x^2-800 g+320 h x^3-848 h x}{3456 \left(x^4-5 x^2+4\right)}+\frac{\log (1-x) (-13 d-16 e-25 f-40 g-61 h)}{1296}+\frac{\log (2-x) (313 d+512 e+820 f+1280 g+1936 h)}{41472}+\frac{\log (x+1) (13 d-16 e+25 f-40 g+61 h)}{1296}+\frac{\log (x+2) (-313 d+512 e-820 f+1280 g-1936 h)}{41472}","-\frac{x \left(-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right)}{3456 \left(x^4-5 x^2+4\right)}+\frac{x \left(-\left(x^2 (5 d+8 f+20 h)\right)+17 d+20 f+32 h\right)}{144 \left(x^4-5 x^2+4\right)^2}-\frac{\tanh ^{-1}\left(\frac{x}{2}\right) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{1}{162} (2 e+5 g) \log \left(1-x^2\right)+\frac{1}{162} (2 e+5 g) \log \left(4-x^2\right)-\frac{\left(5-2 x^2\right) (2 e+5 g)}{108 \left(x^4-5 x^2+4\right)}+\frac{-\left(x^2 (2 e+5 g)\right)+5 e+8 g}{36 \left(x^4-5 x^2+4\right)^2}",1,"(20*e + 32*g + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(144*(4 - 5*x^2 + x^4)^2) + (-320*e - 800*g - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 35*d*x^3 + 140*f*x^3 + 320*h*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-13*d - 16*e - 25*f - 40*g - 61*h)*Log[1 - x])/1296 + ((313*d + 512*e + 820*f + 1280*g + 1936*h)*Log[2 - x])/41472 + ((13*d - 16*e + 25*f - 40*g + 61*h)*Log[1 + x])/1296 + ((-313*d + 512*e - 820*f + 1280*g - 1936*h)*Log[2 + x])/41472","A",1
46,1,261,239,0.1290192,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(4-5 x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(4 - 5*x^2 + x^4)^3,x]","\frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x-68 i x^2+80 i}{144 \left(x^4-5 x^2+4\right)^2}+\frac{35 d x^3-59 d x+128 e x^2-320 e+140 f x^3-380 f x+320 g x^2-800 g+320 h x^3-848 h x+704 i x^2-1760 i}{3456 \left(x^4-5 x^2+4\right)}+\frac{\log (1-x) (-13 d-16 e-25 f-40 g-61 h-88 i)}{1296}+\frac{\log (2-x) (313 d+512 e+820 f+1280 g+1936 h+2816 i)}{41472}+\frac{\log (x+1) (13 d-16 e+25 f-40 g+61 h-88 i)}{1296}+\frac{\log (x+2) (-313 d+512 e-820 f+1280 g-1936 h+2816 i)}{41472}","-\frac{x \left(-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right)}{3456 \left(x^4-5 x^2+4\right)}+\frac{x \left(-\left(x^2 (5 d+8 f+20 h)\right)+17 d+20 f+32 h\right)}{144 \left(x^4-5 x^2+4\right)^2}-\frac{\tanh ^{-1}\left(\frac{x}{2}\right) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{1}{162} \log \left(1-x^2\right) (2 e+5 g+11 i)+\frac{1}{162} \log \left(4-x^2\right) (2 e+5 g+11 i)-\frac{\left(5-2 x^2\right) (2 e+5 g+11 i)}{108 \left(x^4-5 x^2+4\right)}+\frac{-\left(x^2 (2 e+5 g+17 i)\right)+5 e+8 g+20 i}{36 \left(x^4-5 x^2+4\right)^2}",1,"(20*e + 32*g + 80*i + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 68*i*x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(144*(4 - 5*x^2 + x^4)^2) + (-320*e - 800*g - 1760*i - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 704*i*x^2 + 35*d*x^3 + 140*f*x^3 + 320*h*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-13*d - 16*e - 25*f - 40*g - 61*h - 88*i)*Log[1 - x])/1296 + ((313*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*Log[2 - x])/41472 + ((13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*Log[1 + x])/1296 + ((-313*d + 512*e - 820*f + 1280*g - 1936*h + 2816*i)*Log[2 + x])/41472","A",1
47,1,186,185,0.7461169,"\int \frac{d+e x}{\left(1+x^2+x^4\right)^3} \, dx","Integrate[(d + e*x)/(1 + x^2 + x^4)^3,x]","\frac{1}{144} \left(\frac{6 \left(d x \left(2-7 x^2\right)+e \left(8 x^2+4\right)\right)}{x^4+x^2+1}+\frac{12 \left(d \left(x-x^3\right)+2 e x^2+e\right)}{\left(x^4+x^2+1\right)^2}-\frac{\left(7 \sqrt{3}-47 i\right) d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\left(7 \sqrt{3}+47 i\right) d \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-32 \sqrt{3} e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{9}{32} d \log \left(x^2-x+1\right)+\frac{9}{32} d \log \left(x^2+x+1\right)+\frac{d x \left(2-7 x^2\right)}{24 \left(x^4+x^2+1\right)}+\frac{d x \left(1-x^2\right)}{12 \left(x^4+x^2+1\right)^2}-\frac{13 d \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{e \left(2 x^2+1\right)}{6 \left(x^4+x^2+1\right)}+\frac{e \left(2 x^2+1\right)}{12 \left(x^4+x^2+1\right)^2}",1,"((6*(d*x*(2 - 7*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 + d*(x - x^3)))/(1 + x^2 + x^4)^2 - ((-47*I + 7*Sqrt[3])*d*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - ((47*I + 7*Sqrt[3])*d*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 32*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144","C",0
48,1,235,223,0.5919195,"\int \frac{d+e x+f x^2}{\left(1+x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^3,x]","\frac{1}{144} \left(\frac{12 \left(x \left(-d x^2+d+2 f x^2+f\right)+2 e x^2+e\right)}{\left(x^4+x^2+1\right)^2}+\frac{6 \left(-7 d x^3+2 d x+e \left(8 x^2+4\right)+7 f x^3+3 f x\right)}{x^4+x^2+1}-\frac{\left(\left(7 \sqrt{3}-47 i\right) d+\left(-7 \sqrt{3}+17 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\left(\left(7 \sqrt{3}+47 i\right) d-\left(7 \sqrt{3}+17 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-32 \sqrt{3} e \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{32} (9 d-4 f) \log \left(x^2-x+1\right)+\frac{1}{32} (9 d-4 f) \log \left(x^2+x+1\right)+\frac{x \left(-7 x^2 (d-f)+2 d+3 f\right)}{24 \left(x^4+x^2+1\right)}+\frac{x \left(-\left(x^2 (d-2 f)\right)+d+f\right)}{12 \left(x^4+x^2+1\right)^2}-\frac{(13 d+2 f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{e \left(2 x^2+1\right)}{6 \left(x^4+x^2+1\right)}+\frac{e \left(2 x^2+1\right)}{12 \left(x^4+x^2+1\right)^2}",1,"((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 32*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144","C",0
49,1,259,243,0.6579526,"\int \frac{d+e x+f x^2+g x^3}{\left(1+x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(1 + x^2 + x^4)^3,x]","\frac{1}{144} \left(\frac{12 \left(x \left(-d x^2+d+2 f x^2+f\right)+2 e x^2+e-g \left(x^2+2\right)\right)}{\left(x^4+x^2+1\right)^2}+\frac{6 \left(-7 d x^3+2 d x+e \left(8 x^2+4\right)+7 f x^3+3 f x-2 g \left(2 x^2+1\right)\right)}{x^4+x^2+1}-\frac{\left(\left(7 \sqrt{3}-47 i\right) d+\left(-7 \sqrt{3}+17 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\left(\left(7 \sqrt{3}+47 i\right) d-\left(7 \sqrt{3}+17 i\right) f\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-16 \sqrt{3} (2 e-g) \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{32} (9 d-4 f) \log \left(x^2-x+1\right)+\frac{1}{32} (9 d-4 f) \log \left(x^2+x+1\right)+\frac{x \left(-7 x^2 (d-f)+2 d+3 f\right)}{24 \left(x^4+x^2+1\right)}+\frac{x \left(-\left(x^2 (d-2 f)\right)+d+f\right)}{12 \left(x^4+x^2+1\right)^2}-\frac{(13 d+2 f) \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{\left(2 x^2+1\right) (2 e-g)}{12 \left(x^4+x^2+1\right)}+\frac{x^2 (2 e-g)+e-2 g}{12 \left(x^4+x^2+1\right)^2}",1,"((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 16*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144","C",0
50,1,303,263,0.9041428,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(1+x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(1 + x^2 + x^4)^3,x]","\frac{1}{144} \left(-\frac{6 \left(x \left(7 d x^2-2 d-7 f x^2-3 f+4 h x^2+h\right)-4 e \left(2 x^2+1\right)+g \left(4 x^2+2\right)\right)}{x^4+x^2+1}+\frac{12 \left(x \left(-d x^2+d+2 f x^2+f-h \left(x^2+2\right)\right)+2 e x^2+e-g \left(x^2+2\right)\right)}{\left(x^4+x^2+1\right)^2}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(\left(7 \sqrt{3}-47 i\right) d+\left(-7 \sqrt{3}+17 i\right) f+2 \left(2 \sqrt{3}-7 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(\left(7 \sqrt{3}+47 i\right) d-\left(7 \sqrt{3}+17 i\right) f+2 \left(2 \sqrt{3}+7 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-16 \sqrt{3} (2 e-g) \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right)\right)","-\frac{1}{32} \log \left(x^2-x+1\right) (9 d-4 f+3 h)+\frac{1}{32} \log \left(x^2+x+1\right) (9 d-4 f+3 h)+\frac{x \left(-\left(x^2 (7 d-7 f+4 h)\right)+2 d+3 f-h\right)}{24 \left(x^4+x^2+1\right)}+\frac{x \left(-\left(x^2 (d-2 f+h)\right)+d+f-2 h\right)}{12 \left(x^4+x^2+1\right)^2}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{\left(2 x^2+1\right) (2 e-g)}{12 \left(x^4+x^2+1\right)}+\frac{x^2 (2 e-g)+e-2 g}{12 \left(x^4+x^2+1\right)^2}",1,"((-6*(-4*e*(1 + 2*x^2) + g*(2 + 4*x^2) + x*(-2*d - 3*f + h + 7*d*x^2 - 7*f*x^2 + 4*h*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f*x^2 - h*(2 + x^2))))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f + 2*(-7*I + 2*Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f + 2*(7*I + 2*Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 16*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144","C",0
51,1,325,269,0.9773045,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(1+x^2+x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^3,x]","\frac{1}{144} \left(\frac{12 \left(-d x^3+d x+2 e x^2+e+2 f x^3+f x-g \left(x^2+2\right)-h x^3-2 h x-i x^2+i\right)}{\left(x^4+x^2+1\right)^2}+\frac{6 \left(-7 d x^3+2 d x+e \left(8 x^2+4\right)+7 f x^3+3 f x-2 g \left(2 x^2+1\right)-4 h x^3-h x+4 i x^2+2 i\right)}{x^4+x^2+1}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x\right) \left(\left(7 \sqrt{3}-47 i\right) d+\left(-7 \sqrt{3}+17 i\right) f+2 \left(2 \sqrt{3}-7 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x\right) \left(\left(7 \sqrt{3}+47 i\right) d-\left(7 \sqrt{3}+17 i\right) f+2 \left(2 \sqrt{3}+7 i\right) h\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-16 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}}{2 x^2+1}\right) (2 e-g+i)\right)","-\frac{1}{32} \log \left(x^2-x+1\right) (9 d-4 f+3 h)+\frac{1}{32} \log \left(x^2+x+1\right) (9 d-4 f+3 h)+\frac{x \left(-\left(x^2 (7 d-7 f+4 h)\right)+2 d+3 f-h\right)}{24 \left(x^4+x^2+1\right)}+\frac{x \left(-\left(x^2 (d-2 f+h)\right)+d+f-2 h\right)}{12 \left(x^4+x^2+1\right)^2}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right) (13 d+2 f+h)}{48 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right) (2 e-g+i)}{3 \sqrt{3}}+\frac{\left(2 x^2+1\right) (2 e-g+i)}{12 \left(x^4+x^2+1\right)}+\frac{x^2 (2 e-g-i)+e-2 g+i}{12 \left(x^4+x^2+1\right)^2}",1,"((12*(e + i + d*x + f*x - 2*h*x + 2*e*x^2 - i*x^2 - d*x^3 + 2*f*x^3 - h*x^3 - g*(2 + x^2)))/(1 + x^2 + x^4)^2 + (6*(2*i + 2*d*x + 3*f*x - h*x + 4*i*x^2 - 7*d*x^3 + 7*f*x^3 - 4*h*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f + 2*(-7*I + 2*Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f + 2*(7*I + 2*Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 16*Sqrt[3]*(2*e - g + i)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144","C",0
52,1,488,474,1.9107977,"\int \frac{d+e x}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x)/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{16} \left(\frac{8 a^2 c (3 b e+c x (7 d+6 e x))-2 a b c d x \left(25 b+24 c x^2\right)+6 b^3 d x \left(b+c x^2\right)}{a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{2} \sqrt{c} d \left(56 a^2 c^2-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} d \left(56 a^2 c^2-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}-b^3 \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{4 a b e+8 a c x (d+e x)-4 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}\right)","\frac{d x \left(3 b c x^2 \left(b^2-8 a c\right)+\left(b^2-7 a c\right) \left(3 b^2-4 a c\right)\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{3 \sqrt{c} d \left(56 a^2 c^2-10 a b^2 c+b \left(b^2-8 a c\right) \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} d \left(-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-8 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{6 c^2 e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{d x \left(-2 a c+b^2+b c x^2\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 c e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{e \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((4*a*b*e + 8*a*c*x*(d + e*x) - 4*b*d*x*(b + c*x^2))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) - 2*a*b*c*d*x*(25*b + 24*c*x^2) + 8*a^2*c*(3*b*e + c*x*(7*d + 6*e*x)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16","A",1
53,1,625,621,3.60906,"\int \frac{d+e x+f x^2}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{16} \left(\frac{8 a^2 c \left(b (3 e+2 f x)+c x \left(7 d+6 e x+5 f x^2\right)\right)+2 a b x \left(b^2 f-25 b c d+b c f x^2-24 c^2 d x^2\right)+6 b^3 d x \left(b+c x^2\right)}{a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}-30 c d\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}-42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}+30 c d\right)+4 a b c \left(13 a f-6 d \sqrt{b^2-4 a c}\right)+b^3 \left(3 d \sqrt{b^2-4 a c}-a f\right)-3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{4 a b (e+f x)+8 a c x (d+x (e+f x))-4 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}\right)","\frac{x \left(c x^2 \left(20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right)+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)-a b^2 \left(30 c d-f \sqrt{b^2-4 a c}\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 c e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{e \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((4*a*b*(e + f*x) - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) + 2*a*b*x*(-25*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + 8*a^2*c*(b*(3*e + 2*f*x) + c*x*(7*d + 6*e*x + 5*f*x^2)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(-30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d - a*f) + 4*a*b*c*(-6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16","A",1
54,1,661,646,4.2934618,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{16} \left(\frac{-8 a^2 g+4 a b (e+x (f-g x))+8 a c x (d+x (e+f x))-4 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}+\frac{2 \left(a^2 \left(-6 b^2 g+4 b c \left(3 e+2 f x-3 g x^2\right)+4 c^2 x \left(7 d+6 e x+5 f x^2\right)\right)+a b x \left(b^2 f-25 b c d+b c f x^2-24 c^2 d x^2\right)+3 b^3 d x \left(b+c x^2\right)\right)}{a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}-30 c d\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}-42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}+30 c d\right)+4 a b c \left(13 a f-6 d \sqrt{b^2-4 a c}\right)+b^3 \left(3 d \sqrt{b^2-4 a c}-a f\right)-3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{24 c (b g-2 c e) \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{24 c (b g-2 c e) \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}\right)","\frac{x \left(c x^2 \left(20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right)+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)-a b^2 \left(30 c d-f \sqrt{b^2-4 a c}\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{3 c (2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(b+2 c x^2\right) (2 c e-b g)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((-8*a^2*g - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)) + 4*a*b*(e + x*(f - g*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*(3*b^3*d*x*(b + c*x^2) + a*b*x*(-25*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + a^2*(-6*b^2*g + 4*c^2*x*(7*d + 6*e*x + 5*f*x^2) + 4*b*c*(3*e + 2*f*x - 3*g*x^2))))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(-30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d - a*f) + 4*a*b*c*(-6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (24*c*(-2*c*e + b*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + (24*c*(-2*c*e + b*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16","A",1
55,1,845,679,6.5479796,"\int \frac{d+e x+f x^2+g x^3+h x^4}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4)/(a + b*x^2 + c*x^4)^3,x]","-\frac{b c d x^3-2 a c f x^3+a b h x^3-2 a c e x^2+a b g x^2+b^2 d x-2 a c d x-a b f x+2 a^2 h x-a b e+2 a^2 g}{4 a \left(4 a c-b^2\right) \left(c x^4+b x^2+a\right)^2}+\frac{\sqrt{c} \left(3 d b^4+3 \sqrt{b^2-4 a c} d b^3+a f b^3-30 a c d b^2+a \sqrt{b^2-4 a c} f b^2+18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b-52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b+168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f+24 a^3 c h\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-3 d b^4+3 \sqrt{b^2-4 a c} d b^3-a f b^3+30 a c d b^2+a \sqrt{b^2-4 a c} f b^2-18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b+52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b-168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f-24 a^3 c h\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{3 c (2 c e-b g) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}-\frac{3 c (2 c e-b g) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}+\frac{3 d x b^4+3 c d x^3 b^3+a f x b^3+a c f x^3 b^2-6 a^2 g b^2-25 a c d x b^2-7 a^2 h x b^2-24 a c^2 d x^3 b-12 a^2 c h x^3 b-12 a^2 c g x^2 b+12 a^2 c e b+8 a^2 c f x b+20 a^2 c^2 f x^3+24 a^2 c^2 e x^2+28 a^2 c^2 d x+4 a^3 c h x}{8 a^2 \left(4 a c-b^2\right)^2 \left(c x^4+b x^2+a\right)}","\frac{x \left(c x^2 \left(20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left(x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}-\frac{3 c (2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(b+2 c x^2\right) (2 c e-b g)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"-1/4*(-(a*b*e) + 2*a^2*g + b^2*d*x - 2*a*c*d*x - a*b*f*x + 2*a^2*h*x - 2*a*c*e*x^2 + a*b*g*x^2 + b*c*d*x^3 - 2*a*c*f*x^3 + a*b*h*x^3)/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c*e - 6*a^2*b^2*g + 3*b^4*d*x - 25*a*b^2*c*d*x + 28*a^2*c^2*d*x + a*b^3*f*x + 8*a^2*b*c*f*x - 7*a^2*b^2*h*x + 4*a^3*c*h*x + 24*a^2*c^2*e*x^2 - 12*a^2*b*c*g*x^2 + 3*b^3*c*d*x^3 - 24*a*b*c^2*d*x^3 + a*b^2*c*f*x^3 + 20*a^2*c^2*f*x^3 - 12*a^2*b*c*h*x^3)/(8*a^2*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d + a*b^3*f - 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*h + 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(-3*b^4*d + 30*a*b^2*c*d - 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d - a*b^3*f + 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*h - 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (3*c*(2*c*e - b*g)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2)) - (3*c*(2*c*e - b*g)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))","A",1
56,1,980,728,6.6736234,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^3,x]","\frac{-b c^2 d x^3+2 a c^2 f x^3-a b c h x^3+2 a c^2 e x^2-a b c g x^2+a b^2 i x^2-2 a^2 c i x^2+2 a c^2 d x-b^2 c d x+a b c f x-2 a^2 c h x+a b c e-2 a^2 c g+a^2 b i}{4 a c \left(4 a c-b^2\right) \left(c x^4+b x^2+a\right)^2}+\frac{\sqrt{c} \left(3 d b^4+3 \sqrt{b^2-4 a c} d b^3+a f b^3-30 a c d b^2+a \sqrt{b^2-4 a c} f b^2+18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b-52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b+168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f+24 a^3 c h\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-3 d b^4+3 \sqrt{b^2-4 a c} d b^3-a f b^3+30 a c d b^2+a \sqrt{b^2-4 a c} f b^2-18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b+52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b-168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f-24 a^3 c h\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left(i b^2-3 c g b+6 c^2 e+2 a c i\right) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}+\frac{\left(-i b^2+3 c g b-6 c^2 e-2 a c i\right) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}+\frac{3 c d x b^4+3 c^2 d x^3 b^3+2 a^2 i b^3+a c f x b^3+a c^2 f x^3 b^2+4 a^2 c i x^2 b^2-6 a^2 c g b^2-25 a c^2 d x b^2-7 a^2 c h x b^2-24 a c^3 d x^3 b-12 a^2 c^2 h x^3 b-12 a^2 c^2 g x^2 b+12 a^2 c^2 e b+4 a^3 c i b+8 a^2 c^2 f x b+20 a^2 c^3 f x^3+24 a^2 c^3 e x^2+8 a^3 c^2 i x^2+28 a^2 c^3 d x+4 a^3 c^2 h x}{8 a^2 c \left(4 a c-b^2\right)^2 \left(c x^4+b x^2+a\right)}","\frac{x \left(c x^2 \left(20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(2 a c i+b^2 i-3 b c g+6 c^2 e\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{-\left(x^2 \left(-2 a c i+b^2 i-b c g+2 c^2 e\right)\right)-b (a i+c e)+2 a c g}{4 c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{x \left(x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{\left(b+2 c x^2\right) \left(2 a i+\frac{b^2 i}{c}-3 b g+6 c e\right)}{4 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}",1,"(a*b*c*e - 2*a^2*c*g + a^2*b*i - b^2*c*d*x + 2*a*c^2*d*x + a*b*c*f*x - 2*a^2*c*h*x + 2*a*c^2*e*x^2 - a*b*c*g*x^2 + a*b^2*i*x^2 - 2*a^2*c*i*x^2 - b*c^2*d*x^3 + 2*a*c^2*f*x^3 - a*b*c*h*x^3)/(4*a*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^2*e - 6*a^2*b^2*c*g + 2*a^2*b^3*i + 4*a^3*b*c*i + 3*b^4*c*d*x - 25*a*b^2*c^2*d*x + 28*a^2*c^3*d*x + a*b^3*c*f*x + 8*a^2*b*c^2*f*x - 7*a^2*b^2*c*h*x + 4*a^3*c^2*h*x + 24*a^2*c^3*e*x^2 - 12*a^2*b*c^2*g*x^2 + 4*a^2*b^2*c*i*x^2 + 8*a^3*c^2*i*x^2 + 3*b^3*c^2*d*x^3 - 24*a*b*c^3*d*x^3 + a*b^2*c^2*f*x^3 + 20*a^2*c^3*f*x^3 - 12*a^2*b*c^2*h*x^3)/(8*a^2*c*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d + a*b^3*f - 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*h + 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(-3*b^4*d + 30*a*b^2*c*d - 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a*c]*d - a*b^3*f + 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*h - 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((6*c^2*e - 3*b*c*g + b^2*i + 2*a*c*i)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2)) + ((-6*c^2*e + 3*b*c*g - b^2*i - 2*a*c*i)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))","A",1
57,1,1590,1150,7.4796629,"\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^2 + c*x^4)^3,x]","\frac{2 c l a^3+2 c m x a^3-2 c^2 k x^3 a^2+3 b c m x^3 a^2-2 c^2 j x^2 a^2+3 b c l x^2 a^2-2 c^2 g a^2+b c j a^2-b^2 l a^2-2 c^2 h x a^2+b c k x a^2-b^2 m x a^2+2 c^3 f x^3 a-b c^2 h x^3 a+b^2 c k x^3 a-b^3 m x^3 a+2 c^3 e x^2 a-b c^2 g x^2 a+b^2 c j x^2 a-b^3 l x^2 a+b c^2 e a+2 c^3 d x a+b c^2 f x a-b c^3 d x^3-b^2 c^2 d x}{4 a c^2 \left(4 a c-b^2\right) \left(c x^4+b x^2+a\right)^2}+\frac{\left(40 c^2 m a^4+24 c^3 h a^3-36 b c^2 k a^3+12 c^2 \sqrt{b^2-4 a c} k a^3+18 b^2 c m a^3-16 b c \sqrt{b^2-4 a c} m a^3+168 c^4 d a^2-52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2+18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2-3 b^3 c k a^2+3 b^2 c \sqrt{b^2-4 a c} k a^2-b^4 m a^2+b^3 \sqrt{b^2-4 a c} m a^2-30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a+b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a+3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(-40 c^2 m a^4-24 c^3 h a^3+36 b c^2 k a^3+12 c^2 \sqrt{b^2-4 a c} k a^3-18 b^2 c m a^3-16 b c \sqrt{b^2-4 a c} m a^3-168 c^4 d a^2+52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2-18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2+3 b^3 c k a^2+3 b^2 c \sqrt{b^2-4 a c} k a^2+b^4 m a^2+b^3 \sqrt{b^2-4 a c} m a^2+30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a-b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a-3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left(j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}+\frac{\left(-j b^2+3 c g b+3 a l b-6 c^2 e-2 a c j\right) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{2 \left(b^2-4 a c\right)^{5/2}}+\frac{-32 c^2 l a^4-36 c^2 m x a^4+12 c^3 k x^3 a^3-16 b c^2 m x^3 a^3+8 c^3 j x^2 a^3-12 b c^2 l x^2 a^3+4 b c^2 j a^3+10 b^2 c l a^3+4 c^3 h x a^3+4 b c^2 k x a^3+11 b^2 c m x a^3+20 c^4 f x^3 a^2-12 b c^3 h x^3 a^2+3 b^2 c^2 k x^3 a^2+b^3 c m x^3 a^2+24 c^4 e x^2 a^2-12 b c^3 g x^2 a^2+4 b^2 c^2 j x^2 a^2+12 b c^3 e a^2-6 b^2 c^2 g a^2+2 b^3 c j a^2-2 b^4 l a^2+28 c^4 d x a^2+8 b c^3 f x a^2-7 b^2 c^2 h x a^2+2 b^3 c k x a^2-2 b^4 m x a^2-24 b c^4 d x^3 a+b^2 c^3 f x^3 a-25 b^2 c^3 d x a+b^3 c^2 f x a+3 b^3 c^3 d x^3+3 b^4 c^2 d x}{8 a^2 c^2 \left(4 a c-b^2\right)^2 \left(c x^4+b x^2+a\right)}","\frac{-\frac{l b^4}{c^2}+\frac{j b^3}{c}-\left(3 g-\frac{5 a l}{c}\right) b^2+2 (3 c e+a j) b+2 \left(j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right) x^2-16 a^2 l}{4 \left(b^2-4 a c\right)^2 \left(c x^4+b x^2+a\right)}+\frac{\left(\left(m a^2+3 c^2 d\right) b^3+a c (c f+3 a k) b^2-4 a c \left(4 m a^2+3 c h a+6 c^2 d\right) b+4 a^2 c^2 (5 c f+3 a k)+\frac{\left(3 c^2 d-a^2 m\right) b^4+a c (c f-3 a k) b^3-6 a c \left(-3 m a^2-3 c h a+5 c^2 d\right) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left(5 m a^2+3 c h a+21 c^2 d\right)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\left(m a^2+3 c^2 d\right) b^3+a c (c f+3 a k) b^2-4 a c \left(4 m a^2+3 c h a+6 c^2 d\right) b+4 a^2 c^2 (5 c f+3 a k)-\frac{\left(3 c^2 d-a^2 m\right) b^4+a c (c f-3 a k) b^3-6 a c \left(-3 m a^2-3 c h a+5 c^2 d\right) b^2-4 a^2 c^2 (13 c f+9 a k) b+8 a^2 c^2 \left(5 m a^2+3 c h a+21 c^2 d\right)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left(j b^2-3 c g b-3 a l b+6 c^2 e+2 a c j\right) \tanh ^{-1}\left(\frac{2 c x^2+b}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{x \left(\left(3 c^2 d-2 a^2 m\right) b^4+a c (c f+2 a k) b^3-a c \left(-11 m a^2+7 c h a+25 c^2 d\right) b^2+4 a^2 c^2 (2 c f+a k) b+c \left(\left(m a^2+3 c^2 d\right) b^3+a c (c f+3 a k) b^2-4 a c \left(4 m a^2+3 c h a+6 c^2 d\right) b+4 a^2 c^2 (5 c f+3 a k)\right) x^2+4 a^2 c^2 \left(-9 m a^2+c h a+7 c^2 d\right)\right)}{8 a^2 c^2 \left(b^2-4 a c\right)^2 \left(c x^4+b x^2+a\right)}-\frac{-a l b^2+c (c e+a j) b+\left(-l b^3+c (b j+3 a l) b+2 c^3 e-c^2 (b g+2 a j)\right) x^2-2 a c (c g-a l)}{4 c^2 \left(b^2-4 a c\right) \left(c x^4+b x^2+a\right)^2}-\frac{x \left(-\left(\left(m a^2+c^2 d\right) b^2\right)+a c (c f+a k) b+\left(-a m b^3+a c k b^2-c \left(-3 m a^2+c h a+c^2 d\right) b+2 a c^2 (c f-a k)\right) x^2+2 a c \left(m a^2-c h a+c^2 d\right)\right)}{4 a c^2 \left(b^2-4 a c\right) \left(c x^4+b x^2+a\right)^2}",1,"(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*j - a^2*b^2*l + 2*a^3*c*l - b^2*c^2*d*x + 2*a*c^3*d*x + a*b*c^2*f*x - 2*a^2*c^2*h*x + a^2*b*c*k*x - a^2*b^2*m*x + 2*a^3*c*m*x + 2*a*c^3*e*x^2 - a*b*c^2*g*x^2 + a*b^2*c*j*x^2 - 2*a^2*c^2*j*x^2 - a*b^3*l*x^2 + 3*a^2*b*c*l*x^2 - b*c^3*d*x^3 + 2*a*c^3*f*x^3 - a*b*c^2*h*x^3 + a*b^2*c*k*x^3 - 2*a^2*c^2*k*x^3 - a*b^3*m*x^3 + 3*a^2*b*c*m*x^3)/(4*a*c^2*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^3*e - 6*a^2*b^2*c^2*g + 2*a^2*b^3*c*j + 4*a^3*b*c^2*j - 2*a^2*b^4*l + 10*a^3*b^2*c*l - 32*a^4*c^2*l + 3*b^4*c^2*d*x - 25*a*b^2*c^3*d*x + 28*a^2*c^4*d*x + a*b^3*c^2*f*x + 8*a^2*b*c^3*f*x - 7*a^2*b^2*c^2*h*x + 4*a^3*c^3*h*x + 2*a^2*b^3*c*k*x + 4*a^3*b*c^2*k*x - 2*a^2*b^4*m*x + 11*a^3*b^2*c*m*x - 36*a^4*c^2*m*x + 24*a^2*c^4*e*x^2 - 12*a^2*b*c^3*g*x^2 + 4*a^2*b^2*c^2*j*x^2 + 8*a^3*c^3*j*x^2 - 12*a^3*b*c^2*l*x^2 + 3*b^3*c^3*d*x^3 - 24*a*b*c^4*d*x^3 + a*b^2*c^3*f*x^3 + 20*a^2*c^4*f*x^3 - 12*a^2*b*c^3*h*x^3 + 3*a^2*b^2*c^2*k*x^3 + 12*a^3*c^3*k*x^3 + a^2*b^3*c*m*x^3 - 16*a^3*b*c^2*m*x^3)/(8*a^2*c^2*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*c^2*d - 30*a*b^2*c^3*d + 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d + a*b^3*c^2*f - 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*c^2*h + 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h - 3*a^2*b^3*c*k - 36*a^3*b*c^2*k + 3*a^2*b^2*c*Sqrt[b^2 - 4*a*c]*k + 12*a^3*c^2*Sqrt[b^2 - 4*a*c]*k - a^2*b^4*m + 18*a^3*b^2*c*m + 40*a^4*c^2*m + a^2*b^3*Sqrt[b^2 - 4*a*c]*m - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*m)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((-3*b^4*c^2*d + 30*a*b^2*c^3*d - 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d - a*b^3*c^2*f + 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*c^2*h - 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h + 3*a^2*b^3*c*k + 36*a^3*b*c^2*k + 3*a^2*b^2*c*Sqrt[b^2 - 4*a*c]*k + 12*a^3*c^2*Sqrt[b^2 - 4*a*c]*k + a^2*b^4*m - 18*a^3*b^2*c*m - 40*a^4*c^2*m + a^2*b^3*Sqrt[b^2 - 4*a*c]*m - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*m)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((6*c^2*e - 3*b*c*g + b^2*j + 2*a*c*j - 3*a*b*l)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2)) + ((-6*c^2*e + 3*b*c*g - b^2*j - 2*a*c*j + 3*a*b*l)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))","A",1
58,1,775,645,4.4049525,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b*x^2 + c*x^4)^2,x]","\frac{-\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(-b c \left(8 a^2 j+c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}+4 a c f\right)+a b^3 j+2 a c \left(c f \sqrt{b^2-4 a c}+3 a j \sqrt{b^2-4 a c}+2 a c h+6 c^2 d\right)-b^2 \left(a j \sqrt{b^2-4 a c}-a c h+c^2 d\right)\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(b c \left(-8 a^2 j+c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}-4 a c f\right)+a b^3 j+2 a c \left(-c f \sqrt{b^2-4 a c}-3 a j \sqrt{b^2-4 a c}+2 a c h+6 c^2 d\right)+b^2 \left(a j \sqrt{b^2-4 a c}+a c h+c^2 (-d)\right)\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \left(2 a^3 c k+a^2 \left(b^2 (-k)+b c (i+x (j+3 k x))-2 c^2 (g+x (h+x (i+j x)))\right)+a \left(b^3 (-k) x^2+b^2 c x^2 (i+j x)+b c^2 (e+x (f-x (g+h x)))+2 c^3 x (d+x (e+f x))\right)-b c^2 d x \left(b+c x^2\right)\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right) \left(a c \left(6 b k-4 k \sqrt{b^2-4 a c}\right)+b^2 k \left(\sqrt{b^2-4 a c}-b\right)+2 c^2 (b g-2 a i)-4 c^3 e\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right) \left(-2 a c k \left(2 \sqrt{b^2-4 a c}+3 b\right)+b^2 k \left(\sqrt{b^2-4 a c}+b\right)+c^2 (4 a i-2 b g)+4 c^3 e\right)}{\left(b^2-4 a c\right)^{3/2}}}{4 c^2}","\frac{x \left(x^2 \left(-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right)+c \left(-\frac{a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right)\right)}{2 a c \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right) \left(\frac{a b^2 j}{c}+\frac{-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right) \left(\frac{a b^2 j}{c}-\frac{-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt{b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right)}{2 \sqrt{2} a \sqrt{c} \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right) \left(-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right)}{2 c^2 \left(b^2-4 a c\right)^{3/2}}-\frac{x^2 \left(-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right)-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{2 c^2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{k \log \left(a+b x^2+c x^4\right)}{4 c^2}",1,"((2*(2*a^3*c*k - b*c^2*d*x*(b + c*x^2) + a*(-(b^3*k*x^2) + b^2*c*x^2*(i + j*x) + 2*c^3*x*(d + x*(e + f*x)) + b*c^2*(e + x*(f - x*(g + h*x)))) + a^2*(-(b^2*k) + b*c*(i + x*(j + 3*k*x)) - 2*c^2*(g + x*(h + x*(i + j*x))))))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*Sqrt[c]*(a*b^3*j - b*c*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h + 8*a^2*j) - b^2*(c^2*d - a*c*h + a*Sqrt[b^2 - 4*a*c]*j) + 2*a*c*(6*c^2*d + c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(a*b^3*j + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h - 8*a^2*j) + 2*a*c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h - 3*a*Sqrt[b^2 - 4*a*c]*j) + b^2*(-(c^2*d) + a*c*h + a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((-4*c^3*e + 2*c^2*(b*g - 2*a*i) + b^2*(-b + Sqrt[b^2 - 4*a*c])*k + a*c*(6*b*k - 4*Sqrt[b^2 - 4*a*c]*k))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((4*c^3*e + c^2*(-2*b*g + 4*a*i) + b^2*(b + Sqrt[b^2 - 4*a*c])*k - 2*a*c*(3*b + 2*Sqrt[b^2 - 4*a*c])*k)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*c^2)","A",1
59,1,1649,1177,7.3455098,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5+j x^8+k x^{11}}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^8 + k*x^11)/(a + b*x^2 + c*x^4)^3,x]","\frac{-a k x^2 b^5-a^2 k b^4-a c^2 j x^3 b^3+5 a^2 c k x^2 b^3+a c^3 i x^2 b^2+4 a^3 c k b^2-c^4 d x b^2-a^2 c^2 j x b^2-c^5 d x^3 b-a c^4 h x^3 b+3 a^2 c^3 j x^3 b-a c^4 g x^2 b-5 a^3 c^2 k x^2 b+a c^4 e b+a^2 c^3 i b+a c^4 f x b+2 a c^5 f x^3+2 a c^5 e x^2-2 a^2 c^4 i x^2-2 a^2 c^4 g-2 a^4 c^2 k+2 a c^5 d x-2 a^2 c^4 h x+2 a^3 c^3 j x}{4 a c^4 \left(4 a c-b^2\right) \left(c x^4+b x^2+a\right)^2}+\frac{\left(40 c^2 j a^4+24 c^3 h a^3+18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j a^3+168 c^4 d a^2-52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2+18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2-b^4 j a^2+b^3 \sqrt{b^2-4 a c} j a^2-30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a+b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a+3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(-40 c^2 j a^4-24 c^3 h a^3-18 b^2 c j a^3-16 b c \sqrt{b^2-4 a c} j a^3-168 c^4 d a^2+52 b c^3 f a^2+20 c^3 \sqrt{b^2-4 a c} f a^2-18 b^2 c^2 h a^2-12 b c^2 \sqrt{b^2-4 a c} h a^2+b^4 j a^2+b^3 \sqrt{b^2-4 a c} j a^2+30 b^2 c^3 d a-24 b c^3 \sqrt{b^2-4 a c} d a-b^3 c^2 f a+b^2 c^2 \sqrt{b^2-4 a c} f a-3 b^4 c^2 d+3 b^3 c^2 \sqrt{b^2-4 a c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left(-k b^5+\sqrt{b^2-4 a c} k b^4+10 a c k b^3+2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2-6 c^4 g b-30 a^2 c^2 k b+12 c^5 e+4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right) \log \left(-2 c x^2-b+\sqrt{b^2-4 a c}\right)}{4 c^3 \left(b^2-4 a c\right)^{5/2}}+\frac{\left(k b^5+\sqrt{b^2-4 a c} k b^4-10 a c k b^3-2 c^3 i b^2-8 a c \sqrt{b^2-4 a c} k b^2+6 c^4 g b+30 a^2 c^2 k b-12 c^5 e-4 a c^4 i+16 a^2 c^2 \sqrt{b^2-4 a c} k\right) \log \left(2 c x^2+b+\sqrt{b^2-4 a c}\right)}{4 c^3 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a^2 k b^6+8 a^2 c k x^2 b^5+22 a^3 c k b^4+3 c^4 d x b^4-2 a^2 c^2 j x b^4+3 c^5 d x^3 b^3+a^2 c^3 j x^3 b^3-60 a^3 c^2 k x^2 b^3+2 a^2 c^3 i b^3+a c^4 f x b^3+a c^5 f x^3 b^2+4 a^2 c^4 i x^2 b^2-6 a^2 c^4 g b^2-78 a^4 c^2 k b^2-25 a c^5 d x b^2-7 a^2 c^4 h x b^2+11 a^3 c^3 j x b^2-24 a c^6 d x^3 b-12 a^2 c^5 h x^3 b-16 a^3 c^4 j x^3 b-12 a^2 c^5 g x^2 b+100 a^4 c^3 k x^2 b+12 a^2 c^5 e b+4 a^3 c^4 i b+8 a^2 c^5 f x b+20 a^2 c^6 f x^3+24 a^2 c^6 e x^2+8 a^3 c^5 i x^2+64 a^5 c^3 k+28 a^2 c^6 d x+4 a^3 c^5 h x-36 a^4 c^4 j x}{8 a^2 c^4 \left(4 a c-b^2\right)^2 \left(c x^4+b x^2+a\right)}","-\frac{x \left(\left(-\left(\left(\frac{j a^2}{c^2}+d\right) b^2\right)+a f b+2 a \left(\frac{j a^2}{c}-h a+c d\right)\right) c^2+\left(-a j b^3-c \left(-3 j a^2+c h a+c^2 d\right) b+2 a c^3 f\right) x^2\right)}{4 a c^2 \left(b^2-4 a c\right) \left(c x^4+b x^2+a\right)^2}+\frac{\left(\left(j a^2+3 c^2 d\right) b^3+a c^2 f b^2-4 a c \left(4 j a^2+3 c h a+6 c^2 d\right) b+20 a^2 c^3 f+\frac{\left(3 c^2 d-a^2 j\right) b^4+a c^2 f b^3-6 a c \left(-3 j a^2-3 c h a+5 c^2 d\right) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left(5 j a^2+3 c h a+21 c^2 d\right)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(\left(j a^2+3 c^2 d\right) b^3+a c^2 f b^2-4 a c \left(4 j a^2+3 c h a+6 c^2 d\right) b+20 a^2 c^3 f-\frac{\left(3 c^2 d-a^2 j\right) b^4+a c^2 f b^3-6 a c \left(-3 j a^2-3 c h a+5 c^2 d\right) b^2-52 a^2 c^3 f b+8 a^2 c^2 \left(5 j a^2+3 c h a+21 c^2 d\right)}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 c^{3/2} \left(b^2-4 a c\right)^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left(-k b^5+10 a c k b^3+2 c^3 i b^2-30 a^2 c^2 k b+12 c^5 e-c^4 (6 b g-4 a i)\right) \tanh ^{-1}\left(\frac{2 c x^2+b}{\sqrt{b^2-4 a c}}\right)}{2 c^3 \left(b^2-4 a c\right)^{5/2}}+\frac{k \log \left(c x^4+b x^2+a\right)}{4 c^3}+\frac{x \left(\left(\left(j a^2+3 c^2 d\right) b^3+a c^2 f b^2-4 a c \left(4 j a^2+3 c h a+6 c^2 d\right) b+20 a^2 c^3 f\right) x^2+c \left(\left(3 d-\frac{2 a^2 j}{c^2}\right) b^4+a f b^3-a \left(-\frac{11 j a^2}{c}+7 h a+25 c d\right) b^2+8 a^2 c f b+4 a^2 \left(-9 j a^2+c h a+7 c^2 d\right)\right)\right)}{8 a^2 c \left(b^2-4 a c\right)^2 \left(c x^4+b x^2+a\right)}+\frac{-\frac{k b^6}{c}+11 a k b^4+c^2 i b^3-3 \left(g c^3+13 a^2 k c\right) b^2+2 c^3 (3 c e+a i) b+2 \left(2 k b^5-15 a c k b^3+c^3 i b^2+25 a^2 c^2 k b+6 c^5 e-c^4 (3 b g-2 a i)\right) x^2+32 a^3 c^2 k}{4 c^3 \left(b^2-4 a c\right)^2 \left(c x^4+b x^2+a\right)}-\frac{-a k b^4+4 a^2 c k b^2+c^3 (c e+a i) b+\left(-k b^5+5 a c k b^3+c^3 i b^2-5 a^2 c^2 k b+2 c^5 e-c^4 (b g+2 a i)\right) x^2-2 a c^2 \left(k a^2+c^2 g\right)}{4 c^4 \left(b^2-4 a c\right) \left(c x^4+b x^2+a\right)^2}",1,"(a*b*c^4*e - 2*a^2*c^4*g + a^2*b*c^3*i - a^2*b^4*k + 4*a^3*b^2*c*k - 2*a^4*c^2*k - b^2*c^4*d*x + 2*a*c^5*d*x + a*b*c^4*f*x - 2*a^2*c^4*h*x - a^2*b^2*c^2*j*x + 2*a^3*c^3*j*x + 2*a*c^5*e*x^2 - a*b*c^4*g*x^2 + a*b^2*c^3*i*x^2 - 2*a^2*c^4*i*x^2 - a*b^5*k*x^2 + 5*a^2*b^3*c*k*x^2 - 5*a^3*b*c^2*k*x^2 - b*c^5*d*x^3 + 2*a*c^5*f*x^3 - a*b*c^4*h*x^3 - a*b^3*c^2*j*x^3 + 3*a^2*b*c^3*j*x^3)/(4*a*c^4*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (12*a^2*b*c^5*e - 6*a^2*b^2*c^4*g + 2*a^2*b^3*c^3*i + 4*a^3*b*c^4*i - 2*a^2*b^6*k + 22*a^3*b^4*c*k - 78*a^4*b^2*c^2*k + 64*a^5*c^3*k + 3*b^4*c^4*d*x - 25*a*b^2*c^5*d*x + 28*a^2*c^6*d*x + a*b^3*c^4*f*x + 8*a^2*b*c^5*f*x - 7*a^2*b^2*c^4*h*x + 4*a^3*c^5*h*x - 2*a^2*b^4*c^2*j*x + 11*a^3*b^2*c^3*j*x - 36*a^4*c^4*j*x + 24*a^2*c^6*e*x^2 - 12*a^2*b*c^5*g*x^2 + 4*a^2*b^2*c^4*i*x^2 + 8*a^3*c^5*i*x^2 + 8*a^2*b^5*c*k*x^2 - 60*a^3*b^3*c^2*k*x^2 + 100*a^4*b*c^3*k*x^2 + 3*b^3*c^5*d*x^3 - 24*a*b*c^6*d*x^3 + a*b^2*c^5*f*x^3 + 20*a^2*c^6*f*x^3 - 12*a^2*b*c^5*h*x^3 + a^2*b^3*c^3*j*x^3 - 16*a^3*b*c^4*j*x^3)/(8*a^2*c^4*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((3*b^4*c^2*d - 30*a*b^2*c^3*d + 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d + a*b^3*c^2*f - 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*c^2*h + 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h - a^2*b^4*j + 18*a^3*b^2*c*j + 40*a^4*c^2*j + a^2*b^3*Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((-3*b^4*c^2*d + 30*a*b^2*c^3*d - 168*a^2*c^4*d + 3*b^3*c^2*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c^3*Sqrt[b^2 - 4*a*c]*d - a*b^3*c^2*f + 52*a^2*b*c^3*f + a*b^2*c^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c^3*Sqrt[b^2 - 4*a*c]*f - 18*a^2*b^2*c^2*h - 24*a^3*c^3*h - 12*a^2*b*c^2*Sqrt[b^2 - 4*a*c]*h + a^2*b^4*j - 18*a^3*b^2*c*j - 40*a^4*c^2*j + a^2*b^3*Sqrt[b^2 - 4*a*c]*j - 16*a^3*b*c*Sqrt[b^2 - 4*a*c]*j)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((12*c^5*e - 6*b*c^4*g + 2*b^2*c^3*i + 4*a*c^4*i - b^5*k + 10*a*b^3*c*k - 30*a^2*b*c^2*k + b^4*Sqrt[b^2 - 4*a*c]*k - 8*a*b^2*c*Sqrt[b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 - 4*a*c]*k)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2)) + ((-12*c^5*e + 6*b*c^4*g - 2*b^2*c^3*i - 4*a*c^4*i + b^5*k - 10*a*b^3*c*k + 30*a^2*b*c^2*k + b^4*Sqrt[b^2 - 4*a*c]*k - 8*a*b^2*c*Sqrt[b^2 - 4*a*c]*k + 16*a^2*c^2*Sqrt[b^2 - 4*a*c]*k)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*c^3*(b^2 - 4*a*c)^(5/2))","A",1
60,1,416,416,0.1217352,"\int \left(a+b x^2+c x^4\right)^3 \left(a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right) \, dx","Integrate[(a + b*x^2 + c*x^4)^3*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]","a^4 d x+\frac{1}{2} a^4 e x^2+\frac{1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac{2}{5} a^2 x^5 \left(2 a b f+2 a c d+3 b^2 d\right)+\frac{1}{3} a^2 e x^6 \left(2 a c+3 b^2\right)+\frac{1}{10} e x^{10} \left(6 a^2 c^2+12 a b^2 c+b^4\right)+\frac{2}{7} a x^7 \left(2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right)+\frac{1}{11} x^{11} \left(6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right)+\frac{1}{9} x^9 \left(12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right)+\frac{2}{15} c^2 x^{15} \left(2 a c f+3 b^2 f+2 b c d\right)+\frac{1}{7} c^2 e x^{14} \left(2 a c+3 b^2\right)+\frac{1}{3} b c e x^{12} \left(3 a c+b^2\right)+\frac{1}{2} a b e x^8 \left(3 a c+b^2\right)+\frac{2}{13} c x^{13} \left(6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right)+\frac{1}{17} c^3 x^{17} (4 b f+c d)+\frac{1}{4} b c^3 e x^{16}+\frac{1}{18} c^4 e x^{18}+\frac{1}{19} c^4 f x^{19}","a^4 d x+\frac{1}{2} a^4 e x^2+\frac{1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac{2}{5} a^2 x^5 \left(2 a b f+2 a c d+3 b^2 d\right)+\frac{1}{3} a^2 e x^6 \left(2 a c+3 b^2\right)+\frac{1}{10} e x^{10} \left(6 a^2 c^2+12 a b^2 c+b^4\right)+\frac{2}{7} a x^7 \left(2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right)+\frac{1}{11} x^{11} \left(6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right)+\frac{1}{9} x^9 \left(12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right)+\frac{2}{15} c^2 x^{15} \left(2 a c f+3 b^2 f+2 b c d\right)+\frac{1}{7} c^2 e x^{14} \left(2 a c+3 b^2\right)+\frac{1}{3} b c e x^{12} \left(3 a c+b^2\right)+\frac{1}{2} a b e x^8 \left(3 a c+b^2\right)+\frac{2}{13} c x^{13} \left(6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right)+\frac{1}{17} c^3 x^{17} (4 b f+c d)+\frac{1}{4} b c^3 e x^{16}+\frac{1}{18} c^4 e x^{18}+\frac{1}{19} c^4 f x^{19}",1,"a^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d + 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*a*c)*e*x^8)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12*a*b^2*c + 6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2 + 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^14)/7 + (2*c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*f)*x^17)/17 + (c^4*e*x^18)/18 + (c^4*f*x^19)/19","A",1
61,1,259,259,0.0464079,"\int \left(a+b x^2+c x^4\right)^2 \left(a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right) \, dx","Integrate[(a + b*x^2 + c*x^4)^2*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]","a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{7} x^7 \left(3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right)+\frac{1}{3} a^2 x^3 (a f+3 b d)+\frac{3}{4} a^2 b e x^4+\frac{3}{11} c x^{11} \left(a c f+b^2 f+b c d\right)+\frac{3}{5} a x^5 \left(a b f+a c d+b^2 d\right)+\frac{3}{10} c e x^{10} \left(a c+b^2\right)+\frac{1}{8} b e x^8 \left(6 a c+b^2\right)+\frac{1}{2} a e x^6 \left(a c+b^2\right)+\frac{1}{9} x^9 \left(6 a b c f+3 a c^2 d+b^3 f+3 b^2 c d\right)+\frac{1}{13} c^2 x^{13} (3 b f+c d)+\frac{1}{4} b c^2 e x^{12}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15}","a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{7} x^7 \left(3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right)+\frac{1}{3} a^2 x^3 (a f+3 b d)+\frac{3}{4} a^2 b e x^4+\frac{3}{11} c x^{11} \left(a c f+b^2 f+b c d\right)+\frac{3}{5} a x^5 \left(a b f+a c d+b^2 d\right)+\frac{3}{10} c e x^{10} \left(a c+b^2\right)+\frac{1}{8} b e x^8 \left(6 a c+b^2\right)+\frac{1}{2} a e x^6 \left(a c+b^2\right)+\frac{1}{9} x^9 \left(6 a b c f+3 a c^2 d+b^3 f+3 b^2 c d\right)+\frac{1}{13} c^2 x^{13} (3 b f+c d)+\frac{1}{4} b c^2 e x^{12}+\frac{1}{14} c^3 e x^{14}+\frac{1}{15} c^3 f x^{15}",1,"a^3*d*x + (a^3*e*x^2)/2 + (a^2*(3*b*d + a*f)*x^3)/3 + (3*a^2*b*e*x^4)/4 + (3*a*(b^2*d + a*c*d + a*b*f)*x^5)/5 + (a*(b^2 + a*c)*e*x^6)/2 + ((b^3*d + 6*a*b*c*d + 3*a*b^2*f + 3*a^2*c*f)*x^7)/7 + (b*(b^2 + 6*a*c)*e*x^8)/8 + ((3*b^2*c*d + 3*a*c^2*d + b^3*f + 6*a*b*c*f)*x^9)/9 + (3*c*(b^2 + a*c)*e*x^10)/10 + (3*c*(b*c*d + b^2*f + a*c*f)*x^11)/11 + (b*c^2*e*x^12)/4 + (c^2*(c*d + 3*b*f)*x^13)/13 + (c^3*e*x^14)/14 + (c^3*f*x^15)/15","A",1
62,1,154,154,0.032399,"\int \left(a+b x^2+c x^4\right) \left(a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right) \, dx","Integrate[(a + b*x^2 + c*x^4)*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{6} e x^6 \left(2 a c+b^2\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{2} a b e x^4+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{4} b c e x^8+\frac{1}{10} c^2 e x^{10}+\frac{1}{11} c^2 f x^{11}","a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left(2 a c f+b^2 f+2 b c d\right)+\frac{1}{5} x^5 \left(2 a b f+2 a c d+b^2 d\right)+\frac{1}{6} e x^6 \left(2 a c+b^2\right)+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{2} a b e x^4+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{4} b c e x^8+\frac{1}{10} c^2 e x^{10}+\frac{1}{11} c^2 f x^{11}",1,"a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*b*e*x^4)/2 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2 + 2*a*c)*e*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + (b*c*e*x^8)/4 + (c*(c*d + 2*b*f)*x^9)/9 + (c^2*e*x^10)/10 + (c^2*f*x^11)/11","A",1
63,1,20,20,0.001552,"\int \frac{a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{a+b x^2+c x^4} \, dx","Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4),x]","d x+\frac{e x^2}{2}+\frac{f x^3}{3}","d x+\frac{e x^2}{2}+\frac{f x^3}{3}",1,"d*x + (e*x^2)/2 + (f*x^3)/3","A",1
64,1,234,211,0.0587584,"\int \frac{a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left(a+b x^2+c x^4\right)^2} \, dx","Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^2,x]","\frac{\frac{\sqrt{2} \left(f \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(f \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}+e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)-e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{2 \sqrt{b^2-4 a c}}","\frac{\left(\frac{2 c d-b f}{\sqrt{b^2-4 a c}}+f\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(f-\frac{2 c d-b f}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"((Sqrt[2]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(2*Sqrt[b^2 - 4*a*c])","A",1
65,1,398,368,1.1967925,"\int \frac{a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left(a+b x^2+c x^4\right)^3} \, dx","Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^3,x]","\frac{1}{4} \left(\frac{2 a b (e+f x)+4 a c x (d+x (e+f x))-2 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \left(d \sqrt{b^2-4 a c}+4 a f\right)-2 a \left(f \sqrt{b^2-4 a c}+6 c d\right)+b^2 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-4 a b f+12 a c d+b^2 (-d)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 c e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{4 c e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{3/2}}\right)","\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{2 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \left(b^2-4 a c\right) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 c e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)}",1,"((2*a*b*(e + f*x) - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 - 4*a*c]*d + 4*a*f) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*f - 2*a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (4*c*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + (4*c*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4","A",1
66,1,625,621,3.5840584,"\int \frac{a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{\left(a+b x^2+c x^4\right)^4} \, dx","Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4,x]","\frac{1}{16} \left(\frac{8 a^2 c \left(b (3 e+2 f x)+c x \left(7 d+6 e x+5 f x^2\right)\right)+2 a b x \left(b^2 f-25 b c d+b c f x^2-24 c^2 d x^2\right)+6 b^3 d x \left(b+c x^2\right)}{a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}-30 c d\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}-42 c d\right)+a b^2 \left(f \sqrt{b^2-4 a c}+30 c d\right)+4 a b c \left(13 a f-6 d \sqrt{b^2-4 a c}\right)+b^3 \left(3 d \sqrt{b^2-4 a c}-a f\right)-3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}-b-2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{48 c^2 e \log \left(\sqrt{b^2-4 a c}+b+2 c x^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{4 a b (e+f x)+8 a c x (d+x (e+f x))-4 b d x \left(b+c x^2\right)}{a \left(4 a c-b^2\right) \left(a+b x^2+c x^4\right)^2}\right)","\frac{x \left(c x^2 \left(20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right)+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}+\frac{\sqrt{c} \left(-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left(4 a^2 c \left(5 f \sqrt{b^2-4 a c}+42 c d\right)-a b^2 \left(30 c d-f \sqrt{b^2-4 a c}\right)-4 a b c \left(6 d \sqrt{b^2-4 a c}+13 a f\right)+b^3 \left(3 d \sqrt{b^2-4 a c}+a f\right)+3 b^4 d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left(\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{4 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}+\frac{3 c e \left(b+2 c x^2\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x^2+c x^4\right)}-\frac{e \left(b+2 c x^2\right)}{4 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^2}",1,"((4*a*b*(e + f*x) - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) + 2*a*b*x*(-25*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + 8*a^2*c*(b*(3*e + 2*f*x) + c*x*(7*d + 6*e*x + 5*f*x^2)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(-30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d - a*f) + 4*a*b*c*(-6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16","A",1
67,1,4,4,0.0012448,"\int \frac{2-x-2 x^2+x^3}{4-5 x^2+x^4} \, dx","Integrate[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4),x]","\log (x+2)","\log (x+2)",1,"Log[2 + x]","A",1
68,1,16,14,0.0041929,"\int \frac{(d+e x) \left(2-x-2 x^2+x^3\right)}{4-5 x^2+x^4} \, dx","Integrate[((d + e*x)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4),x]","(d-2 e) \log (x+2)+e (x+2)","(d-2 e) \log (x+2)+e x",1,"e*(2 + x) + (d - 2*e)*Log[2 + x]","A",1
69,1,30,31,0.0116146,"\int \frac{\left(d+e x+f x^2\right) \left(2-x-2 x^2+x^3\right)}{4-5 x^2+x^4} \, dx","Integrate[((d + e*x + f*x^2)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4),x]","\log (x+2) (d-2 e+4 f)+\frac{1}{2} (x+2) (2 e+f (x-6))","\log (x+2) (d-2 e+4 f)+x (e-4 f)+\frac{1}{2} f (x+2)^2",1,"((2*e + f*(-6 + x))*(2 + x))/2 + (d - 2*e + 4*f)*Log[2 + x]","A",1
70,1,45,51,0.0245231,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]","\log (x+2) (d-2 e+4 f-8 g)+\frac{1}{6} (x+2) \left(6 e+3 f (x-6)+2 g \left(x^2-5 x+22\right)\right)","\log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3",1,"((2 + x)*(6*e + 3*f*(-6 + x) + 2*g*(22 - 5*x + x^2)))/6 + (d - 2*e + 4*f - 8*g)*Log[2 + x]","A",1
71,1,68,68,0.0183831,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3+h x^4\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4),x]","\log (x+2) (d-2 e+4 f-8 g+16 h)+x (e-2 f+4 g-8 h)+\frac{1}{2} x^2 (f-2 g+4 h)+\frac{1}{3} x^3 (g-2 h)+\frac{h x^4}{4}","\log (x+2) (d-2 e+4 f-8 g+16 h)+x (e-2 f+4 g-8 h)+\frac{1}{2} x^2 (f-2 g+4 h)+\frac{1}{3} x^3 (g-2 h)+\frac{h x^4}{4}",1,"(e - 2*f + 4*g - 8*h)*x + ((f - 2*g + 4*h)*x^2)/2 + ((g - 2*h)*x^3)/3 + (h*x^4)/4 + (d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x]","A",1
72,1,92,92,0.032256,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]","\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (e-2 f+4 g-8 h+16 i)+\frac{1}{2} x^2 (f-2 g+4 h-8 i)+\frac{1}{3} x^3 (g-2 h+4 i)+\frac{1}{4} x^4 (h-2 i)+\frac{i x^5}{5}","\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (e-2 f+4 g-8 h+16 i)+\frac{1}{2} x^2 (f-2 g+4 h-8 i)+\frac{1}{3} x^3 (g-2 h+4 i)+\frac{1}{4} x^4 (h-2 i)+\frac{i x^5}{5}",1,"(e - 2*f + 4*g - 8*h + 16*i)*x + ((f - 2*g + 4*h - 8*i)*x^2)/2 + ((g - 2*h + 4*i)*x^3)/3 + ((h - 2*i)*x^4)/4 + (i*x^5)/5 + (d - 2*e + 4*f - 8*g + 16*h - 32*i)*Log[2 + x]","A",1
73,1,11,11,0.0026871,"\int \frac{2-3 x+x^2}{4-5 x^2+x^4} \, dx","Integrate[(2 - 3*x + x^2)/(4 - 5*x^2 + x^4),x]","\log (x+1)-\log (x+2)","\log (x+1)-\log (x+2)",1,"Log[1 + x] - Log[2 + x]","A",1
74,1,23,22,0.0068679,"\int \frac{(d+e x) \left(2-3 x+x^2\right)}{4-5 x^2+x^4} \, dx","Integrate[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4),x]","(d-e) \log (x+1)+(2 e-d) \log (x+2)","(d-e) \log (x+1)-(d-2 e) \log (x+2)",1,"(d - e)*Log[1 + x] + (-d + 2*e)*Log[2 + x]","A",1
75,1,30,29,0.0125297,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4),x]","\log (x+1) (d-e+f)+\log (x+2) (-d+2 e-4 f)+f x","\log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x",1,"f*x + (d - e + f)*Log[1 + x] + (-d + 2*e - 4*f)*Log[2 + x]","A",1
76,1,44,47,0.0194276,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]","\log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+f x+\frac{1}{2} g (x-6) x","\log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+x (f-3 g)+\frac{g x^2}{2}",1,"f*x + (g*(-6 + x)*x)/2 + (d - e + f - g)*Log[1 + x] - (d - 2*e + 4*f - 8*g)*Log[2 + x]","A",1
77,1,67,66,0.0227107,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3+h x^4\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4),x]","\log (x+1) (d-e+f-g+h)+\log (x+2) (-d+2 e-4 f+8 g-16 h)+x (f-3 g+7 h)+\frac{1}{2} x^2 (g-3 h)+\frac{h x^3}{3}","\log (x+1) (d-e+f-g+h)-\log (x+2) (d-2 e+4 f-8 g+16 h)+x (f-3 g+7 h)+\frac{1}{2} x^2 (g-3 h)+\frac{h x^3}{3}",1,"(f - 3*g + 7*h)*x + ((g - 3*h)*x^2)/2 + (h*x^3)/3 + (d - e + f - g + h)*Log[1 + x] + (-d + 2*e - 4*f + 8*g - 16*h)*Log[2 + x]","A",1
78,1,91,90,0.0364759,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]","\log (x+1) (d-e+f-g+h-i)+\log (x+2) (-d+2 e-4 f+8 g-16 h+32 i)+x (f-3 g+7 h-15 i)+\frac{1}{2} x^2 (g-3 h+7 i)+\frac{1}{3} x^3 (h-3 i)+\frac{i x^4}{4}","\log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (f-3 g+7 h-15 i)+\frac{1}{2} x^2 (g-3 h+7 i)+\frac{1}{3} x^3 (h-3 i)+\frac{i x^4}{4}",1,"(f - 3*g + 7*h - 15*i)*x + ((g - 3*h + 7*i)*x^2)/2 + ((h - 3*i)*x^3)/3 + (i*x^4)/4 + (d - e + f - g + h - i)*Log[1 + x] + (-d + 2*e - 4*f + 8*g - 16*h + 32*i)*Log[2 + x]","A",1
79,1,29,29,0.007256,"\int \frac{2+x}{4-5 x^2+x^4} \, dx","Integrate[(2 + x)/(4 - 5*x^2 + x^4),x]","-\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1)","-\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1)",1,"-1/2*Log[1 - x] + Log[2 - x]/3 + Log[1 + x]/6","A",1
80,1,39,42,0.0185744,"\int \frac{(2+x) (d+e x)}{4-5 x^2+x^4} \, dx","Integrate[((2 + x)*(d + e*x))/(4 - 5*x^2 + x^4),x]","\frac{1}{6} (-3 (d+e) \log (1-x)+2 (d+2 e) \log (2-x)+(d-e) \log (x+1))","-\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1)",1,"(-3*(d + e)*Log[1 - x] + 2*(d + 2*e)*Log[2 - x] + (d - e)*Log[1 + x])/6","A",1
81,1,44,47,0.0210276,"\int \frac{(2+x) \left(d+e x+f x^2\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4),x]","\frac{1}{6} (-3 \log (1-x) (d+e+f)+2 \log (2-x) (d+2 e+4 f)+\log (x+1) (d-e+f))","-\frac{1}{2} \log (1-x) (d+e+f)+\frac{1}{3} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f)",1,"(-3*(d + e + f)*Log[1 - x] + 2*(d + 2*e + 4*f)*Log[2 - x] + (d - e + f)*Log[1 + x])/6","A",1
82,1,55,57,0.0247741,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]","\frac{1}{6} (-3 \log (1-x) (d+e+f+g)+2 \log (2-x) (d+2 e+4 f+8 g)+\log (x+1) (d-e+f-g)+6 g x)","-\frac{1}{2} \log (1-x) (d+e+f+g)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)+g x",1,"(6*g*x - 3*(d + e + f + g)*Log[1 - x] + 2*(d + 2*e + 4*f + 8*g)*Log[2 - x] + (d - e + f - g)*Log[1 + x])/6","A",1
83,1,71,74,0.0334288,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3+h x^4\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4),x]","\frac{1}{6} \left(-3 \log (1-x) (d+e+f+g+h)+2 \log (2-x) (d+2 (e+2 f+4 g+8 h))+\log (x+1) (d-e+f-g+h)+6 x (g+2 h)+3 h x^2\right)","-\frac{1}{2} \log (1-x) (d+e+f+g+h)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac{1}{6} \log (x+1) (d-e+f-g+h)+x (g+2 h)+\frac{h x^2}{2}",1,"(6*(g + 2*h)*x + 3*h*x^2 - 3*(d + e + f + g + h)*Log[1 - x] + 2*(d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] + (d - e + f - g + h)*Log[1 + x])/6","A",1
84,1,91,96,0.0463136,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{4-5 x^2+x^4} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]","\frac{1}{6} \left(-3 \log (1-x) (d+e+f+g+h+i)+2 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+\log (x+1) (d-e+f-g+h-i)+6 x (g+2 h+5 i)+3 x^2 (h+2 i)+2 i x^3\right)","-\frac{1}{2} \log (1-x) (d+e+f+g+h+i)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac{1}{2} x^2 (h+2 i)+\frac{i x^3}{3}",1,"(6*(g + 2*h + 5*i)*x + 3*(h + 2*i)*x^2 + 2*i*x^3 - 3*(d + e + f + g + h + i)*Log[1 - x] + 2*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + (d - e + f - g + h - i)*Log[1 + x])/6","A",1
85,1,42,46,0.0221255,"\int \frac{2-x-2 x^2+x^3}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12}{x+2}+24 \log (-x-1)-8 \log (1-x)+3 \log (2-x)-19 \log (x+2)\right)","\frac{1}{12 (x+2)}-\frac{1}{18} \log (1-x)+\frac{1}{48} \log (2-x)+\frac{1}{6} \log (x+1)-\frac{19}{144} \log (x+2)",1,"(12/(2 + x) + 24*Log[-1 - x] - 8*Log[1 - x] + 3*Log[2 - x] - 19*Log[2 + x])/144","A",1
86,1,66,71,0.046523,"\int \frac{(d+e x) \left(2-x-2 x^2+x^3\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((d + e*x)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (d-2 e)}{x+2}+24 (d-e) \log (-x-1)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(26 e-19 d) \log (x+2)\right)","\frac{d-2 e}{12 (x+2)}-\frac{1}{18} (d+e) \log (1-x)+\frac{1}{48} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1)-\frac{1}{144} (19 d-26 e) \log (x+2)",1,"((12*(d - 2*e))/(2 + x) + 24*(d - e)*Log[-1 - x] - 8*(d + e)*Log[1 - x] + 3*(d + 2*e)*Log[2 - x] + (-19*d + 26*e)*Log[2 + x])/144","A",1
87,1,77,82,0.0606852,"\int \frac{\left(d+e x+f x^2\right) \left(2-x-2 x^2+x^3\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((d + e*x + f*x^2)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (d-2 e+4 f)}{x+2}+24 \log (-x-1) (d-e+f)-8 \log (1-x) (d+e+f)+3 \log (2-x) (d+2 e+4 f)+\log (x+2) (-19 d+26 e-28 f)\right)","\frac{d-2 e+4 f}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f)+\frac{1}{48} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f)",1,"((12*(d - 2*e + 4*f))/(2 + x) + 24*(d - e + f)*Log[-1 - x] - 8*(d + e + f)*Log[1 - x] + 3*(d + 2*e + 4*f)*Log[2 - x] + (-19*d + 26*e - 28*f)*Log[2 + x])/144","A",1
88,1,90,95,0.0494909,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (d-2 e+4 f-8 g)}{x+2}+24 \log (-x-1) (d-e+f-g)-8 \log (1-x) (d+e+f+g)+3 \log (2-x) (d+2 e+4 f+8 g)+\log (x+2) (-19 d+26 e-28 f+8 g)\right)","\frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g)",1,"((12*(d - 2*e + 4*f - 8*g))/(2 + x) + 24*(d - e + f - g)*Log[-1 - x] - 8*(d + e + f + g)*Log[1 - x] + 3*(d + 2*e + 4*f + 8*g)*Log[2 - x] + (-19*d + 26*e - 28*f + 8*g)*Log[2 + x])/144","A",1
89,1,102,106,0.0571352,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3+h x^4\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (d-2 e+4 f-8 g+16 h)}{x+2}+24 \log (-x-1) (d-e+f-g+h)-8 \log (1-x) (d+e+f+g+h)+3 \log (2-x) (d+2 (e+2 f+4 g+8 h))+\log (x+2) (-19 d+26 e-28 f+8 g+80 h)\right)","\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac{1}{6} \log (x+1) (d-e+f-g+h)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h)",1,"((12*(d - 2*e + 4*f - 8*g + 16*h))/(2 + x) + 24*(d - e + f - g + h)*Log[-1 - x] - 8*(d + e + f + g + h)*Log[1 - x] + 3*(d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] + (-19*d + 26*e - 28*f + 8*g + 80*h)*Log[2 + x])/144","A",1
90,1,118,122,0.0631201,"\int \frac{\left(2-x-2 x^2+x^3\right) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (d-2 (e-2 f+4 g-8 h+16 i))}{x+2}-8 \log (1-x) (d+e+f+g+h+i)+3 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+24 \log (x+1) (d-e+f-g+h-i)+\log (x+2) (-19 d+26 e-28 f+8 g+80 h-352 i)+144 i x\right)","\frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h+i)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x",1,"(144*i*x + (12*(d - 2*(e - 2*f + 4*g - 8*h + 16*i)))/(2 + x) - 8*(d + e + f + g + h + i)*Log[1 - x] + 3*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + 24*(d - e + f - g + h - i)*Log[1 + x] + (-19*d + 26*e - 28*f + 8*g + 80*h - 352*i)*Log[2 + x])/144","A",1
91,1,48,56,0.0239591,"\int \frac{2-3 x+x^2}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(2 - 3*x + x^2)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(-\frac{12 (3 x+5)}{x^2+3 x+2}-4 \log (1-x)+\log (2-x)-28 \log (x+1)+31 \log (x+2)\right)","-\frac{3 x+5}{12 \left(x^2+3 x+2\right)}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (x+1)+\frac{31}{144} \log (x+2)",1,"((-12*(5 + 3*x))/(2 + 3*x + x^2) - 4*Log[1 - x] + Log[2 - x] - 28*Log[1 + x] + 31*Log[2 + x])/144","A",1
92,1,80,89,0.0513289,"\int \frac{(d+e x) \left(2-3 x+x^2\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (-3 d x-5 d+4 e x+6 e)}{x^2+3 x+2}-4 (d+e) \log (1-x)+(d+2 e) \log (2-x)+4 (13 e-7 d) \log (x+1)+(31 d-50 e) \log (x+2)\right)","-\frac{x (3 d-4 e)+5 d-6 e}{12 \left(x^2+3 x+2\right)}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (x+1)+\frac{1}{144} (31 d-50 e) \log (x+2)",1,"((12*(-5*d + 6*e - 3*d*x + 4*e*x))/(2 + 3*x + x^2) - 4*(d + e)*Log[1 - x] + (d + 2*e)*Log[2 - x] + 4*(-7*d + 13*e)*Log[1 + x] + (31*d - 50*e)*Log[2 + x])/144","A",1
93,1,97,105,0.0743612,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(-\frac{12 (d (3 x+5)-4 e x-6 e+6 f x+8 f)}{x^2+3 x+2}-4 \log (1-x) (d+e+f)+\log (2-x) (d+2 e+4 f)-4 \log (x+1) (7 d-13 e+19 f)+\log (x+2) (31 d-50 e+76 f)\right)","-\frac{x (3 d-4 e+6 f)+5 d-6 e+8 f}{12 \left(x^2+3 x+2\right)}-\frac{1}{36} \log (1-x) (d+e+f)+\frac{1}{144} \log (2-x) (d+2 e+4 f)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f)",1,"((-12*(-6*e + 8*f - 4*e*x + 6*f*x + d*(5 + 3*x)))/(2 + 3*x + x^2) - 4*(d + e + f)*Log[1 - x] + (d + 2*e + 4*f)*Log[2 - x] - 4*(7*d - 13*e + 19*f)*Log[1 + x] + (31*d - 50*e + 76*f)*Log[2 + x])/144","A",1
94,1,114,117,0.0553921,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (-3 d x-5 d+4 e x+6 e-6 f x-8 f+10 g x+12 g)}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g)+\log (2-x) (d+2 e+4 f+8 g)+4 \log (x+1) (-7 d+13 e-19 f+25 g)+\log (x+2) (31 d-50 e+76 f-104 g)\right)","-\frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{d-e+f-g}{6 (x+1)}-\frac{1}{36} \log (1-x) (d+e+f+g)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g)",1,"((12*(-5*d + 6*e - 8*f + 12*g - 3*d*x + 4*e*x - 6*f*x + 10*g*x))/(2 + 3*x + x^2) - 4*(d + e + f + g)*Log[1 - x] + (d + 2*e + 4*f + 8*g)*Log[2 - x] + 4*(-7*d + 13*e - 19*f + 25*g)*Log[1 + x] + (31*d - 50*e + 76*f - 104*g)*Log[2 + x])/144","A",1
95,1,136,131,0.0622128,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3+h x^4\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(-\frac{12 (d (3 x+5)+2 (-e (2 x+3)+3 f x+4 f-5 g x-6 g+9 h x+10 h))}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g+h)+\log (2-x) (d+2 (e+2 f+4 g+8 h))-4 \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\log (x+2) (31 d-50 e+76 f-104 g+112 h)\right)","-\frac{d-e+f-g+h}{6 (x+1)}-\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{36} \log (1-x) (d+e+f+g+h)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h)",1,"((-12*(d*(5 + 3*x) + 2*(4*f - 6*g + 10*h + 3*f*x - 5*g*x + 9*h*x - e*(3 + 2*x))))/(2 + 3*x + x^2) - 4*(d + e + f + g + h)*Log[1 - x] + (d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] - 4*(7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x] + (31*d - 50*e + 76*f - 104*g + 112*h)*Log[2 + x])/144","A",1
96,1,153,147,0.0829263,"\int \frac{\left(2-3 x+x^2\right) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{144} \left(\frac{12 (2 (e (2 x+3)-3 f x-4 f+5 g x+6 g-9 h x-10 h+17 i x+18 i)-d (3 x+5))}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g+h+i)+\log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+4 \log (x+1) (-7 d+13 e-19 f+25 g-31 h+37 i)+\log (x+2) (31 d-50 e+76 f-104 g+112 h-32 i)\right)","-\frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{d-e+f-g+h-i}{6 (x+1)}-\frac{1}{36} \log (1-x) (d+e+f+g+h+i)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h-37 i)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h-32 i)",1,"((12*(-(d*(5 + 3*x)) + 2*(-4*f + 6*g - 10*h + 18*i - 3*f*x + 5*g*x - 9*h*x + 17*i*x + e*(3 + 2*x))))/(2 + 3*x + x^2) - 4*(d + e + f + g + h + i)*Log[1 - x] + (d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + 4*(-7*d + 13*e - 19*f + 25*g - 31*h + 37*i)*Log[1 + x] + (31*d - 50*e + 76*f - 104*g + 112*h - 32*i)*Log[2 + x])/144","A",1
97,1,60,68,0.0308681,"\int \frac{2+x}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[(2 + x)/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{432} \left(\frac{12 \left(-5 x^2+6 x+5\right)}{x^3-2 x^2-x+2}+24 \log (1-x)-35 \log (2-x)+8 \log (x+1)+3 \log (x+2)\right)","\frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (x+1)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (x+1)+\frac{1}{144} \log (x+2)",1,"((12*(5 + 6*x - 5*x^2))/(2 - x - 2*x^2 + x^3) + 24*Log[1 - x] - 35*Log[2 - x] + 8*Log[1 + x] + 3*Log[2 + x])/432","A",1
98,1,97,105,0.0885798,"\int \frac{(2+x) (d+e x)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 + x)*(d + e*x))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{432} \left(\frac{12 \left(d \left(-5 x^2+6 x+5\right)+2 e \left(5-2 x^2\right)\right)}{x^3-2 x^2-x+2}+12 (2 d+5 e) \log (1-x)-(35 d+58 e) \log (2-x)+4 (2 d+e) \log (x+1)+3 (d-2 e) \log (x+2)\right)","-\frac{d-e}{36 (x+1)}+\frac{d+e}{12 (1-x)}+\frac{d+2 e}{36 (2-x)}+\frac{1}{36} (2 d+5 e) \log (1-x)-\frac{1}{432} (35 d+58 e) \log (2-x)+\frac{1}{108} (2 d+e) \log (x+1)+\frac{1}{144} (d-2 e) \log (x+2)",1,"((12*(d*(5 + 6*x - 5*x^2) + 2*e*(5 - 2*x^2)))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e)*Log[1 - x] - (35*d + 58*e)*Log[2 - x] + 4*(2*d + e)*Log[1 + x] + 3*(d - 2*e)*Log[2 + x])/432","A",1
99,1,121,122,0.0513603,"\int \frac{(2+x) \left(d+e x+f x^2\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{432} \left(\frac{12 \left(d \left(-5 x^2+6 x+5\right)+e \left(10-4 x^2\right)+2 f \left(-4 x^2+3 x+4\right)\right)}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f)-\log (2-x) (35 d+58 e+92 f)+4 \log (x+1) (2 d+e-4 f)+3 \log (x+2) (d-2 e+4 f)\right)","-\frac{d-e+f}{36 (x+1)}+\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f)+\frac{1}{108} \log (x+1) (2 d+e-4 f)+\frac{1}{144} \log (x+2) (d-2 e+4 f)",1,"((12*(d*(5 + 6*x - 5*x^2) + e*(10 - 4*x^2) + 2*f*(4 + 3*x - 4*x^2)))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e + 8*f)*Log[1 - x] - (35*d + 58*e + 92*f)*Log[2 - x] + 4*(2*d + e - 4*f)*Log[1 + x] + 3*(d - 2*e + 4*f)*Log[2 + x])/432","A",1
100,1,144,141,0.0741526,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{432} \left(\frac{12 \left(d \left(-5 x^2+6 x+5\right)+2 \left(e \left(5-2 x^2\right)+f \left(-4 x^2+3 x+4\right)+g \left(8-5 x^2\right)\right)\right)}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f+11 g)-\log (2-x) (35 d+58 e+92 f+136 g)+4 \log (x+1) (2 d+e-4 f+7 g)+3 \log (x+2) (d-2 e+4 f-8 g)\right)","-\frac{d-e+f-g}{36 (x+1)}+\frac{d+e+f+g}{12 (1-x)}+\frac{d+2 e+4 f+8 g}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g)",1,"((12*(d*(5 + 6*x - 5*x^2) + 2*(g*(8 - 5*x^2) + f*(4 + 3*x - 4*x^2) + e*(5 - 2*x^2))))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e + 8*f + 11*g)*Log[1 - x] - (35*d + 58*e + 92*f + 136*g)*Log[2 - x] + 4*(2*d + e - 4*f + 7*g)*Log[1 + x] + 3*(d - 2*e + 4*f - 8*g)*Log[2 + x])/432","A",1
101,1,169,158,0.0895038,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3+h x^4\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]","\frac{1}{432} \left(\frac{12 \left(d \left(-5 x^2+6 x+5\right)+2 \left(e \left(5-2 x^2\right)+f \left(-4 x^2+3 x+4\right)-5 g x^2+8 g-10 h x^2+3 h x+10 h\right)\right)}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f+11 g+14 h)-\log (2-x) (35 d+58 e+92 f+136 g+176 h)+4 \log (x+1) (2 d+e-4 f+7 g-10 h)+3 \log (x+2) (d-2 e+4 f-8 g+16 h)\right)","-\frac{d-e+f-g+h}{36 (x+1)}+\frac{d+e+f+g+h}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h)",1,"((12*(d*(5 + 6*x - 5*x^2) + 2*(8*g + 10*h + 3*h*x - 5*g*x^2 - 10*h*x^2 + f*(4 + 3*x - 4*x^2) + e*(5 - 2*x^2))))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e + 8*f + 11*g + 14*h)*Log[1 - x] - (35*d + 58*e + 92*f + 136*g + 176*h)*Log[2 - x] + 4*(2*d + e - 4*f + 7*g - 10*h)*Log[1 + x] + 3*(d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x])/432","A",1
102,1,195,177,0.1078645,"\int \frac{(2+x) \left(d+e x+f x^2+g x^3+h x^4+i x^5\right)}{\left(4-5 x^2+x^4\right)^2} \, dx","Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4)^2,x]","\frac{-5 d x^2+6 d x+5 d-4 e x^2+10 e-8 f x^2+6 f x+8 f-10 g x^2+16 g-20 h x^2+6 h x+20 h-34 i x^2+40 i}{36 \left(x^3-2 x^2-x+2\right)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h+17 i)+\frac{1}{432} \log (2-x) (-35 d-58 e-92 f-136 g-176 h-160 i)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h+13 i)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h-32 i)","-\frac{d-e+f-g+h-i}{36 (x+1)}+\frac{d+e+f+g+h+i}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h+32 i}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h+17 i)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h+160 i)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h+13 i)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h-32 i)",1,"(5*d + 10*e + 8*f + 16*g + 20*h + 40*i + 6*d*x + 6*f*x + 6*h*x - 5*d*x^2 - 4*e*x^2 - 8*f*x^2 - 10*g*x^2 - 20*h*x^2 - 34*i*x^2)/(36*(2 - x - 2*x^2 + x^3)) + ((2*d + 5*e + 8*f + 11*g + 14*h + 17*i)*Log[1 - x])/36 + ((-35*d - 58*e - 92*f - 136*g - 176*h - 160*i)*Log[2 - x])/432 + ((2*d + e - 4*f + 7*g - 10*h + 13*i)*Log[1 + x])/108 + ((d - 2*e + 4*f - 8*g + 16*h - 32*i)*Log[2 + x])/144","A",1
103,-1,0,717,0,"\int \left(d+e x+f x^2+g x^3\right) \left(a+b x^2+c x^4\right)^{3/2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)*(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","-\frac{x \sqrt{a+b x^2+c x^4} \left(-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right)}{315 c^{5/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(-84 a^2 c^2 f+57 a b^2 c f+\sqrt{a} \sqrt{c} \left(24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right)-144 a b c^2 d-8 b^4 f+18 b^3 c d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{3 \left(b^2-4 a c\right)^2 (2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{512 c^{7/2}}-\frac{3 \left(b^2-4 a c\right) \left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{x \sqrt{a+b x^2+c x^4} \left(3 c x^2 \left(14 a c f-4 b^2 f+9 b c d\right)+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right)}{315 c^2}+\frac{\left(b+2 c x^2\right) \left(a+b x^2+c x^4\right)^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left(a+b x^2+c x^4\right)^{3/2} \left(3 (b f+3 c d)+7 c f x^2\right)}{63 c}+\frac{g \left(a+b x^2+c x^4\right)^{5/2}}{10 c}",1,"$Aborted","F",-1
104,-1,0,505,0,"\int \left(d+e x+f x^2+g x^3\right) \sqrt{a+b x^2+c x^4} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]","\text{\$Aborted}","\frac{x \sqrt{a+b x^2+c x^4} \left(6 a c f-2 b^2 f+5 b c d\right)}{15 c^{3/2} \left(\sqrt{a}+\sqrt{c} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(6 a c f-2 b^2 f+5 b c d\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{15 c^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\left(b^2-4 a c\right) (2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{32 c^{5/2}}+\frac{\sqrt[4]{a} \left(2 \sqrt{a} \sqrt{c}+b\right) \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(3 \sqrt{a} \sqrt{c} f-2 b f+5 c d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{30 c^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{\left(b+2 c x^2\right) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{16 c^2}+\frac{x \sqrt{a+b x^2+c x^4} \left(b f+5 c d+3 c f x^2\right)}{15 c}+\frac{g \left(a+b x^2+c x^4\right)^{3/2}}{6 c}",1,"$Aborted","F",-1
105,1,526,359,1.3818282,"\int \frac{d+e x+f x^2+g x^3}{\sqrt{a+b x^2+c x^4}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/Sqrt[a + b*x^2 + c*x^4],x]","\frac{-i \sqrt{2} \sqrt{c} \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \left(f \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+\sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left(\sqrt{a+b x^2+c x^4} (2 c e-b g) \log \left(2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right)+2 \sqrt{c} g \left(a+b x^2+c x^4\right)\right)+i \sqrt{2} \sqrt{c} f \left(\sqrt{b^2-4 a c}-b\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{4 c^{3/2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d}{\sqrt{a}}+f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{(2 c e-b g) \tanh ^{-1}\left(\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right)}{4 c^{3/2}}-\frac{\sqrt[4]{a} f \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{c^{3/4} \sqrt{a+b x^2+c x^4}}+\frac{f x \sqrt{a+b x^2+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{g \sqrt{a+b x^2+c x^4}}{2 c}",1,"(I*Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*f*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*Sqrt[2]*Sqrt[c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(2*Sqrt[c]*g*(a + b*x^2 + c*x^4) + (2*c*e - b*g)*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]]))/(4*c^(3/2)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])","C",1
106,-1,0,447,0,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} (b d-2 a f) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{a^{3/4} \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d-\sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{2 a^{3/4} \sqrt[4]{c} \left(b-2 \sqrt{a} \sqrt{c}\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} (b d-2 a f)}{a \left(b^2-4 a c\right) \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
107,-1,0,680,0,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^2+c x^4\right)^{5/2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]","\text{\$Aborted}","-\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(6 a^{3/2} \sqrt{c} f-3 \sqrt{a} b \sqrt{c} d+a b f-10 a c d+2 b^2 d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{6 a^{7/4} \left(b-2 \sqrt{a} \sqrt{c}\right) \left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt{c} x \sqrt{a+b x^2+c x^4} \left(12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right)}{3 a^2 \left(b^2-4 a c\right)^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{x \left(c x^2 \left(12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right)+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right)}{3 a^2 \left(b^2-4 a c\right)^2 \sqrt{a+b x^2+c x^4}}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+b x^2+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{4} \left(2-\frac{b}{\sqrt{a} \sqrt{c}}\right)\right)}{3 a^{7/4} \left(b^2-4 a c\right)^2 \sqrt{a+b x^2+c x^4}}+\frac{x \left(c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right)}{3 a \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^{3/2}}+\frac{4 \left(b+2 c x^2\right) (2 c e-b g)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x^2+c x^4}}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{3 \left(b^2-4 a c\right) \left(a+b x^2+c x^4\right)^{3/2}}",1,"$Aborted","F",-1
108,-1,0,19,0,"\int \frac{a g-c g x^4}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(a*g - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{g x}{\sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
109,-1,0,57,0,"\int \frac{a g+e x-c g x^4}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(a*g + e*x - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{e \left(b+2 c x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
110,-1,0,57,0,"\int \frac{a g+f x^3-c g x^4}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(a*g + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{f \left(2 a+b x^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}+\frac{g x}{\sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1
111,-1,0,69,0,"\int \frac{a g+e x+f x^3-c g x^4}{\left(a+b x^2+c x^4\right)^{3/2}} \, dx","Integrate[(a*g + e*x + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]","\text{\$Aborted}","\frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{-2 a f+x^2 (2 c e-b f)+b e}{\left(b^2-4 a c\right) \sqrt{a+b x^2+c x^4}}",1,"$Aborted","F",-1