1,1,1668,0,4.007577,"\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^3+c x^6} \, dx","Int[(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^3 + c*x^6),x]","\frac{m x^3}{3 c}+\frac{l x^2}{2 c}+\frac{k x}{c}-\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}-\frac{\left(m b^2-c j b+2 c^2 f-2 a c m\right) \tanh ^{-1}\left(\frac{2 c x^3+b}{\sqrt{b^2-4 a c}}\right)}{3 c^2 \sqrt{b^2-4 a c}}+\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b-\sqrt{b^2-4 a c}}\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b-\sqrt{b^2-4 a c}}\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b+\sqrt{b^2-4 a c}}\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b-\sqrt{b^2-4 a c}} x+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b-\sqrt{b^2-4 a c}} x+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b+\sqrt{b^2-4 a c}} x+\left(b+\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b+\sqrt{b^2-4 a c}} x+\left(b+\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}+\frac{(c j-b m) \log \left(c x^6+b x^3+a\right)}{6 c^2}","\frac{m x^3}{3 c}+\frac{l x^2}{2 c}+\frac{k x}{c}-\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}-\frac{\left(m b^2-c j b+2 c^2 f-2 a c m\right) \tanh ^{-1}\left(\frac{2 c x^3+b}{\sqrt{b^2-4 a c}}\right)}{3 c^2 \sqrt{b^2-4 a c}}+\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b-\sqrt{b^2-4 a c}}\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b-\sqrt{b^2-4 a c}}\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{2} \sqrt[3]{c} x+\sqrt[3]{b+\sqrt{b^2-4 a c}}\right)}{3\ 2^{2/3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}+\frac{k b^2+2 c^2 d-c (b g+2 a k)}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b-\sqrt{b^2-4 a c}} x+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(h-\frac{b l}{c}+\frac{l b^2+2 c^2 e-c (b h+2 a l)}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b-\sqrt{b^2-4 a c}} x+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(g-\frac{b k}{c}-\frac{k b^2-c g b+2 c^2 d-2 a c k}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b+\sqrt{b^2-4 a c}} x+\left(b+\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \left(b+\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(h-\frac{b l}{c}-\frac{l b^2-c h b+2 c^2 e-2 a c l}{c \sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} c^{2/3} x^2-\sqrt[3]{2} \sqrt[3]{c} \sqrt[3]{b+\sqrt{b^2-4 a c}} x+\left(b+\sqrt{b^2-4 a c}\right)^{2/3}\right)}{6\ 2^{2/3} c^{2/3} \sqrt[3]{b+\sqrt{b^2-4 a c}}}+\frac{(c j-b m) \log \left(c x^6+b x^3+a\right)}{6 c^2}",1,"(k*x)/c + (l*x^2)/(2*c) + (m*x^3)/(3*c) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a*k))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((2*c^2*f - b*c*j + b^2*m - 2*a*c*m)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^2*Sqrt[b^2 - 4*a*c]) + ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a*k))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + ((g - (b*k)/c - (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a*k))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) + ((c*j - b*m)*Log[a + b*x^3 + c*x^6])/(6*c^2)","A",37,16,55,0.2909,1,"{1790, 1789, 1422, 200, 31, 634, 617, 204, 628, 1758, 1510, 292, 1745, 1657, 618, 206}"
2,1,124,0,0.0849048,"\int \frac{1}{a+b x^n+c x^{2 n}} \, dx","Int[(a + b*x^n + c*x^(2*n))^(-1),x]","-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}","-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"(-2*c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (2*c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])","A",3,2,16,0.1250,1,"{1347, 245}"
3,1,263,0,0.255132,"\int \frac{d+e x}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x)/(a + b*x^n + c*x^(2*n)),x]","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"(-2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])","A",9,4,22,0.1818,1,"{1793, 1893, 245, 364}"
4,1,404,0,0.2773739,"\int \frac{d+e x+f x^2}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x + f*x^2)/(a + b*x^n + c*x^(2*n)),x]","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"(-2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (2*c*f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (2*c*f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]))","A",11,4,27,0.1481,1,"{1793, 1893, 245, 364}"
5,1,545,0,0.3511511,"\int \frac{d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)),x]","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c g x^4 \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c g x^4 \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}","-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c e x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c f x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c g x^4 \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c g x^4 \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"(-2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (2*c*d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (c*e*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (2*c*f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (2*c*f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])) - (c*g*x^4*Hypergeometric2F1[1, 4/n, (4 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (c*g*x^4*Hypergeometric2F1[1, 4/n, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]))","A",13,4,32,0.1250,1,"{1793, 1893, 245, 364}"
6,1,283,0,0.3853041,"\int \frac{1}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(a + b*x^n + c*x^(2*n))^(-2),x]","-\frac{c x \left(-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}","-\frac{c x \left(-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (c*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n)","A",4,3,16,0.1875,1,"{1345, 1422, 245}"
7,1,738,0,1.3370404,"\int \frac{d+e x}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x)/(a + b*x^n + c*x^(2*n))^2,x]","-\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n (n+2) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}+\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n (n+2) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}-\frac{c d x \left(-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c d x \left(b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{d x \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{c e x^2 \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c e x^2 \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{e x^2 \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}","-\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n (n+2) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}+\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n (n+2) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}-\frac{c d x \left(-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c d x \left(b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{d x \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{c e x^2 \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c e x^2 \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{e x^2 \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(d*x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e*x^2*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(2 + n)) + (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(2 + n))","A",15,8,22,0.3636,1,"{1796, 1345, 1422, 245, 1384, 1560, 1383, 364}"
8,1,1194,0,2.0531328,"\int \frac{d+e x+f x^2}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x + f*x^2)/(a + b*x^n + c*x^(2*n))^2,x]","-\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+2)}+\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+2)}-\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+3)}+\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+3)}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{f \left(b c x^n+b^2-2 a c\right) x^3}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{e \left(b c x^n+b^2-2 a c\right) x^2}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c d \left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c d \left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{d \left(b c x^n+b^2-2 a c\right) x}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}","-\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+2)}+\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+2)}-\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+3)}+\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+3)}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{f \left(b c x^n+b^2-2 a c\right) x^3}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{e \left(b c x^n+b^2-2 a c\right) x^2}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c d \left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c d \left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{d \left(b c x^n+b^2-2 a c\right) x}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}",1,"(d*x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e*x^2*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (f*x^3*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(2 + n)) + (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(2 + n)) - (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(3 + n)) + (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(3 + n))","A",24,8,27,0.2963,1,"{1796, 1345, 1422, 245, 1384, 1560, 1383, 364}"
9,1,1654,0,2.9093013,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n))^2,x]","-\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+2)}+\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+2)}-\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+3)}+\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+3)}-\frac{2 b c^2 g (4-n) \, _2F_1\left(1,\frac{n+4}{n};2 \left(1+\frac{2}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+4}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+4)}+\frac{2 b c^2 g (4-n) \, _2F_1\left(1,\frac{n+4}{n};2 \left(1+\frac{2}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+4}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+4)}-\frac{c g \left(4 a c (2-n)-b^2 (4-n)\right) \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^4}{2 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c g \left(4 a c (2-n)-b^2 (4-n)\right) \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^4}{2 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{g \left(b c x^n+b^2-2 a c\right) x^4}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{f \left(b c x^n+b^2-2 a c\right) x^3}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{e \left(b c x^n+b^2-2 a c\right) x^2}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c d \left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c d \left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{d \left(b c x^n+b^2-2 a c\right) x}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}","-\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+2)}+\frac{2 b c^2 e (2-n) \, _2F_1\left(1,\frac{n+2}{n};2 \left(1+\frac{1}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+2}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+2)}-\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+3)}+\frac{2 b c^2 f (3-n) \, _2F_1\left(1,\frac{n+3}{n};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+3}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+3)}-\frac{2 b c^2 g (4-n) \, _2F_1\left(1,\frac{n+4}{n};2 \left(1+\frac{2}{n}\right);-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^{n+4}}{a \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right) n (n+4)}+\frac{2 b c^2 g (4-n) \, _2F_1\left(1,\frac{n+4}{n};2 \left(1+\frac{2}{n}\right);-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^{n+4}}{a \left(b^2-4 a c\right)^{3/2} \left(b+\sqrt{b^2-4 a c}\right) n (n+4)}-\frac{c g \left(4 a c (2-n)-b^2 (4-n)\right) \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^4}{2 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c g \left(4 a c (2-n)-b^2 (4-n)\right) \, _2F_1\left(1,\frac{4}{n};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^4}{2 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{g \left(b c x^n+b^2-2 a c\right) x^4}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{2 c f \left(2 a c (3-2 n)-b^2 (3-n)\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^3}{3 a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{f \left(b c x^n+b^2-2 a c\right) x^3}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c e \left(4 a c (1-n)-b^2 (2-n)\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{e \left(b c x^n+b^2-2 a c\right) x^2}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{c d \left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{c d \left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) x}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{d \left(b c x^n+b^2-2 a c\right) x}{a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}",1,"(d*x*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e*x^2*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (f*x^3*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (g*x^4*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*d*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*e*(4*a*c*(1 - n) - b^2*(2 - n))*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (2*c*f*(2*a*c*(3 - 2*n) - b^2*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (c*g*(4*a*c*(2 - n) - b^2*(4 - n))*x^4*Hypergeometric2F1[1, 4/n, (4 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*g*(4*a*c*(2 - n) - b^2*(4 - n))*x^4*Hypergeometric2F1[1, 4/n, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(2 + n)) + (2*b*c^2*e*(2 - n)*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(2 + n)) - (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(3 + n)) + (2*b*c^2*f*(3 - n)*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(3 + n)) - (2*b*c^2*g*(4 - n)*x^(4 + n)*Hypergeometric2F1[1, (4 + n)/n, 2*(1 + 2/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*n*(4 + n)) + (2*b*c^2*g*(4 - n)*x^(4 + n)*Hypergeometric2F1[1, (4 + n)/n, 2*(1 + 2/n), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*n*(4 + n))","A",33,8,32,0.2500,1,"{1796, 1345, 1422, 245, 1384, 1560, 1383, 364}"
10,1,75,0,0.5305684,"\int \frac{-a h x^{-1+\frac{n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac{5 n}{2}}}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Int[(-(a*h*x^(-1 + n/2)) + c*f*x^(-1 + n) + c*g*x^(-1 + 2*n) + c*h*x^(-1 + (5*n)/2))/(a + b*x^n + c*x^(2*n))^(3/2),x]","-\frac{2 \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}","-\frac{2 \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}",1,"(-2*(c*(b*f - 2*a*g) + (b^2 - 4*a*c)*h*x^(n/2) + c*(2*c*f - b*g)*x^n))/((b^2 - 4*a*c)*n*Sqrt[a + b*x^n + c*x^(2*n)])","A",2,2,63,0.03175,1,"{6741, 1753}"
11,1,20,0,0.023506,"\int \left(a+b x^n+c x^{2 n}\right)^p \left(a+b (1+n+n p) x^n+c (1+2 n (1+p)) x^{2 n}\right) \, dx","Int[(a + b*x^n + c*x^(2*n))^p*(a + b*(1 + n + n*p)*x^n + c*(1 + 2*n*(1 + p))*x^(2*n)),x]","x \left(a+b x^n+c x^{2 n}\right)^{p+1}","x \left(a+b x^n+c x^{2 n}\right)^{p+1}",1,"x*(a + b*x^n + c*x^(2*n))^(1 + p)","A",1,1,45,0.02222,1,"{1775}"
12,1,45,0,0.07873,"\int \frac{x^{-1+\frac{n}{4}} \left(-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right)}{\left(a+c x^n\right)^{3/2}} \, dx","Int[(x^(-1 + n/4)*(-(a*h) + c*f*x^(n/4) + c*g*x^((3*n)/4) + c*h*x^n))/(a + c*x^n)^(3/2),x]","-\frac{2 \left(a g+2 a h x^{n/4}-c f x^{n/2}\right)}{a n \sqrt{a+c x^n}}","-\frac{2 \left(a g+2 a h x^{n/4}-c f x^{n/2}\right)}{a n \sqrt{a+c x^n}}",1,"(-2*(a*g + 2*a*h*x^(n/4) - c*f*x^(n/2)))/(a*n*Sqrt[a + c*x^n])","A",1,1,52,0.01923,1,"{1816}"
13,1,65,0,0.1555701,"\int \frac{(d x)^{-1+\frac{n}{4}} \left(-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right)}{\left(a+c x^n\right)^{3/2}} \, dx","Int[((d*x)^(-1 + n/4)*(-(a*h) + c*f*x^(n/4) + c*g*x^((3*n)/4) + c*h*x^n))/(a + c*x^n)^(3/2),x]","-\frac{2 x^{1-\frac{n}{4}} (d x)^{\frac{n-4}{4}} \left(a g+2 a h x^{n/4}-c f x^{n/2}\right)}{a n \sqrt{a+c x^n}}","-\frac{2 x^{1-\frac{n}{4}} (d x)^{\frac{n-4}{4}} \left(a g+2 a h x^{n/4}-c f x^{n/2}\right)}{a n \sqrt{a+c x^n}}",1,"(-2*x^(1 - n/4)*(d*x)^((-4 + n)/4)*(a*g + 2*a*h*x^(n/4) - c*f*x^(n/2)))/(a*n*Sqrt[a + c*x^n])","A",2,2,54,0.03704,1,"{1817, 1816}"
14,1,75,0,0.1081796,"\int \frac{x^{-1+\frac{n}{2}} \left(-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right)}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Int[(x^(-1 + n/2)*(-(a*h) + c*f*x^(n/2) + c*g*x^((3*n)/2) + c*h*x^(2*n)))/(a + b*x^n + c*x^(2*n))^(3/2),x]","-\frac{2 \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}","-\frac{2 \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}",1,"(-2*(c*(b*f - 2*a*g) + (b^2 - 4*a*c)*h*x^(n/2) + c*(2*c*f - b*g)*x^n))/((b^2 - 4*a*c)*n*Sqrt[a + b*x^n + c*x^(2*n)])","A",1,1,61,0.01639,1,"{1753}"
15,1,95,0,0.2172272,"\int \frac{(d x)^{-1+\frac{n}{2}} \left(-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right)}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Int[((d*x)^(-1 + n/2)*(-(a*h) + c*f*x^(n/2) + c*g*x^((3*n)/2) + c*h*x^(2*n)))/(a + b*x^n + c*x^(2*n))^(3/2),x]","-\frac{2 x^{1-\frac{n}{2}} (d x)^{\frac{n-2}{2}} \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}","-\frac{2 x^{1-\frac{n}{2}} (d x)^{\frac{n-2}{2}} \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}",1,"(-2*x^(1 - n/2)*(d*x)^((-2 + n)/2)*(c*(b*f - 2*a*g) + (b^2 - 4*a*c)*h*x^(n/2) + c*(2*c*f - b*g)*x^n))/((b^2 - 4*a*c)*n*Sqrt[a + b*x^n + c*x^(2*n)])","A",2,2,63,0.03175,1,"{1754, 1753}"
16,1,29,0,0.0712211,"\int (g x)^m \left(a+b x^n+c x^{2 n}\right)^p \left(a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right) \, dx","Int[(g*x)^m*(a + b*x^n + c*x^(2*n))^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n + c*(1 + m + 2*n*(1 + p))*x^(2*n)),x]","\frac{(g x)^{m+1} \left(a+b x^n+c x^{2 n}\right)^{p+1}}{g}","\frac{(g x)^{m+1} \left(a+b x^n+c x^{2 n}\right)^{p+1}}{g}",1,"((g*x)^(1 + m)*(a + b*x^n + c*x^(2*n))^(1 + p))/g","A",1,1,56,0.01786,1,"{1747}"
17,1,494,0,1.5840865,"\int \frac{A+B x^n+C x^{2 n}+D x^{3 n}}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(A + B*x^n + C*x^(2*n) + D*x^(3*n))/(a + b*x^n + c*x^(2*n))^2,x]","\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) \left(\frac{A c^2 \left(4 a c (1-2 n)-b^2 (1-n)\right)-a \left(-2 b c (a D (n+2)+B c n)+4 a c^2 C-b^2 c C (1-n)+b^3 D\right)}{\sqrt{b^2-4 a c}}-b c (1-n) (a C+A c)+a b^2 D+2 a c (B c (1-n)-a D (n+1))\right)}{a c n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) \left(-\frac{A c^2 \left(4 a c (1-2 n)-b^2 (1-n)\right)-a \left(-2 b c (a D (n+2)+B c n)+4 a c^2 C-b^2 c C (1-n)+b^3 D\right)}{\sqrt{b^2-4 a c}}-b c (1-n) (a C+A c)+a b^2 D+2 a c (B c (1-n)-a D (n+1))\right)}{a c n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(b c (a C+A c)-a b^2 D-2 a c (B c-a D)\right)+A c \left(b^2-2 a c\right)-a (a b D-2 a c C+b B c)\right)}{a c n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}","\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) \left(\frac{A c^2 \left(4 a c (1-2 n)-b^2 (1-n)\right)-a \left(-2 b c (a D (n+2)+B c n)+4 a c^2 C-b^2 c C (1-n)+b^3 D\right)}{\sqrt{b^2-4 a c}}-b c (1-n) (a C+A c)+a b^2 D+2 a c (B c (1-n)-a D (n+1))\right)}{a c n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) \left(-\frac{A c^2 \left(4 a c (1-2 n)-b^2 (1-n)\right)-a \left(-2 b c (a D (n+2)+B c n)+4 a c^2 C-b^2 c C (1-n)+b^3 D\right)}{\sqrt{b^2-4 a c}}-b c (1-n) (a C+A c)+a b^2 D+2 a c (B c (1-n)-a D (n+1))\right)}{a c n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(b c (a C+A c)-a b^2 D-2 a c (B c-a D)\right)+A c \left(b^2-2 a c\right)-a (a b D-2 a c C+b B c)\right)}{a c n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(x*(A*c*(b^2 - 2*a*c) - a*(b*B*c - 2*a*c*C + a*b*D) + (b*c*(A*c + a*C) - a*b^2*D - 2*a*c*(B*c - a*D))*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + ((a*b^2*D - b*c*(A*c + a*C)*(1 - n) + 2*a*c*(B*c*(1 - n) - a*D*(1 + n)) + (A*c^2*(4*a*c*(1 - 2*n) - b^2*(1 - n)) - a*(4*a*c^2*C + b^3*D - b^2*c*C*(1 - n) - 2*b*c*(B*c*n + a*D*(2 + n))))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) + ((a*b^2*D - b*c*(A*c + a*C)*(1 - n) + 2*a*c*(B*c*(1 - n) - a*D*(1 + n)) - (A*c^2*(4*a*c*(1 - 2*n) - b^2*(1 - n)) - a*(4*a*c^2*C + b^3*D - b^2*c*C*(1 - n) - 2*b*c*(B*c*n + a*D*(2 + n))))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)","A",4,3,38,0.07895,1,"{1794, 1422, 245}"