1,1,223,1668,1.6810772,"\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","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^3 + c*x^6),x]","\frac{-2 \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^5 b m \log (x-\text{$\#$1})+\text{$\#$1}^5 (-c) j \log (x-\text{$\#$1})+\text{$\#$1}^4 b l \log (x-\text{$\#$1})-\text{$\#$1}^4 c h \log (x-\text{$\#$1})+\text{$\#$1}^3 b k \log (x-\text{$\#$1})-\text{$\#$1}^3 c g \log (x-\text{$\#$1})+\text{$\#$1}^2 a m \log (x-\text{$\#$1})-\text{$\#$1}^2 c f \log (x-\text{$\#$1})+a k \log (x-\text{$\#$1})+\text{$\#$1} a l \log (x-\text{$\#$1})-c d \log (x-\text{$\#$1})-\text{$\#$1} c e \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]+6 k x+3 l x^2+2 m x^3}{6 c}","\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,"(6*k*x + 3*l*x^2 + 2*m*x^3 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (-(c*d*Log[x - #1]) + a*k*Log[x - #1] - c*e*Log[x - #1]*#1 + a*l*Log[x - #1]*#1 - c*f*Log[x - #1]*#1^2 + a*m*Log[x - #1]*#1^2 - c*g*Log[x - #1]*#1^3 + b*k*Log[x - #1]*#1^3 - c*h*Log[x - #1]*#1^4 + b*l*Log[x - #1]*#1^4 - c*j*Log[x - #1]*#1^5 + b*m*Log[x - #1]*#1^5)/(b*#1^2 + 2*c*#1^5) & ])/(6*c)","C",1
2,1,261,124,0.2723817,"\int \frac{1}{a+b x^n+c x^{2 n}} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(-1),x]","-2 c x \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{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{1-2^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)","-\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*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])))","B",1
3,1,525,263,1.1912074,"\int \frac{d+e x}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x)/(a + b*x^n + c*x^(2*n)),x]","c x \left(-2 d \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{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{1-2^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)-e x \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{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{1-4^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)\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}",1,"c*x*(-(e*x*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])))) - 2*d*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))","A",1
4,1,834,404,1.1263545,"\int \frac{d+e x+f x^2}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(2 f \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)+\left(-b^2+\sqrt{b^2-4 a c} b+4 a c\right) \left(1-8^{-1/n} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right) x^2+3 e \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)+\left(-b^2+\sqrt{b^2-4 a c} b+4 a c\right) \left(1-4^{-1/n} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right) x+6 d \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)-2^{-1/n} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right) \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-1/n} \left(2^{\frac{1}{n}} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{\frac{1}{n}}-\, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right)\right)}{12 a \left(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,"(x*(2*f*x^2*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(3/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(3/n)))) + 3*e*x*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))) + 6*d*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1)) - (Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])*(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]))/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))))/(12*a*(-b^2 + 4*a*c))","B",1
5,1,1093,545,1.5751083,"\int \frac{d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(3 g \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-4/n} \, _2F_1\left(-\frac{4}{n},-\frac{4}{n};\frac{n-4}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)+\left(-b^2+\sqrt{b^2-4 a c} b+4 a c\right) \left(1-2^{-4/n} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-4/n} \, _2F_1\left(-\frac{4}{n},-\frac{4}{n};\frac{n-4}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right) x^3+4 f \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)+\left(-b^2+\sqrt{b^2-4 a c} b+4 a c\right) \left(1-8^{-1/n} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right) x^2+6 e \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)+\left(-b^2+\sqrt{b^2-4 a c} b+4 a c\right) \left(1-4^{-1/n} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right) x+12 d \left(\left(-b^2-\sqrt{b^2-4 a c} b+4 a c\right) \left(1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)-2^{-1/n} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right) \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{-1/n} \left(2^{\frac{1}{n}} \left(\frac{c x^n}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)^{\frac{1}{n}}-\, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)\right)\right)}{24 a \left(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,"(x*(3*g*x^3*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-4/n, -4/n, (-4 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(4/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-4/n, -4/n, (-4 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^(4/n)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(4/n)))) + 4*f*x^2*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(3/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(3/n)))) + 6*e*x*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))) + 12*d*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1)) - (Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])*(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]))/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))))/(24*a*(-b^2 + 4*a*c))","B",1
6,1,456,283,3.5316048,"\int \frac{1}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(-2),x]","-\frac{x \left(\frac{4 a^2 c n+a \left(-b^2 n+b c (4 n-3) x^n+2 c^2 (2 n-1) x^{2 n}\right)-b^2 (n-1) x^n \left(b+c x^n\right)}{a+x^n \left(b+c x^n\right)}+\frac{a c 2^{-1/n} \left(b (n-1) \sqrt{b^2-4 a c}+4 a c (2 n-1)-\left(b^2 (n-1)\right)\right) \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{a c 2^{-1/n} \left(b^2 (n-1) \sqrt{b^2-4 a c}+4 a c (1-2 n) \sqrt{b^2-4 a c}-4 a b c (n-1)+b^3 (n-1)\right) \left(\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(b \sqrt{b^2-4 a c}+4 a c-b^2\right)}\right)}{a^2 n \left(b^2-4 a c\right)}","-\frac{c x \left(-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)-\left(b^2 (1-n)\right)\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)-\left(b^2 (1-n)\right)\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*((4*a^2*c*n - b^2*(-1 + n)*x^n*(b + c*x^n) + a*(-(b^2*n) + b*c*(-3 + 4*n)*x^n + 2*c^2*(-1 + 2*n)*x^(2*n)))/(a + x^n*(b + c*x^n)) + (a*c*(4*a*c*Sqrt[b^2 - 4*a*c]*(1 - 2*n) + b^3*(-1 + n) - 4*a*b*c*(-1 + n) + b^2*Sqrt[b^2 - 4*a*c]*(-1 + n))*Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(2^n^(-1)*Sqrt[b^2 - 4*a*c]*(-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)) + (a*c*(-(b^2*(-1 + n)) + b*Sqrt[b^2 - 4*a*c]*(-1 + n) + 4*a*c*(-1 + 2*n))*Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(2^n^(-1)*Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1))))/(a^2*(b^2 - 4*a*c)*n))","A",1
7,1,4162,738,6.3929096,"\int \frac{d+e x}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x)/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","-\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)-\left(b^2 (1-n)\right)\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)-\left(b^2 (1-n)\right)\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,"(x*(d + e*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))) - (b*c*e*x^(2 + n)*(x^n)^(2/n - (2 + n)/n)*(-(Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))) + Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))))/(2*a*(-b^2 + 4*a*c)) + (b*c*e*x^(2 + n)*(x^n)^(2/n - (2 + n)/n)*(-(Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))) + Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))))/(a*(-b^2 + 4*a*c)*n) + (b^2*e*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(2*a*(-b^2 + 4*a*c)) - (2*c*e*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(-b^2 + 4*a*c) - (b^2*e*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n) + (2*c*e*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*n) - (b*c*d*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))))/(a*(-b^2 + 4*a*c)) + (b*c*d*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))))/(a*(-b^2 + 4*a*c)*n) + (b^2*d*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)) - (4*c*d*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(-b^2 + 4*a*c) - (b^2*d*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*n) + (2*c*d*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*n)","B",1
8,1,6525,1194,6.5131253,"\int \frac{d+e x+f x^2}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","-\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(-\left((1-n) b^2\right)-\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(-\left((1-n) b^2\right)+\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,"Result too large to show","B",0
9,1,8737,1654,6.6232487,"\int \frac{d+e x+f x^2+g x^3}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","-\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(-\left((1-n) b^2\right)-\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(-\left((1-n) b^2\right)+\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,"Result too large to show","B",0
10,-1,0,75,0,"\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","Integrate[(-(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]","\text{\$Aborted}","-\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,"$Aborted","F",-1
11,1,19,20,0.3332713,"\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","Integrate[(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+x^n \left(b+c x^n\right)\right)^{p+1}","x \left(a+b x^n+c x^{2 n}\right)^{p+1}",1,"x*(a + x^n*(b + c*x^n))^(1 + p)","A",1
12,1,45,45,0.2142935,"\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","Integrate[(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 c f x^{n/2}-2 a \left(g+2 h x^{n/4}\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*c*f*x^(n/2) - 2*a*(g + 2*h*x^(n/4)))/(a*n*Sqrt[a + c*x^n])","A",1
13,1,64,65,0.1441194,"\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","Integrate[((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^{-n/4} (d x)^{n/4} \left(c f x^{n/2}-a \left(g+2 h x^{n/4}\right)\right)}{a d 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*(d*x)^(n/4)*(c*f*x^(n/2) - a*(g + 2*h*x^(n/4))))/(a*d*n*x^(n/4)*Sqrt[a + c*x^n])","A",1
14,-1,0,75,0,"\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","Integrate[(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]","\text{\$Aborted}","-\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,"$Aborted","F",-1
15,-1,0,95,0,"\int \frac{(d x)^{-1+\frac{n}{2}} \left(-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right)}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[((d*x)^(-1 + n/2)*(-(a*h) + c*f*x^(n/2) + c*g*x^((3*n)/2) + c*h*x^(2*n)))/(a + b*x^n + c*x^(2*n))^(3/2),x]","\text{\$Aborted}","-\frac{2 x^{1-\frac{n}{2}} (d x)^{\frac{n-2}{2}} \left(h x^{n/2} \left(b^2-4 a c\right)+c (b f-2 a g)+c x^n (2 c f-b g)\right)}{n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}",1,"$Aborted","F",-1
16,1,24,29,0.4338566,"\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","Integrate[(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]","x (g x)^m \left(a+x^n \left(b+c x^n\right)\right)^{p+1}","\frac{(g x)^{m+1} \left(a+b x^n+c x^{2 n}\right)^{p+1}}{g}",1,"x*(g*x)^m*(a + x^n*(b + c*x^n))^(1 + p)","A",1
17,1,5439,494,6.8935199,"\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","Integrate[(A + B*x^n + C*x^(2*n) + D*x^(3*n))/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","\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^3 D-b^2 c C (1-n)\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^3 D-b^2 c C (1-n)\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)}",1,"Result too large to show","B",1