1,1,334,305,0.1004956,"\int \frac{d+e x^3}{a+c x^6} \, dx","Integrate[(d + e*x^3)/(a + c*x^6),x]","-\frac{\left(\sqrt{3} \sqrt[6]{a} \sqrt{c} d-a^{2/3} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a c^{2/3}}-\frac{\left(-a^{2/3} e-\sqrt{3} \sqrt[6]{a} \sqrt{c} d\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a c^{2/3}}+\frac{\left(\sqrt{3} a^{2/3} e+\sqrt[6]{a} \sqrt{c} d\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x-\sqrt{3} \sqrt[6]{a}}{\sqrt[6]{a}}\right)}{6 a c^{2/3}}+\frac{\left(\sqrt[6]{a} \sqrt{c} d-\sqrt{3} a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a}+2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 a c^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt[6]{c}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}","-\frac{\left(\sqrt{3} \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{3} \sqrt{c} d\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{3} \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{c} d-\sqrt{3} \sqrt{a} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt[6]{c}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}",1,"(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) + ((a^(1/6)*Sqrt[c]*d + Sqrt[3]*a^(2/3)*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(2/3)) + ((a^(1/6)*Sqrt[c]*d - Sqrt[3]*a^(2/3)*e)*ArcTan[(Sqrt[3]*a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(2/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3]*a^(1/6)*Sqrt[c]*d - a^(2/3)*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(2/3)) - ((-(Sqrt[3]*a^(1/6)*Sqrt[c]*d) - a^(2/3)*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(2/3))","A",1
2,1,337,323,0.1161087,"\int \frac{d+e x^3}{a-c x^6} \, dx","Integrate[(d + e*x^3)/(a - c*x^6),x]","\frac{-2 \sqrt{3} \left(\sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}}{\sqrt{3}}\right)+2 \sqrt{3} \left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+1}{\sqrt{3}}\right)-\sqrt{c} d \log \left(-\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)+\sqrt{c} d \log \left(\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)-2 \sqrt{c} d \log \left(\sqrt[6]{a}-\sqrt[6]{c} x\right)+2 \sqrt{c} d \log \left(\sqrt[6]{a}+\sqrt[6]{c} x\right)+\sqrt{a} e \log \left(-\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)+\sqrt{a} e \log \left(\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)-2 \sqrt{a} e \log \left(\sqrt[6]{a}-\sqrt[6]{c} x\right)-2 \sqrt{a} e \log \left(\sqrt[6]{a}+\sqrt[6]{c} x\right)}{12 a^{5/6} c^{2/3}}","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a}-\sqrt[6]{c} x\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}+2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} \sqrt[6]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt[6]{a}+\sqrt[6]{c} x\right)}{6 a^{5/6} \sqrt[6]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}-2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} \sqrt[6]{c}}",1,"(-2*Sqrt[3]*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(1 - (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]] + 2*Sqrt[3]*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(1 + (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]] - 2*Sqrt[c]*d*Log[a^(1/6) - c^(1/6)*x] - 2*Sqrt[a]*e*Log[a^(1/6) - c^(1/6)*x] + 2*Sqrt[c]*d*Log[a^(1/6) + c^(1/6)*x] - 2*Sqrt[a]*e*Log[a^(1/6) + c^(1/6)*x] - Sqrt[c]*d*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[a]*e*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[c]*d*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[a]*e*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))","A",1
3,1,534,754,0.6279759,"\int \frac{d+e x^4}{a+c x^8} \, dx","Integrate[(d + e*x^4)/(a + c*x^8),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[8]{c} x \sec \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}-\tan \left(\frac{\pi }{8}\right)\right) \left(\sqrt[8]{a} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)-a^{5/8} e \sin \left(\frac{\pi }{8}\right)\right)+2 \tan ^{-1}\left(\frac{\sqrt[8]{c} x \sec \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}+\tan \left(\frac{\pi }{8}\right)\right) \left(\sqrt[8]{a} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)-a^{5/8} e \sin \left(\frac{\pi }{8}\right)\right)-\sqrt[8]{a} \log \left(-2 \sqrt[8]{a} \sqrt[8]{c} x \sin \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(\sqrt{a} e \cos \left(\frac{\pi }{8}\right)+\sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right)+\sqrt[8]{a} \log \left(2 \sqrt[8]{a} \sqrt[8]{c} x \sin \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(\sqrt{a} e \cos \left(\frac{\pi }{8}\right)+\sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right)+\sqrt[8]{a} \log \left(-2 \sqrt[8]{a} \sqrt[8]{c} x \cos \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(\sqrt{a} e \sin \left(\frac{\pi }{8}\right)-\sqrt{c} d \cos \left(\frac{\pi }{8}\right)\right)-\sqrt[8]{a} \log \left(2 \sqrt[8]{a} \sqrt[8]{c} x \cos \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(\sqrt{a} e \sin \left(\frac{\pi }{8}\right)-\sqrt{c} d \cos \left(\frac{\pi }{8}\right)\right)-2 \sqrt[8]{a} \left(\sqrt{a} e \cos \left(\frac{\pi }{8}\right)+\sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\cot \left(\frac{\pi }{8}\right)-\frac{\sqrt[8]{c} x \csc \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}\right)+2 \sqrt[8]{a} \left(\sqrt{a} e \cos \left(\frac{\pi }{8}\right)+\sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x \csc \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}+\cot \left(\frac{\pi }{8}\right)\right)}{8 a c^{5/8}}","\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\left(-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} \sqrt[8]{c}}",1,"(-2*a^(1/8)*ArcTan[Cot[Pi/8] - (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + 2*a^(1/8)*ArcTan[Cot[Pi/8] + (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) - a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(-(Sqrt[c]*d*Cos[Pi/8]) + Sqrt[a]*e*Sin[Pi/8]) - a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(-(Sqrt[c]*d*Cos[Pi/8]) + Sqrt[a]*e*Sin[Pi/8]) + 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*(a^(1/8)*Sqrt[c]*d*Cos[Pi/8] - a^(5/8)*e*Sin[Pi/8]) + 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*(a^(1/8)*Sqrt[c]*d*Cos[Pi/8] - a^(5/8)*e*Sin[Pi/8]))/(8*a*c^(5/8))","A",1
4,1,425,329,0.1338182,"\int \frac{d+e x^4}{a-c x^8} \, dx","Integrate[(d + e*x^4)/(a - c*x^8),x]","\frac{\left(a^{5/8} e-\sqrt[8]{a} \sqrt{c} d\right) \log \left(-\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a c^{5/8}}-\frac{\left(a^{5/8} e-\sqrt[8]{a} \sqrt{c} d\right) \log \left(\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a c^{5/8}}-\frac{\left(a^{5/8} e+\sqrt[8]{a} \sqrt{c} d\right) \log \left(\sqrt[8]{a}-\sqrt[8]{c} x\right)}{8 a c^{5/8}}-\frac{\left(-a^{5/8} e-\sqrt[8]{a} \sqrt{c} d\right) \log \left(\sqrt[8]{a}+\sqrt[8]{c} x\right)}{8 a c^{5/8}}+\frac{\left(a^{5/8} e+\sqrt[8]{a} \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a c^{5/8}}-\frac{\left(a^{5/8} e-\sqrt[8]{a} \sqrt{c} d\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{c} x-\sqrt{2} \sqrt[8]{a}}{\sqrt{2} \sqrt[8]{a}}\right)}{4 \sqrt{2} a c^{5/8}}-\frac{\left(a^{5/8} e-\sqrt[8]{a} \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2} \sqrt[8]{a}}\right)}{4 \sqrt{2} a c^{5/8}}","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}+1\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}",1,"((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcTan[(-(Sqrt[2]*a^(1/8)) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcTan[(Sqrt[2]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*Log[a^(1/8) - c^(1/8)*x])/(8*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) - a^(5/8)*e)*Log[a^(1/8) + c^(1/8)*x])/(8*a*c^(5/8)) + ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8))","A",1
5,1,67,791,0.0448265,"\int \frac{d+e x^4}{d^2+b x^4+e^2 x^8} \, dx","Integrate[(d + e*x^4)/(d^2 + b*x^4 + e^2*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 e^2+\text{$\#$1}^4 b+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2+\text{$\#$1}^3 b}\&\right]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}",1,"RootSum[d^2 + b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(b*#1^3 + 2*e^2*#1^7) & ]/4","C",1
6,1,67,791,0.0353656,"\int \frac{d+e x^4}{d^2+f x^4+e^2 x^8} \, dx","Integrate[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 e^2+\text{$\#$1}^4 f+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2+\text{$\#$1}^3 f}\&\right]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}",1,"RootSum[d^2 + f*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(f*#1^3 + 2*e^2*#1^7) & ]/4","C",1
7,1,69,349,0.0448285,"\int \frac{d+e x^4}{d^2-b x^4+e^2 x^8} \, dx","Integrate[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 e^2-\text{$\#$1}^4 b+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2-\text{$\#$1}^3 b}\&\right]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}",1,"RootSum[d^2 - b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(-(b*#1^3) + 2*e^2*#1^7) & ]/4","C",1
8,1,69,751,0.0390735,"\int \frac{d+e x^4}{d^2-f x^4+e^2 x^8} \, dx","Integrate[(d + e*x^4)/(d^2 - f*x^4 + e^2*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 e^2-\text{$\#$1}^4 f+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2-\text{$\#$1}^3 f}\&\right]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}",1,"RootSum[d^2 - f*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(-(f*#1^3) + 2*e^2*#1^7) & ]/4","C",1
9,1,55,411,0.0259627,"\int \frac{1+x^4}{1+b x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 + b*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+\text{$\#$1}^4 b+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+\text{$\#$1}^3 b}\&\right]","-\frac{\log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}+\frac{\log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}-\frac{\log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}+\frac{\log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}",1,"RootSum[1 + b*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(b*#1^3 + 2*#1^7) & ]/4","C",1
10,1,55,451,0.0150584,"\int \frac{1+x^4}{1+3 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 + 3*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}",1,"RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(3*#1^3 + 2*#1^7) & ]/4","C",1
11,1,64,85,0.0201077,"\int \frac{1+x^4}{1+2 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 + 2*x^4 + x^8),x]","\frac{-\log \left(x^2-\sqrt{2} x+1\right)+\log \left(x^2+\sqrt{2} x+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} x\right)+2 \tan ^{-1}\left(\sqrt{2} x+1\right)}{4 \sqrt{2}}","-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{4 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{4 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{2 \sqrt{2}}",1,"(-2*ArcTan[1 - Sqrt[2]*x] + 2*ArcTan[1 + Sqrt[2]*x] - Log[1 - Sqrt[2]*x + x^2] + Log[1 + Sqrt[2]*x + x^2])/(4*Sqrt[2])","A",1
12,1,135,140,0.1740759,"\int \frac{1+x^4}{1+x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 + x^4 + x^8),x]","\frac{1}{48} \left(-6 \log \left(x^2-x+1\right)+6 \log \left(x^2+x+1\right)+4 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)-4 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"((4*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (4*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] + 4*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 6*Log[1 - x + x^2] + 6*Log[1 + x + x^2])/48","C",0
13,1,258,347,0.1881543,"\int \frac{1+x^4}{1+x^8} \, dx","Integrate[(1 + x^4)/(1 + x^8),x]","\frac{1}{8} \left(-\left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2-2 x \sin \left(\frac{\pi }{8}\right)+1\right)+\left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2+2 x \sin \left(\frac{\pi }{8}\right)+1\right)+\left(\sin \left(\frac{\pi }{8}\right)-\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2-2 x \cos \left(\frac{\pi }{8}\right)+1\right)+\left(\cos \left(\frac{\pi }{8}\right)-\sin \left(\frac{\pi }{8}\right)\right) \log \left(x^2+2 x \cos \left(\frac{\pi }{8}\right)+1\right)+2 \left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\csc \left(\frac{\pi }{8}\right) \left(x-\cos \left(\frac{\pi }{8}\right)\right)\right)+2 \left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\csc \left(\frac{\pi }{8}\right) \left(x+\cos \left(\frac{\pi }{8}\right)\right)\right)+2 \left(\cos \left(\frac{\pi }{8}\right)-\sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\sec \left(\frac{\pi }{8}\right) \left(x+\sin \left(\frac{\pi }{8}\right)\right)\right)+2 \left(\cos \left(\frac{\pi }{8}\right)-\sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(x \sec \left(\frac{\pi }{8}\right)-\tan \left(\frac{\pi }{8}\right)\right)\right)","-\frac{\log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}-\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)",1,"(2*ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*(Cos[Pi/8] - Sin[Pi/8]) + 2*ArcTan[x*Sec[Pi/8] - Tan[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + Log[1 + x^2 + 2*x*Cos[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + Log[1 + x^2 - 2*x*Cos[Pi/8]]*(-Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[(x - Cos[Pi/8])*Csc[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) - Log[1 + x^2 - 2*x*Sin[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) + Log[1 + x^2 + 2*x*Sin[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]))/8","A",1
14,1,55,331,0.0158842,"\int \frac{1+x^4}{1-x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
15,1,31,27,0.0134434,"\int \frac{1+x^4}{1-2 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - 2*x^4 + x^8),x]","\frac{1}{8} \left(-\frac{4 x}{x^4-1}-\log (1-x)+\log (x+1)+2 \tan ^{-1}(x)\right)","\frac{x}{2 \left(1-x^4\right)}+\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x)",1,"((-4*x)/(-1 + x^4) + 2*ArcTan[x] - Log[1 - x] + Log[1 + x])/8","A",1
16,1,131,131,0.0772623,"\int \frac{1+x^4}{1-3 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - 3*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}",1,"ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])]","A",1
17,1,53,157,0.0130545,"\int \frac{1+x^4}{1-4 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - 4*x^4 + x^8),x]","\frac{1}{8} \text{RootSum}\left[\text{$\#$1}^8-4 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{\text{$\#$1}^7-2 \text{$\#$1}^3}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}",1,"RootSum[1 - 4*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-2*#1^3 + #1^7) & ]/8","C",1
18,1,55,171,0.0134607,"\int \frac{1+x^4}{1-5 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - 5*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-5 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\&\right]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}",1,"RootSum[1 - 5*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-5*#1^3 + 2*#1^7) & ]/4","C",1
19,1,111,117,0.0539124,"\int \frac{1+x^4}{1-6 x^4+x^8} \, dx","Integrate[(1 + x^4)/(1 - 6*x^4 + x^8),x]","\frac{1}{4} \left(\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)-\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)\right)","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}",1,"(Sqrt[1 + Sqrt[2]]*ArcTan[x/Sqrt[-1 + Sqrt[2]]] - Sqrt[-1 + Sqrt[2]]*ArcTan[x/Sqrt[1 + Sqrt[2]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[x/Sqrt[1 + Sqrt[2]]])/4","A",1
20,1,57,511,0.0254134,"\int \frac{1-x^4}{1+b x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 + b*x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+\text{$\#$1}^4 b+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+\text{$\#$1}^3 b}\&\right]","\frac{\sqrt{2-\sqrt{2-b}} \log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}",1,"-1/4*RootSum[1 + b*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(b*#1^3 + 2*#1^7) & ]","C",1
21,1,57,411,0.0145563,"\int \frac{1-x^4}{1+3 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 + 3*x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4}}",1,"-1/4*RootSum[1 + 3*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(3*#1^3 + 2*#1^7) & ]","C",1
22,1,90,97,0.0647682,"\int \frac{1-x^4}{1+2 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 + 2*x^4 + x^8),x]","\frac{1}{16} \left(\frac{8 x}{x^4+1}-\sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)+\sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{x}{2 \left(x^4+1\right)}-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{8 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{8 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{4 \sqrt{2}}",1,"((8*x)/(1 + x^4) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/16","A",1
23,1,129,140,0.1732429,"\int \frac{1-x^4}{1+x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 + x^4 + x^8),x]","\frac{1}{8} \left(\log \left(x^2-x+1\right)-\log \left(x^2+x+1\right)-2 \sqrt{-2-2 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)-2 \sqrt{-2+2 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)-\frac{1}{8} \sqrt{3} \log \left(x^2-\sqrt{3} x+1\right)+\frac{1}{8} \sqrt{3} \log \left(x^2+\sqrt{3} x+1\right)-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"(-2*Sqrt[-2 - (2*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - 2*Sqrt[-2 + (2*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + Log[1 - x + x^2] - Log[1 + x + x^2])/8","C",0
24,1,257,347,0.1638769,"\int \frac{1-x^4}{1+x^8} \, dx","Integrate[(1 - x^4)/(1 + x^8),x]","\frac{1}{8} \left(-\left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2-2 x \cos \left(\frac{\pi }{8}\right)+1\right)+\left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2+2 x \cos \left(\frac{\pi }{8}\right)+1\right)+\left(\sin \left(\frac{\pi }{8}\right)-\cos \left(\frac{\pi }{8}\right)\right) \log \left(x^2+2 x \sin \left(\frac{\pi }{8}\right)+1\right)+\left(\cos \left(\frac{\pi }{8}\right)-\sin \left(\frac{\pi }{8}\right)\right) \log \left(x^2-2 x \sin \left(\frac{\pi }{8}\right)+1\right)+2 \left(\sin \left(\frac{\pi }{8}\right)-\cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\csc \left(\frac{\pi }{8}\right) \left(x+\cos \left(\frac{\pi }{8}\right)\right)\right)+2 \left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\sec \left(\frac{\pi }{8}\right) \left(x+\sin \left(\frac{\pi }{8}\right)\right)\right)+2 \left(\sin \left(\frac{\pi }{8}\right)+\cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(x \sec \left(\frac{\pi }{8}\right)-\tan \left(\frac{\pi }{8}\right)\right)+2 \left(\cos \left(\frac{\pi }{8}\right)-\sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\cot \left(\frac{\pi }{8}\right)-x \csc \left(\frac{\pi }{8}\right)\right)\right)","\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}",1,"(2*ArcTan[Cot[Pi/8] - x*Csc[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + Log[1 + x^2 - 2*x*Sin[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + 2*ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*(-Cos[Pi/8] + Sin[Pi/8]) + Log[1 + x^2 + 2*x*Sin[Pi/8]]*(-Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*(Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[x*Sec[Pi/8] - Tan[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) - Log[1 + x^2 - 2*x*Cos[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) + Log[1 + x^2 + 2*x*Cos[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]))/8","A",1
25,1,57,355,0.0163004,"\int \frac{1-x^4}{1-x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"-1/4*RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]","C",1
26,1,25,13,0.0050266,"\int \frac{1-x^4}{1-2 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - 2*x^4 + x^8),x]","-\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1)+\frac{1}{2} \tan ^{-1}(x)","\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"ArcTan[x]/2 - Log[1 - x]/4 + Log[1 + x]/4","A",1
27,1,129,129,0.0773282,"\int \frac{1-x^4}{1-3 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - 3*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}",1,"ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]","A",1
28,1,55,165,0.0133845,"\int \frac{1-x^4}{1-4 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - 4*x^4 + x^8),x]","-\frac{1}{8} \text{RootSum}\left[\text{$\#$1}^8-4 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{\text{$\#$1}^7-2 \text{$\#$1}^3}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}",1,"-1/8*RootSum[1 - 4*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-2*#1^3 + #1^7) & ]","C",1
29,1,57,169,0.0138249,"\int \frac{1-x^4}{1-5 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - 5*x^4 + x^8),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-5 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\&\right]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}",1,"-1/4*RootSum[1 - 5*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-5*#1^3 + 2*#1^7) & ]","C",1
30,1,114,125,0.0539987,"\int \frac{1-x^4}{1-6 x^4+x^8} \, dx","Integrate[(1 - x^4)/(1 - 6*x^4 + x^8),x]","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)+\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)+\sqrt{\sqrt{2}-1} \tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}",1,"(Sqrt[1 + Sqrt[2]]*ArcTan[x/Sqrt[-1 + Sqrt[2]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[x/Sqrt[1 + Sqrt[2]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]] + Sqrt[-1 + Sqrt[2]]*ArcTanh[x/Sqrt[1 + Sqrt[2]]])/(4*Sqrt[2])","A",1
31,1,71,135,0.0344743,"\int \frac{-1+\sqrt{3}+2 x^4}{1-x^4+x^8} \, dx","Integrate[(-1 + Sqrt[3] + 2*x^4)/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{2 \text{$\#$1}^4 \log (x-\text{$\#$1})+\sqrt{3} \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}",1,"RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Sqrt[3]*Log[x - #1] + 2*Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
32,1,72,164,0.0379192,"\int \frac{1+\left(1+\sqrt{3}\right) x^4}{1-x^4+x^8} \, dx","Integrate[(1 + (1 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\sqrt{3} \text{$\#$1}^4 \log (x-\text{$\#$1})+\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","-\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4 + Sqrt[3]*Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
33,1,89,180,0.0455131,"\int \frac{3-2 \sqrt{3}+\left(-3+\sqrt{3}\right) x^4}{1-x^4+x^8} \, dx","Integrate[(3 - 2*Sqrt[3] + (-3 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\sqrt{3} \text{$\#$1}^4 \log (x-\text{$\#$1})-3 \text{$\#$1}^4 \log (x-\text{$\#$1})-2 \sqrt{3} \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)-\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"RootSum[1 - #1^4 + #1^8 & , (3*Log[x - #1] - 2*Sqrt[3]*Log[x - #1] - 3*Log[x - #1]*#1^4 + Sqrt[3]*Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
34,1,49,49,0.023877,"\int \frac{d+\frac{e}{x}}{c+\frac{a}{x^2}} \, dx","Integrate[(d + e/x)/(c + a/x^2),x]","-\frac{\sqrt{a} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2}}+\frac{e \log \left(a+c x^2\right)}{2 c}+\frac{d x}{c}","-\frac{\sqrt{a} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2}}+\frac{e \log \left(a+c x^2\right)}{2 c}+\frac{d x}{c}",1,"(d*x)/c - (Sqrt[a]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/c^(3/2) + (e*Log[a + c*x^2])/(2*c)","A",1
35,1,86,86,0.089802,"\int \frac{d+\frac{e}{x}}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Integrate[(d + e/x)/(c + a/x^2 + b/x),x]","\frac{\frac{2 \left(-2 a c d+b^2 d-b c e\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+(c e-b d) \log (a+x (b+c x))+2 c d x}{2 c^2}","-\frac{\left(-2 a c d+b^2 d-b c e\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{(b d-c e) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{d x}{c}",1,"(2*c*d*x + (2*(b^2*d - 2*a*c*d - b*c*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-(b*d) + c*e)*Log[a + x*(b + c*x)])/(2*c^2)","A",1
36,1,293,253,0.0963722,"\int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}} \, dx","Integrate[(d + e/x^2)/(c + a/x^4),x]","\frac{\left(a^{5/4} \sqrt{c} d+a^{3/4} c e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a c^{7/4}}-\frac{\left(a^{5/4} \sqrt{c} d+a^{3/4} c e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} a c^{7/4}}+\frac{\left(a^{3/4} c e-a^{5/4} \sqrt{c} d\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{c} x-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{2 \sqrt{2} a c^{7/4}}+\frac{\left(a^{3/4} c e-a^{5/4} \sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{c} x}{\sqrt{2} \sqrt[4]{a}}\right)}{2 \sqrt{2} a c^{7/4}}+\frac{d x}{c}","\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{d x}{c}",1,"(d*x)/c + ((-(a^(5/4)*Sqrt[c]*d) + a^(3/4)*c*e)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*a*c^(7/4)) + ((-(a^(5/4)*Sqrt[c]*d) + a^(3/4)*c*e)*ArcTan[(Sqrt[2]*a^(1/4) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*a*c^(7/4)) + ((a^(5/4)*Sqrt[c]*d + a^(3/4)*c*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4)) - ((a^(5/4)*Sqrt[c]*d + a^(3/4)*c*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4))","A",1
37,1,251,208,0.1729272,"\int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}+\frac{b}{x^2}} \, dx","Integrate[(d + e/x^2)/(c + a/x^4 + b/x^2),x]","-\frac{\left(b d \sqrt{b^2-4 a c}-c e \sqrt{b^2-4 a c}+2 a c d+b^2 (-d)+b c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(b d \sqrt{b^2-4 a c}-c e \sqrt{b^2-4 a c}-2 a c d+b^2 d-b c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{d x}{c}","-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{d x}{c}",1,"(d*x)/c - ((-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + b*c*e - c*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^2*d - 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - b*c*e - c*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
38,1,346,311,0.1124979,"\int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}} \, dx","Integrate[(d + e/x^3)/(c + a/x^6),x]","-\frac{\left(-\sqrt{3} a^{7/6} \sqrt{c} d-a^{2/3} c e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a c^{5/3}}-\frac{\left(\sqrt{3} a^{7/6} \sqrt{c} d-a^{2/3} c e\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a c^{5/3}}+\frac{\left(\sqrt{3} a^{2/3} c e-a^{7/6} \sqrt{c} d\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x-\sqrt{3} \sqrt[6]{a}}{\sqrt[6]{a}}\right)}{6 a c^{5/3}}+\frac{\left(a^{7/6} \left(-\sqrt{c}\right) d-\sqrt{3} a^{2/3} c e\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a}+2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 a c^{5/3}}-\frac{\sqrt[6]{a} d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 c^{7/6}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}+\frac{d x}{c}","\frac{\left(\sqrt{3} \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{3} \sqrt{a} d-\sqrt{c} e\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}+\frac{\left(\sqrt{a} d-\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{a} d+\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\sqrt[6]{a} d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 c^{7/6}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}+\frac{d x}{c}",1,"(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((-(a^(7/6)*Sqrt[c]*d) + Sqrt[3]*a^(2/3)*c*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) + ((-(a^(7/6)*Sqrt[c]*d) - Sqrt[3]*a^(2/3)*c*e)*ArcTan[(Sqrt[3]*a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((-(Sqrt[3]*a^(7/6)*Sqrt[c]*d) - a^(2/3)*c*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(5/3)) - ((Sqrt[3]*a^(7/6)*Sqrt[c]*d - a^(2/3)*c*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(5/3))","A",1
39,1,88,716,0.05385,"\int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}+\frac{b}{x^3}} \, dx","Integrate[(d + e/x^3)/(c + a/x^6 + b/x^3),x]","\frac{d x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b d \log (x-\text{$\#$1})-\text{$\#$1}^3 c e \log (x-\text{$\#$1})+a d \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 c}","\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{d x}{c}",1,"(d*x)/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*d*Log[x - #1] + b*d*Log[x - #1]*#1^3 - c*e*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*c)","C",1
40,1,551,753,0.9033708,"\int \frac{d+\frac{e}{x^4}}{c+\frac{a}{x^8}} \, dx","Integrate[(d + e/x^4)/(c + a/x^8),x]","\frac{\log \left(2 \sqrt[8]{a} \sqrt[8]{c} x \sin \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(a^{5/8} c e \cos \left(\frac{\pi }{8}\right)-a^{9/8} \sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right)+\log \left(-2 \sqrt[8]{a} \sqrt[8]{c} x \sin \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(a^{9/8} \sqrt{c} d \sin \left(\frac{\pi }{8}\right)-a^{5/8} c e \cos \left(\frac{\pi }{8}\right)\right)+\log \left(-2 \sqrt[8]{a} \sqrt[8]{c} x \cos \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(a^{9/8} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)+a^{5/8} c e \sin \left(\frac{\pi }{8}\right)\right)-\log \left(2 \sqrt[8]{a} \sqrt[8]{c} x \cos \left(\frac{\pi }{8}\right)+\sqrt[4]{a}+\sqrt[4]{c} x^2\right) \left(a^{9/8} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)+a^{5/8} c e \sin \left(\frac{\pi }{8}\right)\right)-2 \tan ^{-1}\left(\frac{\sqrt[8]{c} x \sec \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}-\tan \left(\frac{\pi }{8}\right)\right) \left(a^{9/8} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)+a^{5/8} c e \sin \left(\frac{\pi }{8}\right)\right)-2 \tan ^{-1}\left(\frac{\sqrt[8]{c} x \sec \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}+\tan \left(\frac{\pi }{8}\right)\right) \left(a^{9/8} \sqrt{c} d \cos \left(\frac{\pi }{8}\right)+a^{5/8} c e \sin \left(\frac{\pi }{8}\right)\right)+2 \left(a^{5/8} c e \cos \left(\frac{\pi }{8}\right)-a^{9/8} \sqrt{c} d \sin \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x \csc \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}+\cot \left(\frac{\pi }{8}\right)\right)+2 \left(a^{9/8} \sqrt{c} d \sin \left(\frac{\pi }{8}\right)-a^{5/8} c e \cos \left(\frac{\pi }{8}\right)\right) \tan ^{-1}\left(\cot \left(\frac{\pi }{8}\right)-\frac{\sqrt[8]{c} x \csc \left(\frac{\pi }{8}\right)}{\sqrt[8]{a}}\right)+8 a c^{5/8} d x}{8 a c^{13/8}}","-\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{d x}{c}",1,"(8*a*c^(5/8)*d*x + 2*ArcTan[Cot[Pi/8] + (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(a^(5/8)*c*e*Cos[Pi/8] - a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(a^(5/8)*c*e*Cos[Pi/8] - a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) + 2*ArcTan[Cot[Pi/8] - (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(-(a^(5/8)*c*e*Cos[Pi/8]) + a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(-(a^(5/8)*c*e*Cos[Pi/8]) + a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) - 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]) - 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]) - Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]))/(8*a*c^(13/8))","A",1
41,1,88,433,0.0746699,"\int \frac{d+\frac{e}{x^4}}{c+\frac{a}{x^8}+\frac{b}{x^4}} \, dx","Integrate[(d + e/x^4)/(c + a/x^8 + b/x^4),x]","\frac{d x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b d \log (x-\text{$\#$1})-\text{$\#$1}^4 c e \log (x-\text{$\#$1})+a d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 c}","\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{d x}{c}",1,"(d*x)/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*d*Log[x - #1] + b*d*Log[x - #1]*#1^4 - c*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)","C",1
42,1,127,141,0.241808,"\int \frac{\left(d+e x^n\right)^3}{a+c x^{2 n}} \, dx","Integrate[(d + e*x^n)^3/(a + c*x^(2*n)),x]","\frac{x \left(d (n+1) \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+e \left(x^n \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+a e \left(3 d (n+1)+e x^n\right)\right)\right)}{a c (n+1)}","\frac{e x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}+\frac{d x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)}",1,"(x*(d*(c*d^2 - 3*a*e^2)*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + e*(a*e*(3*d*(1 + n) + e*x^n) + (3*c*d^2 - a*e^2)*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])))/(a*c*(1 + n))","A",1
43,1,107,107,0.1496523,"\int \frac{\left(d+e x^n\right)^2}{a+c x^{2 n}} \, dx","Integrate[(d + e*x^n)^2/(a + c*x^(2*n)),x]","\frac{x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^2 x}{c}","\frac{x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^2 x}{c}",1,"(e^2*x)/c + ((c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (2*d*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))","A",1
44,1,83,83,0.0386277,"\int \frac{d+e x^n}{a+c x^{2 n}} \, dx","Integrate[(d + e*x^n)/(a + c*x^(2*n)),x]","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}",1,"(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))","A",1
45,1,131,152,0.1368447,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)} \, dx","Integrate[1/((d + e*x^n)*(a + c*x^(2*n))),x]","\frac{x \left(c d^2 (n+1) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+e \left(a e (n+1) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)-c d x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)\right)}{a d (n+1) \left(a e^2+c d^2\right)}","-\frac{c e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)}+\frac{c d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)}",1,"(x*(c*d^2*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + e*(a*e*(1 + n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] - c*d*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])))/(a*d*(c*d^2 + a*e^2)*(1 + n))","A",1
46,1,186,205,0.2553206,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)} \, dx","Integrate[1/((d + e*x^n)^2*(a + c*x^(2*n))),x]","\frac{x \left(e \left(-2 c^2 d^3 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+a e (n+1) \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)+2 a c d^2 e (n+1) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)\right)+c d^2 (n+1) \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)}{a (n+1) \left(a d e^2+c d^3\right)^2}","-\frac{2 c^2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}+\frac{c x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{2 c e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)}",1,"(x*(c*d^2*(c*d^2 - a*e^2)*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + e*(2*a*c*d^2*e*(1 + n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] - 2*c^2*d^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)] + a*e*(c*d^2 + a*e^2)*(1 + n)*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])))/(a*(c*d^3 + a*d*e^2)^2*(1 + n))","A",1
47,1,81,81,0.059394,"\int \frac{d+e x^n}{a-c x^{2 n}} \, dx","Integrate[(d + e*x^n)/(a - c*x^(2*n)),x]","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a (n+1)}","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a (n+1)}",1,"(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, (c*x^(2*n))/a])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, (c*x^(2*n))/a])/(a*(1 + n))","A",1
48,1,188,288,0.2705432,"\int \frac{\left(d+e x^n\right)^3}{\left(a+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)^3/(a + c*x^(2*n))^2,x]","\frac{x \left(d \left(c d^2-3 a e^2\right) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{e x^n \left(3 c d^2-a e^2\right) \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+3 a d e^2 \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{a e^3 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}\right)}{a^2 c}","-\frac{e (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{3 d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}",1,"(x*(3*a*d*e^2*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (a*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n) + d*(c*d^2 - 3*a*e^2)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (e*(3*c*d^2 - a*e^2)*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n)))/(a^2*c)","A",1
49,1,136,203,0.1572422,"\int \frac{\left(d+e x^n\right)^2}{\left(a+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)^2/(a + c*x^(2*n))^2,x]","\frac{x \left((n+1) \left(c d^2-a e^2\right) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+2 c d e x^n \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+a e^2 (n+1) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)}{a^2 c (n+1)}","-\frac{(1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1)}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}",1,"(x*(a*e^2*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (c*d^2 - a*e^2)*(1 + n)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + 2*c*d*e*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]))/(a^2*c*(1 + n))","A",1
50,1,83,134,0.0420396,"\int \frac{d+e x^n}{\left(a+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)/(a + c*x^(2*n))^2,x]","\frac{d x \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2}+\frac{e x^{n+1} \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 (n+1)}","-\frac{d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n}-\frac{e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1)}+\frac{x \left(d+e x^n\right)}{2 a n \left(a+c x^{2 n}\right)}",1,"(d*x*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^2 + (e*x^(1 + n)*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n))","A",1
51,1,227,333,0.2670789,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)^2} \, dx","Integrate[1/((d + e*x^n)*(a + c*x^(2*n))^2),x]","\frac{x \left(a^2 e^4 (n+1) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)+a c d^2 e^2 (n+1) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+c d \left(\left(a e^2+c d^2\right) \left(d (n+1) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)-e x^n \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)-a e^3 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)\right)}{a^2 d (n+1) \left(a e^2+c d^2\right)^2}","\frac{c e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)}-\frac{c d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)}+\frac{c d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{c x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^2}-\frac{c e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}",1,"(x*(a*c*d^2*e^2*(1 + n)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + a^2*e^4*(1 + n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] + c*d*(-(a*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]) + (c*d^2 + a*e^2)*(d*(1 + n)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] - e*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]))))/(a^2*d*(c*d^2 + a*e^2)^2*(1 + n))","A",1
52,1,298,410,0.4747746,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^2} \, dx","Integrate[1/((d + e*x^n)^2*(a + c*x^(2*n))^2),x]","\frac{x \left(-\frac{2 c^2 d e x^n \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 (n+1)}+\frac{c \left(c d^2-a e^2\right) \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2}-\frac{4 c^2 d e^3 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{c e^2 \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+\frac{e^4 \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2}+4 c e^4 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)\right)}{\left(a e^2+c d^2\right)^3}","\frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^2}-\frac{c (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}+\frac{4 c e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^3}+\frac{e^4 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^2}",1,"(x*((c*e^2*(3*c*d^2 - a*e^2)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + 4*c*e^4*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] - (4*c^2*d*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n)) + (c*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^2 + (e^4*(c*d^2 + a*e^2)*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^2 - (2*c^2*d*e*(c*d^2 + a*e^2)*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n))))/(c*d^2 + a*e^2)^3","A",1
53,1,188,424,0.2909051,"\int \frac{\left(d+e x^n\right)^3}{\left(a+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)^3/(a + c*x^(2*n))^3,x]","\frac{x \left(d \left(c d^2-3 a e^2\right) \, _2F_1\left(3,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{e x^n \left(3 c d^2-a e^2\right) \, _2F_1\left(3,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+3 a d e^2 \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{a e^3 x^n \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}\right)}{a^3 c}","\frac{e (1-3 n) (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2 (n+1)}+\frac{d (1-4 n) (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}-\frac{x \left(e (1-3 n) x^n \left(3 c d^2-a e^2\right)+d (1-4 n) \left(c d^2-3 a e^2\right)\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}-\frac{3 d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{4 a c n \left(a+c x^{2 n}\right)^2}+\frac{e^2 x \left(3 d+e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}",1,"(x*(3*a*d*e^2*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (a*e^3*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n) + d*(c*d^2 - 3*a*e^2)*Hypergeometric2F1[3, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (e*(3*c*d^2 - a*e^2)*x^n*Hypergeometric2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n)))/(a^3*c)","A",1
54,1,136,272,0.1648897,"\int \frac{\left(d+e x^n\right)^2}{\left(a+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)^2/(a + c*x^(2*n))^3,x]","\frac{x \left((n+1) \left(c d^2-a e^2\right) \, _2F_1\left(3,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+2 c d e x^n \, _2F_1\left(3,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+a e^2 (n+1) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)}{a^3 c (n+1)}","\frac{(1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}+\frac{d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1)}-\frac{x \left((1-4 n) \left(c d^2-a e^2\right)+2 c d e (1-3 n) x^n\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 c}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{4 a c n \left(a+c x^{2 n}\right)^2}",1,"(x*(a*e^2*(1 + n)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (c*d^2 - a*e^2)*(1 + n)*Hypergeometric2F1[3, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)] + 2*c*d*e*x^n*Hypergeometric2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)]))/(a^3*c*(1 + n))","A",1
55,1,83,184,0.0442683,"\int \frac{d+e x^n}{\left(a+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)/(a + c*x^(2*n))^3,x]","\frac{d x \, _2F_1\left(3,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3}+\frac{e x^{n+1} \, _2F_1\left(3,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3 (n+1)}","\frac{d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2}+\frac{e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1)}-\frac{x \left(d (1-4 n)+e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a+c x^{2 n}\right)}+\frac{x \left(d+e x^n\right)}{4 a n \left(a+c x^{2 n}\right)^2}",1,"(d*x*Hypergeometric2F1[3, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^3 + (e*x^(1 + n)*Hypergeometric2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*(1 + n))","A",1
56,1,346,582,0.4289978,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)^3} \, dx","Integrate[1/((d + e*x^n)*(a + c*x^(2*n))^3),x]","\frac{x \left(\frac{c d \left(a e^2+c d^2\right)^2 \, _2F_1\left(3,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3}-\frac{c e x^n \left(a e^2+c d^2\right)^2 \, _2F_1\left(3,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3 (n+1)}+\frac{c d e^2 \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2}-\frac{c e^3 x^n \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 (n+1)}+\frac{c d e^4 \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}-\frac{c e^5 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^6 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d}\right)}{\left(a e^2+c d^2\right)^3}","-\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1) \left(a e^2+c d^2\right)}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)}-\frac{c d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{c x \left(d (1-4 n)-e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)^2}+\frac{c e^2 x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}+\frac{c x \left(d-e x^n\right)}{4 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)^2}+\frac{e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^3}-\frac{c e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c d e^4 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}",1,"(x*((c*d*e^4*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e^6*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d - (c*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n)) + (c*d*e^2*(c*d^2 + a*e^2)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^2 - (c*e^3*(c*d^2 + a*e^2)*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n)) + (c*d*(c*d^2 + a*e^2)^2*Hypergeometric2F1[3, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^3 - (c*e*(c*d^2 + a*e^2)^2*x^n*Hypergeometric2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*(1 + n))))/(c*d^2 + a*e^2)^3","A",1
57,1,426,701,0.7043505,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^3} \, dx","Integrate[1/((d + e*x^n)^2*(a + c*x^(2*n))^3),x]","\frac{x \left(-\frac{2 c^2 d e x^n \left(a e^2+c d^2\right)^2 \, _2F_1\left(3,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3 (n+1)}+\frac{c \left(c d^2-a e^2\right) \left(a e^2+c d^2\right)^2 \, _2F_1\left(3,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^3}-\frac{4 c^2 d e^3 x^n \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 (n+1)}+\frac{c e^2 \left(3 c d^2-a e^2\right) \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2}-\frac{6 c^2 d e^5 x^n \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^6 \left(a e^2+c d^2\right) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2}+\frac{c e^4 \left(5 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+6 c e^6 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)\right)}{\left(a e^2+c d^2\right)^4}","-\frac{c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1) \left(a e^2+c d^2\right)^2}+\frac{c (1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)^2}+\frac{2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^3}-\frac{c e^2 (1-2 n) x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^3}-\frac{c x \left((1-4 n) \left(c d^2-a e^2\right)-2 c d e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}-\frac{6 c^2 d e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^4}+\frac{c e^2 x \left(-a e^2+3 c d^2-4 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^3 \left(a+c x^{2 n}\right)}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{4 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)^2}+\frac{6 c e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^4}+\frac{e^6 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^3}+\frac{c e^4 x \left(5 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^4}",1,"(x*((c*e^4*(5*c*d^2 - a*e^2)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + 6*c*e^6*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] - (6*c^2*d*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n)) + (c*e^2*(3*c*d^2 - a*e^2)*(c*d^2 + a*e^2)*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^2 + (e^6*(c*d^2 + a*e^2)*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^2 - (4*c^2*d*e^3*(c*d^2 + a*e^2)*x^n*Hypergeometric2F1[2, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n)) + (c*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^2*Hypergeometric2F1[3, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a^3 - (2*c^2*d*e*(c*d^2 + a*e^2)^2*x^n*Hypergeometric2F1[3, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*(1 + n))))/(c*d^2 + a*e^2)^4","A",1
58,0,0,171,0.1521297,"\int \frac{1}{\left(d+e x^n\right) \sqrt{a+c x^{2 n}}} \, dx","Integrate[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]),x]","\int \frac{1}{\left(d+e x^n\right) \sqrt{a+c x^{2 n}}} \, dx","\frac{x \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d \sqrt{a+c x^{2 n}}}-\frac{e x^{n+1} \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{n+1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1) \sqrt{a+c x^{2 n}}}",1,"Integrate[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]), x]","F",-1
59,0,0,24,0.1552934,"\int \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^q*(a + c*x^(2*n))^p,x]","\int \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","\text{Int}\left(\left(a+c x^{2 n}\right)^p \left(d+e x^n\right)^q,x\right)",0,"Integrate[(d + e*x^n)^q*(a + c*x^(2*n))^p, x]","A",-1
60,1,213,299,0.1976567,"\int \left(d+e x^n\right)^3 \left(a+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^3*(a + c*x^(2*n))^p,x]","x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(d^2 \left(d \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{3 e x^n \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}\right)+\frac{3 d e^2 x^{2 n} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}+\frac{e^3 x^{3 n} \, _2F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right),-p;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{3 n+1}\right)","d^3 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{3 d^2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+\frac{3 d e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right),-p;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{3 n+1}",1,"(x*(a + c*x^(2*n))^p*((3*d*e^2*x^(2*n)*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + 2*n) + (e^3*x^(3*n)*Hypergeometric2F1[(3 + n^(-1))/2, -p, (5 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + 3*n) + d^2*(d*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + (3*e*x^n*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + n))))/(1 + (c*x^(2*n))/a)^p","A",1
61,1,171,217,0.1039979,"\int \left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(d (2 n+1) \left(d (n+1) \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+2 e x^n \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)+e^2 (n+1) x^{2 n} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)}{(n+1) (2 n+1)}","d^2 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{2 d e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}",1,"(x*(a + c*x^(2*n))^p*(e^2*(1 + n)*x^(2*n)*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^(-1))/2, -((c*x^(2*n))/a)] + d*(1 + 2*n)*(d*(1 + n)*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + 2*e*x^n*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])))/((1 + n)*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p)","A",1
62,1,110,135,0.0474603,"\int \left(d+e x^n\right) \left(a+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)*(a + c*x^(2*n))^p,x]","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \left(d (n+1) \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+e x^n \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)\right)}{n+1}","d x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}",1,"(x*(a + c*x^(2*n))^p*(d*(1 + n)*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + e*x^n*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)]))/((1 + n)*(1 + (c*x^(2*n))/a)^p)","A",1
63,0,0,167,0.0704283,"\int \frac{\left(a+c x^{2 n}\right)^p}{d+e x^n} \, dx","Integrate[(a + c*x^(2*n))^p/(d + e*x^n),x]","\int \frac{\left(a+c x^{2 n}\right)^p}{d+e x^n} \, dx","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d}-\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1)}",1,"Integrate[(a + c*x^(2*n))^p/(d + e*x^n), x]","F",-1
64,0,0,261,0.1007416,"\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]","\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,2;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,2;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (2 n+1)}-\frac{2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,2;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3 (n+1)}",1,"Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^2, x]","F",-1
65,0,0,357,0.3163604,"\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^3,x]","\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","-\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^6 (3 n+1)}+\frac{3 e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^5 (2 n+1)}-\frac{3 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,3;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (n+1)}+\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,3;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3}",1,"Integrate[(a + c*x^(2*n))^p/(d + e*x^n)^3, x]","F",-1
66,1,57,62,0.1521263,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right) \, dx","Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n)),x]","x \left(\frac{x^n (a e+b d)}{n+1}+a d+\frac{x^{2 n} (b e+c d)}{2 n+1}+\frac{c e x^{3 n}}{3 n+1}\right)","\frac{x^{n+1} (a e+b d)}{n+1}+a d x+\frac{x^{2 n+1} (b e+c d)}{2 n+1}+\frac{c e x^{3 n+1}}{3 n+1}",1,"x*(a*d + ((b*d + a*e)*x^n)/(1 + n) + ((c*d + b*e)*x^(2*n))/(1 + 2*n) + (c*e*x^(3*n))/(1 + 3*n))","A",1
67,1,123,132,0.2475912,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^2 \, dx","Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^2,x]","x \left(a^2 d+\frac{x^{2 n} \left(2 a b e+2 a c d+b^2 d\right)}{2 n+1}+\frac{x^{3 n} \left(2 a c e+b^2 e+2 b c d\right)}{3 n+1}+\frac{a x^n (a e+2 b d)}{n+1}+\frac{c x^{4 n} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n}}{5 n+1}\right)","a^2 d x+\frac{x^{2 n+1} \left(2 a b e+2 a c d+b^2 d\right)}{2 n+1}+\frac{x^{3 n+1} \left(2 a c e+b^2 e+2 b c d\right)}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1}",1,"x*(a^2*d + (a*(2*b*d + a*e)*x^n)/(1 + n) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^(2*n))/(1 + 2*n) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^(3*n))/(1 + 3*n) + (c*(c*d + 2*b*e)*x^(4*n))/(1 + 4*n) + (c^2*e*x^(5*n))/(1 + 5*n))","A",1
68,1,205,218,0.4264023,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^3 \, dx","Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^3,x]","x \left(a^3 d+\frac{x^{3 n} \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{3 n+1}+\frac{a^2 x^n (a e+3 b d)}{n+1}+\frac{3 a x^{2 n} \left(a b e+a c d+b^2 d\right)}{2 n+1}+\frac{3 c x^{5 n} \left(a c e+b^2 e+b c d\right)}{5 n+1}+\frac{x^{4 n} \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)}{4 n+1}+\frac{c^2 x^{6 n} (3 b e+c d)}{6 n+1}+\frac{c^3 e x^{7 n}}{7 n+1}\right)","a^3 d x+\frac{x^{3 n+1} \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{3 n+1}+\frac{a^2 x^{n+1} (a e+3 b d)}{n+1}+\frac{3 a x^{2 n+1} \left(a b e+a c d+b^2 d\right)}{2 n+1}+\frac{3 c x^{5 n+1} \left(a c e+b^2 e+b c d\right)}{5 n+1}+\frac{x^{4 n+1} \left(6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right)}{4 n+1}+\frac{c^2 x^{6 n+1} (3 b e+c d)}{6 n+1}+\frac{c^3 e x^{7 n+1}}{7 n+1}",1,"x*(a^3*d + (a^2*(3*b*d + a*e)*x^n)/(1 + n) + (3*a*(b^2*d + a*c*d + a*b*e)*x^(2*n))/(1 + 2*n) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^(3*n))/(1 + 3*n) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^(4*n))/(1 + 4*n) + (3*c*(b*c*d + b^2*e + a*c*e)*x^(5*n))/(1 + 5*n) + (c^2*(c*d + 3*b*e)*x^(6*n))/(1 + 6*n) + (c^3*e*x^(7*n))/(1 + 7*n))","A",1
69,1,295,308,0.8434982,"\int \frac{\left(d+e x^n\right)^3}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\frac{\left(\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{b-\sqrt{b^2-4 a c}}+\frac{\left(\frac{(b e-2 c d) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\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)}{\sqrt{b^2-4 a c}+b}+e^2 (3 c d-b e)+\frac{c e^3 x^n}{n+1}\right)}{c^2}","\frac{x \left(\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\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)}{c^2 \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\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)}{c^2 \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)}",1,"(x*(e^2*(3*c*d - b*e) + (c*e^3*x^n)/(1 + n) + ((3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3 + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c]) + ((3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3 + ((-2*c*d + b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c])))/c^2","A",1
70,1,216,224,0.524056,"\int \frac{\left(d+e x^n\right)^2}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\frac{\left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{b-\sqrt{b^2-4 a c}}+\frac{\left(-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\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)}{\sqrt{b^2-4 a c}+b}+e^2\right)}{c}","\frac{x \left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\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)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\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)}{c \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x}{c}",1,"(x*(e^2 + ((2*c*d*e - b*e^2 + (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c]) + ((2*c*d*e - b*e^2 - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c])))/c","A",1
71,1,134,154,0.0739653,"\int \frac{d+e x^n}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x^n)/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+\left(d \sqrt{b^2-4 a c}+2 a e-b d\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)\right)}{2 a \sqrt{b^2-4 a c}}","\frac{x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\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)}{b-\sqrt{b^2-4 a c}}+\frac{x \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\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)}{\sqrt{b^2-4 a c}+b}",1,"(x*((b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]))/(2*a*Sqrt[b^2 - 4*a*c])","A",1
72,1,200,243,0.4378173,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]","\frac{x \left(-\frac{c \left(\frac{b e-2 c d}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{b-\sqrt{b^2-4 a c}}-\frac{c \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\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)}{\sqrt{b^2-4 a c}+b}+\frac{e^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d}\right)}{e (a e-b d)+c d^2}","-\frac{c x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\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)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{c x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\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)}{\left(\sqrt{b^2-4 a c}+b\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)}",1,"(x*(-((c*(e + (-2*c*d + b*e)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c])) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c]) + (e^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d))/(c*d^2 + e*(-(b*d) + a*e))","A",1
73,1,327,368,0.8934597,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))),x]","\frac{x \left(\frac{c \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{b \sqrt{b^2-4 a c}+4 a c-b^2}+\frac{c \left(2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)-2 c^2 d^2\right) \, _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{e^2 \left(e (a e-b d)+c d^2\right) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2}+\frac{e^2 (2 c d-b e) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d}\right)}{\left(e (a e-b d)+c d^2\right)^2}","-\frac{c x \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \, _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(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c x \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \, _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(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)}",1,"(x*((c*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*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*(-2*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*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]) + (e^2*(2*c*d - b*e)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d + (e^2*(c*d^2 + e*(-(b*d) + a*e))*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^2))/(c*d^2 + e*(-(b*d) + a*e))^2","A",1
74,1,509,552,1.7442409,"\int \frac{1}{\left(d+e x^n\right)^3 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/((d + e*x^n)^3*(a + b*x^n + c*x^(2*n))),x]","\frac{x \left(\frac{e^2 \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d}+\frac{c \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)-c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)-2 c^3 d^3\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}-2 a e-b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}-b\right)+2 c^3 d^3\right) \, _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{e^2 (2 c d-b e) \left(e (a e-b d)+c d^2\right) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2}+\frac{e^2 \left(e (a e-b d)+c d^2\right)^2 \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^3}\right)}{\left(e (a e-b d)+c d^2\right)^3}","\frac{e^2 x \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \, _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(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(-d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)-b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+2 c^3 d^3\right) \, _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(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^3 \left(a e^2-b d e+c d^2\right)}",1,"(x*((c*(-2*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) - c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*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*(2*c^3*d^3 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*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]) + (e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d + (e^2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^2 + (e^2*(c*d^2 + e*(-(b*d) + a*e))^2*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d^3))/(c*d^2 + e*(-(b*d) + a*e))^3","A",1
75,1,5537,750,6.9571546,"\int \frac{\left(d+e x^n\right)^3}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","\frac{x \left(-\left(x^n \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right)-a b e \left(a e^2+3 c d^2\right)-2 a c d \left(c d^2-3 a e^2\right)+b^2 c d^3\right)}{a c n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}+\frac{e^2 x \left(\frac{6 c d-3 b e}{\sqrt{b^2-4 a c}}+e\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)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{e^2 x \left(e-\frac{3 (2 c d-b e)}{\sqrt{b^2-4 a c}}\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)}{c \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left((1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)+\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}\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 c n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left((1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)-\frac{-a b^3 e^3 (1-3 n)+b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}\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 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
76,1,2980,543,4.5167235,"\int \frac{\left(d+e x^n\right)^2}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","-\frac{x \left((1-n) \left(a b e^2-4 a c d e+b c d^2\right)-\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}\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}\right)}-\frac{x \left(\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}+(1-n) \left(a b e^2-4 a c d e+b c d^2\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(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(a b e^2-4 a c d e+b c d^2\right)-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{2 e^2 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 e^2 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,"-((x*(-(a*Sqrt[b^2 - 4*a*c]*(b^2*d^2 + 2*a^2*e^2 + b*c*d^2*x^n + a*b*e*(-2*d + e*x^n) - 2*a*c*d*(d + 2*e*x^n))) + (a*b*c*d^2*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) - 2^(2 - n^(-1))*a^2*c*d*e*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)) + (a^2*b*e^2*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) - (a*b*c*d^2*n*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) + 2^(2 - n^(-1))*a^2*c*d*e*n*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)) - (a^2*b*e^2*n*(a + x^n*(b + c*x^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)]/((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)]/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) + (b^2*d^2*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^((1 + n)/n) - (a*c*d^2*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) - (a*b*d*e*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) + (a^2*e^2*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^n^(-1) - (b^2*d^2*n*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/2^((1 + n)/n) + 2^((-1 + n)/n)*a*c*d^2*n*(a + x^n*(b + c*x^n))*(2^(1 + n^(-1))*Sqrt[b^2 - 4*a*c] - ((b + Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1) + ((b - Sqrt[b^2 - 4*a*c])*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)])/((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1))))/(a^2*(b^2 - 4*a*c)^(3/2)*n*(a + x^n*(b + c*x^n))))","B",1
77,1,603,362,5.6903409,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^2,x]","\frac{c x \left(\frac{4 \left(b^2-4 a c\right) \left(-2 a^2 c \left(2 d n+e x^n\right)+a \left(b^2 \left(d n+e x^n\right)+b c x^n \left(-4 d n+3 d+e x^n\right)-2 c^2 d (2 n-1) x^{2 n}\right)+b^2 d (n-1) x^n \left(b+c x^n\right)\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a+x^n \left(b+c x^n\right)\right)}+\frac{2^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \left(b d (n-1) \sqrt{b^2-4 a c}-2 a e (n-1) \sqrt{b^2-4 a c}-2 a b e n+4 a c d (2 n-1)+b^2 (d-d n)\right) \, _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{2^{-1/n} \left(\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \left(b^2 (n-1) \left(d \sqrt{b^2-4 a c}-2 a e\right)+2 a b \left(e n \sqrt{b^2-4 a c}-2 c d (n-1)\right)+4 a c \left(d (1-2 n) \sqrt{b^2-4 a c}+2 a e (n-1)\right)+b^3 d (n-1)\right) \, _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 n \left(4 a c-b^2\right)}","-\frac{c x \left(-b \left(d (1-n) \sqrt{b^2-4 a c}-2 a e n\right)+2 a \left(e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right)-\left(b^2 (d-d 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 \left(d (1-n) \sqrt{b^2-4 a c}+2 a e n\right)+2 a \left(c d (2-4 n)-e (1-n) \sqrt{b^2-4 a c}\right)+b^2 (-d) (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(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(c*x*((4*(b^2 - 4*a*c)*(b^2*d*(-1 + n)*x^n*(b + c*x^n) - 2*a^2*c*(2*d*n + e*x^n) + a*(-2*c^2*d*(-1 + 2*n)*x^(2*n) + b*c*x^n*(3*d - 4*d*n + e*x^n) + b^2*(d*n + e*x^n))))/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(a + x^n*(b + c*x^n))) + ((4*a*c*(Sqrt[b^2 - 4*a*c]*d*(1 - 2*n) + 2*a*e*(-1 + n)) + b^3*d*(-1 + n) + b^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)*(-1 + n) + 2*a*b*(-2*c*d*(-1 + n) + Sqrt[b^2 - 4*a*c]*e*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)) + ((b*Sqrt[b^2 - 4*a*c]*d*(-1 + n) - 2*a*Sqrt[b^2 - 4*a*c]*e*(-1 + n) - 2*a*b*e*n + 4*a*c*d*(-1 + 2*n) + b^2*(d - d*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*(-b^2 + 4*a*c)*n)","A",1
78,1,11767,726,7.222903,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))^2),x]","\text{Result too large to show}","-\frac{c e^2 x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\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)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c e^2 x \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\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)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c x \left((1-n) \left(2 a c e+b^2 (-e)+b c d\right)+\frac{2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^3 (-e) (1-n)+b^2 c d (1-n)}{\sqrt{b^2-4 a c}}\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}\right) \left(a e^2-b d e+c d^2\right)}+\frac{x \left(c x^n \left(2 a c e+b^2 (-e)+b c d\right)+3 a b c e-2 a c^2 d-b^3 e+b^2 c d\right)}{a n \left(b^2-4 a c\right) \left(a e^2-b d e+c d^2\right) \left(a+b x^n+c x^{2 n}\right)}+\frac{c x \left(b^2 (1-n) \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(2 a e (2-3 n)-d (1-n) \sqrt{b^2-4 a c}\right)-2 a c \left(e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right)+b^3 (-e) (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) \left(a e^2-b d e+c d^2\right)}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}",1,"Result too large to show","B",0
79,1,16855,1129,8.0223427,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2),x]","\text{Result too large to show}","\frac{2 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d \left(c d^2-b e d+a e^2\right)^3}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d^2 \left(c d^2-b e d+a e^2\right)^2}-\frac{2 c \left(3 c^2 d^2+b \left(b+\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d+2 \sqrt{b^2-4 a c} d+a e\right)\right) 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) e^2}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{2 c \left(3 c^2 d^2+b \left(b-\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d-2 \sqrt{b^2-4 a c} d+a e\right)\right) 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) e^2}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right)-3 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) 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)}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(3 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) 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)}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}",1,"Result too large to show","B",0
80,1,13018,1707,7.7910108,"\int \frac{\left(d+e x^n\right)^3}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^3,x]","\text{Result too large to show}","\frac{\left(-e (1-n) b^3+\left(3 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)+3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) 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) e^2}{a c \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{\left(-e (1-n) b^3+\left(3 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) 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) e^2}{a c \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{x \left(c \left(-e b^2+3 c d b-2 a c e\right) x^n-6 a c^2 d+3 b^2 c d-b^3 e+a b c e\right) e^2}{a c^2 \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)-\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)+\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}-\frac{x \left(c \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right) x^n+a b^2 c^2 d \left(3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)+4 a^2 c^3 d \left(c d^2-3 a e^2\right) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left(3 c d^2 (2-3 n)-5 a e^2 n\right)-3 a b^3 c e \left(c d^2-3 a e^2 n\right)+b^4 c d \left(c (1-2 n) d^2+6 a e^2 n\right)\right)}{2 a^2 c^2 \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(-\left(\left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right) x^n\right)+b^2 c d^3-2 a c d \left(c d^2-3 a e^2\right)-a b e \left(3 c d^2+a e^2\right)\right)}{2 a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"Result too large to show","B",1
81,1,10910,1191,6.8694488,"\int \frac{\left(d+e x^n\right)^2}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^3,x]","\text{Result too large to show}","-\frac{\left(-\left((1-n) b^2\right)-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) 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) e^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{\left(-\left((1-n) b^2\right)+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) 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) e^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{x \left(b c x^n+b^2-2 a c\right) e^2}{a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{\left((1-n) \left(-\left(\left(c (1-2 n) d^2+2 a e^2 n\right) b^3\right)+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)+\frac{-\left((1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4\right)+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}-\frac{\left((1-n) \left(-\left(\left(c (1-2 n) d^2+2 a e^2 n\right) b^3\right)+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)-\frac{-\left((1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4\right)+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}+\frac{x \left(c \left(-\left(\left(c (1-2 n) d^2+2 a e^2 n\right) b^3\right)+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right) x^n+2 a b^3 c d e-a b^2 c \left(a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)-4 a^2 c^2 \left(c d^2-a e^2\right) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left(c (1-2 n) d^2+2 a e^2 n\right)\right)}{2 a^2 c \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(\left(b c d^2-4 a c e d+a b e^2\right) x^n+b^2 d^2-2 a b d e-2 a \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"Result too large to show","B",1
82,1,8593,713,6.5957429,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^3,x]","\text{Result too large to show}","\frac{x \left(c x^n \left(-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right)-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(a+b x^n+c x^{2 n}\right)}+\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}+2 c d \left(8 n^2-6 n+1\right)\right)-2 a b c \left(2 a e \left(-3 n^2-n+1\right)-d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right)+b^3 (1-n) \left(a e-d (1-2 n) \sqrt{b^2-4 a c}\right)+b^4 (-d) \left(2 n^2-3 n+1\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)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}-2 c d \left(8 n^2-6 n+1\right)\right)+2 a b c \left(d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}+2 a e \left(-3 n^2-n+1\right)\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right)-b^3 (1-n) \left(d (1-2 n) \sqrt{b^2-4 a c}+a e\right)+b^4 d \left(2 n^2-3 n+1\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)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{2 a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)^2}",1,"Result too large to show","B",1
83,1,43535,1708,8.5318146,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))^3),x]","\text{Result too large to show}","\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) 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) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) 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) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(-e (1-n) b^3+\left(c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) 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) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(-e (1-n) b^3+\left(c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) 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) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}-\frac{c \left(-e \left(2 n^2-3 n+1\right) b^5+\left(c d-\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4+c \left(a e (7-18 n)+\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)+2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)-2 c d \left(8 n^2-6 n+1\right)\right)\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{c \left(e \left(2 n^2-3 n+1\right) b^5-\left(c d+\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4-c \left(a e (7-18 n)-\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)-2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)+2 c d \left(8 n^2-6 n+1\right)\right)\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{x \left(-c \left(-e (1-2 n) b^4+c d (1-2 n) b^3+a c e (5-14 n) b^2-2 a c^2 d (2-7 n) b-4 a^2 c^2 e (1-3 n)\right) x^n+2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right) n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"Result too large to show","B",0
84,1,56566,2446,9.9296581,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3),x]","\text{Result too large to show}","\frac{3 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^4}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d^2 \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(10 c^2 d^2+3 b \left(b+\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d+3 \sqrt{b^2-4 a c} d+a e\right)\right) 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) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}-\frac{c \left(10 c^2 d^2+3 b \left(b-\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d-3 \sqrt{b^2-4 a c} d+a e\right)\right) 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) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d-2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right)-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) 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) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d+2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(5 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) 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) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}-\frac{x \left(c \left(-2 e^2 b^3+5 c d e b^2-c \left(3 c d^2-5 a e^2\right) b-8 a c^2 d e\right) x^n-2 b^4 e^2-14 a b c^2 d e+5 b^3 c d e-b^2 c \left(3 c d^2-7 a e^2\right)+2 a c^2 \left(3 c d^2-a e^2\right)\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n \left(b x^n+c x^{2 n}+a\right)}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)-\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)+\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) 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)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}-\frac{x \left(c \left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) x^n+a b^2 c^2 \left(a e^2 (13-37 n)-5 c d^2 (1-3 n)\right)-b^4 c \left(a e^2 (7-17 n)-c d^2 (1-2 n)\right)-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left(c d^2-a e^2\right) (1-4 n)+b^6 e^2 (1-2 n)-2 b^5 c d e (1-2 n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)^2}",1,"Result too large to show","B",0
85,1,424,292,1.6102439,"\int \left(d+e x^n\right) \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[(d + e*x^n)*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x \left(2 (n+1) \left(a n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} (2 c (2 d n+d)-b e) F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+\left(a+x^n \left(b+c x^n\right)\right) \left(b e n+2 c \left(2 d n+d+e (n+1) x^n\right)\right)\right)-n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right) \left(-4 a c e (n+1)+b^2 e (n+2)-2 b c d (2 n+1)\right)\right)}{4 (n+1)^2 (2 c n+c) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{1}{2},-\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(-(n*(-4*a*c*e*(1 + n) + b^2*e*(2 + n) - 2*b*c*d*(1 + 2*n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]) + 2*(1 + n)*((a + x^n*(b + c*x^n))*(b*e*n + 2*c*(d + 2*d*n + e*(1 + n)*x^n)) + a*n*(-(b*e) + 2*c*(d + 2*d*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(4*(1 + n)^2*(c + 2*c*n)*Sqrt[a + x^n*(b + c*x^n)])","A",0
86,1,690,294,4.525237,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x \left(3 n^2 x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right) \left(16 a^2 c^2 e \left(3 n^2+4 n+1\right)-4 a b^2 c e \left(6 n^2+14 n+5\right)+8 a b c^2 d \left(12 n^2+11 n+2\right)+b^4 e \left(3 n^2+8 n+4\right)-2 b^3 c d \left(4 n^2+9 n+2\right)\right)+2 (n+1) \left(3 a n^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right) \left(-4 a b c e (5 n+2)+8 a c^2 d \left(8 n^2+6 n+1\right)+b^3 e (3 n+2)-2 b^2 c d (4 n+1)\right)+\left(a+x^n \left(b+c x^n\right)\right) \left(4 a c \left(3 b e (5 n+2) n^2+2 c \left(d (2 n+1) (4 n+1)^2+e \left(15 n^3+23 n^2+9 n+1\right) x^n\right)\right)-3 b^3 e n^2 (3 n+2)+6 b^2 c n^2 \left(4 d n+d+e (n+1) x^n\right)+4 b c^2 (n+1) x^n \left(d \left(28 n^2+15 n+2\right)+e \left(18 n^2+13 n+2\right) x^n\right)+8 c^3 \left(2 n^2+3 n+1\right) x^{2 n} \left(4 d n+d+e (3 n+1) x^n\right)\right)\right)\right)}{16 c^2 (n+1)^2 (2 n+1) (3 n+1) (4 n+1) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{a e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{3}{2},-\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(3*n^2*(16*a^2*c^2*e*(1 + 4*n + 3*n^2) + b^4*e*(4 + 8*n + 3*n^2) - 2*b^3*c*d*(2 + 9*n + 4*n^2) - 4*a*b^2*c*e*(5 + 14*n + 6*n^2) + 8*a*b*c^2*d*(2 + 11*n + 12*n^2))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(1 + n)*((a + x^n*(b + c*x^n))*(-3*b^3*e*n^2*(2 + 3*n) + 6*b^2*c*n^2*(d + 4*d*n + e*(1 + n)*x^n) + 8*c^3*(1 + 3*n + 2*n^2)*x^(2*n)*(d + 4*d*n + e*(1 + 3*n)*x^n) + 4*b*c^2*(1 + n)*x^n*(d*(2 + 15*n + 28*n^2) + e*(2 + 13*n + 18*n^2)*x^n) + 4*a*c*(3*b*e*n^2*(2 + 5*n) + 2*c*(d*(1 + 2*n)*(1 + 4*n)^2 + e*(1 + 9*n + 23*n^2 + 15*n^3)*x^n))) + 3*a*n^2*(b^3*e*(2 + 3*n) - 2*b^2*c*d*(1 + 4*n) - 4*a*b*c*e*(2 + 5*n) + 8*a*c^2*d*(1 + 6*n + 8*n^2))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(16*c^2*(1 + n)^2*(1 + 2*n)*(1 + 3*n)*(1 + 4*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
87,1,245,292,0.33014,"\int \frac{d+e x^n}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[(d + e*x^n)/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} \left(d (n+1) F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e x^n F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{(n+1) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(x*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*(e*x^n*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + d*(1 + n)*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/((1 + n)*Sqrt[a + x^n*(b + c*x^n)])","A",0
88,1,414,298,1.479024,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x \left(2 c x^n (b d-2 a e) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-(n+1) \left(\sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right) \left(2 a b e-4 a c d (n-1)+b^2 d (n-2)\right)+2 \left(b \left(c d x^n-a e\right)-2 a c \left(d+e x^n\right)+b^2 d\right)\right)\right)}{a n (n+1) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{3}{2},\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{3}{2},\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a (n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(x*(2*c*(b*d - 2*a*e)*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (1 + n)*(2*(b^2*d + b*(-(a*e) + c*d*x^n) - 2*a*c*(d + e*x^n)) + (2*a*b*e + b^2*d*(-2 + n) - 4*a*c*d*(-1 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(a*(-b^2 + 4*a*c)*n*(1 + n)*Sqrt[a + x^n*(b + c*x^n)])","A",0
89,1,6752,298,6.5664856,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^{5/2}} \, dx","Integrate[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^(5/2),x]","\text{Result too large to show}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{5}{2},\frac{5}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{5}{2},\frac{5}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 (n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"Result too large to show","B",0
90,0,0,29,0.2859338,"\int \left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p,x]","\int \left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p \, dx","\text{Int}\left(\left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p,x\right)",0,"Integrate[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]","A",-1
91,1,438,606,1.0687872,"\int \left(d+e x^n\right)^3 \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^3*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p \left((n+1) \left((2 n+1) \left(d^3 (3 n+1) F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e^3 x^{3 n} F_1\left(3+\frac{1}{n};-p,-p;4+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)+3 d e^2 (3 n+1) x^{2 n} F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)+3 d^2 e \left(6 n^2+5 n+1\right) x^n F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{(n+1) (2 n+1) (3 n+1)}","d^3 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{3 d^2 e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+\frac{3 d e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(3+\frac{1}{n};-p,-p;4+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 n+1}",1,"(x*(a + x^n*(b + c*x^n))^p*(3*d^2*e*(1 + 5*n + 6*n^2)*x^n*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + n)*(3*d*e^2*(1 + 3*n)*x^(2*n)*AppellF1[2 + n^(-1), -p, -p, 3 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + 2*n)*(e^3*x^(3*n)*AppellF1[3 + n^(-1), -p, -p, 4 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + d^3*(1 + 3*n)*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))))/((1 + n)*(1 + 2*n)*(1 + 3*n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
92,1,338,447,0.7454682,"\int \left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p \left((n+1) \left(d^2 (2 n+1) F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e^2 x^{2 n} F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)+2 d e (2 n+1) x^n F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{(n+1) (2 n+1)}","d^2 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{2 d e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}",1,"(x*(a + x^n*(b + c*x^n))^p*(2*d*e*(1 + 2*n)*x^n*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + n)*(e^2*x^(2*n)*AppellF1[2 + n^(-1), -p, -p, 3 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + d^2*(1 + 2*n)*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/((1 + n)*(1 + 2*n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
93,1,243,288,0.4584672,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p \left(d (n+1) F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+e x^n F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{n+1}","d x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}",1,"(x*(a + x^n*(b + c*x^n))^p*(e*x^n*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + d*(1 + n)*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/((1 + n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
94,0,0,29,0.1511075,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n),x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n},x\right)",0,"Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n), x]","A",-1
95,0,0,29,0.1894417,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^2,x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2},x\right)",0,"Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^2, x]","A",-1
96,0,0,29,0.7278403,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^3,x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3},x\right)",0,"Integrate[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^3, x]","A",-1