1,1,53,72,0.1574773,"\int \frac{c+d x+e x^2}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2)/Sqrt[a + b*x],x]","\frac{2 \sqrt{a+b x} \left(8 a^2 e-2 a b (5 d+2 e x)+b^2 (15 c+x (5 d+3 e x))\right)}{15 b^3}","\frac{2 \sqrt{a+b x} \left(a^2 e-a b d+b^2 c\right)}{b^3}+\frac{2 (a+b x)^{3/2} (b d-2 a e)}{3 b^3}+\frac{2 e (a+b x)^{5/2}}{5 b^3}",1,"(2*Sqrt[a + b*x]*(8*a^2*e - 2*a*b*(5*d + 2*e*x) + b^2*(15*c + x*(5*d + 3*e*x))))/(15*b^3)","A",1
2,1,155,161,0.291845,"\int \frac{\left(c+d x+e x^2\right)^2}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2)^2/Sqrt[a + b*x],x]","\frac{2 \sqrt{a+b x} \left(128 a^4 e^2-32 a^3 b e (9 d+2 e x)+24 a^2 b^2 \left(2 e \left(7 c+e x^2\right)+7 d^2+6 d e x\right)-4 a b^3 \left(21 c (5 d+2 e x)+x \left(21 d^2+27 d e x+10 e^2 x^2\right)\right)+b^4 \left(315 c^2+42 c x (5 d+3 e x)+x^2 \left(63 d^2+90 d e x+35 e^2 x^2\right)\right)\right)}{315 b^5}","-\frac{2 (a+b x)^{5/2} \left(-6 a^2 e^2+6 a b d e-\left(b^2 \left(2 c e+d^2\right)\right)\right)}{5 b^5}+\frac{4 (a+b x)^{3/2} (b d-2 a e) \left(a^2 e-a b d+b^2 c\right)}{3 b^5}+\frac{2 \sqrt{a+b x} \left(a^2 e-a b d+b^2 c\right)^2}{b^5}+\frac{4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac{2 e^2 (a+b x)^{9/2}}{9 b^5}",1,"(2*Sqrt[a + b*x]*(128*a^4*e^2 - 32*a^3*b*e*(9*d + 2*e*x) + 24*a^2*b^2*(7*d^2 + 6*d*e*x + 2*e*(7*c + e*x^2)) - 4*a*b^3*(21*c*(5*d + 2*e*x) + x*(21*d^2 + 27*d*e*x + 10*e^2*x^2)) + b^4*(315*c^2 + 42*c*x*(5*d + 3*e*x) + x^2*(63*d^2 + 90*d*e*x + 35*e^2*x^2))))/(315*b^5)","A",1
3,1,294,274,1.0292546,"\int \frac{\left(c+d x+e x^2\right)^3}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2)^3/Sqrt[a + b*x],x]","\frac{2 \sqrt{a+b x} (c+x (d+e x))^3}{b}-\frac{4 (a+b x)^{3/2} \left(-2560 a^5 e^3+640 a^4 b e^2 (13 d+6 e x)-64 a^3 b^2 e \left(e \left(143 c+75 e x^2\right)+143 d^2+195 d e x\right)+8 a^2 b^3 \left(78 d e \left(33 c+25 e x^2\right)+4 e^2 x \left(429 c+175 e x^2\right)+429 d^3+1716 d^2 e x\right)-4 a b^4 \left(3003 c^2 e+429 c \left(7 d^2+18 d e x+10 e^2 x^2\right)+x \left(1287 d^3+4290 d^2 e x+4550 d e^2 x^2+1575 e^3 x^3\right)\right)+b^5 \left(3003 c^2 (5 d+6 e x)+286 c x \left(63 d^2+135 d e x+70 e^2 x^2\right)+5 x^2 \left(1287 d^3+4004 d^2 e x+4095 d e^2 x^2+1386 e^3 x^3\right)\right)\right)}{15015 b^7}","-\frac{2 e (a+b x)^{9/2} \left(-5 a^2 e^2+5 a b d e-\left(b^2 \left(c e+d^2\right)\right)\right)}{3 b^7}-\frac{2 (a+b x)^{7/2} (b d-2 a e) \left(-10 a^2 e^2+10 a b d e-\left(b^2 \left(6 c e+d^2\right)\right)\right)}{7 b^7}-\frac{6 (a+b x)^{5/2} \left(a^2 e-a b d+b^2 c\right) \left(-5 a^2 e^2+5 a b d e-\left(b^2 \left(c e+d^2\right)\right)\right)}{5 b^7}+\frac{2 (a+b x)^{3/2} (b d-2 a e) \left(a^2 e-a b d+b^2 c\right)^2}{b^7}+\frac{2 \sqrt{a+b x} \left(a^2 e-a b d+b^2 c\right)^3}{b^7}+\frac{6 e^2 (a+b x)^{11/2} (b d-2 a e)}{11 b^7}+\frac{2 e^3 (a+b x)^{13/2}}{13 b^7}",1,"(2*Sqrt[a + b*x]*(c + x*(d + e*x))^3)/b - (4*(a + b*x)^(3/2)*(-2560*a^5*e^3 + 640*a^4*b*e^2*(13*d + 6*e*x) - 64*a^3*b^2*e*(143*d^2 + 195*d*e*x + e*(143*c + 75*e*x^2)) + 8*a^2*b^3*(429*d^3 + 1716*d^2*e*x + 78*d*e*(33*c + 25*e*x^2) + 4*e^2*x*(429*c + 175*e*x^2)) + b^5*(3003*c^2*(5*d + 6*e*x) + 286*c*x*(63*d^2 + 135*d*e*x + 70*e^2*x^2) + 5*x^2*(1287*d^3 + 4004*d^2*e*x + 4095*d*e^2*x^2 + 1386*e^3*x^3)) - 4*a*b^4*(3003*c^2*e + 429*c*(7*d^2 + 18*d*e*x + 10*e^2*x^2) + x*(1287*d^3 + 4290*d^2*e*x + 4550*d*e^2*x^2 + 1575*e^3*x^3))))/(15015*b^7)","A",1
4,1,82,114,0.1750292,"\int \frac{c+d x+e x^2+f x^3}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x],x]","\frac{2 \sqrt{a+b x} \left(-48 a^3 f+8 a^2 b (7 e+3 f x)-2 a b^2 (35 d+x (14 e+9 f x))+b^3 (105 c+x (35 d+3 x (7 e+5 f x)))\right)}{105 b^4}","\frac{2 (a+b x)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}+\frac{2 \sqrt{a+b x} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^4}+\frac{2 (a+b x)^{5/2} (b e-3 a f)}{5 b^4}+\frac{2 f (a+b x)^{7/2}}{7 b^4}",1,"(2*Sqrt[a + b*x]*(-48*a^3*f + 8*a^2*b*(7*e + 3*f*x) - 2*a*b^2*(35*d + x*(14*e + 9*f*x)) + b^3*(105*c + x*(35*d + 3*x*(7*e + 5*f*x)))))/(105*b^4)","A",1
5,1,303,320,0.5650552,"\int \frac{\left(c+d x+e x^2+f x^3\right)^2}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]","\frac{2 \left(-\frac{1}{9} (a+b x)^{9/2} \left(-15 a^2 f^2+10 a b e f-\left(b^2 \left(2 d f+e^2\right)\right)\right)+\frac{2}{7} (a+b x)^{7/2} \left(-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left(2 d f+e^2\right)+b^3 (c f+d e)\right)+\frac{2}{3} (a+b x)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+\sqrt{a+b x} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^2+\frac{1}{5} (a+b x)^{5/2} \left(15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left(2 d f+e^2\right)-6 a b^3 (c f+d e)+b^4 \left(2 c e+d^2\right)\right)+\frac{2}{11} f (a+b x)^{11/2} (b e-3 a f)+\frac{1}{13} f^2 (a+b x)^{13/2}\right)}{b^7}","-\frac{2 (a+b x)^{9/2} \left(-15 a^2 f^2+10 a b e f-\left(b^2 \left(2 d f+e^2\right)\right)\right)}{9 b^7}+\frac{4 (a+b x)^{7/2} \left(-10 a^3 f^2+10 a^2 b e f-2 a b^2 \left(2 d f+e^2\right)+b^3 (c f+d e)\right)}{7 b^7}+\frac{4 (a+b x)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^7}+\frac{2 \sqrt{a+b x} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^2}{b^7}+\frac{2 (a+b x)^{5/2} \left(15 a^4 f^2-20 a^3 b e f+6 a^2 b^2 \left(2 d f+e^2\right)-6 a b^3 (c f+d e)+b^4 \left(2 c e+d^2\right)\right)}{5 b^7}+\frac{4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7}",1,"(2*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x] + (2*(b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a + b*x)^(3/2))/3 + ((b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e + c*f) + 6*a^2*b^2*(e^2 + 2*d*f))*(a + b*x)^(5/2))/5 + (2*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d*e + c*f) - 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2))/7 - ((10*a*b*e*f - 15*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)^(9/2))/9 + (2*f*(b*e - 3*a*f)*(a + b*x)^(11/2))/11 + (f^2*(a + b*x)^(13/2))/13))/b^7","A",1
6,1,678,708,2.8935737,"\int \frac{\left(c+d x+e x^2+f x^3\right)^3}{\sqrt{a+b x}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)^3/Sqrt[a + b*x],x]","\frac{2 \left(\frac{1}{5} f (a+b x)^{15/2} \left(12 a^2 f^2-8 a b e f+b^2 \left(d f+e^2\right)\right)+\frac{1}{13} (a+b x)^{13/2} \left(-84 a^3 f^3+84 a^2 b e f^2-21 a b^2 f \left(d f+e^2\right)+b^3 \left(3 c f^2+6 d e f+e^3\right)\right)+(a+b x)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^2+\sqrt{a+b x} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^3+\frac{3}{5} (a+b x)^{5/2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right) \left(12 a^4 f^2-16 a^3 b e f+a^2 b^2 \left(9 d f+5 e^2\right)-a b^3 (3 c f+5 d e)+b^4 \left(c e+d^2\right)\right)+\frac{3}{11} (a+b x)^{11/2} \left(42 a^4 f^3-56 a^3 b e f^2+21 a^2 b^2 f \left(d f+e^2\right)-2 a b^3 \left(3 c f^2+6 d e f+e^3\right)+b^4 \left(2 c e f+d^2 f+d e^2\right)\right)+\frac{1}{3} (a+b x)^{9/2} \left(-42 a^5 f^3+70 a^4 b e f^2-35 a^3 b^2 f \left(d f+e^2\right)+5 a^2 b^3 \left(3 c f^2+6 d e f+e^3\right)-5 a b^4 \left(2 c e f+d^2 f+d e^2\right)+b^5 \left(2 c d f+c e^2+d^2 e\right)\right)+\frac{1}{7} (a+b x)^{7/2} \left(84 a^6 f^3-168 a^5 b e f^2+105 a^4 b^2 f \left(d f+e^2\right)-20 a^3 b^3 \left(3 c f^2+6 d e f+e^3\right)+30 a^2 b^4 \left(2 c e f+d^2 f+d e^2\right)-12 a b^5 \left(2 c d f+c e^2+d^2 e\right)+b^6 \left(3 c^2 f+6 c d e+d^3\right)\right)+\frac{3}{17} f^2 (a+b x)^{17/2} (b e-3 a f)+\frac{1}{19} f^3 (a+b x)^{19/2}\right)}{b^{10}}","-\frac{2 f (a+b x)^{15/2} \left(-12 a^2 f^2+8 a b e f-\left(b^2 \left(d f+e^2\right)\right)\right)}{5 b^{10}}+\frac{2 (a+b x)^{13/2} \left(-84 a^3 f^3+84 a^2 b e f^2-21 a b^2 f \left(d f+e^2\right)+b^3 \left(3 c f^2+6 d e f+e^3\right)\right)}{13 b^{10}}+\frac{2 (a+b x)^{3/2} \left(3 a^2 f-2 a b e+b^2 d\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^2}{b^{10}}+\frac{2 \sqrt{a+b x} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)^3}{b^{10}}+\frac{6 (a+b x)^{5/2} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right) \left(12 a^4 f^2-16 a^3 b e f+a^2 b^2 \left(9 d f+5 e^2\right)-a b^3 (3 c f+5 d e)+b^4 \left(c e+d^2\right)\right)}{5 b^{10}}-\frac{6 (a+b x)^{11/2} \left(-42 a^4 f^3+56 a^3 b e f^2-21 a^2 b^2 f \left(d f+e^2\right)+2 a b^3 \left(3 c f^2+6 d e f+e^3\right)-b^4 \left(2 c e f+d^2 f+d e^2\right)\right)}{11 b^{10}}+\frac{2 (a+b x)^{9/2} \left(-42 a^5 f^3+70 a^4 b e f^2-35 a^3 b^2 f \left(d f+e^2\right)+5 a^2 b^3 \left(3 c f^2+6 d e f+e^3\right)-5 a b^4 \left(2 c e f+d^2 f+d e^2\right)+b^5 \left(2 c d f+c e^2+d^2 e\right)\right)}{3 b^{10}}-\frac{2 (a+b x)^{7/2} \left(-84 a^6 f^3+168 a^5 b e f^2-105 a^4 b^2 f \left(d f+e^2\right)+20 a^3 b^3 \left(3 c f^2+6 d e f+e^3\right)-30 a^2 b^4 \left(2 c e f+d^2 f+d e^2\right)+12 a b^5 \left(2 c d f+c e^2+d^2 e\right)-b^6 \left(3 c^2 f+6 c d e+d^3\right)\right)}{7 b^{10}}+\frac{6 f^2 (a+b x)^{17/2} (b e-3 a f)}{17 b^{10}}+\frac{2 f^3 (a+b x)^{19/2}}{19 b^{10}}",1,"(2*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)^3*Sqrt[a + b*x] + (b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*(a + b*x)^(3/2) + (3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(b^4*(d^2 + c*e) - 16*a^3*b*e*f + 12*a^4*f^2 - a*b^3*(5*d*e + 3*c*f) + a^2*b^2*(5*e^2 + 9*d*f))*(a + b*x)^(5/2))/5 + ((-168*a^5*b*e*f^2 + 84*a^6*f^3 + b^6*(d^3 + 6*c*d*e + 3*c^2*f) + 105*a^4*b^2*f*(e^2 + d*f) - 12*a*b^5*(d^2*e + c*e^2 + 2*c*d*f) + 30*a^2*b^4*(d*e^2 + d^2*f + 2*c*e*f) - 20*a^3*b^3*(e^3 + 6*d*e*f + 3*c*f^2))*(a + b*x)^(7/2))/7 + ((70*a^4*b*e*f^2 - 42*a^5*f^3 - 35*a^3*b^2*f*(e^2 + d*f) + b^5*(d^2*e + c*e^2 + 2*c*d*f) - 5*a*b^4*(d*e^2 + d^2*f + 2*c*e*f) + 5*a^2*b^3*(e^3 + 6*d*e*f + 3*c*f^2))*(a + b*x)^(9/2))/3 + (3*(-56*a^3*b*e*f^2 + 42*a^4*f^3 + 21*a^2*b^2*f*(e^2 + d*f) + b^4*(d*e^2 + d^2*f + 2*c*e*f) - 2*a*b^3*(e^3 + 6*d*e*f + 3*c*f^2))*(a + b*x)^(11/2))/11 + ((84*a^2*b*e*f^2 - 84*a^3*f^3 - 21*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f + 3*c*f^2))*(a + b*x)^(13/2))/13 + (f*(-8*a*b*e*f + 12*a^2*f^2 + b^2*(e^2 + d*f))*(a + b*x)^(15/2))/5 + (3*f^2*(b*e - 3*a*f)*(a + b*x)^(17/2))/17 + (f^3*(a + b*x)^(19/2))/19))/b^10","A",1
7,1,124,161,0.0833946,"\int \frac{c+d x}{a+b x^3} \, dx","Integrate[(c + d*x)/(a + b*x^3),x]","\frac{\left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \left(2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)\right)-2 \sqrt{3} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 a^{2/3} b^{2/3}}","-\frac{\left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{2/3}}",1,"(-2*Sqrt[3]*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (b^(1/3)*c - a^(1/3)*d)*(2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*a^(2/3)*b^(2/3))","A",1
8,1,180,189,0.2304649,"\int \frac{c+d x}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x)/(a + b*x^3)^2,x]","\frac{\frac{\left(a^{2/3} d-2 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 \left(2 \sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 a x (c+d x)}{a+b x^3}}{18 a^2}","-\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{2/3}}+\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left(a+b x^3\right)}",1,"((6*a*x*(c + d*x))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(2*a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-2*a^(1/3)*b^(1/3)*c + a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(18*a^2)","A",1
9,1,205,215,0.2541314,"\int \frac{c+d x}{\left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x)/(a + b*x^3)^3,x]","\frac{\frac{\left(2 a^{2/3} d-5 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 \left(5 \sqrt[3]{a} \sqrt[3]{b} c-2 a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{9 a^2 x (c+d x)}{\left(a+b x^3\right)^2}-\frac{2 \sqrt{3} \sqrt[3]{a} \left(2 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{3 a x (5 c+4 d x)}{a+b x^3}}{54 a^3}","-\frac{\left(5 \sqrt[3]{b} c-2 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{8/3} b^{2/3}}+\frac{\left(5 \sqrt[3]{b} c-2 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{8/3} b^{2/3}}-\frac{\left(2 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left(a+b x^3\right)}+\frac{x (c+d x)}{6 a \left(a+b x^3\right)^2}",1,"((9*a^2*x*(c + d*x))/(a + b*x^3)^2 + (3*a*x*(5*c + 4*d*x))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(5*b^(1/3)*c + 2*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(5*a^(1/3)*b^(1/3)*c - 2*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-5*a^(1/3)*b^(1/3)*c + 2*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(54*a^3)","A",1
10,1,229,240,0.2285088,"\int \frac{c+d x}{\left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x)/(a + b*x^3)^4,x]","\frac{\frac{2 \left(7 a^{2/3} d-20 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{4 \left(20 \sqrt[3]{a} \sqrt[3]{b} c-7 a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{54 a^3 x (c+d x)}{\left(a+b x^3\right)^3}+\frac{9 a^2 x (8 c+7 d x)}{\left(a+b x^3\right)^2}-\frac{4 \sqrt{3} \sqrt[3]{a} \left(7 \sqrt[3]{a} d+20 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4}","-\frac{\left(20 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{11/3} b^{2/3}}+\frac{2 \left(20 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{11/3} b^{2/3}}-\frac{2 \left(7 \sqrt[3]{a} d+20 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left(a+b x^3\right)}+\frac{x (8 c+7 d x)}{54 a^2 \left(a+b x^3\right)^2}+\frac{x (c+d x)}{9 a \left(a+b x^3\right)^3}",1,"((54*a^3*x*(c + d*x))/(a + b*x^3)^3 + (9*a^2*x*(8*c + 7*d*x))/(a + b*x^3)^2 + (12*a*x*(10*c + 7*d*x))/(a + b*x^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(20*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/3)*c + 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(486*a^4)","A",1
11,1,125,161,0.0732769,"\int \frac{a+b x}{d+e x^3} \, dx","Integrate[(a + b*x)/(d + e*x^3),x]","\frac{-\left(b \sqrt[3]{d}-a \sqrt[3]{e}\right) \left(2 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)-\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)\right)-2 \sqrt{3} \left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right)}{6 d^{2/3} e^{2/3}}","-\frac{\left(a-\frac{b \sqrt[3]{d}}{\sqrt[3]{e}}\right) \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)}{6 d^{2/3} \sqrt[3]{e}}-\frac{\left(b \sqrt[3]{d}-a \sqrt[3]{e}\right) \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{3 d^{2/3} e^{2/3}}-\frac{\left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right)}{\sqrt{3} d^{2/3} e^{2/3}}",1,"(-2*Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - (b*d^(1/3) - a*e^(1/3))*(2*Log[d^(1/3) + e^(1/3)*x] - Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2]))/(6*d^(2/3)*e^(2/3))","A",1
12,1,125,161,0.0582407,"\int \frac{a+b x}{d-e x^3} \, dx","Integrate[(a + b*x)/(d - e*x^3),x]","\frac{-\left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \left(2 \log \left(\sqrt[3]{d}-\sqrt[3]{e} x\right)-\log \left(d^{2/3}+\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)\right)-2 \sqrt{3} \left(b \sqrt[3]{d}-a \sqrt[3]{e}\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}+1}{\sqrt{3}}\right)}{6 d^{2/3} e^{2/3}}","\frac{\left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \log \left(d^{2/3}+\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)}{6 d^{2/3} e^{2/3}}-\frac{\left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \log \left(\sqrt[3]{d}-\sqrt[3]{e} x\right)}{3 d^{2/3} e^{2/3}}-\frac{\left(b \sqrt[3]{d}-a \sqrt[3]{e}\right) \tan ^{-1}\left(\frac{\sqrt[3]{d}+2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right)}{\sqrt{3} d^{2/3} e^{2/3}}",1,"(-2*Sqrt[3]*(b*d^(1/3) - a*e^(1/3))*ArcTan[(1 + (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - (b*d^(1/3) + a*e^(1/3))*(2*Log[d^(1/3) - e^(1/3)*x] - Log[d^(2/3) + d^(1/3)*e^(1/3)*x + e^(2/3)*x^2]))/(6*d^(2/3)*e^(2/3))","A",1
13,1,19,19,0.0071788,"\int \frac{1+x}{1+x^3} \, dx","Integrate[(1 + x)/(1 + x^3),x]","\frac{2 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{2 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(2*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3]","A",1
14,1,19,19,0.0061138,"\int \frac{1-x}{1-x^3} \, dx","Integrate[(1 - x)/(1 - x^3),x]","\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(2*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3]","A",1
15,1,22,22,0.0043577,"\int \frac{1+x}{1-x^3} \, dx","Integrate[(1 + x)/(1 - x^3),x]","\frac{1}{3} \log \left(x^2+x+1\right)-\frac{2}{3} \log (1-x)","\frac{1}{3} \log \left(x^2+x+1\right)-\frac{2}{3} \log (1-x)",1,"(-2*Log[1 - x])/3 + Log[1 + x + x^2]/3","A",1
16,1,22,22,0.0051615,"\int \frac{1-x}{1+x^3} \, dx","Integrate[(1 - x)/(1 + x^3),x]","\frac{2}{3} \log (x+1)-\frac{1}{3} \log \left(x^2-x+1\right)","\frac{2}{3} \log (x+1)-\frac{1}{3} \log \left(x^2-x+1\right)",1,"(2*Log[1 + x])/3 - Log[1 - x + x^2]/3","A",1
17,1,41,41,0.0094075,"\int \frac{3-x}{1-x^3} \, dx","Integrate[(3 - x)/(1 - x^3),x]","\frac{1}{3} \log \left(x^2+x+1\right)-\frac{2}{3} \log (1-x)+\frac{4 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{3} \log \left(x^2+x+1\right)-\frac{2}{3} \log (1-x)+\frac{4 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(4*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] - (2*Log[1 - x])/3 + Log[1 + x + x^2]/3","A",1
18,1,31,29,0.011768,"\int \frac{c+d x}{c^3+d^3 x^3} \, dx","Integrate[(c + d*x)/(c^3 + d^3*x^3),x]","\frac{2 \tan ^{-1}\left(\frac{2 d x-c}{\sqrt{3} c}\right)}{\sqrt{3} c d}","-\frac{2 \tan ^{-1}\left(\frac{c-2 d x}{\sqrt{3} c}\right)}{\sqrt{3} c d}",1,"(2*ArcTan[(-c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)","A",1
19,1,29,29,0.0253663,"\int \frac{c-d x}{c^3-d^3 x^3} \, dx","Integrate[(c - d*x)/(c^3 - d^3*x^3),x]","\frac{2 \tan ^{-1}\left(\frac{c+2 d x}{\sqrt{3} c}\right)}{\sqrt{3} c d}","\frac{2 \tan ^{-1}\left(\frac{c+2 d x}{\sqrt{3} c}\right)}{\sqrt{3} c d}",1,"(2*ArcTan[(c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)","A",1
20,1,35,39,0.0157996,"\int \frac{\sqrt[3]{a} \sqrt[3]{b} B+b^{2/3} B x}{a+b x^3} \, dx","Integrate[(a^(1/3)*b^(1/3)*B + b^(2/3)*B*x)/(a + b*x^3),x]","-\frac{2 B \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{a}}","-\frac{2 B \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a}}",1,"(-2*B*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(1/3))","A",1
21,1,129,41,0.0661241,"\int \frac{\sqrt[3]{a} \sqrt[3]{-b} B-(-b)^{2/3} B x}{a+b x^3} \, dx","Integrate[(a^(1/3)*(-b)^(1/3)*B - (-b)^(2/3)*B*x)/(a + b*x^3),x]","\frac{\sqrt[3]{-b} B \left(\left(\sqrt[3]{-b}+\sqrt[3]{b}\right) \left(2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)\right)+2 \sqrt{3} \left(\sqrt[3]{-b}-\sqrt[3]{b}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 \sqrt[3]{a} b^{2/3}}","\frac{2 B \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{-b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a}}",1,"((-b)^(1/3)*B*(2*Sqrt[3]*((-b)^(1/3) - b^(1/3))*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + ((-b)^(1/3) + b^(1/3))*(2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])))/(6*a^(1/3)*b^(2/3))","B",1
22,1,90,118,0.033279,"\int \left(-\frac{C x^2}{a+b x^3}+\frac{B x+C x^2}{a+b x^3}\right) \, dx","Integrate[-((C*x^2)/(a + b*x^3)) + (B*x + C*x^2)/(a + b*x^3),x]","\frac{B \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 \sqrt[3]{a} b^{2/3}}","\frac{B \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} b^{2/3}}-\frac{B \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{B \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3}}",1,"(B*(-2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*x] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*a^(1/3)*b^(2/3))","A",1
23,1,90,118,0.0159083,"\int \left(-\frac{C x^2}{a+b x^3}+\frac{A+C x^2}{a+b x^3}\right) \, dx","Integrate[-((C*x^2)/(a + b*x^3)) + (A + C*x^2)/(a + b*x^3),x]","-\frac{A \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 a^{2/3} \sqrt[3]{b}}","-\frac{A \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{A \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} \sqrt[3]{b}}-\frac{A \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b}}",1,"-1/6*(A*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*x] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(2/3)*b^(1/3))","A",1
24,1,124,161,0.0471781,"\int \left(-\frac{C x^2}{a+b x^3}+\frac{A+B x+C x^2}{a+b x^3}\right) \, dx","Integrate[-((C*x^2)/(a + b*x^3)) + (A + B*x + C*x^2)/(a + b*x^3),x]","\frac{\left(A \sqrt[3]{b}-\sqrt[3]{a} B\right) \left(2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)\right)-2 \sqrt{3} \left(\sqrt[3]{a} B+A \sqrt[3]{b}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 a^{2/3} b^{2/3}}","-\frac{\left(A-\frac{\sqrt[3]{a} B}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\left(A \sqrt[3]{b}-\sqrt[3]{a} B\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} B+A \sqrt[3]{b}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{2/3}}",1,"(-2*Sqrt[3]*(A*b^(1/3) + a^(1/3)*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (A*b^(1/3) - a^(1/3)*B)*(2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*a^(2/3)*b^(2/3))","A",1
25,1,122,134,0.0260439,"\int \frac{b x+c x^2}{d+e x^3} \, dx","Integrate[(b*x + c*x^2)/(d + e*x^3),x]","\frac{b \sqrt[3]{e} \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)-2 b \sqrt[3]{e} \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)-2 \sqrt{3} b \sqrt[3]{e} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right)+2 c \sqrt[3]{d} \log \left(d+e x^3\right)}{6 \sqrt[3]{d} e}","\frac{b \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)}{6 \sqrt[3]{d} e^{2/3}}-\frac{b \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right)}{3 \sqrt[3]{d} e^{2/3}}-\frac{b \tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right)}{\sqrt{3} \sqrt[3]{d} e^{2/3}}+\frac{c \log \left(d+e x^3\right)}{3 e}",1,"(-2*Sqrt[3]*b*e^(1/3)*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - 2*b*e^(1/3)*Log[d^(1/3) + e^(1/3)*x] + b*e^(1/3)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2] + 2*c*d^(1/3)*Log[d + e*x^3])/(6*d^(1/3)*e)","A",1
26,1,123,134,0.0385723,"\int \frac{a+c x^2}{d-e x^3} \, dx","Integrate[(a + c*x^2)/(d - e*x^3),x]","\frac{a e^{2/3} \log \left(d^{2/3}+\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)-2 a e^{2/3} \log \left(\sqrt[3]{d}-\sqrt[3]{e} x\right)+2 \sqrt{3} a e^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}+1}{\sqrt{3}}\right)-2 c d^{2/3} \log \left(d-e x^3\right)}{6 d^{2/3} e}","\frac{a \log \left(d^{2/3}+\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right)}{6 d^{2/3} \sqrt[3]{e}}-\frac{a \log \left(\sqrt[3]{d}-\sqrt[3]{e} x\right)}{3 d^{2/3} \sqrt[3]{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt[3]{d}+2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right)}{\sqrt{3} d^{2/3} \sqrt[3]{e}}-\frac{c \log \left(d-e x^3\right)}{3 e}",1,"(2*Sqrt[3]*a*e^(2/3)*ArcTan[(1 + (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - 2*a*e^(2/3)*Log[d^(1/3) - e^(1/3)*x] + a*e^(2/3)*Log[d^(2/3) + d^(1/3)*e^(1/3)*x + e^(2/3)*x^2] - 2*c*d^(2/3)*Log[d - e*x^3])/(6*d^(2/3)*e)","A",1
27,1,72,37,0.0233641,"\int \frac{2 a^2+b^2 x^2}{a^3+b^3 x^3} \, dx","Integrate[(2*a^2 + b^2*x^2)/(a^3 + b^3*x^3),x]","\frac{\log \left(a^3+b^3 x^3\right)-\log \left(a^2-a b x+b^2 x^2\right)+2 \log (a+b x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 b x-a}{\sqrt{3} a}\right)}{3 b}","\frac{\log (a+b x)}{b}-\frac{2 \tan ^{-1}\left(\frac{a-2 b x}{\sqrt{3} a}\right)}{\sqrt{3} b}",1,"(2*Sqrt[3]*ArcTan[(-a + 2*b*x)/(Sqrt[3]*a)] + 2*Log[a + b*x] - Log[a^2 - a*b*x + b^2*x^2] + Log[a^3 + b^3*x^3])/(3*b)","A",1
28,1,71,39,0.0262864,"\int \frac{2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx","Integrate[(2*a^2 + b^2*x^2)/(a^3 - b^3*x^3),x]","\frac{-\log \left(a^3-b^3 x^3\right)+\log \left(a^2+a b x+b^2 x^2\right)-2 \log (a-b x)+2 \sqrt{3} \tan ^{-1}\left(\frac{a+2 b x}{\sqrt{3} a}\right)}{3 b}","\frac{2 \tan ^{-1}\left(\frac{a+2 b x}{\sqrt{3} a}\right)}{\sqrt{3} b}-\frac{\log (a-b x)}{b}",1,"(2*Sqrt[3]*ArcTan[(a + 2*b*x)/(Sqrt[3]*a)] - 2*Log[a - b*x] + Log[a^2 + a*b*x + b^2*x^2] - Log[a^3 - b^3*x^3])/(3*b)","A",1
29,1,76,48,0.0220416,"\int \frac{8 C+b^{2/3} C x^2}{8+b x^3} \, dx","Integrate[(8*C + b^(2/3)*C*x^2)/(8 + b*x^3),x]","\frac{C \left(-\log \left(b^{2/3} x^2-2 \sqrt[3]{b} x+4\right)+\log \left(b x^3+8\right)+2 \log \left(\sqrt[3]{b} x+2\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{b} x-1}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{b}}","\frac{C \log \left(\sqrt[3]{b} x+2\right)}{\sqrt[3]{b}}-\frac{2 C \tan ^{-1}\left(\frac{1-\sqrt[3]{b} x}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}",1,"(C*(2*Sqrt[3]*ArcTan[(-1 + b^(1/3)*x)/Sqrt[3]] + 2*Log[2 + b^(1/3)*x] - Log[4 - 2*b^(1/3)*x + b^(2/3)*x^2] + Log[8 + b*x^3]))/(3*b^(1/3))","A",1
30,1,72,47,0.0291562,"\int \frac{a^{2/3} C+2 C x^2}{a+8 x^3} \, dx","Integrate[(a^(2/3)*C + 2*C*x^2)/(a + 8*x^3),x]","\frac{1}{12} C \left(-\log \left(a^{2/3}-2 \sqrt[3]{a} x+4 x^2\right)+\log \left(a+8 x^3\right)+2 \log \left(\sqrt[3]{a}+2 x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{4 x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)","\frac{1}{4} C \log \left(\sqrt[3]{a}+2 x\right)-\frac{C \tan ^{-1}\left(\frac{\sqrt[3]{a}-4 x}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt{3}}",1,"(C*(-2*Sqrt[3]*ArcTan[(1 - (4*x)/a^(1/3))/Sqrt[3]] + 2*Log[a^(1/3) + 2*x] - Log[a^(2/3) - 2*a^(1/3)*x + 4*x^2] + Log[a + 8*x^3]))/12","A",1
31,1,99,57,0.0331605,"\int \frac{8 C+(-b)^{2/3} C x^2}{-8+b x^3} \, dx","Integrate[(8*C + (-b)^(2/3)*C*x^2)/(-8 + b*x^3),x]","\frac{C \left(-b^{2/3} \log \left(b^{2/3} x^2+2 \sqrt[3]{b} x+4\right)+2 b^{2/3} \log \left(2-\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b} x+1}{\sqrt{3}}\right)+(-b)^{2/3} \log \left(8-b x^3\right)\right)}{3 b}","\frac{2 C \tan ^{-1}\left(\frac{1-\sqrt[3]{-b} x}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{-b}}-\frac{C \log \left(\sqrt[3]{-b} x+2\right)}{\sqrt[3]{-b}}",1,"(C*(-2*Sqrt[3]*b^(2/3)*ArcTan[(1 + b^(1/3)*x)/Sqrt[3]] + 2*b^(2/3)*Log[2 - b^(1/3)*x] - b^(2/3)*Log[4 + 2*b^(1/3)*x + b^(2/3)*x^2] + (-b)^(2/3)*Log[8 - b*x^3]))/(3*b)","A",1
32,1,106,47,0.0434897,"\int \frac{(-a)^{2/3} C+2 C x^2}{a-8 x^3} \, dx","Integrate[((-a)^(2/3)*C + 2*C*x^2)/(a - 8*x^3),x]","\frac{C \left(-a^{2/3} \log \left(8 x^3-a\right)+(-a)^{2/3} \log \left(a^{2/3}+2 \sqrt[3]{a} x+4 x^2\right)-2 (-a)^{2/3} \log \left(\sqrt[3]{a}-2 x\right)+2 \sqrt{3} (-a)^{2/3} \tan ^{-1}\left(\frac{\frac{4 x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{12 a^{2/3}}","\frac{C \tan ^{-1}\left(\frac{1-\frac{4 x}{\sqrt[3]{-a}}}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{1}{4} C \log \left(\sqrt[3]{-a}+2 x\right)",1,"(C*(2*Sqrt[3]*(-a)^(2/3)*ArcTan[(1 + (4*x)/a^(1/3))/Sqrt[3]] - 2*(-a)^(2/3)*Log[a^(1/3) - 2*x] + (-a)^(2/3)*Log[a^(2/3) + 2*a^(1/3)*x + 4*x^2] - a^(2/3)*Log[-a + 8*x^3]))/(12*a^(2/3))","B",1
33,1,146,50,0.0538304,"\int \frac{2 \left(\frac{a}{b}\right)^{2/3} C+C x^2}{a+b x^3} \, dx","Integrate[(2*(a/b)^(2/3)*C + C*x^2)/(a + b*x^3),x]","\frac{C \left(-b^{2/3} \left(\frac{a}{b}\right)^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+a^{2/3} \log \left(a+b x^3\right)+2 b^{2/3} \left(\frac{a}{b}\right)^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} \left(\frac{a}{b}\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{3 a^{2/3} b}","\frac{C \log \left(\sqrt[3]{\frac{a}{b}}+x\right)}{b}-\frac{2 C \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"(C*(-2*Sqrt[3]*(a/b)^(2/3)*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(a/b)^(2/3)*b^(2/3)*Log[a^(1/3) + b^(1/3)*x] - (a/b)^(2/3)*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(2/3)*Log[a + b*x^3]))/(3*a^(2/3)*b)","B",1
34,1,150,53,0.1017447,"\int \frac{2 \left(-\frac{a}{b}\right)^{2/3} C+C x^2}{a-b x^3} \, dx","Integrate[(2*(-(a/b))^(2/3)*C + C*x^2)/(a - b*x^3),x]","\frac{C \left(b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-a^{2/3} \log \left(a-b x^3\right)-2 b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)+2 \sqrt{3} b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{3 a^{2/3} b}","\frac{2 C \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} b}-\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}+x\right)}{b}",1,"(C*(2*Sqrt[3]*(-(a/b))^(2/3)*b^(2/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(-(a/b))^(2/3)*b^(2/3)*Log[a^(1/3) - b^(1/3)*x] + (-(a/b))^(2/3)*b^(2/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - a^(2/3)*Log[a - b*x^3]))/(3*a^(2/3)*b)","B",1
35,1,149,54,0.0672926,"\int \frac{2 \left(-\frac{a}{b}\right)^{2/3} C+C x^2}{a+b x^3} \, dx","Integrate[(2*(-(a/b))^(2/3)*C + C*x^2)/(a + b*x^3),x]","\frac{C \left(-b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+a^{2/3} \log \left(a+b x^3\right)+2 b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} \left(-\frac{a}{b}\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{3 a^{2/3} b}","\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}-x\right)}{b}-\frac{2 C \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"(C*(-2*Sqrt[3]*(-(a/b))^(2/3)*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-(a/b))^(2/3)*b^(2/3)*Log[a^(1/3) + b^(1/3)*x] - (-(a/b))^(2/3)*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(2/3)*Log[a + b*x^3]))/(3*a^(2/3)*b)","B",1
36,1,147,53,0.052472,"\int \frac{2 \left(\frac{a}{b}\right)^{2/3} C+C x^2}{a-b x^3} \, dx","Integrate[(2*(a/b)^(2/3)*C + C*x^2)/(a - b*x^3),x]","\frac{C \left(b^{2/3} \left(\frac{a}{b}\right)^{2/3} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-a^{2/3} \log \left(a-b x^3\right)-2 b^{2/3} \left(\frac{a}{b}\right)^{2/3} \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)+2 \sqrt{3} b^{2/3} \left(\frac{a}{b}\right)^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{3 a^{2/3} b}","\frac{2 C \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} b}-\frac{C \log \left(\sqrt[3]{\frac{a}{b}}-x\right)}{b}",1,"(C*(2*Sqrt[3]*(a/b)^(2/3)*b^(2/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(a/b)^(2/3)*b^(2/3)*Log[a^(1/3) - b^(1/3)*x] + (a/b)^(2/3)*b^(2/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - a^(2/3)*Log[a - b*x^3]))/(3*a^(2/3)*b)","B",1
37,1,95,61,0.0205807,"\int \frac{2 a^{2/3} C+b^{2/3} C x^2}{a+b x^3} \, dx","Integrate[(2*a^(2/3)*C + b^(2/3)*C*x^2)/(a + b*x^3),x]","\frac{C \left(-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\log \left(a+b x^3\right)+2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{b}}","\frac{C \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}-\frac{2 C \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{b}}",1,"(C*(-2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + Log[a + b*x^3]))/(3*b^(1/3))","A",1
38,1,116,70,0.0317098,"\int \frac{-2 a^{2/3} C-(-b)^{2/3} C x^2}{a+b x^3} \, dx","Integrate[(-2*a^(2/3)*C - (-b)^(2/3)*C*x^2)/(a + b*x^3),x]","-\frac{C \left(-b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+(-b)^{2/3} \log \left(a+b x^3\right)\right)}{3 b}","\frac{C \log \left(\sqrt[3]{a}-\sqrt[3]{-b} x\right)}{\sqrt[3]{-b}}-\frac{2 C \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{-b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{-b}}",1,"-1/3*(C*(-2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(2/3)*Log[a^(1/3) + b^(1/3)*x] - b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + (-b)^(2/3)*Log[a + b*x^3]))/b","A",1
39,1,50,40,0.0302206,"\int \frac{-3+x^2}{-1+x^3} \, dx","Integrate[(-3 + x^2)/(-1 + x^3),x]","\frac{1}{3} \log \left(1-x^3\right)+\frac{1}{2} \log \left(x^2+x+1\right)-\log (1-x)+\sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)","\frac{5}{6} \log \left(x^2+x+1\right)-\frac{2}{3} \log (1-x)+\sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)",1,"Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - Log[1 - x] + Log[1 + x + x^2]/2 + Log[1 - x^3]/3","A",1
40,1,122,70,0.0519035,"\int \frac{\sqrt[3]{a} \sqrt[3]{b} B+2 a^{2/3} C+b^{2/3} B x+b^{2/3} C x^2}{a+b x^3} \, dx","Integrate[(a^(1/3)*b^(1/3)*B + 2*a^(2/3)*C + b^(2/3)*B*x + b^(2/3)*C*x^2)/(a + b*x^3),x]","\frac{\sqrt[3]{a} C \left(-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\log \left(a+b x^3\right)+2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)\right)-2 \sqrt{3} \left(\sqrt[3]{a} C+\sqrt[3]{b} B\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b}}","\frac{C \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}-\frac{2 \left(\frac{B}{\sqrt[3]{a}}+\frac{C}{\sqrt[3]{b}}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3}}",1,"(-2*Sqrt[3]*(b^(1/3)*B + a^(1/3)*C)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + a^(1/3)*C*(2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + Log[a + b*x^3]))/(3*a^(1/3)*b^(1/3))","A",1
41,1,238,88,0.6554582,"\int \frac{\sqrt[3]{a} \sqrt[3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx","Integrate[(a^(1/3)*(-b)^(1/3)*B - 2*a^(2/3)*C - (-b)^(2/3)*B*x - (-b)^(2/3)*C*x^2)/(a + b*x^3),x]","\frac{\frac{\left(2 \sqrt[3]{a} b \sqrt[3]{-b} C+b^{5/3} B+(-b)^{5/3} B\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 b \left(2 \sqrt[3]{a} \sqrt[3]{-b} C+\left(b^{2/3}-(-b)^{2/3}\right) B\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt[3]{a} (-b)^{2/3} \sqrt[3]{-b^2} C \log \left(a+b x^3\right)}{\sqrt[3]{-b^2}}+2 \sqrt{3} \sqrt[3]{b} \left(2 \sqrt[3]{a} \sqrt[3]{b} C+\left((-b)^{2/3}-\sqrt[3]{-b^2}\right) B\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 \sqrt[3]{a} b}","\frac{2 \left(\sqrt[3]{a} (-b)^{2/3} C+b B\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{-b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b}+\frac{C \log \left(\sqrt[3]{a}-\sqrt[3]{-b} x\right)}{\sqrt[3]{-b}}",1,"(2*Sqrt[3]*b^(1/3)*(((-b)^(2/3) - (-b^2)^(1/3))*B + 2*a^(1/3)*b^(1/3)*C)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (-2*b*((-(-b)^(2/3) + b^(2/3))*B + 2*a^(1/3)*(-b)^(1/3)*C)*Log[a^(1/3) + b^(1/3)*x] + ((-b)^(5/3)*B + b^(5/3)*B + 2*a^(1/3)*(-b)^(1/3)*b*C)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a^(1/3)*(-b)^(2/3)*(-b^2)^(1/3)*C*Log[a + b*x^3])/(-b^2)^(1/3))/(6*a^(1/3)*b)","B",0
42,1,12,11,0.0017472,"\int \frac{B^2+B C x+C^2 x^2}{-B^3+C^3 x^3} \, dx","Integrate[(B^2 + B*C*x + C^2*x^2)/(-B^3 + C^3*x^3),x]","\frac{\log (C x-B)}{C}","\frac{\log (B-C x)}{C}",1,"Log[-B + C*x]/C","A",1
43,1,21,21,0.0032647,"\int \frac{a^{2/3} C-\sqrt[3]{a} \sqrt[3]{b} C x+b^{2/3} C x^2}{a+b x^3} \, dx","Integrate[(a^(2/3)*C - a^(1/3)*b^(1/3)*C*x + b^(2/3)*C*x^2)/(a + b*x^3),x]","\frac{C \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}","\frac{C \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}",1,"(C*Log[a^(1/3) + b^(1/3)*x])/b^(1/3)","A",1
44,1,247,71,0.3342872,"\int \frac{\sqrt[3]{\frac{a}{b}} B+2 \left(\frac{a}{b}\right)^{2/3} C+B x+C x^2}{a+b x^3} \, dx","Integrate[((a/b)^(1/3)*B + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]","\frac{\sqrt[3]{b} \left(a^{2/3} B-\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(2 C \sqrt[3]{\frac{a}{b}}+B\right)\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(2 C \sqrt[3]{\frac{a}{b}}+B\right)-a^{2/3} B\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(2 C \sqrt[3]{\frac{a}{b}}+B\right)+\sqrt[3]{a} B\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)+2 a C \log \left(a+b x^3\right)}{6 a b}","\frac{C \log \left(\sqrt[3]{\frac{a}{b}}+x\right)}{b}-\frac{2 \left(\frac{a}{b}\right)^{2/3} \left(C \sqrt[3]{\frac{a}{b}}+B\right) \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} a}",1,"(2*Sqrt[3]*a^(1/3)*b^(1/3)*(a^(1/3)*B + (a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 2*b^(1/3)*(-(a^(2/3)*B) + a^(1/3)*(a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(a^(2/3)*B - a^(1/3)*(a/b)^(1/3)*b^(1/3)*(B + 2*(a/b)^(1/3)*C))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*C*Log[a + b*x^3])/(6*a*b)","B",1
45,1,288,76,0.2537219,"\int \frac{\sqrt[3]{-\frac{a}{b}} B+2 \left(-\frac{a}{b}\right)^{2/3} C+B x+C x^2}{a-b x^3} \, dx","Integrate[((-(a/b))^(1/3)*B + 2*(-(a/b))^(2/3)*C + B*x + C*x^2)/(a - b*x^3),x]","-\frac{\left(-a^{2/3} B-\sqrt[3]{a} \sqrt[3]{b} B \sqrt[3]{-\frac{a}{b}}-2 \sqrt[3]{a} \sqrt[3]{b} C \left(-\frac{a}{b}\right)^{2/3}\right) \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a b^{2/3}}-\frac{\left(a^{2/3} B+\sqrt[3]{a} \sqrt[3]{b} B \sqrt[3]{-\frac{a}{b}}+2 \sqrt[3]{a} \sqrt[3]{b} C \left(-\frac{a}{b}\right)^{2/3}\right) \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{3 a b^{2/3}}-\frac{\left(a^{2/3} B-\sqrt[3]{a} \sqrt[3]{b} B \sqrt[3]{-\frac{a}{b}}-2 \sqrt[3]{a} \sqrt[3]{b} C \left(-\frac{a}{b}\right)^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a b^{2/3}}-\frac{C \log \left(a-b x^3\right)}{3 b}","\frac{2 \left(C \sqrt[3]{-\frac{a}{b}}+B\right) \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} b \sqrt[3]{-\frac{a}{b}}}-\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}+x\right)}{b}",1,"-(((a^(2/3)*B - a^(1/3)*(-(a/b))^(1/3)*b^(1/3)*B - 2*a^(1/3)*(-(a/b))^(2/3)*b^(1/3)*C)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a*b^(2/3))) - ((a^(2/3)*B + a^(1/3)*(-(a/b))^(1/3)*b^(1/3)*B + 2*a^(1/3)*(-(a/b))^(2/3)*b^(1/3)*C)*Log[a^(1/3) - b^(1/3)*x])/(3*a*b^(2/3)) - ((-(a^(2/3)*B) - a^(1/3)*(-(a/b))^(1/3)*b^(1/3)*B - 2*a^(1/3)*(-(a/b))^(2/3)*b^(1/3)*C)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a*b^(2/3)) - (C*Log[a - b*x^3])/(3*b)","B",1
46,1,253,78,0.3563506,"\int \frac{-\sqrt[3]{-\frac{a}{b}} B+2 \left(-\frac{a}{b}\right)^{2/3} C+B x+C x^2}{a+b x^3} \, dx","Integrate[(-((-(a/b))^(1/3)*B) + 2*(-(a/b))^(2/3)*C + B*x + C*x^2)/(a + b*x^3),x]","\frac{\sqrt[3]{b} \left(a^{2/3} B+\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \left(B-2 C \sqrt[3]{-\frac{a}{b}}\right)\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt[3]{b} \left(a^{2/3} B+\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \left(B-2 C \sqrt[3]{-\frac{a}{b}}\right)\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \left(2 C \sqrt[3]{-\frac{a}{b}}-B\right)+\sqrt[3]{a} B\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)+2 a C \log \left(a+b x^3\right)}{6 a b}","\frac{2 \left(B-C \sqrt[3]{-\frac{a}{b}}\right) \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} b \sqrt[3]{-\frac{a}{b}}}+\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}-x\right)}{b}",1,"(2*Sqrt[3]*a^(1/3)*b^(1/3)*(a^(1/3)*B + (-(a/b))^(1/3)*b^(1/3)*(-B + 2*(-(a/b))^(1/3)*C))*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 2*b^(1/3)*(a^(2/3)*B + a^(1/3)*(-(a/b))^(1/3)*b^(1/3)*(B - 2*(-(a/b))^(1/3)*C))*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(a^(2/3)*B + a^(1/3)*(-(a/b))^(1/3)*b^(1/3)*(B - 2*(-(a/b))^(1/3)*C))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*C*Log[a + b*x^3])/(6*a*b)","B",1
47,1,244,75,0.3218286,"\int \frac{-\sqrt[3]{\frac{a}{b}} B+2 \left(\frac{a}{b}\right)^{2/3} C+B x+C x^2}{a-b x^3} \, dx","Integrate[(-((a/b)^(1/3)*B) + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a - b*x^3),x]","\frac{\sqrt[3]{b} \left(a^{2/3} B+\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(2 C \sqrt[3]{\frac{a}{b}}-B\right)\right) \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt[3]{b} \left(a^{2/3} B+\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(2 C \sqrt[3]{\frac{a}{b}}-B\right)\right) \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)-2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \left(B-2 C \sqrt[3]{\frac{a}{b}}\right)+\sqrt[3]{a} B\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-2 a C \log \left(a-b x^3\right)}{6 a b}","-\frac{2 \left(\frac{a}{b}\right)^{2/3} \left(B-C \sqrt[3]{\frac{a}{b}}\right) \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} a}-\frac{C \log \left(\sqrt[3]{\frac{a}{b}}-x\right)}{b}",1,"(-2*Sqrt[3]*a^(1/3)*b^(1/3)*(a^(1/3)*B + (a/b)^(1/3)*b^(1/3)*(B - 2*(a/b)^(1/3)*C))*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*b^(1/3)*(a^(2/3)*B + a^(1/3)*(a/b)^(1/3)*b^(1/3)*(-B + 2*(a/b)^(1/3)*C))*Log[a^(1/3) - b^(1/3)*x] + b^(1/3)*(a^(2/3)*B + a^(1/3)*(a/b)^(1/3)*b^(1/3)*(-B + 2*(a/b)^(1/3)*C))*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*C*Log[a - b*x^3])/(6*a*b)","B",1
48,1,31,32,0.0142392,"\int \frac{a+a x+c x^2}{1-x^3} \, dx","Integrate[(a + a*x + c*x^2)/(1 - x^3),x]","\frac{1}{3} \left((a-c) \log \left(x^2+x+1\right)-(2 a+c) \log (1-x)\right)","\frac{1}{3} (a-c) \log \left(x^2+x+1\right)-\frac{1}{3} (2 a+c) \log (1-x)",1,"(-((2*a + c)*Log[1 - x]) + (a - c)*Log[1 + x + x^2])/3","A",1
49,1,62,55,0.038391,"\int \frac{a+b x+c x^2}{1-x^3} \, dx","Integrate[(a + b*x + c*x^2)/(1 - x^3),x]","\frac{1}{6} \left((a+b) \log \left(x^2+x+1\right)-2 (a+b) \log (1-x)+2 \sqrt{3} (a-b) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)-2 c \log \left(1-x^3\right)\right)","\frac{1}{6} \log \left(x^2+x+1\right) (a+b-2 c)-\frac{1}{3} \log (1-x) (a+b+c)+\frac{(a-b) \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(2*Sqrt[3]*(a - b)*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*(a + b)*Log[1 - x] + (a + b)*Log[1 + x + x^2] - 2*c*Log[1 - x^3])/6","A",1
50,1,8,8,0.0011269,"\int \frac{1+x+x^2}{1-x^3} \, dx","Integrate[(1 + x + x^2)/(1 - x^3),x]","-\log (1-x)","-\log (1-x)",1,"-Log[1 - x]","A",1
51,1,30,30,0.0101666,"\int \frac{1-x+3 x^2}{1-x^3} \, dx","Integrate[(1 - x + 3*x^2)/(1 - x^3),x]","\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}-\log \left(1-x^3\right)","\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}-\log \left(1-x^3\right)",1,"(2*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] - Log[1 - x^3]","A",1
52,1,18,18,0.0058595,"\int \frac{1+x+4 x^2}{1-x^3} \, dx","Integrate[(1 + x + 4*x^2)/(1 - x^3),x]","-\log \left(x^2+x+1\right)-2 \log (1-x)","-\log \left(x^2+x+1\right)-2 \log (1-x)",1,"-2*Log[1 - x] - Log[1 + x + x^2]","A",1
53,1,113,113,0.0046557,"\int \left(a+b x^3\right)^3 \left(a c+a d x+b c x^3+b d x^4\right) \, dx","Integrate[(a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]","a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}","a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}",1,"a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14","A",1
54,1,88,88,0.0034704,"\int \left(a+b x^3\right)^2 \left(a c+a d x+b c x^3+b d x^4\right) \, dx","Integrate[(a + b*x^3)^2*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}",1,"a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3*c*x^10)/10 + (b^3*d*x^11)/11","A",1
55,1,60,60,0.0027845,"\int \left(a+b x^3\right) \left(a c+a d x+b c x^3+b d x^4\right) \, dx","Integrate[(a + b*x^3)*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8",1,"a^2*c*x + (a^2*d*x^2)/2 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (b^2*c*x^7)/7 + (b^2*d*x^8)/8","A",1
56,1,12,12,0.0008047,"\int \frac{a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x]","c x+\frac{d x^2}{2}","c x+\frac{d x^2}{2}",1,"c*x + (d*x^2)/2","A",1
57,1,124,161,0.0513658,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^2} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^2,x]","\frac{\left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \left(2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)\right)-2 \sqrt{3} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 a^{2/3} b^{2/3}}","-\frac{\left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{2/3}}",1,"(-2*Sqrt[3]*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (b^(1/3)*c - a^(1/3)*d)*(2*Log[a^(1/3) + b^(1/3)*x] - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*a^(2/3)*b^(2/3))","A",1
58,1,180,189,0.1727241,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^3} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^3,x]","\frac{\frac{\left(a^{2/3} d-2 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 \left(2 \sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 a x (c+d x)}{a+b x^3}}{18 a^2}","-\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{2/3}}+\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left(a+b x^3\right)}",1,"((6*a*x*(c + d*x))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(2*a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-2*a^(1/3)*b^(1/3)*c + a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(18*a^2)","A",1
59,1,78,585,0.0539112,"\int \left(a+b x^3\right)^{3/2} \left(a c+a d x+b c x^3+b d x^4\right) \, dx","Integrate[(a + b*x^3)^(3/2)*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]","\frac{a^2 x \sqrt{a+b x^3} \left(2 c \, _2F_1\left(-\frac{5}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(-\frac{5}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{405 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{10/3} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{810 a^3 d \sqrt{a+b x^3}}{1729 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{54 a^2 \sqrt{a+b x^3} \left(1729 c x+935 d x^2\right)}{323323}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(1729 \sqrt[3]{b} c-935 \left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{323323 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{30 a \left(a+b x^3\right)^{3/2} \left(247 c x+187 d x^2\right)}{46189}+\frac{2}{323} \left(a+b x^3\right)^{5/2} \left(19 c x+17 d x^2\right)",1,"(a^2*x*Sqrt[a + b*x^3]*(2*c*Hypergeometric2F1[-5/2, 1/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[-5/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[1 + (b*x^3)/a])","C",1
60,1,76,556,0.0348834,"\int \sqrt{a+b x^3} \left(a c+a d x+b c x^3+b d x^4\right) \, dx","Integrate[Sqrt[a + b*x^3]*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]","\frac{a x \sqrt{a+b x^3} \left(2 c \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{54 a^2 d \sqrt{a+b x^3}}{91 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(91 \sqrt[3]{b} c-55 \left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{5005 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{18 a \sqrt{a+b x^3} \left(91 c x+55 d x^2\right)}{5005}+\frac{2}{143} \left(a+b x^3\right)^{3/2} \left(13 c x+11 d x^2\right)",1,"(a*x*Sqrt[a + b*x^3]*(2*c*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[1 + (b*x^3)/a])","C",1
61,1,75,525,0.0359089,"\int \frac{a c+a d x+b c x^3+b d x^4}{\sqrt{a+b x^3}} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{a+b x^3} \left(2 c \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{\frac{b x^3}{a}+1}}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(7 \sqrt[3]{b} c-5 \left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{35 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a d \sqrt{a+b x^3}}{7 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2}{35} \sqrt{a+b x^3} \left(7 c x+5 d x^2\right)",1,"(x*Sqrt[a + b*x^3]*(2*c*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[1 + (b*x^3)/a])","C",1
62,1,75,490,0.0676658,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^(3/2),x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} c-\left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 d \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
63,1,96,522,0.0652775,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^(5/2),x]","\frac{x \left(2 c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+3 d x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+4 c\right)}{6 a \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a} d+\sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 d \sqrt{a+b x^3}}{3 a b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x (c+d x)}{3 a \sqrt{a+b x^3}}",1,"(x*(4*c + 2*c*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*d*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(6*a*Sqrt[a + b*x^3])","C",1
64,1,123,554,0.105291,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^{7/2}} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^(7/2),x]","\frac{14 c x \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+4 c x \left(10 a+7 b x^3\right)+27 d x^2 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)}{54 a^2 \left(a+b x^3\right)^{3/2}}","\frac{5 \sqrt{2-\sqrt{3}} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} a^{5/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{10 d \sqrt{a+b x^3}}{27 a^2 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x (7 c+5 d x)}{27 a^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(5 \left(1-\sqrt{3}\right) \sqrt[3]{a} d+7 \sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a^2 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x (c+d x)}{9 a \left(a+b x^3\right)^{3/2}}",1,"(4*c*x*(10*a + 7*b*x^3) + 14*c*x*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 27*d*x^2*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)])/(54*a^2*(a + b*x^3)^(3/2))","C",1
65,1,138,581,0.2038865,"\int \frac{a c+a d x+b c x^3+b d x^4}{\left(a+b x^3\right)^{9/2}} \, dx","Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^(9/2),x]","\frac{4 c x \left(157 a^2+221 a b x^3+91 b^2 x^6\right)+182 c x \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+405 d x^2 \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)^2 \, _2F_1\left(\frac{2}{3},\frac{7}{2};\frac{5}{3};-\frac{b x^3}{a}\right)}{810 a^3 \left(a+b x^3\right)^{5/2}}","\frac{11 \sqrt{2-\sqrt{3}} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27\ 3^{3/4} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{22 d \sqrt{a+b x^3}}{81 a^3 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x (91 c+55 d x)}{405 a^3 \sqrt{a+b x^3}}+\frac{2 x (13 c+11 d x)}{135 a^2 \left(a+b x^3\right)^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(55 \left(1-\sqrt{3}\right) \sqrt[3]{a} d+91 \sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{405 \sqrt[4]{3} a^3 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x (c+d x)}{15 a \left(a+b x^3\right)^{5/2}}",1,"(4*c*x*(157*a^2 + 221*a*b*x^3 + 91*b^2*x^6) + 182*c*x*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 405*d*x^2*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 7/2, 5/3, -((b*x^3)/a)])/(810*a^3*(a + b*x^3)^(5/2))","C",1
66,1,135,590,0.1624015,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\sqrt{a+b x^3}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/Sqrt[a + b*x^3],x]","\frac{42 x \sqrt{\frac{b x^3}{a}+1} (5 b c-2 a f) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+15 x^2 \sqrt{\frac{b x^3}{a}+1} (7 b d-4 a g) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 \left(a+b x^3\right) (35 e+3 x (7 f+5 g x))}{210 b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(7 \sqrt[3]{b} (5 b c-2 a f)-5 \left(1-\sqrt{3}\right) \sqrt[3]{a} (7 b d-4 a g)\right)}{35 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 b d-4 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (7 b d-4 a g)}{7 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 e \sqrt{a+b x^3}}{3 b}+\frac{2 f x \sqrt{a+b x^3}}{5 b}+\frac{2 g x^2 \sqrt{a+b x^3}}{7 b}",1,"(4*(a + b*x^3)*(35*e + 3*x*(7*f + 5*g*x)) + 42*(5*b*c - 2*a*f)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 15*(7*b*d - 4*a*g)*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(210*b*Sqrt[a + b*x^3])","C",1
67,1,130,594,0.1449087,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(3/2),x]","\frac{2 x \sqrt{\frac{b x^3}{a}+1} (2 a f+b c) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+3 x^2 \sqrt{\frac{b x^3}{a}+1} (b d-4 a g) \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-4 a (e+x (f-3 g x))+4 b c x}{6 a b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(\sqrt[3]{b} (2 a f+b c)+\left(1-\sqrt{3}\right) \sqrt[3]{a} (b d-4 a g)\right)}{3 \sqrt[4]{3} a b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (b d-4 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (b d-4 a g)}{3 a b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x \left(x (b d-a g)-a f+b c+b e x^2\right)}{3 a b \sqrt{a+b x^3}}-\frac{2 e \sqrt{a+b x^3}}{3 a b}",1,"(4*b*c*x - 4*a*(e + x*(f - 3*g*x)) + 2*(b*c + 2*a*f)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*(b*d - 4*a*g)*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)])/(6*a*b*Sqrt[a + b*x^3])","C",1
68,1,170,628,0.2216419,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(5/2),x]","\frac{-4 a^2 (15 e+x (5 f+27 g x))+10 x \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (2 a f+7 b c) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+40 a b x \left(5 c+f x^3\right)+27 x^2 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (4 a g+5 b d) \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+140 b^2 c x^4}{270 a^2 b \left(a+b x^3\right)^{3/2}}","\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a g+5 b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (4 a g+5 b d)}{27 a^2 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 (3 a e-x (x (4 a g+5 b d)+2 a f+7 b c))}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(\sqrt[3]{b} (2 a f+7 b c)+\left(1-\sqrt{3}\right) \sqrt[3]{a} (4 a g+5 b d)\right)}{27 \sqrt[4]{3} a^2 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x \left(x (b d-a g)-a f+b c+b e x^2\right)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(140*b^2*c*x^4 + 40*a*b*x*(5*c + f*x^3) - 4*a^2*(15*e + x*(5*f + 27*g*x)) + 10*(7*b*c + 2*a*f)*x*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 27*(5*b*d + 4*a*g)*x^2*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)])/(270*a^2*b*(a + b*x^3)^(3/2))","C",1
69,1,196,676,0.3956847,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^3\right)^{7/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(7/2),x]","\frac{-4 a^3 (297 e+x (77 f+405 g x))+44 a^2 b x \left(157 c+34 f x^3\right)+44 a b^2 x^4 \left(221 c+14 f x^3\right)+154 x \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} (2 a f+13 b c) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+405 x^2 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} (4 a g+11 b d) \, _2F_1\left(\frac{2}{3},\frac{7}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+4004 b^3 c x^7}{8910 a^3 b \left(a+b x^3\right)^{5/2}}","\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a g+11 b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27\ 3^{3/4} a^{8/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (4 a g+11 b d)}{81 a^3 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{405 a^3 b \sqrt{a+b x^3}}-\frac{2 (9 a e-x (x (4 a g+11 b d)+2 a f+13 b c))}{135 a^2 b \left(a+b x^3\right)^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(7 \sqrt[3]{b} (2 a f+13 b c)+5 \left(1-\sqrt{3}\right) \sqrt[3]{a} (4 a g+11 b d)\right)}{405 \sqrt[4]{3} a^3 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x \left(x (b d-a g)-a f+b c+b e x^2\right)}{15 a b \left(a+b x^3\right)^{5/2}}",1,"(4004*b^3*c*x^7 + 44*a*b^2*x^4*(221*c + 14*f*x^3) + 44*a^2*b*x*(157*c + 34*f*x^3) - 4*a^3*(297*e + x*(77*f + 405*g*x)) + 154*(13*b*c + 2*a*f)*x*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 405*(11*b*d + 4*a*g)*x^2*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 7/2, 5/3, -((b*x^3)/a)])/(8910*a^3*b*(a + b*x^3)^(5/2))","C",1
70,1,200,186,0.0876843,"\int \frac{(a+b x)^2}{c+d x^3} \, dx","Integrate[(a + b*x)^2/(c + d*x^3),x]","-\frac{\left(a^2 \sqrt[3]{c} \sqrt[3]{d}-2 a b c^{2/3}\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c d^{2/3}}+\frac{\left(a^2 \sqrt[3]{c} \sqrt[3]{d}-2 a b c^{2/3}\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c d^{2/3}}+\frac{\left(a^2 \sqrt[3]{c} \sqrt[3]{d}+2 a b c^{2/3}\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{d} x-\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c d^{2/3}}+\frac{b^2 \log \left(c+d x^3\right)}{3 d}","\frac{a \left(2 b \sqrt[3]{c}-a \sqrt[3]{d}\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} d^{2/3}}-\frac{a \left(2 b \sqrt[3]{c}-a \sqrt[3]{d}\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} d^{2/3}}-\frac{a \left(a \sqrt[3]{d}+2 b \sqrt[3]{c}\right) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} d^{2/3}}+\frac{b^2 \log \left(c+d x^3\right)}{3 d}",1,"((2*a*b*c^(2/3) + a^2*c^(1/3)*d^(1/3))*ArcTan[(-c^(1/3) + 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c*d^(2/3)) + ((-2*a*b*c^(2/3) + a^2*c^(1/3)*d^(1/3))*Log[c^(1/3) + d^(1/3)*x])/(3*c*d^(2/3)) - ((-2*a*b*c^(2/3) + a^2*c^(1/3)*d^(1/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c*d^(2/3)) + (b^2*Log[c + d*x^3])/(3*d)","A",1
71,1,214,222,0.2384251,"\int \frac{(a+b x)^3}{c+d x^3} \, dx","Integrate[(a + b*x)^3/(c + d*x^3),x]","\frac{\left(a^3 (-d)+3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)-2 \left(a^3 (-d)+3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)+2 \sqrt{3} \left(a^3 (-d)-3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)+6 a b^2 c^{2/3} \sqrt[3]{d} \log \left(c+d x^3\right)+6 b^3 c^{2/3} \sqrt[3]{d} x}{6 c^{2/3} d^{4/3}}","\frac{\left(a^3 (-d)+3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} d^{4/3}}-\frac{\left(a^3 (-d)+3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} d^{4/3}}+\frac{\left(a^3 (-d)-3 a^2 b \sqrt[3]{c} d^{2/3}+b^3 c\right) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{a b^2 \log \left(c+d x^3\right)}{d}+\frac{b^3 x}{d}",1,"(6*b^3*c^(2/3)*d^(1/3)*x + 2*Sqrt[3]*(b^3*c - 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] - 2*(b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(1/3) + d^(1/3)*x] + (b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2] + 6*a*b^2*c^(2/3)*d^(1/3)*Log[c + d*x^3])/(6*c^(2/3)*d^(4/3))","A",1
72,1,277,282,0.3650465,"\int \frac{(a+b x)^4}{c+d x^3} \, dx","Integrate[(a + b*x)^4/(c + d*x^3),x]","\frac{12 a^2 b^2 d^{2/3} \log \left(c+d x^3\right)-\frac{\left(a^4 d^{4/3}-4 a^3 b \sqrt[3]{c} d-4 a b^3 c \sqrt[3]{d}+b^4 c^{4/3}\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{2/3}}+\frac{2 \left(a^4 d^{4/3}-4 a^3 b \sqrt[3]{c} d-4 a b^3 c \sqrt[3]{d}+b^4 c^{4/3}\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{2/3}}+\frac{2 \sqrt{3} \left(a^4 \left(-d^{4/3}\right)-4 a^3 b \sqrt[3]{c} d+4 a b^3 c \sqrt[3]{d}+b^4 c^{4/3}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{2/3}}+24 a b^3 d^{2/3} x+3 b^4 d^{2/3} x^2}{6 d^{5/3}}","\frac{2 a^2 b^2 \log \left(c+d x^3\right)}{d}-\frac{\left(b \sqrt[3]{c} \left(b^3 c-4 a^3 d\right)-\sqrt[3]{d} \left(4 a b^3 c-a^4 d\right)\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} d^{5/3}}+\frac{\left(b \sqrt[3]{c} \left(b^3 c-4 a^3 d\right)-\sqrt[3]{d} \left(4 a b^3 c-a^4 d\right)\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} d^{5/3}}+\frac{\left(a^4 \left(-d^{4/3}\right)-4 a^3 b \sqrt[3]{c} d+4 a b^3 c \sqrt[3]{d}+b^4 c^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} d^{5/3}}+\frac{4 a b^3 x}{d}+\frac{b^4 x^2}{2 d}",1,"(24*a*b^3*d^(2/3)*x + 3*b^4*d^(2/3)*x^2 + (2*Sqrt[3]*(b^4*c^(4/3) + 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d - a^4*d^(4/3))*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(2/3) + (2*(b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d + a^4*d^(4/3))*Log[c^(1/3) + d^(1/3)*x])/c^(2/3) - ((b^4*c^(4/3) - 4*a*b^3*c*d^(1/3) - 4*a^3*b*c^(1/3)*d + a^4*d^(4/3))*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(2/3) + 12*a^2*b^2*d^(2/3)*Log[c + d*x^3])/(6*d^(5/3))","A",1
73,1,269,272,0.4223906,"\int \frac{\left(a+b x+c x^2\right)^2}{d+e x^3} \, dx","Integrate[(a + b*x + c*x^2)^2/(d + e*x^3),x]","\frac{2 e^{2/3} \left(2 a c+b^2\right) \log \left(d+e x^3\right)-\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(a e \left(a \sqrt[3]{e}-2 b \sqrt[3]{d}\right)-2 b c d \sqrt[3]{e}+c^2 d^{4/3}\right)}{d^{2/3}}+\frac{2 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(a e \left(a \sqrt[3]{e}-2 b \sqrt[3]{d}\right)-2 b c d \sqrt[3]{e}+c^2 d^{4/3}\right)}{d^{2/3}}+\frac{2 \sqrt{3} \left(c d^{2/3}-a e^{2/3}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(a e^{2/3}+2 b \sqrt[3]{d} \sqrt[3]{e}+c d^{2/3}\right)}{d^{2/3}}+12 b c e^{2/3} x+3 c^2 e^{2/3} x^2}{6 e^{5/3}}","\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(\sqrt[3]{e} \left(2 b c d-a^2 e\right)-\sqrt[3]{d} \left(c^2 d-2 a b e\right)\right)}{6 d^{2/3} e^{5/3}}-\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(\sqrt[3]{e} \left(2 b c d-a^2 e\right)-\sqrt[3]{d} \left(c^2 d-2 a b e\right)\right)}{3 d^{2/3} e^{5/3}}+\frac{\left(2 a c+b^2\right) \log \left(d+e x^3\right)}{3 e}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(-a e \left(a \sqrt[3]{e}+2 b \sqrt[3]{d}\right)+2 b c d \sqrt[3]{e}+c^2 d^{4/3}\right)}{\sqrt{3} d^{2/3} e^{5/3}}+\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}",1,"(12*b*c*e^(2/3)*x + 3*c^2*e^(2/3)*x^2 + (2*Sqrt[3]*(c*d^(2/3) - a*e^(2/3))*(c*d^(2/3) + 2*b*d^(1/3)*e^(1/3) + a*e^(2/3))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (2*(c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^(1/3) + a*e^(1/3))*e)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - ((c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^(1/3) + a*e^(1/3))*e)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 2*(b^2 + 2*a*c)*e^(2/3)*Log[d + e*x^3])/(6*e^(5/3))","A",1
74,1,439,416,0.5774574,"\int \frac{\left(a+b x+c x^2\right)^3}{d+e x^3} \, dx","Integrate[(a + b*x + c*x^2)^3/(d + e*x^3),x]","\frac{12 \sqrt[3]{e} \log \left(d+e x^3\right) \left(a^2 c e+a b^2 e-b c^2 d\right)-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(e \left(a^3 e+3 a^2 b \sqrt[3]{d} e^{2/3}-b^3 d\right)-3 c \left(2 a b d e+b^2 d^{4/3} e^{2/3}\right)-3 a c^2 d^{4/3} e^{2/3}+c^3 d^2\right)}{d^{2/3}}-\frac{2 \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(a^3 e^2-3 a^2 b \sqrt[3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right)}{d^{2/3}}+\frac{4 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(a^3 e^2-3 a^2 b \sqrt[3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right)}{d^{2/3}}+12 \sqrt[3]{e} x \left(6 a b c e+b^3 e-c^3 d\right)+18 c e^{4/3} x^2 \left(a c+b^2\right)+12 b c^2 e^{4/3} x^3+3 c^3 e^{4/3} x^4}{12 e^{7/3}}","-\frac{\log \left(d+e x^3\right) \left(a^2 (-c) e-a b^2 e+b c^2 d\right)}{e^2}-\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(-e \left(b^3 d-a^3 e\right)+3 \sqrt[3]{d} e^{2/3} \left(a^2 (-b) e+a c^2 d+b^2 c d\right)-6 a b c d e+c^3 d^2\right)}{6 d^{2/3} e^{7/3}}+\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(-e \left(b^3 d-a^3 e\right)+3 \sqrt[3]{d} e^{2/3} \left(a^2 (-b) e+a c^2 d+b^2 c d\right)-6 a b c d e+c^3 d^2\right)}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(a^3 e^2+3 a^2 b \sqrt[3]{d} e^{5/3}-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right)}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x \left(-6 a b c e+b^3 (-e)+c^3 d\right)}{e^2}+\frac{3 c x^2 \left(a c+b^2\right)}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}",1,"(12*e^(1/3)*(-(c^3*d) + b^3*e + 6*a*b*c*e)*x + 18*c*(b^2 + a*c)*e^(4/3)*x^2 + 12*b*c^2*e^(4/3)*x^3 + 3*c^3*e^(4/3)*x^4 - (4*Sqrt[3]*(c^3*d^2 - 3*a*c^2*d^(4/3)*e^(2/3) + e*(-(b^3*d) + 3*a^2*b*d^(1/3)*e^(2/3) + a^3*e) - 3*c*(b^2*d^(4/3)*e^(2/3) + 2*a*b*d*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (4*(c^3*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (2*(c^3*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e - 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 12*e^(1/3)*(-(b*c^2*d) + a*b^2*e + a^2*c*e)*Log[d + e*x^3])/(12*e^(7/3))","A",1
75,1,678,645,0.4764749,"\int \frac{\left(a+b x+c x^2\right)^4}{d+e x^3} \, dx","Integrate[(a + b*x + c*x^2)^4/(d + e*x^3),x]","\frac{15 e^{2/3} x^2 \left(6 a^2 c^2 e+12 a b^2 c e+b^4 e-4 b c^3 d\right)+60 e^{2/3} x \left(6 a^2 b c e+2 a b^3 e-2 a c^3 d-3 b^2 c^2 d\right)+\frac{10 \log \left(d+e x^3\right) \left(4 c e \left(a^3 e-b^3 d\right)+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right)}{\sqrt[3]{e}}+\frac{10 \sqrt{3} \left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{e} x}{\sqrt[3]{d}}}{\sqrt{3}}\right) \left(e \left(a^3 (-e)-3 a^2 b \sqrt[3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt[3]{e}+b^3 d\right)+c^2 \left(6 a d^{4/3} e^{2/3}-6 b d^{5/3} \sqrt[3]{e}\right)+12 a b c d e-4 c^3 d^2\right)}{d^{2/3}}-\frac{5 \log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(a^4 e^{7/3}+6 a^2 c^2 d^{4/3} e-4 b \left(a^3 \sqrt[3]{d} e^2+3 a^2 c d e^{4/3}+c^3 d^{7/3}\right)-4 a b^3 d e^{4/3}+6 b^2 \left(2 a c d^{4/3} e+c^2 d^2 \sqrt[3]{e}\right)+4 a c^3 d^2 \sqrt[3]{e}+b^4 d^{4/3} e\right)}{d^{2/3}}+\frac{10 \log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(a^4 e^{7/3}+6 a^2 c^2 d^{4/3} e-4 b \left(a^3 \sqrt[3]{d} e^2+3 a^2 c d e^{4/3}+c^3 d^{7/3}\right)-4 a b^3 d e^{4/3}+6 b^2 \left(2 a c d^{4/3} e+c^2 d^2 \sqrt[3]{e}\right)+4 a c^3 d^2 \sqrt[3]{e}+b^4 d^{4/3} e\right)}{d^{2/3}}+10 c e^{2/3} x^3 \left(12 a b c e+4 b^3 e-c^3 d\right)+15 c^2 e^{5/3} x^4 \left(2 a c+3 b^2\right)+24 b c^3 e^{5/3} x^5+5 c^4 e^{5/3} x^6}{30 e^{8/3}}","-\frac{x^2 \left(-6 a^2 c^2 e-12 a b^2 c e+b^4 (-e)+4 b c^3 d\right)}{2 e^2}-\frac{2 x \left(-6 a^2 b c e-2 a b^3 e+2 a c^3 d+3 b^2 c^2 d\right)}{e^2}+\frac{\log \left(d+e x^3\right) \left(-4 c e \left(b^3 d-a^3 e\right)+6 a^2 b^2 e^2-12 a b c^2 d e+c^4 d^2\right)}{3 e^3}-\frac{\left(a \sqrt[3]{e}+b \sqrt[3]{d}\right) \tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{e} x}{\sqrt{3} \sqrt[3]{d}}\right) \left(-e \left(a^3 (-e)-3 a^2 b \sqrt[3]{d} e^{2/3}+3 a b^2 d^{2/3} \sqrt[3]{e}+b^3 d\right)+6 c^2 \left(b d^{5/3} \sqrt[3]{e}-a d^{4/3} e^{2/3}\right)-12 a b c d e+4 c^3 d^2\right)}{\sqrt{3} d^{2/3} e^{8/3}}-\frac{\log \left(d^{2/3}-\sqrt[3]{d} \sqrt[3]{e} x+e^{2/3} x^2\right) \left(\sqrt[3]{e} \left(a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right)+\sqrt[3]{d} \left(-4 b \left(a^3 e^2+c^3 d^2\right)+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right)\right)}{6 d^{2/3} e^{8/3}}+\frac{\log \left(\sqrt[3]{d}+\sqrt[3]{e} x\right) \left(\sqrt[3]{e} \left(a^4 e^2-12 a^2 b c d e-4 a b^3 d e+4 a c^3 d^2+6 b^2 c^2 d^2\right)+\sqrt[3]{d} \left(-4 b \left(a^3 e^2+c^3 d^2\right)+6 a^2 c^2 d e+12 a b^2 c d e+b^4 d e\right)\right)}{3 d^{2/3} e^{8/3}}-\frac{c x^3 \left(-12 a b c e-4 b^3 e+c^3 d\right)}{3 e^2}+\frac{c^2 x^4 \left(2 a c+3 b^2\right)}{2 e}+\frac{4 b c^3 x^5}{5 e}+\frac{c^4 x^6}{6 e}",1,"(60*e^(2/3)*(-3*b^2*c^2*d - 2*a*c^3*d + 2*a*b^3*e + 6*a^2*b*c*e)*x + 15*e^(2/3)*(-4*b*c^3*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^2 + 10*c*e^(2/3)*(-(c^3*d) + 4*b^3*e + 12*a*b*c*e)*x^3 + 15*c^2*(3*b^2 + 2*a*c)*e^(5/3)*x^4 + 24*b*c^3*e^(5/3)*x^5 + 5*c^4*e^(5/3)*x^6 + (10*Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*(-4*c^3*d^2 + c^2*(-6*b*d^(5/3)*e^(1/3) + 6*a*d^(4/3)*e^(2/3)) + 12*a*b*c*d*e + e*(b^3*d + 3*a*b^2*d^(2/3)*e^(1/3) - 3*a^2*b*d^(1/3)*e^(2/3) - a^3*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (10*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (5*(4*a*c^3*d^2*e^(1/3) + b^4*d^(4/3)*e + 6*a^2*c^2*d^(4/3)*e - 4*a*b^3*d*e^(4/3) + a^4*e^(7/3) + 6*b^2*(c^2*d^2*e^(1/3) + 2*a*c*d^(4/3)*e) - 4*b*(c^3*d^(7/3) + 3*a^2*c*d*e^(4/3) + a^3*d^(1/3)*e^2))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + (10*(c^4*d^2 - 12*a*b*c^2*d*e + 6*a^2*b^2*e^2 + 4*c*e*(-(b^3*d) + a^3*e))*Log[d + e*x^3])/e^(1/3))/(30*e^(8/3))","A",1
76,1,54,43,0.0161732,"\int \frac{2 x^2+x^4}{1+x^3} \, dx","Integrate[(2*x^2 + x^4)/(1 + x^3),x]","\frac{1}{6} \left(4 \log \left(x^3+1\right)+3 x^2-\log \left(x^2-x+1\right)+2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{x^2}{2}+\frac{1}{2} \log \left(x^2-x+1\right)+\log (x+1)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(3*x^2 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - Log[1 - x + x^2] + 4*Log[1 + x^3])/6","A",1
77,1,54,46,0.0287763,"\int \frac{2 x^2+x^4}{1-x^3} \, dx","Integrate[(2*x^2 + x^4)/(1 - x^3),x]","\frac{1}{6} \left(-4 \log \left(1-x^3\right)-3 x^2+\log \left(x^2+x+1\right)-2 \log (1-x)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{x^2}{2}-\frac{1}{2} \log \left(x^2+x+1\right)-\log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(-3*x^2 - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Log[1 - x] + Log[1 + x + x^2] - 4*Log[1 - x^3])/6","A",1
78,1,44,44,0.0099717,"\int \frac{1-x+4 x^3}{1+x^3} \, dx","Integrate[(1 - x + 4*x^3)/(1 + x^3),x]","\frac{1}{3} \log \left(x^2-x+1\right)+4 x-\frac{2}{3} \log (x+1)-\frac{4 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{3} \log \left(x^2-x+1\right)+4 x-\frac{2}{3} \log (x+1)+\frac{4 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"4*x - (4*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] - (2*Log[1 + x])/3 + Log[1 - x + x^2]/3","A",1
79,1,47,230,0.0409321,"\int \frac{1+\sqrt{3}+x}{\sqrt{1+x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/Sqrt[1 + x^3],x]","\left(1+\sqrt{3}\right) x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)+\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)","\frac{2 \sqrt{x^3+1}}{x+\sqrt{3}+1}+\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(1 + Sqrt[3])*x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + (x^2*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3])/2","C",1
80,1,43,257,0.0172009,"\int \frac{1+\sqrt{3}-x}{\sqrt{1-x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/Sqrt[1 - x^3],x]","\left(1+\sqrt{3}\right) x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)","-\frac{2 \sqrt{1-x^3}}{-x+\sqrt{3}+1}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(1 + Sqrt[3])*x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] - (x^2*Hypergeometric2F1[1/2, 2/3, 5/3, x^3])/2","C",1
81,1,63,144,0.0391779,"\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/Sqrt[-1 + x^3],x]","-\frac{x \sqrt{1-x^3} \left(x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)-2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)\right)}{2 \sqrt{x^3-1}}","\frac{2 \sqrt{x^3-1}}{-x-\sqrt{3}+1}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"-1/2*(x*Sqrt[1 - x^3]*(-2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, x^3]))/Sqrt[-1 + x^3]","C",1
82,1,67,135,0.0290742,"\int \frac{1+\sqrt{3}+x}{\sqrt{-1-x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/Sqrt[-1 - x^3],x]","\frac{x \sqrt{x^3+1} \left(2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)+x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)\right)}{2 \sqrt{-x^3-1}}","\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{2 \sqrt{-x^3-1}}{x-\sqrt{3}+1}",1,"(x*Sqrt[1 + x^3]*(2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/(2*Sqrt[-1 - x^3])","C",1
83,1,90,468,0.0974752,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt{a+b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 \left(1+\sqrt{3}\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{\sqrt[3]{b} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*(1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
84,1,91,481,0.0874538,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\sqrt{a-b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[a - b*x^3],x]","\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(1+\sqrt{3}\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)-\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{a-b x^3}}","-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \sqrt{a-b x^3}}{\sqrt[3]{b} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 - (b*x^3)/a]*(2*(1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] - b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[a - b*x^3])","C",1
85,1,92,271,0.0441938,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\sqrt{-a+b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[-a + b*x^3],x]","\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(1+\sqrt{3}\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)-\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{b x^3-a}}","\frac{2 \sqrt{b x^3-a}}{\sqrt[3]{b} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}",1,"(x*Sqrt[1 - (b*x^3)/a]*(2*(1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] - b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[-a + b*x^3])","C",1
86,1,93,266,0.0498998,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt{-a-b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/Sqrt[-a - b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 \left(1+\sqrt{3}\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{-a-b x^3}}","\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt{-a-b x^3}}{\sqrt[3]{b} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*(1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])","C",1
87,1,89,520,0.0544254,"\int \frac{1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\sqrt{a+b x^3}} \, dx","Integrate[(1 + Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\left(1+\sqrt{3}\right) \sqrt[3]{b}-\left(1-\sqrt{3}\right) \sqrt[3]{a} \sqrt[3]{\frac{b}{a}}\right) \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{\frac{b}{a}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt[3]{\frac{b}{a}} \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
88,1,89,533,0.0552345,"\int \frac{1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\sqrt{a-b x^3}} \, dx","Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[a - b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)-2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{a-b x^3}}","-\frac{2 \sqrt{2+\sqrt{3}} \left(\left(1+\sqrt{3}\right) \sqrt[3]{b}-\left(1-\sqrt{3}\right) \sqrt[3]{a} \sqrt[3]{\frac{b}{a}}\right) \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{\frac{b}{a}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \sqrt[3]{\frac{b}{a}} \sqrt{a-b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(-2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[a - b*x^3]","C",1
89,1,90,256,0.0430288,"\int \frac{1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\sqrt{-a+b x^3}} \, dx","Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[-a + b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)-2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{b x^3-a}}","\frac{2 \left(\frac{b}{a}\right)^{2/3} \sqrt{b x^3-a}}{b \left(x \left(-\sqrt[3]{\frac{b}{a}}\right)-\sqrt{3}+1\right)}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-x \sqrt[3]{\frac{b}{a}}\right) \sqrt{\frac{x^2 \left(\frac{b}{a}\right)^{2/3}+x \sqrt[3]{\frac{b}{a}}+1}{\left(x \left(-\sqrt[3]{\frac{b}{a}}\right)-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b}{a}} x+\sqrt{3}+1}{-\sqrt[3]{\frac{b}{a}} x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{\frac{b}{a}} \sqrt{-\frac{1-x \sqrt[3]{\frac{b}{a}}}{\left(x \left(-\sqrt[3]{\frac{b}{a}}\right)-\sqrt{3}+1\right)^2}} \sqrt{b x^3-a}}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(-2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[-a + b*x^3]","C",1
90,1,92,251,0.0405016,"\int \frac{1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\sqrt{-a-b x^3}} \, dx","Integrate[(1 + Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[-a - b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 \left(1+\sqrt{3}\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{-a-b x^3}}","\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(x \sqrt[3]{\frac{b}{a}}+1\right) \sqrt{\frac{x^2 \left(\frac{b}{a}\right)^{2/3}-x \sqrt[3]{\frac{b}{a}}+1}{\left(x \sqrt[3]{\frac{b}{a}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{\frac{b}{a}} x+\sqrt{3}+1}{\sqrt[3]{\frac{b}{a}} x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{\frac{b}{a}} \sqrt{-\frac{x \sqrt[3]{\frac{b}{a}}+1}{\left(x \sqrt[3]{\frac{b}{a}}-\sqrt{3}+1\right)^2}} \sqrt{-a-b x^3}}-\frac{2 \left(\frac{b}{a}\right)^{2/3} \sqrt{-a-b x^3}}{b \left(x \sqrt[3]{\frac{b}{a}}-\sqrt{3}+1\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*(1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])","C",1
91,1,49,127,0.0262995,"\int \frac{1-\sqrt{3}+x}{\sqrt{1+x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/Sqrt[1 + x^3],x]","\left(1-\sqrt{3}\right) x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)+\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)","\frac{2 \sqrt{x^3+1}}{x+\sqrt{3}+1}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(1 - Sqrt[3])*x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + (x^2*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3])/2","C",1
92,1,45,142,0.017551,"\int \frac{1-\sqrt{3}-x}{\sqrt{1-x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/Sqrt[1 - x^3],x]","\left(1-\sqrt{3}\right) x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)","\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{1-x^3}}{-x+\sqrt{3}+1}",1,"(1 - Sqrt[3])*x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] - (x^2*Hypergeometric2F1[1/2, 2/3, 5/3, x^3])/2","C",1
93,1,63,264,0.0325208,"\int \frac{1-\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/Sqrt[-1 + x^3],x]","-\frac{x \sqrt{1-x^3} \left(2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)+x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)\right)}{2 \sqrt{x^3-1}}","\frac{2 \sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"-1/2*(x*Sqrt[1 - x^3]*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, x^3]))/Sqrt[-1 + x^3]","C",1
94,1,67,247,0.0318468,"\int \frac{1-\sqrt{3}+x}{\sqrt{-1-x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/Sqrt[-1 - x^3],x]","\frac{x \sqrt{x^3+1} \left(x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)-2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)\right)}{2 \sqrt{-x^3-1}}","-\frac{2 \sqrt{-x^3-1}}{x-\sqrt{3}+1}-\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(x*Sqrt[1 + x^3]*(-2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/(2*Sqrt[-1 - x^3])","C",1
95,1,47,126,0.0350271,"\int \frac{-1+\sqrt{3}-x}{\sqrt{1+x^3}} \, dx","Integrate[(-1 + Sqrt[3] - x)/Sqrt[1 + x^3],x]","\left(\sqrt{3}-1\right) x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)","\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2 \sqrt{x^3+1}}{x+\sqrt{3}+1}",1,"(-1 + Sqrt[3])*x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] - (x^2*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3])/2","C",1
96,1,43,143,0.012017,"\int \frac{-1+\sqrt{3}+x}{\sqrt{1-x^3}} \, dx","Integrate[(-1 + Sqrt[3] + x)/Sqrt[1 - x^3],x]","\frac{1}{2} x \left(2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)+x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)\right)","\frac{2 \sqrt{1-x^3}}{-x+\sqrt{3}+1}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(x*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, x^3]))/2","C",1
97,1,63,263,0.0230924,"\int \frac{-1+\sqrt{3}+x}{\sqrt{-1+x^3}} \, dx","Integrate[(-1 + Sqrt[3] + x)/Sqrt[-1 + x^3],x]","\frac{x \sqrt{1-x^3} \left(2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)+x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)\right)}{2 \sqrt{x^3-1}}","-\frac{2 \sqrt{x^3-1}}{-x-\sqrt{3}+1}-\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(x*Sqrt[1 - x^3]*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, x^3]))/(2*Sqrt[-1 + x^3])","C",1
98,1,67,248,0.0241188,"\int \frac{-1+\sqrt{3}-x}{\sqrt{-1-x^3}} \, dx","Integrate[(-1 + Sqrt[3] - x)/Sqrt[-1 - x^3],x]","-\frac{x \sqrt{x^3+1} \left(x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)-2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)\right)}{2 \sqrt{-x^3-1}}","\frac{2 \sqrt{-x^3-1}}{x-\sqrt{3}+1}+\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"-1/2*(x*Sqrt[1 + x^3]*(-2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + x*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/Sqrt[-1 - x^3]","C",1
99,1,90,256,0.0735951,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt{a+b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-2 \left(\sqrt{3}-1\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{2 \sqrt{a+b x^3}}{\sqrt[3]{b} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}",1,"(x*Sqrt[1 + (b*x^3)/a]*(-2*(-1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
100,1,90,263,0.0933436,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\sqrt{a-b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[a - b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(\sqrt{3}-1\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{a-b x^3}}","\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \sqrt{a-b x^3}}{\sqrt[3]{b} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[a - b*x^3]","C",1
101,1,91,497,0.0677937,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\sqrt{-a+b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)/Sqrt[-a + b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(\sqrt{3}-1\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{b x^3-a}}","\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt{b x^3-a}}{\sqrt[3]{b} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[-a + b*x^3]","C",1
102,1,93,488,0.0897079,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt{-a-b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/Sqrt[-a - b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(\sqrt[3]{b} x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-2 \left(\sqrt{3}-1\right) \sqrt[3]{a} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{-a-b x^3}}","-\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt{-a-b x^3}}{\sqrt[3]{b} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(-2*(-1 + Sqrt[3])*a^(1/3)*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + b^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])","C",1
103,1,89,241,0.0774392,"\int \frac{1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\sqrt{a+b x^3}} \, dx","Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{2 \left(\frac{b}{a}\right)^{2/3} \sqrt{a+b x^3}}{b \left(x \sqrt[3]{\frac{b}{a}}+\sqrt{3}+1\right)}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(x \sqrt[3]{\frac{b}{a}}+1\right) \sqrt{\frac{x^2 \left(\frac{b}{a}\right)^{2/3}-x \sqrt[3]{\frac{b}{a}}+1}{\left(x \sqrt[3]{\frac{b}{a}}+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{\frac{b}{a}} x-\sqrt{3}+1}{\sqrt[3]{\frac{b}{a}} x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{\frac{b}{a}} \sqrt{\frac{x \sqrt[3]{\frac{b}{a}}+1}{\left(x \sqrt[3]{\frac{b}{a}}+\sqrt{3}+1\right)^2}} \sqrt{a+b x^3}}",1,"(x*Sqrt[1 + (b*x^3)/a]*(-2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
104,1,89,248,0.0614113,"\int \frac{1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\sqrt{a-b x^3}} \, dx","Integrate[(1 - Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[a - b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{a-b x^3}}","\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(1-x \sqrt[3]{\frac{b}{a}}\right) \sqrt{\frac{x^2 \left(\frac{b}{a}\right)^{2/3}+x \sqrt[3]{\frac{b}{a}}+1}{\left(x \left(-\sqrt[3]{\frac{b}{a}}\right)+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{b}{a}} x-\sqrt{3}+1}{-\sqrt[3]{\frac{b}{a}} x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[3]{\frac{b}{a}} \sqrt{\frac{1-x \sqrt[3]{\frac{b}{a}}}{\left(x \left(-\sqrt[3]{\frac{b}{a}}\right)+\sqrt{3}+1\right)^2}} \sqrt{a-b x^3}}-\frac{2 \left(\frac{b}{a}\right)^{2/3} \sqrt{a-b x^3}}{b \left(x \left(-\sqrt[3]{\frac{b}{a}}\right)+\sqrt{3}+1\right)}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[a - b*x^3]","C",1
105,1,90,549,0.0450858,"\int \frac{1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\sqrt{-a+b x^3}} \, dx","Integrate[(1 - Sqrt[3] - (b/a)^(1/3)*x)/Sqrt[-a + b*x^3],x]","-\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{b x^3-a}}","-\frac{2 \sqrt{2-\sqrt{3}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{b}-\left(1+\sqrt{3}\right) \sqrt[3]{a} \sqrt[3]{\frac{b}{a}}\right) \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \sqrt[3]{\frac{b}{a}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt[3]{\frac{b}{a}} \sqrt{b x^3-a}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"-1/2*(x*Sqrt[1 - (b*x^3)/a]*(2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/Sqrt[-a + b*x^3]","C",1
106,1,92,540,0.0478651,"\int \frac{1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\sqrt{-a-b x^3}} \, dx","Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/Sqrt[-a - b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(x \sqrt[3]{\frac{b}{a}} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-2 \left(\sqrt{3}-1\right) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{-a-b x^3}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{b}-\left(1+\sqrt{3}\right) \sqrt[3]{a} \sqrt[3]{\frac{b}{a}}\right) \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \sqrt[3]{\frac{b}{a}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt[3]{\frac{b}{a}} \sqrt{-a-b x^3}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(-2*(-1 + Sqrt[3])*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + (b/a)^(1/3)*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])","C",1
107,1,75,490,0.031777,"\int \frac{c+d x}{\sqrt{a+b x^3}} \, dx","Integrate[(c + d*x)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} c-\left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 d \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[a + b*x^3])","C",1
108,1,75,503,0.0442798,"\int \frac{c+d x}{\sqrt{a-b x^3}} \, dx","Integrate[(c + d*x)/Sqrt[a - b*x^3],x]","\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{a-b x^3}}","-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a} d+\sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{2 d \sqrt{a-b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 - (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[a - b*x^3])","C",1
109,1,76,515,0.027503,"\int \frac{c+d x}{\sqrt{-a+b x^3}} \, dx","Integrate[(c + d*x)/Sqrt[-a + b*x^3],x]","\frac{x \sqrt{1-\frac{b x^3}{a}} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right)\right)}{2 \sqrt{b x^3-a}}","-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a} d+\sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2 d \sqrt{b x^3-a}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 - (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[-a + b*x^3])","C",1
110,1,78,508,0.0327023,"\int \frac{c+d x}{\sqrt{-a-b x^3}} \, dx","Integrate[(c + d*x)/Sqrt[-a - b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt{-a-b x^3}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} c-\left(1+\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2 d \sqrt{-a-b x^3}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(x*Sqrt[1 + (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[-a - b*x^3])","C",1
111,1,42,246,0.0102834,"\int \frac{c+d x}{\sqrt{1+x^3}} \, dx","Integrate[(c + d*x)/Sqrt[1 + x^3],x]","c x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)+\frac{1}{2} d x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)","\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(c-\left(1-\sqrt{3}\right) d\right) F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}+\frac{2 d \sqrt{x^3+1}}{x+\sqrt{3}+1}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"c*x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + (d*x^2*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3])/2","C",1
112,1,38,271,0.0108869,"\int \frac{c+d x}{\sqrt{1-x^3}} \, dx","Integrate[(c + d*x)/Sqrt[1 - x^3],x]","c x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)+\frac{1}{2} d x^2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)","-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(c-\sqrt{3} d+d\right) F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}+\frac{2 d \sqrt{1-x^3}}{-x+\sqrt{3}+1}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"c*x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + (d*x^2*Hypergeometric2F1[1/2, 2/3, 5/3, x^3])/2","C",1
113,1,58,275,0.0303494,"\int \frac{c+d x}{\sqrt{-1+x^3}} \, dx","Integrate[(c + d*x)/Sqrt[-1 + x^3],x]","\frac{x \sqrt{1-x^3} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};x^3\right)\right)}{2 \sqrt{x^3-1}}","-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(c+\sqrt{3} d+d\right) F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{2 d \sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} d (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(x*Sqrt[1 - x^3]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, x^3] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, x^3]))/(2*Sqrt[-1 + x^3])","C",1
114,1,62,261,0.0221537,"\int \frac{c+d x}{\sqrt{-1-x^3}} \, dx","Integrate[(c + d*x)/Sqrt[-1 - x^3],x]","\frac{x \sqrt{x^3+1} \left(2 c \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)+d x \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)\right)}{2 \sqrt{-x^3-1}}","\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(c-\left(1+\sqrt{3}\right) d\right) F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{2 d \sqrt{-x^3-1}}{x-\sqrt{3}+1}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(x*Sqrt[1 + x^3]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/(2*Sqrt[-1 - x^3])","C",1
115,1,134,87,0.0394656,"\int \frac{c+d x}{a-b x^4} \, dx","Integrate[(c + d*x)/(a - b*x^4),x]","\frac{-\left(\sqrt[4]{a} d+\sqrt[4]{b} c\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)+\sqrt[4]{b} c \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+2 \sqrt[4]{b} c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)+\sqrt[4]{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)-\sqrt[4]{a} d \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{4 a^{3/4} \sqrt{b}}","\frac{c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b}}+\frac{c \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"(2*b^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)] - (b^(1/4)*c + a^(1/4)*d)*Log[a^(1/4) - b^(1/4)*x] + b^(1/4)*c*Log[a^(1/4) + b^(1/4)*x] - a^(1/4)*d*Log[a^(1/4) + b^(1/4)*x] + a^(1/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(4*a^(3/4)*Sqrt[b])","A",1
116,1,184,219,0.0895332,"\int \frac{c+d x}{a+b x^4} \, dx","Integrate[(c + d*x)/(a + b*x^4),x]","\frac{-2 \left(2 \sqrt[4]{a} d+\sqrt{2} \sqrt[4]{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \left(\sqrt{2} \sqrt[4]{b} c-2 \sqrt[4]{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)+\sqrt{2} \sqrt[4]{b} c \left(\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)\right)}{8 a^{3/4} \sqrt{b}}","-\frac{c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{b}}+\frac{c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{c \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}+\frac{c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"(-2*(Sqrt[2]*b^(1/4)*c + 2*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[2]*b^(1/4)*c - 2*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*c*(-Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(8*a^(3/4)*Sqrt[b])","A",1
117,1,168,110,0.201922,"\int \frac{c+d x}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x)/(a - b*x^4)^2,x]","\frac{\frac{4 a x (c+d x)}{a-b x^4}-\frac{\left(3 \sqrt[4]{a} \sqrt[4]{b} c+2 \sqrt{a} d\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{\left(3 \sqrt[4]{a} \sqrt[4]{b} c-2 \sqrt{a} d\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{6 \sqrt[4]{a} c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{2 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{16 a^2}","\frac{3 c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} \sqrt[4]{b}}+\frac{3 c \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} \sqrt[4]{b}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left(a-b x^4\right)}",1,"((4*a*x*(c + d*x))/(a - b*x^4) + (6*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/4) - ((3*a^(1/4)*b^(1/4)*c + 2*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + ((3*a^(1/4)*b^(1/4)*c - 2*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (2*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)","A",1
118,1,224,241,0.2794358,"\int \frac{c+d x}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x)/(a + b*x^4)^2,x]","\frac{\frac{8 a^{3/4} x (c+d x)}{a+b x^4}-\frac{2 \left(4 \sqrt[4]{a} d+3 \sqrt{2} \sqrt[4]{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt{b}}+\frac{2 \left(3 \sqrt{2} \sqrt[4]{b} c-4 \sqrt[4]{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{\sqrt{b}}-\frac{3 \sqrt{2} c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}+\frac{3 \sqrt{2} c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}}{32 a^{7/4}}","-\frac{3 c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{3 c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{3 c \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{3 c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left(a+b x^4\right)}",1,"((8*a^(3/4)*x*(c + d*x))/(a + b*x^4) - (2*(3*Sqrt[2]*b^(1/4)*c + 4*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] + (2*(3*Sqrt[2]*b^(1/4)*c - 4*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (3*Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (3*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(32*a^(7/4))","A",1
119,1,193,136,0.208316,"\int \frac{c+d x}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x)/(a - b*x^4)^3,x]","\frac{\frac{16 a^2 x (c+d x)}{\left(a-b x^4\right)^2}+\frac{4 a x (7 c+6 d x)}{a-b x^4}-\frac{3 \left(7 \sqrt[4]{a} \sqrt[4]{b} c+4 \sqrt{a} d\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{3 \left(7 \sqrt[4]{a} \sqrt[4]{b} c-4 \sqrt{a} d\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{42 \sqrt[4]{a} c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{12 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{128 a^3}","\frac{21 c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} \sqrt[4]{b}}+\frac{21 c \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} \sqrt[4]{b}}+\frac{3 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left(a-b x^4\right)}+\frac{x (c+d x)}{8 a \left(a-b x^4\right)^2}",1,"((16*a^2*x*(c + d*x))/(a - b*x^4)^2 + (4*a*x*(7*c + 6*d*x))/(a - b*x^4) + (42*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/4) - (3*(7*a^(1/4)*b^(1/4)*c + 4*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + (3*(7*a^(1/4)*b^(1/4)*c - 4*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(128*a^3)","A",1
120,1,249,266,0.263858,"\int \frac{c+d x}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x)/(a + b*x^4)^3,x]","\frac{\frac{32 a^{7/4} x (c+d x)}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} x (7 c+6 d x)}{a+b x^4}-\frac{6 \left(8 \sqrt[4]{a} d+7 \sqrt{2} \sqrt[4]{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt{b}}+\frac{6 \left(7 \sqrt{2} \sqrt[4]{b} c-8 \sqrt[4]{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{\sqrt{b}}-\frac{21 \sqrt{2} c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}+\frac{21 \sqrt{2} c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}}{256 a^{11/4}}","-\frac{21 c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{21 c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{21 c \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{21 c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{3 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left(a+b x^4\right)}+\frac{x (c+d x)}{8 a \left(a+b x^4\right)^2}",1,"((32*a^(7/4)*x*(c + d*x))/(a + b*x^4)^2 + (8*a^(3/4)*x*(7*c + 6*d*x))/(a + b*x^4) - (6*(7*Sqrt[2]*b^(1/4)*c + 8*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] + (6*(7*Sqrt[2]*b^(1/4)*c - 8*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (21*Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (21*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(256*a^(11/4))","A",1
121,1,217,162,0.2151222,"\int \frac{c+d x}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x)/(a - b*x^4)^4,x]","\frac{\frac{128 a^3 x (c+d x)}{\left(a-b x^4\right)^3}+\frac{16 a^2 x (11 c+10 d x)}{\left(a-b x^4\right)^2}+\frac{4 a x (77 c+60 d x)}{a-b x^4}-\frac{3 \left(77 \sqrt[4]{a} \sqrt[4]{b} c+40 \sqrt{a} d\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{3 \left(77 \sqrt[4]{a} \sqrt[4]{b} c-40 \sqrt{a} d\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{\sqrt{b}}+\frac{462 \sqrt[4]{a} c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{120 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{1536 a^4}","\frac{77 c \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} \sqrt[4]{b}}+\frac{77 c \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} \sqrt[4]{b}}+\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left(a-b x^4\right)}+\frac{x (11 c+10 d x)}{96 a^2 \left(a-b x^4\right)^2}+\frac{x (c+d x)}{12 a \left(a-b x^4\right)^3}",1,"((128*a^3*x*(c + d*x))/(a - b*x^4)^3 + (16*a^2*x*(11*c + 10*d*x))/(a - b*x^4)^2 + (4*a*x*(77*c + 60*d*x))/(a - b*x^4) + (462*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/4) - (3*(77*a^(1/4)*b^(1/4)*c + 40*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + (3*(77*a^(1/4)*b^(1/4)*c - 40*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)","A",1
122,1,274,291,0.3383481,"\int \frac{c+d x}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x)/(a + b*x^4)^4,x]","\frac{\frac{256 a^{11/4} x (c+d x)}{\left(a+b x^4\right)^3}+\frac{32 a^{7/4} x (11 c+10 d x)}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} x (77 c+60 d x)}{a+b x^4}-\frac{6 \left(80 \sqrt[4]{a} d+77 \sqrt{2} \sqrt[4]{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{\sqrt{b}}+\frac{6 \left(77 \sqrt{2} \sqrt[4]{b} c-80 \sqrt[4]{a} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{\sqrt{b}}-\frac{231 \sqrt{2} c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}+\frac{231 \sqrt{2} c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{b}}}{3072 a^{15/4}}","-\frac{77 c \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{77 c \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} \sqrt[4]{b}}-\frac{77 c \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 \sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{77 c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{256 \sqrt{2} a^{15/4} \sqrt[4]{b}}+\frac{5 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x (77 c+60 d x)}{384 a^3 \left(a+b x^4\right)}+\frac{x (11 c+10 d x)}{96 a^2 \left(a+b x^4\right)^2}+\frac{x (c+d x)}{12 a \left(a+b x^4\right)^3}",1,"((256*a^(11/4)*x*(c + d*x))/(a + b*x^4)^3 + (32*a^(7/4)*x*(11*c + 10*d*x))/(a + b*x^4)^2 + (8*a^(3/4)*x*(77*c + 60*d*x))/(a + b*x^4) - (6*(77*Sqrt[2]*b^(1/4)*c + 80*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] + (6*(77*Sqrt[2]*b^(1/4)*c - 80*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (231*Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (231*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(3072*a^(15/4))","A",1
123,1,42,24,0.0211035,"\int \frac{c+d x}{1-x^4} \, dx","Integrate[(c + d*x)/(1 - x^4),x]","\frac{1}{4} \left(-(c+d) \log (1-x)+c \log (x+1)+2 c \tan ^{-1}(x)+d \log \left(x^2+1\right)-d \log (x+1)\right)","\frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left(x^2\right)",1,"(2*c*ArcTan[x] - (c + d)*Log[1 - x] + c*Log[1 + x] - d*Log[1 + x] + d*Log[1 + x^2])/4","A",1
124,1,99,98,0.1252705,"\int \frac{c+d x}{1+x^4} \, dx","Integrate[(c + d*x)/(1 + x^4),x]","\frac{1}{4} \left(-\left(\left(\sqrt[4]{-1} c+i d\right) \log \left(\sqrt[4]{-1}-x\right)\right)+\left(-(-1)^{3/4} c+i d\right) \log \left((-1)^{3/4}-x\right)+\left(\sqrt[4]{-1} c-i d\right) \log \left(x+\sqrt[4]{-1}\right)+\left((-1)^{3/4} c+i d\right) \log \left(x+(-1)^{3/4}\right)\right)","-\frac{c \log \left(x^2-\sqrt{2} x+1\right)}{4 \sqrt{2}}+\frac{c \log \left(x^2+\sqrt{2} x+1\right)}{4 \sqrt{2}}-\frac{c \tan ^{-1}\left(1-\sqrt{2} x\right)}{2 \sqrt{2}}+\frac{c \tan ^{-1}\left(\sqrt{2} x+1\right)}{2 \sqrt{2}}+\frac{1}{2} d \tan ^{-1}\left(x^2\right)",1,"(-(((-1)^(1/4)*c + I*d)*Log[(-1)^(1/4) - x]) + (-((-1)^(3/4)*c) + I*d)*Log[(-1)^(3/4) - x] + ((-1)^(1/4)*c - I*d)*Log[(-1)^(1/4) + x] + ((-1)^(3/4)*c + I*d)*Log[(-1)^(3/4) + x])/4","C",1
125,1,187,116,0.0618318,"\int \frac{c+d x+e x^2}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2)/(a - b*x^4),x]","\frac{-\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(\sqrt[4]{a} \sqrt[4]{b} d+\sqrt{a} e+\sqrt{b} c\right)+2 \left(\sqrt{b} c-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)+\sqrt{b} c \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+\sqrt[4]{a} \sqrt[4]{b} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)-\sqrt[4]{a} \sqrt[4]{b} d \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+\sqrt{a} e \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{4 a^{3/4} b^{3/4}}","\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"(2*(Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + a^(1/4)*b^(1/4)*d + Sqrt[a]*e)*Log[a^(1/4) - b^(1/4)*x] + Sqrt[b]*c*Log[a^(1/4) + b^(1/4)*x] - a^(1/4)*b^(1/4)*d*Log[a^(1/4) + b^(1/4)*x] + Sqrt[a]*e*Log[a^(1/4) + b^(1/4)*x] + a^(1/4)*b^(1/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(4*a^(3/4)*b^(3/4))","A",1
126,1,229,277,0.1150891,"\int \frac{c+d x+e x^2}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^4),x]","\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+\sqrt{2} \sqrt{b} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-2 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+\sqrt{2} \sqrt{b} c\right)-\sqrt{2} \left(\sqrt{b} c-\sqrt{a} e\right) \left(\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)\right)}{8 a^{3/4} b^{3/4}}","-\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"(-2*(Sqrt[2]*Sqrt[b]*c + 2*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[2]*Sqrt[b]*c - 2*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*(Sqrt[b]*c - Sqrt[a]*e)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(8*a^(3/4)*b^(3/4))","A",1
127,1,211,146,0.2826307,"\int \frac{c+d x+e x^2}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(a - b*x^4)^2,x]","\frac{-\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{3/4} e+3 \sqrt[4]{a} \sqrt{b} c+2 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(a^{3/4} e+3 \sqrt[4]{a} \sqrt{b} c-2 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}-\frac{2 \sqrt[4]{a} \left(\sqrt{a} e-3 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{3/4}}+\frac{4 a x (c+x (d+e x))}{a-b x^4}+\frac{2 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{16 a^2}","\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} b^{3/4}}+\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x \left(c+d x+e x^2\right)}{4 a \left(a-b x^4\right)}",1,"((4*a*x*(c + x*(d + e*x)))/(a - b*x^4) - (2*a^(1/4)*(-3*Sqrt[b]*c + Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - ((3*a^(1/4)*Sqrt[b]*c + 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + ((3*a^(1/4)*Sqrt[b]*c - 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (2*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)","A",1
128,1,305,308,0.5453017,"\int \frac{c+d x+e x^2}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^4)^2,x]","\frac{\frac{\sqrt{2} \left(a^{3/4} e-3 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{\sqrt{2} \left(3 \sqrt[4]{a} \sqrt{b} c-a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{8 a x (c+x (d+e x))}{a+b x^4}}{32 a^2}","-\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x \left(c+d x+e x^2\right)}{4 a \left(a+b x^4\right)}",1,"((8*a*x*(c + x*(d + e*x)))/(a + b*x^4) - (2*a^(1/4)*(3*Sqrt[2]*Sqrt[b]*c + 4*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (2*a^(1/4)*(3*Sqrt[2]*Sqrt[b]*c - 4*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqrt[2]*(-3*a^(1/4)*Sqrt[b]*c + a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(3*a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(32*a^2)","A",1
129,1,244,179,0.2894818,"\int \frac{c+d x+e x^2}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(a - b*x^4)^3,x]","\frac{-\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(5 a^{3/4} e+21 \sqrt[4]{a} \sqrt{b} c+12 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(5 a^{3/4} e+21 \sqrt[4]{a} \sqrt{b} c-12 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{16 a^2 x (c+x (d+e x))}{\left(a-b x^4\right)^2}+\frac{2 \sqrt[4]{a} \left(21 \sqrt{b} c-5 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{3/4}}+\frac{4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac{12 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{128 a^3}","\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} b^{3/4}}+\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x \left(7 c+6 d x+5 e x^2\right)}{32 a^2 \left(a-b x^4\right)}+\frac{x \left(c+d x+e x^2\right)}{8 a \left(a-b x^4\right)^2}",1,"((16*a^2*x*(c + x*(d + e*x)))/(a - b*x^4)^2 + (4*a*x*(7*c + x*(6*d + 5*e*x)))/(a - b*x^4) + (2*a^(1/4)*(21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - ((21*a^(1/4)*Sqrt[b]*c + 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + ((21*a^(1/4)*Sqrt[b]*c - 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(128*a^3)","A",1
130,1,337,341,0.4167764,"\int \frac{c+d x+e x^2}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^4)^3,x]","\frac{\frac{\sqrt{2} \left(5 a^{3/4} e-21 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{\sqrt{2} \left(21 \sqrt[4]{a} \sqrt{b} c-5 a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{32 a^2 x (c+x (d+e x))}{\left(a+b x^4\right)^2}-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(24 \sqrt[4]{a} \sqrt[4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-24 \sqrt[4]{a} \sqrt[4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3}","-\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x \left(7 c+6 d x+5 e x^2\right)}{32 a^2 \left(a+b x^4\right)}+\frac{x \left(c+d x+e x^2\right)}{8 a \left(a+b x^4\right)^2}",1,"((32*a^2*x*(c + x*(d + e*x)))/(a + b*x^4)^2 + (8*a*x*(7*c + x*(6*d + 5*e*x)))/(a + b*x^4) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c + 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c - 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[b]*c + 5*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[b]*c - 5*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(256*a^3)","A",1
131,1,276,211,0.2839939,"\int \frac{c+d x+e x^2}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(a - b*x^4)^4,x]","\frac{-\frac{3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(15 a^{3/4} e+77 \sqrt[4]{a} \sqrt{b} c+40 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(15 a^{3/4} e+77 \sqrt[4]{a} \sqrt{b} c-40 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{128 a^3 x (c+x (d+e x))}{\left(a-b x^4\right)^3}+\frac{16 a^2 x (11 c+x (10 d+9 e x))}{\left(a-b x^4\right)^2}+\frac{6 \sqrt[4]{a} \left(77 \sqrt{b} c-15 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{3/4}}+\frac{4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac{120 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{1536 a^4}","\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} b^{3/4}}+\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(77 c+60 d x+45 e x^2\right)}{384 a^3 \left(a-b x^4\right)}+\frac{x \left(11 c+10 d x+9 e x^2\right)}{96 a^2 \left(a-b x^4\right)^2}+\frac{x \left(c+d x+e x^2\right)}{12 a \left(a-b x^4\right)^3}",1,"((128*a^3*x*(c + x*(d + e*x)))/(a - b*x^4)^3 + (4*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a - b*x^4)^2 + (6*a^(1/4)*(77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - (3*(77*a^(1/4)*Sqrt[b]*c + 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + (3*(77*a^(1/4)*Sqrt[b]*c - 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)","A",1
132,1,369,372,0.5849441,"\int \frac{c+d x+e x^2}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^4)^4,x]","\frac{\frac{3 \sqrt{2} \left(15 a^{3/4} e-77 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{3 \sqrt{2} \left(77 \sqrt[4]{a} \sqrt{b} c-15 a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{256 a^3 x (c+x (d+e x))}{\left(a+b x^4\right)^3}+\frac{32 a^2 x (11 c+x (10 d+9 e x))}{\left(a+b x^4\right)^2}-\frac{6 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(80 \sqrt[4]{a} \sqrt[4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{6 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-80 \sqrt[4]{a} \sqrt[4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}}{3072 a^4}","-\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} b^{3/4}}-\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{5 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(77 c+60 d x+45 e x^2\right)}{384 a^3 \left(a+b x^4\right)}+\frac{x \left(11 c+10 d x+9 e x^2\right)}{96 a^2 \left(a+b x^4\right)^2}+\frac{x \left(c+d x+e x^2\right)}{12 a \left(a+b x^4\right)^3}",1,"((256*a^3*x*(c + x*(d + e*x)))/(a + b*x^4)^3 + (8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c - 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-77*a^(1/4)*Sqrt[b]*c + 15*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (3*Sqrt[2]*(77*a^(1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(3072*a^4)","A",1
133,1,27,28,0.0021556,"\int a \left(e+f x^4\right)^2 \, dx","Integrate[a*(e + f*x^4)^2,x]","a \left(e^2 x+\frac{2}{5} e f x^5+\frac{f^2 x^9}{9}\right)","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9",1,"a*(e^2*x + (2*e*f*x^5)/5 + (f^2*x^9)/9)","A",1
134,1,32,33,0.0015628,"\int b x \left(e+f x^4\right)^2 \, dx","Integrate[b*x*(e + f*x^4)^2,x]","b \left(\frac{e^2 x^2}{2}+\frac{1}{3} e f x^6+\frac{f^2 x^{10}}{10}\right)","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}",1,"b*((e^2*x^2)/2 + (e*f*x^6)/3 + (f^2*x^10)/10)","A",1
135,1,60,60,0.0024307,"\int (a+b x) \left(e+f x^4\right)^2 \, dx","Integrate[(a + b*x)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}",1,"a*e^2*x + (b*e^2*x^2)/2 + (2*a*e*f*x^5)/5 + (b*e*f*x^6)/3 + (a*f^2*x^9)/9 + (b*f^2*x^10)/10","A",1
136,1,33,33,0.0012473,"\int c x^2 \left(e+f x^4\right)^2 \, dx","Integrate[c*x^2*(e + f*x^4)^2,x]","\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}","\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}",1,"(c*e^2*x^3)/3 + (2*c*e*f*x^7)/7 + (c*f^2*x^11)/11","A",1
137,1,60,60,0.0028644,"\int \left(a+c x^2\right) \left(e+f x^4\right)^2 \, dx","Integrate[(a + c*x^2)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}",1,"a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c*f^2*x^11)/11","A",1
138,1,65,65,0.0029394,"\int \left(b x+c x^2\right) \left(e+f x^4\right)^2 \, dx","Integrate[(b*x + c*x^2)*(e + f*x^4)^2,x]","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}",1,"(b*e^2*x^2)/2 + (c*e^2*x^3)/3 + (b*e*f*x^6)/3 + (2*c*e*f*x^7)/7 + (b*f^2*x^10)/10 + (c*f^2*x^11)/11","A",1
139,1,92,92,0.0039543,"\int \left(a+b x+c x^2\right) \left(e+f x^4\right)^2 \, dx","Integrate[(a + b*x + c*x^2)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}",1,"a*e^2*x + (b*e^2*x^2)/2 + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (b*e*f*x^6)/3 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (b*f^2*x^10)/10 + (c*f^2*x^11)/11","A",1
140,1,33,17,0.0009466,"\int d x^3 \left(e+f x^4\right)^2 \, dx","Integrate[d*x^3*(e + f*x^4)^2,x]","\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","\frac{d \left(e+f x^4\right)^3}{12 f}",1,"(d*e^2*x^4)/4 + (d*e*f*x^8)/4 + (d*f^2*x^12)/12","A",1
141,1,60,45,0.0028691,"\int \left(a+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(a + d*x^3)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"a*e^2*x + (d*e^2*x^4)/4 + (2*a*e*f*x^5)/5 + (d*e*f*x^8)/4 + (a*f^2*x^9)/9 + (d*f^2*x^12)/12","A",1
142,1,65,50,0.0034342,"\int \left(b x+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(b*x + d*x^3)*(e + f*x^4)^2,x]","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"(b*e^2*x^2)/2 + (d*e^2*x^4)/4 + (b*e*f*x^6)/3 + (d*e*f*x^8)/4 + (b*f^2*x^10)/10 + (d*f^2*x^12)/12","A",1
143,1,92,77,0.0036037,"\int \left(a+b x+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(a + b*x + d*x^3)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"a*e^2*x + (b*e^2*x^2)/2 + (d*e^2*x^4)/4 + (2*a*e*f*x^5)/5 + (b*e*f*x^6)/3 + (d*e*f*x^8)/4 + (a*f^2*x^9)/9 + (b*f^2*x^10)/10 + (d*f^2*x^12)/12","A",1
144,1,65,50,0.0035901,"\int \left(c x^2+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(c*x^2 + d*x^3)*(e + f*x^4)^2,x]","\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"(c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (c*f^2*x^11)/11 + (d*f^2*x^12)/12","A",1
145,1,92,77,0.0051469,"\int \left(a+c x^2+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(a + c*x^2 + d*x^3)*(e + f*x^4)^2,x]","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"a*e^2*x + (c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (a*f^2*x^9)/9 + (c*f^2*x^11)/11 + (d*f^2*x^12)/12","A",1
146,1,97,82,0.0049116,"\int \left(b x+c x^2+d x^3\right) \left(e+f x^4\right)^2 \, dx","Integrate[(b*x + c*x^2 + d*x^3)*(e + f*x^4)^2,x]","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12}","\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left(e+f x^4\right)^3}{12 f}",1,"(b*e^2*x^2)/2 + (c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (b*e*f*x^6)/3 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (b*f^2*x^10)/10 + (c*f^2*x^11)/11 + (d*f^2*x^12)/12","A",1
147,1,124,109,0.0076109,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^2 \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^2,x]","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{4} a^2 f x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{4} a b f x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{1}{12} b^2 f x^{12}","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left(a+b x^4\right)^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}",1,"a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a^2*f*x^4)/4 + (2*a*b*c*x^5)/5 + (a*b*d*x^6)/3 + (2*a*b*e*x^7)/7 + (a*b*f*x^8)/4 + (b^2*c*x^9)/9 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11 + (b^2*f*x^12)/12","A",1
148,1,180,151,0.0050871,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^3 \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^3,x]","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{1}{4} a^3 f x^4+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a^2 b f x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{4} a b^2 f x^{12}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}+\frac{1}{16} b^3 f x^{16}","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{f \left(a+b x^4\right)^4}{16 b}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}",1,"a^3*c*x + (a^3*d*x^2)/2 + (a^3*e*x^3)/3 + (a^3*f*x^4)/4 + (3*a^2*b*c*x^5)/5 + (a^2*b*d*x^6)/2 + (3*a^2*b*e*x^7)/7 + (3*a^2*b*f*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)/11 + (a*b^2*f*x^12)/4 + (b^3*c*x^13)/13 + (b^3*d*x^14)/14 + (b^3*e*x^15)/15 + (b^3*f*x^16)/16","A",1
149,1,220,155,0.2289957,"\int \frac{c+d x+e x^2+f x^3}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4)^2,x]","\frac{-\sqrt[4]{b} \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{3/4} e+3 \sqrt[4]{a} \sqrt{b} c+2 \sqrt{a} \sqrt[4]{b} d\right)+\sqrt[4]{b} \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(a^{3/4} e+3 \sqrt[4]{a} \sqrt{b} c-2 \sqrt{a} \sqrt[4]{b} d\right)+\frac{4 a (a f+b x (c+x (d+e x)))}{a-b x^4}-2 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a} e-3 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt{a} \sqrt{b} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{16 a^2 b}","\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} b^{3/4}}+\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{a f+b x \left(c+d x+e x^2\right)}{4 a b \left(a-b x^4\right)}",1,"((4*a*(a*f + b*x*(c + x*(d + e*x))))/(a - b*x^4) - 2*a^(1/4)*b^(1/4)*(-3*Sqrt[b]*c + Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)] - b^(1/4)*(3*a^(1/4)*Sqrt[b]*c + 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x] + b^(1/4)*(3*a^(1/4)*Sqrt[b]*c - 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x] + 2*Sqrt[a]*Sqrt[b]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(16*a^2*b)","A",1
150,1,253,188,0.261854,"\int \frac{c+d x+e x^2+f x^3}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4)^3,x]","\frac{-\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(5 a^{3/4} e+21 \sqrt[4]{a} \sqrt{b} c+12 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(5 a^{3/4} e+21 \sqrt[4]{a} \sqrt{b} c-12 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{16 a^2 (a f+b x (c+x (d+e x)))}{b \left(a-b x^4\right)^2}+\frac{2 \sqrt[4]{a} \left(21 \sqrt{b} c-5 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{3/4}}+\frac{4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac{12 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{128 a^3}","\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} b^{3/4}}+\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x \left(7 c+6 d x+5 e x^2\right)}{32 a^2 \left(a-b x^4\right)}+\frac{a f+b x \left(c+d x+e x^2\right)}{8 a b \left(a-b x^4\right)^2}",1,"((4*a*x*(7*c + x*(6*d + 5*e*x)))/(a - b*x^4) + (16*a^2*(a*f + b*x*(c + x*(d + e*x))))/(b*(a - b*x^4)^2) + (2*a^(1/4)*(21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - ((21*a^(1/4)*Sqrt[b]*c + 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + ((21*a^(1/4)*Sqrt[b]*c - 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(128*a^3)","A",1
151,1,286,220,0.5018531,"\int \frac{c+d x+e x^2+f x^3}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4)^4,x]","\frac{-\frac{3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(15 a^{3/4} e+77 \sqrt[4]{a} \sqrt{b} c+40 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}+\frac{3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(15 a^{3/4} e+77 \sqrt[4]{a} \sqrt{b} c-40 \sqrt{a} \sqrt[4]{b} d\right)}{b^{3/4}}-\frac{128 a^3 (a f+b x (c+x (d+e x)))}{b \left(b x^4-a\right)^3}+\frac{16 a^2 x (11 c+x (10 d+9 e x))}{\left(a-b x^4\right)^2}+\frac{6 \sqrt[4]{a} \left(77 \sqrt{b} c-15 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{3/4}}+\frac{4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac{120 \sqrt{a} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{b}}}{1536 a^4}","\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} b^{3/4}}+\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 a^{15/4} b^{3/4}}+\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(77 c+60 d x+45 e x^2\right)}{384 a^3 \left(a-b x^4\right)}+\frac{x \left(11 c+10 d x+9 e x^2\right)}{96 a^2 \left(a-b x^4\right)^2}+\frac{a f+b x \left(c+d x+e x^2\right)}{12 a b \left(a-b x^4\right)^3}",1,"((4*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a - b*x^4)^2 - (128*a^3*(a*f + b*x*(c + x*(d + e*x))))/(b*(-a + b*x^4)^3) + (6*a^(1/4)*(77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - (3*(77*a^(1/4)*Sqrt[b]*c + 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + (3*(77*a^(1/4)*Sqrt[b]*c - 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)","A",1
152,1,78,101,0.0344902,"\int \frac{a}{2+3 x^4} \, dx","Integrate[a/(2 + 3*x^4),x]","\frac{a \left(-\log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)+\log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-2 \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \tan ^{-1}\left(\sqrt[4]{6} x+1\right)\right)}{8 \sqrt[4]{6}}","-\frac{a \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}+\frac{a \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}-\frac{a \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4 \sqrt[4]{6}}+\frac{a \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4 \sqrt[4]{6}}",1,"(a*(-2*ArcTan[1 - 6^(1/4)*x] + 2*ArcTan[1 + 6^(1/4)*x] - Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(1/4))","A",1
153,1,22,22,0.0107614,"\int \frac{b x}{2+3 x^4} \, dx","Integrate[(b*x)/(2 + 3*x^4),x]","\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}","\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}",1,"(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6])","A",1
154,1,107,123,0.0553547,"\int \frac{a+b x}{2+3 x^4} \, dx","Integrate[(a + b*x)/(2 + 3*x^4),x]","\frac{-2 \left(\sqrt[4]{6} a+2 b\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \left(\sqrt[4]{6} a-2 b\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+\sqrt[4]{6} a \left(\log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-\log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)\right)}{8 \sqrt{6}}","-\frac{a \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}+\frac{a \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}-\frac{a \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4 \sqrt[4]{6}}+\frac{a \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4 \sqrt[4]{6}}+\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}",1,"(-2*(6^(1/4)*a + 2*b)*ArcTan[1 - 6^(1/4)*x] + 2*(6^(1/4)*a - 2*b)*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*a*(-Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*Sqrt[6])","A",1
155,1,78,101,0.0166109,"\int \frac{c x^2}{2+3 x^4} \, dx","Integrate[(c*x^2)/(2 + 3*x^4),x]","\frac{c \left(\log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-\log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-2 \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \tan ^{-1}\left(\sqrt[4]{6} x+1\right)\right)}{4\ 6^{3/4}}","\frac{c \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{2\ 6^{3/4}}",1,"(c*(-2*ArcTan[1 - 6^(1/4)*x] + 2*ArcTan[1 + 6^(1/4)*x] + Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(4*6^(3/4))","A",1
156,1,113,141,0.0611294,"\int \frac{a+c x^2}{2+3 x^4} \, dx","Integrate[(a + c*x^2)/(2 + 3*x^4),x]","\frac{-\left(\sqrt{6} a-2 c\right) \left(\log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-\log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)\right)-2 \left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{8\ 6^{3/4}}","-\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}+\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}-\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4\ 6^{3/4}}+\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4\ 6^{3/4}}",1,"(-2*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*(Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x] - (Sqrt[6]*a - 2*c)*(Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(3/4))","A",1
157,1,99,123,0.0538676,"\int \frac{b x+c x^2}{2+3 x^4} \, dx","Integrate[(b*x + c*x^2)/(2 + 3*x^4),x]","\frac{-2 \left(\sqrt[4]{6} b+c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \left(c-\sqrt[4]{6} b\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+c \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-c \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)}{4\ 6^{3/4}}","\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}+\frac{c \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{2\ 6^{3/4}}",1,"(-2*(6^(1/4)*b + c)*ArcTan[1 - 6^(1/4)*x] + 2*(-(6^(1/4)*b) + c)*ArcTan[1 + 6^(1/4)*x] + c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2])/(4*6^(3/4))","A",1
158,1,129,163,0.0825462,"\int \frac{a+b x+c x^2}{2+3 x^4} \, dx","Integrate[(a + b*x + c*x^2)/(2 + 3*x^4),x]","\frac{-2 \tan ^{-1}\left(1-\sqrt[4]{6} x\right) \left(\sqrt{6} a+2 \left(\sqrt[4]{6} b+c\right)\right)+2 \tan ^{-1}\left(\sqrt[4]{6} x+1\right) \left(\sqrt{6} a-2 \sqrt[4]{6} b+2 c\right)-\left(\sqrt{6} a-2 c\right) \left(\log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-\log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)\right)}{8\ 6^{3/4}}","-\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}+\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}-\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4\ 6^{3/4}}+\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}",1,"(-2*(Sqrt[6]*a + 2*(6^(1/4)*b + c))*ArcTan[1 - 6^(1/4)*x] + 2*(Sqrt[6]*a - 2*6^(1/4)*b + 2*c)*ArcTan[1 + 6^(1/4)*x] - (Sqrt[6]*a - 2*c)*(Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(3/4))","A",1
159,1,13,13,0.0036665,"\int \frac{d x^3}{2+3 x^4} \, dx","Integrate[(d*x^3)/(2 + 3*x^4),x]","\frac{1}{12} d \log \left(3 x^4+2\right)","\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(d*Log[2 + 3*x^4])/12","A",1
160,1,108,114,0.0340264,"\int \frac{a+d x^3}{2+3 x^4} \, dx","Integrate[(a + d*x^3)/(2 + 3*x^4),x]","\frac{1}{48} \left(-6^{3/4} a \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)+6^{3/4} a \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-2\ 6^{3/4} a \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2\ 6^{3/4} a \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+4 d \log \left(3 x^4+2\right)\right)","-\frac{a \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}+\frac{a \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}-\frac{a \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4 \sqrt[4]{6}}+\frac{a \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4 \sqrt[4]{6}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*6^(3/4)*a*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*a*ArcTan[1 + 6^(1/4)*x] - 6^(3/4)*a*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*a*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48","A",1
161,1,65,36,0.0448115,"\int \frac{b x+d x^3}{2+3 x^4} \, dx","Integrate[(b*x + d*x^3)/(2 + 3*x^4),x]","\frac{1}{24} \left(2 d+i \sqrt{6} b\right) \log \left(\sqrt{6}-3 i x^2\right)+\frac{1}{24} \left(2 d-i \sqrt{6} b\right) \log \left(\sqrt{6}+3 i x^2\right)","\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"((I*Sqrt[6]*b + 2*d)*Log[Sqrt[6] - (3*I)*x^2])/24 + (((-I)*Sqrt[6]*b + 2*d)*Log[Sqrt[6] + (3*I)*x^2])/24","C",1
162,1,128,136,0.0957929,"\int \frac{a+b x+d x^3}{2+3 x^4} \, dx","Integrate[(a + b*x + d*x^3)/(2 + 3*x^4),x]","\frac{1}{48} \left(-2 \sqrt{6} \left(\sqrt[4]{6} a+2 b\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \sqrt{6} \left(\sqrt[4]{6} a-2 b\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)-6^{3/4} a \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)+6^{3/4} a \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)+4 d \log \left(3 x^4+2\right)\right)","-\frac{a \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}+\frac{a \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8 \sqrt[4]{6}}-\frac{a \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4 \sqrt[4]{6}}+\frac{a \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4 \sqrt[4]{6}}+\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*Sqrt[6]*(6^(1/4)*a + 2*b)*ArcTan[1 - 6^(1/4)*x] + 2*Sqrt[6]*(6^(1/4)*a - 2*b)*ArcTan[1 + 6^(1/4)*x] - 6^(3/4)*a*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*a*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48","A",1
163,1,108,114,0.0300832,"\int \frac{c x^2+d x^3}{2+3 x^4} \, dx","Integrate[(c*x^2 + d*x^3)/(2 + 3*x^4),x]","\frac{1}{24} \left(\sqrt[4]{6} c \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-\sqrt[4]{6} c \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-2 \sqrt[4]{6} c \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \sqrt[4]{6} c \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+2 d \log \left(3 x^4+2\right)\right)","\frac{c \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{2\ 6^{3/4}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*6^(1/4)*c*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*c*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/24","A",1
164,1,148,154,0.1062233,"\int \frac{a+c x^2+d x^3}{2+3 x^4} \, dx","Integrate[(a + c*x^2 + d*x^3)/(2 + 3*x^4),x]","\frac{1}{48} \left(-\sqrt[4]{6} \left(\sqrt{6} a-2 c\right) \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)+\sqrt[4]{6} \left(\sqrt{6} a-2 c\right) \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)-2 \sqrt[4]{6} \left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \sqrt[4]{6} \left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+4 d \log \left(3 x^4+2\right)\right)","-\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}+\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}-\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4\ 6^{3/4}}+\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4\ 6^{3/4}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48","A",1
165,1,125,136,0.083217,"\int \frac{b x+c x^2+d x^3}{2+3 x^4} \, dx","Integrate[(b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]","\frac{1}{24} \left(-2 \sqrt[4]{6} \left(\sqrt[4]{6} b+c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)+2 \sqrt[4]{6} \left(c-\sqrt[4]{6} b\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)+\sqrt[4]{6} c \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)-\sqrt[4]{6} c \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)+2 d \log \left(3 x^4+2\right)\right)","\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}+\frac{c \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{2\ 6^{3/4}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*6^(1/4)*(6^(1/4)*b + c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(-(6^(1/4)*b) + c)*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/24","A",1
166,1,164,176,0.1868784,"\int \frac{a+b x+c x^2+d x^3}{2+3 x^4} \, dx","Integrate[(a + b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]","\frac{1}{48} \left(-2 \sqrt[4]{6} \tan ^{-1}\left(1-\sqrt[4]{6} x\right) \left(\sqrt{6} a+2 \left(\sqrt[4]{6} b+c\right)\right)+2 \sqrt[4]{6} \tan ^{-1}\left(\sqrt[4]{6} x+1\right) \left(\sqrt{6} a-2 \sqrt[4]{6} b+2 c\right)-\sqrt[4]{6} \left(\sqrt{6} a-2 c\right) \log \left(\sqrt{6} x^2-2 \sqrt[4]{6} x+2\right)+\sqrt[4]{6} \left(\sqrt{6} a-2 c\right) \log \left(\sqrt{6} x^2+2 \sqrt[4]{6} x+2\right)+4 d \log \left(3 x^4+2\right)\right)","-\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2-6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}+\frac{\left(\sqrt{6} a-2 c\right) \log \left(3 x^2+6^{3/4} x+\sqrt{6}\right)}{8\ 6^{3/4}}-\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(1-\sqrt[4]{6} x\right)}{4\ 6^{3/4}}+\frac{\left(\sqrt{6} a+2 c\right) \tan ^{-1}\left(\sqrt[4]{6} x+1\right)}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left(\sqrt{\frac{3}{2}} x^2\right)}{2 \sqrt{6}}+\frac{1}{12} d \log \left(3 x^4+2\right)",1,"(-2*6^(1/4)*(Sqrt[6]*a + 2*(6^(1/4)*b + c))*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a - 2*6^(1/4)*b + 2*c)*ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48","A",1
167,1,8,8,0.0025639,"\int \frac{1+x+x^2+x^3}{1-x^4} \, dx","Integrate[(1 + x + x^2 + x^3)/(1 - x^4),x]","-\log (1-x)","-\log (1-x)",1,"-Log[1 - x]","A",1
168,1,50,53,0.0421104,"\int \frac{1+x+x^2+x^3}{1+x^4} \, dx","Integrate[(1 + x + x^2 + x^3)/(1 + x^4),x]","\frac{1}{4} \left(\log \left(x^4+1\right)-2 \left(1+\sqrt{2}\right) \tan ^{-1}\left(1-\sqrt{2} x\right)+2 \left(\sqrt{2}-1\right) \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{1}{4} \log \left(x^4+1\right)+\frac{1}{2} \tan ^{-1}\left(x^2\right)-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{\sqrt{2}}",1,"(-2*(1 + Sqrt[2])*ArcTan[1 - Sqrt[2]*x] + 2*(-1 + Sqrt[2])*ArcTan[1 + Sqrt[2]*x] + Log[1 + x^4])/4","A",1
169,1,203,124,0.0641028,"\int \frac{1+x+x^2+x^3}{a-b x^4} \, dx","Integrate[(1 + x + x^2 + x^3)/(a - b*x^4),x]","-\frac{\left(a^{3/4}+\sqrt{a} \sqrt[4]{b}+\sqrt[4]{a} \sqrt{b}\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right)}{4 a b^{3/4}}-\frac{\left(-a^{3/4}+\sqrt{a} \sqrt[4]{b}-\sqrt[4]{a} \sqrt{b}\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{4 a b^{3/4}}+\frac{\left(\sqrt[4]{a} \sqrt{b}-a^{3/4}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a b^{3/4}}-\frac{\log \left(a-b x^4\right)}{4 b}+\frac{\log \left(\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{a} \sqrt{b}}","-\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}-\frac{\log \left(a-b x^4\right)}{4 b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"((-a^(3/4) + a^(1/4)*Sqrt[b])*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a*b^(3/4)) - ((a^(3/4) + Sqrt[a]*b^(1/4) + a^(1/4)*Sqrt[b])*Log[a^(1/4) - b^(1/4)*x])/(4*a*b^(3/4)) - ((-a^(3/4) + Sqrt[a]*b^(1/4) - a^(1/4)*Sqrt[b])*Log[a^(1/4) + b^(1/4)*x])/(4*a*b^(3/4)) + Log[Sqrt[a] + Sqrt[b]*x^2]/(4*Sqrt[a]*Sqrt[b]) - Log[a - b*x^4]/(4*b)","A",1
170,1,283,277,0.2396419,"\int \frac{1+x+x^2+x^3}{a+b x^4} \, dx","Integrate[(1 + x + x^2 + x^3)/(a + b*x^4),x]","\frac{\sqrt{2} \sqrt[4]{b} \left(a^{3/4}-\sqrt[4]{a} \sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} \sqrt[4]{b} \left(\sqrt[4]{a} \sqrt{b}-a^{3/4}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+2 a \log \left(a+b x^4\right)-2 \sqrt[4]{a} \sqrt[4]{b} \left(2 \sqrt[4]{a} \sqrt[4]{b}+\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt[4]{a} \sqrt[4]{b} \left(-2 \sqrt[4]{a} \sqrt[4]{b}+\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 a b}","\frac{\left(\sqrt{a}-\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{a}-\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\log \left(a+b x^4\right)}{4 b}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}",1,"(-2*a^(1/4)*(Sqrt[2]*Sqrt[a] + 2*a^(1/4)*b^(1/4) + Sqrt[2]*Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*(Sqrt[2]*Sqrt[a] - 2*a^(1/4)*b^(1/4) + Sqrt[2]*Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(a^(3/4) - a^(1/4)*Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(-a^(3/4) + a^(1/4)*Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 2*a*Log[a + b*x^4])/(8*a*b)","A",1
171,1,249,148,0.092108,"\int \frac{c+d x+e x^2+f x^3+g x^4}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4),x]","\frac{-a^{3/4} \sqrt[4]{b} f \log \left(a-b x^4\right)-4 a^{3/4} \sqrt[4]{b} g x-\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(\sqrt[4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e+a g+b c\right)+\sqrt[4]{a} b^{3/4} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)-\sqrt[4]{a} b^{3/4} d \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e+a g+b c\right)+b c \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+\sqrt{a} \sqrt{b} e \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)+a g \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right)}{4 a^{3/4} b^{5/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e+a g+b c\right)}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g+b c\right)}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left(a-b x^4\right)}{4 b}-\frac{g x}{b}",1,"(-4*a^(3/4)*b^(1/4)*g*x + 2*(b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (b*c + a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e + a*g)*Log[a^(1/4) - b^(1/4)*x] + b*c*Log[a^(1/4) + b^(1/4)*x] - a^(1/4)*b^(3/4)*d*Log[a^(1/4) + b^(1/4)*x] + Sqrt[a]*Sqrt[b]*e*Log[a^(1/4) + b^(1/4)*x] + a*g*Log[a^(1/4) + b^(1/4)*x] + a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2] - a^(3/4)*b^(1/4)*f*Log[a - b*x^4])/(4*a^(3/4)*b^(5/4))","A",1
172,1,221,172,0.425117,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]","\frac{\frac{4 a^{3/4} \sqrt[4]{b} (a (f+g x)+b x (c+x (d+e x)))}{a-b x^4}-\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(2 \sqrt[4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-2 \sqrt[4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right)+2 \sqrt[4]{a} b^{3/4} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)-2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g-3 b c\right)}{16 a^{7/4} b^{5/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e-a g+3 b c\right)}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e-a g+3 b c\right)}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x \left(a g+b c+b d x+b e x^2+b f x^3\right)}{4 a b \left(a-b x^4\right)}",1,"((4*a^(3/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*x^4) - 2*(-3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (3*b*c + 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) - b^(1/4)*x] + (3*b*c - 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) + b^(1/4)*x] + 2*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(5/4))","A",1
173,1,263,221,0.7720205,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x]","\frac{\frac{4 a^{3/4} \sqrt[4]{b} \left(a^2 (4 f+3 g x)+a b x \left(11 c+x \left(10 d+9 e x+g x^3\right)\right)-b^2 x^5 (7 c+x (6 d+5 e x))\right)}{\left(a-b x^4\right)^2}-\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(12 \sqrt[4]{a} b^{3/4} d+5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-12 \sqrt[4]{a} b^{3/4} d+5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)+12 \sqrt[4]{a} b^{3/4} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)}{128 a^{11/4} b^{5/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)}{64 a^{11/4} b^{5/4}}+\frac{3 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x \left(-a g+7 b c+6 b d x+5 b e x^2\right)+4 a f}{32 a^2 b \left(a-b x^4\right)}+\frac{x \left(a g+b c+b d x+b e x^2+b f x^3\right)}{8 a b \left(a-b x^4\right)^2}",1,"((4*a^(3/4)*b^(1/4)*(a^2*(4*f + 3*g*x) - b^2*x^5*(7*c + x*(6*d + 5*e*x)) + a*b*x*(11*c + x*(10*d + 9*e*x + g*x^3))))/(a - b*x^4)^2 + 2*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (21*b*c + 12*a^(1/4)*b^(3/4)*d + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[a^(1/4) - b^(1/4)*x] + (21*b*c - 12*a^(1/4)*b^(3/4)*d + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[a^(1/4) + b^(1/4)*x] + 12*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^(5/4))","A",1
174,1,313,266,0.3904944,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^4,x]","\frac{\frac{128 a^{11/4} \sqrt[4]{b} (a (f+g x)+b x (c+x (d+e x)))}{\left(a-b x^4\right)^3}+\frac{16 a^{7/4} \sqrt[4]{b} x (-a g+11 b c+b x (10 d+9 e x))}{\left(a-b x^4\right)^2}+\frac{4 a^{3/4} \sqrt[4]{b} x (-7 a g+77 b c+15 b x (4 d+3 e x))}{a-b x^4}-3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(40 \sqrt[4]{a} b^{3/4} d+15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)+3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-40 \sqrt[4]{a} b^{3/4} d+15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)+120 \sqrt[4]{a} b^{3/4} d \log \left(\sqrt{a}+\sqrt{b} x^2\right)+6 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)}{1536 a^{15/4} b^{5/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)}{256 a^{15/4} b^{5/4}}+\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(7 (11 b c-a g)+60 b d x+45 b e x^2\right)}{384 a^3 b \left(a-b x^4\right)}+\frac{x \left(-a g+11 b c+10 b d x+9 b e x^2\right)+8 a f}{96 a^2 b \left(a-b x^4\right)^2}+\frac{x \left(a g+b c+b d x+b e x^2+b f x^3\right)}{12 a b \left(a-b x^4\right)^3}",1,"((4*a^(3/4)*b^(1/4)*x*(77*b*c - 7*a*g + 15*b*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^(7/4)*b^(1/4)*x*(11*b*c - a*g + b*x*(10*d + 9*e*x)))/(a - b*x^4)^2 + (128*a^(11/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*x^4)^3 + 6*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(77*b*c + 40*a^(1/4)*b^(3/4)*d + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*Log[a^(1/4) - b^(1/4)*x] + 3*(77*b*c - 40*a^(1/4)*b^(3/4)*d + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*Log[a^(1/4) + b^(1/4)*x] + 120*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*b^(5/4))","A",1
175,1,311,319,0.4072191,"\int \frac{c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4),x]","\frac{2 a^{3/4} \sqrt[4]{b} f \log \left(a+b x^4\right)+8 a^{3/4} \sqrt[4]{b} g x-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 \sqrt[4]{a} b^{3/4} d+\sqrt{2} \sqrt{a} \sqrt{b} e-\sqrt{2} a g+\sqrt{2} b c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-2 \sqrt[4]{a} b^{3/4} d+\sqrt{2} \sqrt{a} \sqrt{b} e-\sqrt{2} a g+\sqrt{2} b c\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{a} \sqrt{b} e+a g-b c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)}{8 a^{3/4} b^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e-a g+b c\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{a} \sqrt{b} e-a g+b c\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}+\frac{f \log \left(a+b x^4\right)}{4 b}+\frac{g x}{b}",1,"(8*a^(3/4)*b^(1/4)*g*x - 2*(Sqrt[2]*b*c + 2*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e - Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[2]*b*c - 2*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e - Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-(b*c) + Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(b*c - Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 2*a^(3/4)*b^(1/4)*f*Log[a + b*x^4])/(8*a^(3/4)*b^(5/4))","A",1
176,1,319,341,0.2315634,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^2,x]","\frac{-\frac{8 a^{3/4} \sqrt[4]{b} (a (f+g x)-b x (c+x (d+e x)))}{a+b x^4}-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 \sqrt[4]{a} b^{3/4} d+\sqrt{2} \sqrt{a} \sqrt{b} e+\sqrt{2} a g+3 \sqrt{2} b c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 \sqrt[4]{a} b^{3/4} d+\sqrt{2} \sqrt{a} \sqrt{b} e+\sqrt{2} a g+3 \sqrt{2} b c\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{a} \sqrt{b} e-a g-3 b c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{32 a^{7/4} b^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}+\frac{x \left(-a g+b c+b d x+b e x^2+b f x^3\right)}{4 a b \left(a+b x^4\right)}",1,"((-8*a^(3/4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4) - 2*(3*Sqrt[2]*b*c + 4*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e + Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b*c - 4*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e + Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b*c - Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(5/4))","A",1
177,1,366,394,0.3921068,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^3,x]","\frac{-\frac{32 a^{7/4} \sqrt[4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} \sqrt[4]{b} x (a g+7 b c+b x (6 d+5 e x))}{a+b x^4}-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(24 \sqrt[4]{a} b^{3/4} d+5 \sqrt{2} \sqrt{a} \sqrt{b} e+3 \sqrt{2} a g+21 \sqrt{2} b c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-24 \sqrt[4]{a} b^{3/4} d+5 \sqrt{2} \sqrt{a} \sqrt{b} e+3 \sqrt{2} a g+21 \sqrt{2} b c\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(5 \sqrt{a} \sqrt{b} e-3 a g-21 b c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{256 a^{11/4} b^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}-\frac{4 a f-x \left(a g+7 b c+6 b d x+5 b e x^2\right)}{32 a^2 b \left(a+b x^4\right)}+\frac{x \left(-a g+b c+b d x+b e x^2+b f x^3\right)}{8 a b \left(a+b x^4\right)^2}",1,"((8*a^(3/4)*b^(1/4)*x*(7*b*c + a*g + b*x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^(7/4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*b*c + 24*a^(1/4)*b^(3/4)*d + 5*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 3*Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(21*Sqrt[2]*b*c - 24*a^(1/4)*b^(3/4)*d + 5*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 3*Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(5/4))","A",1
178,1,411,437,0.5199043,"\int \frac{c+d x+e x^2+f x^3+g x^4}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^4)^4,x]","\frac{-\frac{256 a^{11/4} \sqrt[4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left(a+b x^4\right)^3}+\frac{32 a^{7/4} \sqrt[4]{b} x (a g+11 b c+b x (10 d+9 e x))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} \sqrt[4]{b} x (7 a g+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(80 \sqrt[4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right)+6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-80 \sqrt[4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right)-3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)+3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{3072 a^{15/4} b^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{5 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(7 (a g+11 b c)+60 b d x+45 b e x^2\right)}{384 a^3 b \left(a+b x^4\right)}-\frac{8 a f-x \left(a g+11 b c+10 b d x+9 b e x^2\right)}{96 a^2 b \left(a+b x^4\right)^2}+\frac{x \left(-a g+b c+b d x+b e x^2+b f x^3\right)}{12 a b \left(a+b x^4\right)^3}",1,"((8*a^(3/4)*b^(1/4)*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^(7/4)*b^(1/4)*x*(11*b*c + a*g + b*x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (256*a^(11/4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4)^3 - 6*(77*Sqrt[2]*b*c + 80*a^(1/4)*b^(3/4)*d + 15*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 7*Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b*c - 80*a^(1/4)*b^(3/4)*d + 15*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 7*Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^(5/4))","A",1
179,1,9,11,0.0022849,"\int \frac{\left(1-x^4\right)^3}{\left(1+x+x^2+x^3\right)^3} \, dx","Integrate[(1 - x^4)^3/(1 + x + x^2 + x^3)^3,x]","-\frac{1}{4} (x-1)^4","-\frac{1}{4} (1-x)^4",1,"-1/4*(-1 + x)^4","A",1
180,1,14,11,0.000662,"\int \frac{\left(1-x^4\right)^2}{\left(1+x+x^2+x^3\right)^2} \, dx","Integrate[(1 - x^4)^2/(1 + x + x^2 + x^3)^2,x]","\frac{x^3}{3}-x^2+x","-\frac{1}{3} (1-x)^3",1,"x - x^2 + x^3/3","A",1
181,1,9,9,0.0007672,"\int \frac{1-x^4}{1+x+x^2+x^3} \, dx","Integrate[(1 - x^4)/(1 + x + x^2 + x^3),x]","x-\frac{x^2}{2}","x-\frac{x^2}{2}",1,"x - x^2/2","A",1
182,1,8,8,0.0010568,"\int \frac{1+x+x^2+x^3}{1-x^4} \, dx","Integrate[(1 + x + x^2 + x^3)/(1 - x^4),x]","-\log (1-x)","-\log (1-x)",1,"-Log[1 - x]","A",1
183,1,7,7,0.0011766,"\int \frac{\left(1+x+x^2+x^3\right)^2}{\left(1-x^4\right)^2} \, dx","Integrate[(1 + x + x^2 + x^3)^2/(1 - x^4)^2,x]","-\frac{1}{x-1}","\frac{1}{1-x}",1,"-(-1 + x)^(-1)","A",1
184,1,9,11,0.0025312,"\int \frac{\left(1+x+x^2+x^3\right)^3}{\left(1-x^4\right)^3} \, dx","Integrate[(1 + x + x^2 + x^3)^3/(1 - x^4)^3,x]","\frac{1}{2 (x-1)^2}","\frac{1}{2 (1-x)^2}",1,"1/(2*(-1 + x)^2)","A",1
185,1,9,11,0.001261,"\int \frac{\left(1+x+x^2+x^3\right)^4}{\left(1-x^4\right)^4} \, dx","Integrate[(1 + x + x^2 + x^3)^4/(1 - x^4)^4,x]","-\frac{1}{3 (x-1)^3}","\frac{1}{3 (1-x)^3}",1,"-1/3*1/(-1 + x)^3","A",1
186,1,256,165,0.4268646,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4),x]","\frac{-\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{5/4} h+\sqrt{a} b^{3/4} e+\sqrt[4]{a} b d+a \sqrt[4]{b} g+b^{5/4} c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(a^{5/4} (-h)+\sqrt{a} b^{3/4} e-\sqrt[4]{a} b d+a \sqrt[4]{b} g+b^{5/4} c\right)-a^{3/4} \sqrt{b} f \log \left(a-b x^4\right)-4 a^{3/4} \sqrt{b} g x-2 a^{3/4} \sqrt{b} h x^2+2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e+a g+b c\right)+\sqrt[4]{a} (a h+b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{4 a^{3/4} b^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e+a g+b c\right)}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g+b c\right)}{2 a^{3/4} b^{5/4}}+\frac{(a h+b d) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{f \log \left(a-b x^4\right)}{4 b}-\frac{g x}{b}-\frac{h x^2}{2 b}",1,"(-4*a^(3/4)*Sqrt[b]*g*x - 2*a^(3/4)*Sqrt[b]*h*x^2 + 2*b^(1/4)*(b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (b^(5/4)*c + a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g + a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + (b^(5/4)*c - a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g - a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] + a^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2] - a^(3/4)*Sqrt[b]*f*Log[a - b*x^4])/(4*a^(3/4)*b^(3/2))","A",1
187,1,301,188,0.5473888,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4),x]","\frac{-\frac{3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{5/4} \sqrt[4]{b} h+a^{3/2} i+\sqrt[4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-a^{5/4} \sqrt[4]{b} h+a^{3/2} i-\sqrt[4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{6 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(a^{3/2} (-i)-\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}-3 b^{3/4} f \log \left(a-b x^4\right)+\frac{3 \sqrt[4]{b} (a h+b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{a}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3}{12 b^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right)}{2 \sqrt[4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right)}{2 \sqrt[4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{f \log \left(a-b x^4\right)}{4 b}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}",1,"(-12*b^(3/4)*g*x - 6*b^(3/4)*h*x^2 - 4*b^(3/4)*i*x^3 + (6*(b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*(b^(3/2)*c + a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g + a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x])/a^(3/4) + (3*(b^(3/2)*c - a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g - a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x])/a^(3/4) + (3*b^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[a] - 3*b^(3/4)*f*Log[a - b*x^4])/(12*b^(7/4))","A",1
188,1,318,205,0.512166,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4),x]","\frac{-\frac{3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{5/4} \sqrt[4]{b} h+a^{3/2} i+\sqrt[4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-a^{5/4} \sqrt[4]{b} h+a^{3/2} i-\sqrt[4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{6 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(a^{3/2} (-i)-\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{3 \sqrt[4]{b} (a h+b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt{a}}-\frac{3 (a j+b f) \log \left(a-b x^4\right)}{\sqrt[4]{b}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3-3 b^{3/4} j x^4}{12 b^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right)}{2 \sqrt[4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right)}{2 \sqrt[4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{(a j+b f) \log \left(a-b x^4\right)}{4 b^2}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}-\frac{j x^4}{4 b}",1,"(-12*b^(3/4)*g*x - 6*b^(3/4)*h*x^2 - 4*b^(3/4)*i*x^3 - 3*b^(3/4)*j*x^4 + (6*(b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*(b^(3/2)*c + a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g + a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x])/a^(3/4) + (3*(b^(3/2)*c - a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g - a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x])/a^(3/4) + (3*b^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[a] - (3*(b*f + a*j)*Log[a - b*x^4])/b^(1/4))/(12*b^(7/4))","A",1
189,1,342,337,0.4871423,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4),x]","\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-2 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e+2 \sqrt[4]{a} b d-\sqrt{2} a \sqrt[4]{b} g+\sqrt{2} b^{5/4} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(2 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e-2 \sqrt[4]{a} b d-\sqrt{2} a \sqrt[4]{b} g+\sqrt{2} b^{5/4} c\right)+\sqrt[4]{b} \left(2 a^{3/4} \sqrt[4]{b} \left(f \log \left(a+b x^4\right)+2 x (2 g+h x)\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{a} \sqrt{b} e+a g-b c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)\right)}{8 a^{3/4} b^{3/2}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e-a g+b c\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e-a g+b c\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{a} \sqrt{b} e-a g+b c\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(b d-a h) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}+\frac{f \log \left(a+b x^4\right)}{4 b}+\frac{g x}{b}+\frac{h x^2}{2 b}",1,"(-2*(Sqrt[2]*b^(5/4)*c + 2*a^(1/4)*b*d + Sqrt[2]*Sqrt[a]*b^(3/4)*e - Sqrt[2]*a*b^(1/4)*g - 2*a^(5/4)*h)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[2]*b^(5/4)*c - 2*a^(1/4)*b*d + Sqrt[2]*Sqrt[a]*b^(3/4)*e - Sqrt[2]*a*b^(1/4)*g + 2*a^(5/4)*h)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + b^(1/4)*(Sqrt[2]*(-(b*c) + Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(b*c - Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 2*a^(3/4)*b^(1/4)*(2*x*(2*g + h*x) + f*Log[a + b*x^4])))/(8*a^(3/4)*b^(3/2))","A",1
190,1,427,384,0.370899,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4),x]","\frac{\frac{6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 a^{5/4} \sqrt[4]{b} h+\sqrt{2} a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d-\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g-\sqrt{2} b^{3/2} c\right)}{a^{3/4}}+\frac{6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(2 a^{5/4} \sqrt[4]{b} h-\sqrt{2} a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e-\sqrt{2} a \sqrt{b} g+\sqrt{2} b^{3/2} c\right)}{a^{3/4}}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+6 b^{3/4} f \log \left(a+b x^4\right)+24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3}{24 b^{7/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right)}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right)}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right)}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right)}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}+\frac{f \log \left(a+b x^4\right)}{4 b}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}",1,"(24*b^(3/4)*g*x + 12*b^(3/4)*h*x^2 + 8*b^(3/4)*i*x^3 + (6*(-(Sqrt[2]*b^(3/2)*c) - 2*a^(1/4)*b^(5/4)*d - Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (6*(Sqrt[2]*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e - Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + 6*b^(3/4)*f*Log[a + b*x^4])/(24*b^(7/4))","A",1
191,1,445,402,0.4183759,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4),x]","\frac{\frac{6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 a^{5/4} \sqrt[4]{b} h+\sqrt{2} a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d-\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g-\sqrt{2} b^{3/2} c\right)}{a^{3/4}}+\frac{6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(2 a^{5/4} \sqrt[4]{b} h-\sqrt{2} a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e-\sqrt{2} a \sqrt{b} g+\sqrt{2} b^{3/2} c\right)}{a^{3/4}}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right)}{a^{3/4}}+\frac{6 (b f-a j) \log \left(a+b x^4\right)}{\sqrt[4]{b}}+24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+6 b^{3/4} j x^4}{24 b^{7/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right)}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right)}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right)}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right)}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}+\frac{(b f-a j) \log \left(a+b x^4\right)}{4 b^2}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}+\frac{j x^4}{4 b}",1,"(24*b^(3/4)*g*x + 12*b^(3/4)*h*x^2 + 8*b^(3/4)*i*x^3 + 6*b^(3/4)*j*x^4 + (6*(-(Sqrt[2]*b^(3/2)*c) - 2*a^(1/4)*b^(5/4)*d - Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (6*(Sqrt[2]*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e - Sqrt[2]*a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) - (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (3*Sqrt[2]*(b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (6*(b*f - a*j)*Log[a + b*x^4])/b^(1/4))/(24*b^(7/4))","A",1
192,1,257,184,0.2836338,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4)^2,x]","\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(2 a^{5/4} h-\sqrt{a} b^{3/4} e-2 \sqrt[4]{a} b d+a \sqrt[4]{b} g-3 b^{5/4} c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(2 a^{5/4} h+\sqrt{a} b^{3/4} e-2 \sqrt[4]{a} b d-a \sqrt[4]{b} g+3 b^{5/4} c\right)+\frac{4 a^{3/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{a-b x^4}-2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g-3 b c\right)-2 \sqrt[4]{a} (a h-b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{16 a^{7/4} b^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\sqrt{a} \sqrt{b} e-a g+3 b c\right)}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e-a g+3 b c\right)}{8 a^{7/4} b^{5/4}}+\frac{(b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{x \left(x (a h+b d)+a g+b c+b e x^2+b f x^3\right)}{4 a b \left(a-b x^4\right)}",1,"((4*a^(3/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4) - 2*b^(1/4)*(-3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-3*b^(5/4)*c - 2*a^(1/4)*b*d - Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g + 2*a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + (3*b^(5/4)*c - 2*a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e - a*b^(1/4)*g + 2*a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] - 2*a^(1/4)*(-(b*d) + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(3/2))","A",1
193,1,302,203,0.2790003,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^2,x]","\frac{\frac{4 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{a-b x^4}+\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(2 a^{5/4} \sqrt[4]{b} h+3 a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d-\sqrt{a} b e+a \sqrt{b} g-3 b^{3/2} c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(2 a^{5/4} \sqrt[4]{b} h-3 a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d+\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right)-2 \sqrt[4]{a} \sqrt[4]{b} (a h-b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{16 a^{7/4} b^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right)}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right)}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right)}{4 a b \left(a-b x^4\right)}",1,"((4*a^(3/4)*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4) + 2*(3*b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-3*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d - Sqrt[a]*b*e + a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + (3*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e - a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 2*a^(1/4)*b^(1/4)*(-(b*d) + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(7/4))","A",1
194,1,338,225,0.2509284,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a-b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^2,x]","\frac{\frac{\sqrt[4]{b} \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(2 a^{5/4} \sqrt[4]{b} h+3 a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d-\sqrt{a} b e+a \sqrt{b} g-3 b^{3/2} c\right)}{a^{7/4}}+\frac{\sqrt[4]{b} \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(2 a^{5/4} \sqrt[4]{b} h-3 a^{3/2} i-2 \sqrt[4]{a} b^{5/4} d+\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right)}{a^{7/4}}+\frac{2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right)}{a^{7/4}}+\frac{2 \sqrt{b} (b d-a h) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{a^{3/2}}+\frac{4 \left(a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{a \left(a-b x^4\right)}+4 j \log \left(a-b x^4\right)}{16 b^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right)}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right)}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{j \log \left(a-b x^4\right)}{4 b^2}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right)}{4 a b \left(a-b x^4\right)}",1,"((4*(a^2*j + b^2*x*(c + x*(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a*(a - b*x^4)) + (2*b^(1/4)*(3*b^(3/2)*c - Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/a^(7/4) + (b^(1/4)*(-3*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d - Sqrt[a]*b*e + a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x])/a^(7/4) + (b^(1/4)*(3*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e - a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x])/a^(7/4) + (2*Sqrt[b]*(b*d - a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/a^(3/2) + 4*j*Log[a - b*x^4])/(16*b^2)","A",1
195,1,359,353,0.3024061,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4)^2,x]","\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e+4 \sqrt[4]{a} b d+\sqrt{2} a \sqrt[4]{b} g+3 \sqrt{2} b^{5/4} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e-4 \sqrt[4]{a} b d+\sqrt{2} a \sqrt[4]{b} g+3 \sqrt{2} b^{5/4} c\right)-\frac{8 a^{3/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{a+b x^4}+\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{a} \sqrt{b} e-a g-3 b c\right)+\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{32 a^{7/4} b^{3/2}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{a} \sqrt{b} e+a g+3 b c\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a h+b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{x \left(x (b d-a h)-a g+b c+b e x^2+b f x^3\right)}{4 a b \left(a+b x^4\right)}",1,"((-8*a^(3/4)*Sqrt[b]*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(a + b*x^4) - 2*(3*Sqrt[2]*b^(5/4)*c + 4*a^(1/4)*b*d + Sqrt[2]*Sqrt[a]*b^(3/4)*e + Sqrt[2]*a*b^(1/4)*g + 4*a^(5/4)*h)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b^(5/4)*c - 4*a^(1/4)*b*d + Sqrt[2]*Sqrt[a]*b^(3/4)*e + Sqrt[2]*a*b^(1/4)*g - 4*a^(5/4)*h)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*(-3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(3*b*c - Sqrt[a]*Sqrt[b]*e + a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(3/2))","A",1
196,1,415,395,0.465077,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^2,x]","\frac{-\frac{8 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{a+b x^4}-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i+4 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i-4 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 a^{3/2} i+\sqrt{a} b e-a \sqrt{b} g-3 b^{3/2} c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-3 a^{3/2} i-\sqrt{a} b e+a \sqrt{b} g+3 b^{3/2} c\right)}{32 a^{7/4} b^{7/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right)}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right)}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right)}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right)}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right)}{4 a b \left(a+b x^4\right)}",1,"((-8*a^(3/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4) - 2*(3*Sqrt[2]*b^(3/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b^(3/2)*c - 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-3*b^(3/2)*c + Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(7/4))","A",1
197,1,460,417,0.4397581,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^2,x]","\frac{-\frac{2 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i+4 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right)}{a^{7/4}}+\frac{2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i-4 \sqrt[4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right)}{a^{7/4}}+\frac{\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 a^{3/2} i+\sqrt{a} b e-a \sqrt{b} g-3 b^{3/2} c\right)}{a^{7/4}}+\frac{\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-3 a^{3/2} i-\sqrt{a} b e+a \sqrt{b} g+3 b^{3/2} c\right)}{a^{7/4}}+\frac{8 \left(a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{a \left(a+b x^4\right)}+8 j \log \left(a+b x^4\right)}{32 b^2}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right)}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right)}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right)}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right)}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} b^{3/2}}+\frac{j \log \left(a+b x^4\right)}{4 b^2}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right)}{4 a b \left(a+b x^4\right)}",1,"((8*(a^2*j + b^2*x*(c + x*(d + e*x)) - a*b*(f + x*(g + x*(h + i*x)))))/(a*(a + b*x^4)) - (2*b^(1/4)*(3*Sqrt[2]*b^(3/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (2*b^(1/4)*(3*Sqrt[2]*b^(3/2)*c - 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (Sqrt[2]*b^(1/4)*(-3*b^(3/2)*c + Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (Sqrt[2]*b^(1/4)*(3*b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + 8*j*Log[a + b*x^4])/(32*b^2)","A",1
198,1,309,241,0.4163056,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4)^3,x]","\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(4 a^{5/4} h-5 \sqrt{a} b^{3/4} e-12 \sqrt[4]{a} b d+3 a \sqrt[4]{b} g-21 b^{5/4} c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(4 a^{5/4} h+5 \sqrt{a} b^{3/4} e-12 \sqrt[4]{a} b d-3 a \sqrt[4]{b} g+21 b^{5/4} c\right)+\frac{16 a^{7/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left(a-b x^4\right)^2}+\frac{4 a^{3/4} \sqrt{b} x (-a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a-b x^4}+2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)-4 \sqrt[4]{a} (a h-3 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{128 a^{11/4} b^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right)}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}+\frac{x \left(2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right)+4 a f}{32 a^2 b \left(a-b x^4\right)}+\frac{x \left(x (a h+b d)+a g+b c+b e x^2+b f x^3\right)}{8 a b \left(a-b x^4\right)^2}",1,"((4*a^(3/4)*Sqrt[b]*x*(7*b*c + b*x*(6*d + 5*e*x) - a*(g + 2*h*x)))/(a - b*x^4) + (16*a^(7/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4)^2 + 2*b^(1/4)*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-21*b^(5/4)*c - 12*a^(1/4)*b*d - 5*Sqrt[a]*b^(3/4)*e + 3*a*b^(1/4)*g + 4*a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + (21*b^(5/4)*c - 12*a^(1/4)*b*d + 5*Sqrt[a]*b^(3/4)*e - 3*a*b^(1/4)*g + 4*a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^(3/2))","A",1
199,1,359,268,0.3999311,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^3,x]","\frac{\frac{16 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left(a-b x^4\right)^2}-\frac{4 a^{3/4} b^{3/4} x (a (g+x (2 h+3 i x))-b (7 c+x (6 d+5 e x)))}{a-b x^4}+\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(4 a^{5/4} \sqrt[4]{b} h+3 a^{3/2} i-12 \sqrt[4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right)+\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(4 a^{5/4} \sqrt[4]{b} h-3 a^{3/2} i-12 \sqrt[4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right)-4 \sqrt[4]{a} \sqrt[4]{b} (a h-3 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{128 a^{11/4} b^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right)}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right)}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}+\frac{x \left(2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right)+4 a f}{32 a^2 b \left(a-b x^4\right)}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right)}{8 a b \left(a-b x^4\right)^2}",1,"((-4*a^(3/4)*b^(3/4)*x*(-(b*(7*c + x*(6*d + 5*e*x))) + a*(g + x*(2*h + 3*i*x))))/(a - b*x^4) + (16*a^(7/4)*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^2 + 2*(21*b^(3/2)*c - 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + (21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*b^(1/4)*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^(7/4))","A",1
200,1,380,285,0.3379025,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a-b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^3,x]","\frac{\sqrt[4]{b} \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(4 a^{5/4} \sqrt[4]{b} h+3 a^{3/2} i-12 \sqrt[4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right)+\sqrt[4]{b} \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(4 a^{5/4} \sqrt[4]{b} h-3 a^{3/2} i-12 \sqrt[4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right)+2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right)+\frac{16 a^{7/4} \left(a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{\left(a-b x^4\right)^2}-\frac{4 a^{3/4} \left(8 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (7 c+x (6 d+5 e x))\right)}{a-b x^4}-4 \sqrt[4]{a} \sqrt{b} (a h-3 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{128 a^{11/4} b^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right)}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right)}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}+\frac{x \left(b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right)+4 a (b f-a j)}{32 a^2 b^2 \left(a-b x^4\right)}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right)}{8 a b \left(a-b x^4\right)^2}",1,"((-4*a^(3/4)*(8*a^2*j - b^2*x*(7*c + x*(6*d + 5*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4) + (16*a^(7/4)*(a^2*j + b^2*x*(c + x*(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^2 + 2*b^(1/4)*(21*b^(3/2)*c - 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + b^(1/4)*(-21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + b^(1/4)*(21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*Sqrt[b]*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^2)","A",1
201,1,411,413,0.4251207,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4)^3,x]","\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e+24 \sqrt[4]{a} b d+3 \sqrt{2} a \sqrt[4]{b} g+21 \sqrt{2} b^{5/4} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e-24 \sqrt[4]{a} b d+3 \sqrt{2} a \sqrt[4]{b} g+21 \sqrt{2} b^{5/4} c\right)-\frac{32 a^{7/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} \sqrt{b} x (a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a+b x^4}+\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(5 \sqrt{a} \sqrt{b} e-3 a g-21 b c\right)+\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{256 a^{11/4} b^{3/2}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right)}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(a h+3 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left(2 x (a h+3 b d)+a g+7 b c+5 b e x^2\right)}{32 a^2 b \left(a+b x^4\right)}+\frac{x \left(x (b d-a h)-a g+b c+b e x^2+b f x^3\right)}{8 a b \left(a+b x^4\right)^2}",1,"((8*a^(3/4)*Sqrt[b]*x*(7*b*c + b*x*(6*d + 5*e*x) + a*(g + 2*h*x)))/(a + b*x^4) - (32*a^(7/4)*Sqrt[b]*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*b^(5/4)*c + 24*a^(1/4)*b*d + 5*Sqrt[2]*Sqrt[a]*b^(3/4)*e + 3*Sqrt[2]*a*b^(1/4)*g + 8*a^(5/4)*h)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(21*Sqrt[2]*b^(5/4)*c - 24*a^(1/4)*b*d + 5*Sqrt[2]*Sqrt[a]*b^(3/4)*e + 3*Sqrt[2]*a*b^(1/4)*g - 8*a^(5/4)*h)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*(-21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(3/2))","A",1
202,1,473,463,0.6771346,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^3,x]","\frac{-\frac{32 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} b^{3/4} x (a g+a x (2 h+3 i x)+7 b c+b x (6 d+5 e x))}{a+b x^4}-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(8 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i+24 \sqrt[4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-8 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i-24 \sqrt[4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right)+\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 a^{3/2} i+5 \sqrt{a} b e-3 a \sqrt{b} g-21 b^{3/2} c\right)+\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-3 a^{3/2} i-5 \sqrt{a} b e+3 a \sqrt{b} g+21 b^{3/2} c\right)}{256 a^{11/4} b^{7/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right)}{128 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right)}{128 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right)}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right)}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{(a h+3 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left(2 x (a h+3 b d)+x^2 (3 a i+5 b e)+a g+7 b c\right)}{32 a^2 b \left(a+b x^4\right)}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right)}{8 a b \left(a+b x^4\right)^2}",1,"((8*a^(3/4)*b^(3/4)*x*(7*b*c + a*g + b*x*(6*d + 5*e*x) + a*x*(2*h + 3*i*x)))/(a + b*x^4) - (32*a^(7/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*b^(3/2)*c + 24*a^(1/4)*b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sqrt[2]*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(21*Sqrt[2]*b^(3/2)*c - 24*a^(1/4)*b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sqrt[2]*a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-21*b^(3/2)*c + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(21*b^(3/2)*c - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(7/4))","A",1
203,1,500,480,0.5084046,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^3,x]","\frac{-2 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(8 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i+24 \sqrt[4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right)+2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-8 a^{5/4} \sqrt[4]{b} h+3 \sqrt{2} a^{3/2} i-24 \sqrt[4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right)+\sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 a^{3/2} i+5 \sqrt{a} b e-3 a \sqrt{b} g-21 b^{3/2} c\right)+\sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-3 a^{3/2} i-5 \sqrt{a} b e+3 a \sqrt{b} g+21 b^{3/2} c\right)+\frac{32 a^{7/4} \left(a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} \left(-8 a^2 j+a b x (g+x (2 h+3 i x))+b^2 x (7 c+x (6 d+5 e x))\right)}{a+b x^4}}{256 a^{11/4} b^2}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right)}{128 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right)}{128 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right)}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right)}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{(a h+3 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} b^{3/2}}-\frac{4 a (a j+b f)-x \left(b (a g+7 b c)+2 b x (a h+3 b d)+b x^2 (3 a i+5 b e)\right)}{32 a^2 b^2 \left(a+b x^4\right)}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right)}{8 a b \left(a+b x^4\right)^2}",1,"((8*a^(3/4)*(-8*a^2*j + b^2*x*(7*c + x*(6*d + 5*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a + b*x^4) + (32*a^(7/4)*(a^2*j + b^2*x*(c + x*(d + e*x)) - a*b*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^2 - 2*b^(1/4)*(21*Sqrt[2]*b^(3/2)*c + 24*a^(1/4)*b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sqrt[2]*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*b^(1/4)*(21*Sqrt[2]*b^(3/2)*c - 24*a^(1/4)*b^(5/4)*d + 5*Sqrt[2]*Sqrt[a]*b*e + 3*Sqrt[2]*a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*(-21*b^(3/2)*c + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(21*b^(3/2)*c - 5*Sqrt[a]*b*e + 3*a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^2)","A",1
204,1,360,293,0.4872127,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a - b*x^4)^4,x]","\frac{-3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(-8 a^{5/4} h+15 \sqrt{a} b^{3/4} e+40 \sqrt[4]{a} b d-7 a \sqrt[4]{b} g+77 b^{5/4} c\right)+3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(8 a^{5/4} h+15 \sqrt{a} b^{3/4} e-40 \sqrt[4]{a} b d-7 a \sqrt[4]{b} g+77 b^{5/4} c\right)+\frac{128 a^{11/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left(a-b x^4\right)^3}+\frac{16 a^{7/4} \sqrt{b} x (-a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left(a-b x^4\right)^2}+\frac{4 a^{3/4} \sqrt{b} x \left(-7 a g-12 a h x+77 b c+60 b d x+45 b e x^2\right)}{a-b x^4}+6 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)-24 \sqrt[4]{a} (a h-5 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{1536 a^{15/4} b^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right)}{256 a^{15/4} b^{5/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right)}{384 a^3 b \left(a-b x^4\right)}+\frac{x \left(2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right)+8 a f}{96 a^2 b \left(a-b x^4\right)^2}+\frac{x \left(x (a h+b d)+a g+b c+b e x^2+b f x^3\right)}{12 a b \left(a-b x^4\right)^3}",1,"((4*a^(3/4)*Sqrt[b]*x*(77*b*c - 7*a*g + 60*b*d*x - 12*a*h*x + 45*b*e*x^2))/(a - b*x^4) + (16*a^(7/4)*Sqrt[b]*x*(11*b*c + b*x*(10*d + 9*e*x) - a*(g + 2*h*x)))/(a - b*x^4)^2 + (128*a^(11/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4)^3 + 6*b^(1/4)*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(77*b^(5/4)*c + 40*a^(1/4)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a*b^(1/4)*g - 8*a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + 3*(77*b^(5/4)*c - 40*a^(1/4)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a*b^(1/4)*g + 8*a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] - 24*a^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*b^(3/2))","A",1
205,1,422,331,0.5449641,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^4,x]","\frac{3 \sqrt[4]{a} \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(8 a^{5/4} \sqrt[4]{b} h+5 a^{3/2} i-40 \sqrt[4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right)-3 \sqrt[4]{a} \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-8 a^{5/4} \sqrt[4]{b} h+5 a^{3/2} i+40 \sqrt[4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right)+6 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right)+\frac{128 a^3 b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left(a-b x^4\right)^3}-\frac{16 a^2 b^{3/4} x (a (g+x (2 h+3 i x))-b (11 c+x (10 d+9 e x)))}{\left(a-b x^4\right)^2}-\frac{4 a b^{3/4} x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt[4]{b} (a h-5 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{1536 a^4 b^{7/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right)}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right)}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right)}{384 a^3 b \left(a-b x^4\right)}+\frac{x \left(2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right)+8 a f}{96 a^2 b \left(a-b x^4\right)^2}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right)}{12 a b \left(a-b x^4\right)^3}",1,"((-4*a*b^(3/4)*x*(-77*b*c + 7*a*g - 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a - b*x^4) - (16*a^2*b^(3/4)*x*(-(b*(11*c + x*(10*d + 9*e*x))) + a*(g + x*(2*h + 3*i*x))))/(a - b*x^4)^2 + (128*a^3*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*(77*b^(3/2)*c - 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + 3*a^(1/4)*(-77*b^(3/2)*c - 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] - 3*a^(1/4)*(-77*b^(3/2)*c + 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 24*Sqrt[a]*b^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^4*b^(7/4))","A",1
206,1,439,349,0.5246441,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a-b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^4,x]","\frac{3 \sqrt[4]{a} \sqrt[4]{b} \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(8 a^{5/4} \sqrt[4]{b} h+5 a^{3/2} i-40 \sqrt[4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right)+3 \sqrt[4]{a} \sqrt[4]{b} \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(8 a^{5/4} \sqrt[4]{b} h-5 a^{3/2} i-40 \sqrt[4]{a} b^{5/4} d+15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right)+6 \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right)-\frac{16 a^2 \left(12 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right)}{\left(a-b x^4\right)^2}+\frac{128 a^3 \left(a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{\left(a-b x^4\right)^3}-\frac{4 a b x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt{b} (a h-5 b d) \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{1536 a^4 b^2}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right)}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right)}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right)}{384 a^3 b \left(a-b x^4\right)}+\frac{x \left(b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right)+4 a (2 b f-a j)}{96 a^2 b^2 \left(a-b x^4\right)^2}+\frac{x \left(x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right)}{12 a b \left(a-b x^4\right)^3}",1,"((-4*a*b*x*(-77*b*c + 7*a*g - 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a - b*x^4) - (16*a^2*(12*a^2*j - b^2*x*(11*c + x*(10*d + 9*e*x)) + a*b*x*(g + x*(2*h + 3*i*x))))/(a - b*x^4)^2 + (128*a^3*(a^2*j + b^2*x*(c + x*(d + e*x)) + a*b*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*b^(1/4)*(77*b^(3/2)*c - 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + 3*a^(1/4)*b^(1/4)*(-77*b^(3/2)*c - 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + 3*a^(1/4)*b^(1/4)*(77*b^(3/2)*c - 40*a^(1/4)*b^(5/4)*d + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*h - 5*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 24*Sqrt[a]*Sqrt[b]*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^4*b^2)","A",1
207,1,461,462,0.5812349,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4)^4,x]","\frac{-6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(16 a^{5/4} h+15 \sqrt{2} \sqrt{a} b^{3/4} e+80 \sqrt[4]{a} b d+7 \sqrt{2} a \sqrt[4]{b} g+77 \sqrt{2} b^{5/4} c\right)+6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-16 a^{5/4} h+15 \sqrt{2} \sqrt{a} b^{3/4} e-80 \sqrt[4]{a} b d+7 \sqrt{2} a \sqrt[4]{b} g+77 \sqrt{2} b^{5/4} c\right)-\frac{256 a^{11/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{\left(a+b x^4\right)^3}+\frac{32 a^{7/4} \sqrt{b} x (a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} \sqrt{b} x \left(7 a g+12 a h x+77 b c+60 b d x+45 b e x^2\right)}{a+b x^4}-3 \sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)+3 \sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{3072 a^{15/4} b^{3/2}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right)}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(a h+5 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (a g+11 b c)+12 x (a h+5 b d)+45 b e x^2\right)}{384 a^3 b \left(a+b x^4\right)}-\frac{8 a f-x \left(2 x (a h+5 b d)+a g+11 b c+9 b e x^2\right)}{96 a^2 b \left(a+b x^4\right)^2}+\frac{x \left(x (b d-a h)-a g+b c+b e x^2+b f x^3\right)}{12 a b \left(a+b x^4\right)^3}",1,"((8*a^(3/4)*Sqrt[b]*x*(77*b*c + 7*a*g + 60*b*d*x + 12*a*h*x + 45*b*e*x^2))/(a + b*x^4) + (32*a^(7/4)*Sqrt[b]*x*(11*b*c + b*x*(10*d + 9*e*x) + a*(g + 2*h*x)))/(a + b*x^4)^2 - (256*a^(11/4)*Sqrt[b]*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(a + b*x^4)^3 - 6*(77*Sqrt[2]*b^(5/4)*c + 80*a^(1/4)*b*d + 15*Sqrt[2]*Sqrt[a]*b^(3/4)*e + 7*Sqrt[2]*a*b^(1/4)*g + 16*a^(5/4)*h)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b^(5/4)*c - 80*a^(1/4)*b*d + 15*Sqrt[2]*Sqrt[a]*b^(3/4)*e + 7*Sqrt[2]*a*b^(1/4)*g - 16*a^(5/4)*h)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*b^(1/4)*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*b^(1/4)*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^(3/2))","A",1
208,1,530,516,1.0097788,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^4,x]","\frac{-\frac{256 a^{11/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left(a+b x^4\right)^3}+\frac{32 a^{7/4} b^{3/4} x (a g+a x (2 h+3 i x)+11 b c+b x (10 d+9 e x))}{\left(a+b x^4\right)^2}+\frac{8 a^{3/4} b^{3/4} x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(16 a^{5/4} \sqrt[4]{b} h+5 \sqrt{2} a^{3/2} i+80 \sqrt[4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right)+6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-16 a^{5/4} \sqrt[4]{b} h+5 \sqrt{2} a^{3/2} i-80 \sqrt[4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right)+3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(5 a^{3/2} i+15 \sqrt{a} b e-7 a \sqrt{b} g-77 b^{3/2} c\right)+3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 a^{3/2} i-15 \sqrt{a} b e+7 a \sqrt{b} g+77 b^{3/2} c\right)}{3072 a^{15/4} b^{7/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right)}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right)}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right)}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right)}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right)}{384 a^3 b \left(a+b x^4\right)}-\frac{8 a f-x \left(2 x (a h+5 b d)+3 x^2 (a i+3 b e)+a g+11 b c\right)}{96 a^2 b \left(a+b x^4\right)^2}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right)}{12 a b \left(a+b x^4\right)^3}",1,"((32*a^(7/4)*b^(3/4)*x*(11*b*c + a*g + b*x*(10*d + 9*e*x) + a*x*(2*h + 3*i*x)))/(a + b*x^4)^2 + (8*a^(3/4)*b^(3/4)*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a + b*x^4) - (256*a^(11/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^3 - 6*(77*Sqrt[2]*b^(3/2)*c + 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g + 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b^(3/2)*c - 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g - 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*(-77*b^(3/2)*c + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*(77*b^(3/2)*c - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 5*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^(7/4))","A",1
209,1,555,534,0.7055346,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^4,x]","\frac{-6 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(16 a^{5/4} \sqrt[4]{b} h+5 \sqrt{2} a^{3/2} i+80 \sqrt[4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right)+6 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-16 a^{5/4} \sqrt[4]{b} h+5 \sqrt{2} a^{3/2} i-80 \sqrt[4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right)+3 \sqrt{2} \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(5 a^{3/2} i+15 \sqrt{a} b e-7 a \sqrt{b} g-77 b^{3/2} c\right)+3 \sqrt{2} \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(-5 a^{3/2} i-15 \sqrt{a} b e+7 a \sqrt{b} g+77 b^{3/2} c\right)+\frac{8 a^{3/4} b x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}+\frac{256 a^{11/4} \left(a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right)}{\left(a+b x^4\right)^3}-\frac{32 a^{7/4} \left(12 a^2 j-a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right)}{\left(a+b x^4\right)^2}}{3072 a^{15/4} b^2}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right)}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right) \left(7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right)}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right)}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right)}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} b^{3/2}}+\frac{x \left(7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right)}{384 a^3 b \left(a+b x^4\right)}-\frac{4 a (a j+2 b f)-x \left(b (a g+11 b c)+2 b x (a h+5 b d)+3 b x^2 (a i+3 b e)\right)}{96 a^2 b^2 \left(a+b x^4\right)^2}+\frac{x \left(x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right)}{12 a b \left(a+b x^4\right)^3}",1,"((8*a^(3/4)*b*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a + b*x^4) - (32*a^(7/4)*(12*a^2*j - b^2*x*(11*c + x*(10*d + 9*e*x)) - a*b*x*(g + x*(2*h + 3*i*x))))/(a + b*x^4)^2 + (256*a^(11/4)*(a^2*j + b^2*x*(c + x*(d + e*x)) - a*b*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^3 - 6*b^(1/4)*(77*Sqrt[2]*b^(3/2)*c + 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g + 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*b^(1/4)*(77*Sqrt[2]*b^(3/2)*c - 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g - 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*b^(1/4)*(-77*b^(3/2)*c + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*b^(1/4)*(77*b^(3/2)*c - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 5*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^2)","A",1
210,1,79,121,0.0613664,"\int \frac{c+d x}{\sqrt{a+b x^4}} \, dx","Integrate[(c + d*x)/Sqrt[a + b*x^4],x]","\frac{c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}","\frac{c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}",1,"(d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4]","C",1
211,1,81,87,0.0529973,"\int \frac{c+d x}{\sqrt{a-b x^4}} \, dx","Integrate[(c + d*x)/Sqrt[a - b*x^4],x]","\frac{c x \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b x^4}{a}\right)}{\sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right)}{2 \sqrt{b}}","\frac{\sqrt[4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{b} \sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right)}{2 \sqrt{b}}",1,"(d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a - b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a])/Sqrt[a - b*x^4]","C",1
212,1,83,89,0.0491502,"\int \frac{c+d x}{\sqrt{-a+b x^4}} \, dx","Integrate[(c + d*x)/Sqrt[-a + b*x^4],x]","\frac{c x \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b x^4}{a}\right)}{\sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right)}{2 \sqrt{b}}","\frac{\sqrt[4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{\sqrt[4]{b} \sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right)}{2 \sqrt{b}}",1,"(d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[-a + b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a])/Sqrt[-a + b*x^4]","C",1
213,1,85,127,0.0525572,"\int \frac{c+d x}{\sqrt{-a-b x^4}} \, dx","Integrate[(c + d*x)/Sqrt[-a - b*x^4],x]","\frac{c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right)}{2 \sqrt{b}}","\frac{c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right)}{2 \sqrt{b}}",1,"(d*ArcTan[(Sqrt[b]*x^2)/Sqrt[-a - b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[-a - b*x^4]","C",1
214,1,131,257,0.1398336,"\int \frac{c+d x+e x^2}{\sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2)/Sqrt[a + b*x^4],x]","\frac{c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{3 \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\frac{\sqrt{b} c}{\sqrt{a}}+e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{a} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{e x \sqrt{a+b x^4}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4] + (e*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(3*Sqrt[a + b*x^4])","C",1
215,1,14,14,0.0095287,"\int \frac{a g-b g x^4}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(a*g - b*g*x^4)/(a + b*x^4)^(3/2),x]","\frac{g x}{\sqrt{a+b x^4}}","\frac{g x}{\sqrt{a+b x^4}}",1,"(g*x)/Sqrt[a + b*x^4]","A",1
216,1,27,29,0.1034954,"\int \frac{a g+e x-b g x^4}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(a*g + e*x - b*g*x^4)/(a + b*x^4)^(3/2),x]","\frac{x (2 a g+e x)}{2 a \sqrt{a+b x^4}}","\frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}}",1,"(x*(2*a*g + e*x))/(2*a*Sqrt[a + b*x^4])","A",1
217,1,27,25,0.0427639,"\int \frac{a g+f x^3-b g x^4}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(a*g + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]","\frac{2 b g x-f}{2 b \sqrt{a+b x^4}}","-\frac{f-2 b g x}{2 b \sqrt{a+b x^4}}",1,"(-f + 2*b*g*x)/(2*b*Sqrt[a + b*x^4])","A",1
218,1,38,38,0.0457107,"\int \frac{a g+e x+f x^3-b g x^4}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(a*g + e*x + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]","\frac{2 a b g x-a f+b e x^2}{2 a b \sqrt{a+b x^4}}","-\frac{-2 a b g x+a f-b e x^2}{2 a b \sqrt{a+b x^4}}",1,"(-(a*f) + 2*a*b*g*x + b*e*x^2)/(2*a*b*Sqrt[a + b*x^4])","A",1
219,1,12,12,0.005771,"\int \frac{-1+x^4}{\left(1+x^4\right)^{3/2}} \, dx","Integrate[(-1 + x^4)/(1 + x^4)^(3/2),x]","-\frac{x}{\sqrt{x^4+1}}","-\frac{x}{\sqrt{x^4+1}}",1,"-(x/Sqrt[1 + x^4])","A",1
220,1,281,385,0.2561463,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/Sqrt[a + b*x^4],x]","\frac{-20 \sqrt{b} x \sqrt{\frac{b x^4}{a}+1} (a g-3 b c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+30 b d \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+4 \sqrt{b} x^3 \sqrt{\frac{b x^4}{a}+1} (5 b e-3 a i) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+30 a \sqrt{b} f+20 a \sqrt{b} g x+15 a \sqrt{b} h x^2-15 a h \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+12 a \sqrt{b} i x^3+30 b^{3/2} f x^4+20 b^{3/2} g x^5+15 b^{3/2} h x^6+12 b^{3/2} i x^7}{60 b^{3/2} \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\frac{5 \sqrt{b} (3 b c-a g)}{\sqrt{a}}-9 a i+15 b e\right)}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{(2 b d-a h) \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 b^{3/2}}+\frac{x \sqrt{a+b x^4} (5 b e-3 a i)}{5 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} (5 b e-3 a i) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{2 b}+\frac{g x \sqrt{a+b x^4}}{3 b}+\frac{h x^2 \sqrt{a+b x^4}}{4 b}+\frac{i x^3 \sqrt{a+b x^4}}{5 b}",1,"(30*a*Sqrt[b]*f + 20*a*Sqrt[b]*g*x + 15*a*Sqrt[b]*h*x^2 + 12*a*Sqrt[b]*i*x^3 + 30*b^(3/2)*f*x^4 + 20*b^(3/2)*g*x^5 + 15*b^(3/2)*h*x^6 + 12*b^(3/2)*i*x^7 + 30*b*d*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 15*a*h*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 20*Sqrt[b]*(-3*b*c + a*g)*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 4*Sqrt[b]*(5*b*e - 3*a*i)*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(60*b^(3/2)*Sqrt[a + b*x^4])","C",1
221,1,51,109,0.0119937,"\int \frac{1+x}{1+x^5} \, dx","Integrate[(1 + x)/(1 + x^5),x]","\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^3+\text{$\#$1}^2-\text{$\#$1}+1\&,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3-3 \text{$\#$1}^2+2 \text{$\#$1}-1}\&\right]","-\frac{1}{5} \sqrt[5]{-1} \left(1+\sqrt[5]{-1}\right) \log \left(\sqrt[5]{-1}-x\right)+\frac{1}{5} (-1)^{4/5} \left(1-(-1)^{4/5}\right) \log \left(-x-(-1)^{4/5}\right)+\frac{1}{5} (-1)^{2/5} \left(1-(-1)^{2/5}\right) \log \left(x+(-1)^{2/5}\right)-\frac{1}{5} (-1)^{3/5} \left(1+(-1)^{3/5}\right) \log \left(x-(-1)^{3/5}\right)",1,"RootSum[1 - #1 + #1^2 - #1^3 + #1^4 & , Log[x - #1]/(-1 + 2*#1 - 3*#1^2 + 4*#1^3) & ]","C",1
222,1,47,109,0.0106029,"\int \frac{1-x}{1-x^5} \, dx","Integrate[(1 - x)/(1 - x^5),x]","\text{RootSum}\left[\text{$\#$1}^4+\text{$\#$1}^3+\text{$\#$1}^2+\text{$\#$1}+1\&,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3+3 \text{$\#$1}^2+2 \text{$\#$1}+1}\&\right]","-\frac{1}{5} (-1)^{2/5} \left(1-(-1)^{2/5}\right) \log \left((-1)^{2/5}-x\right)+\frac{1}{5} (-1)^{3/5} \left(1+(-1)^{3/5}\right) \log \left(-x-(-1)^{3/5}\right)+\frac{1}{5} \sqrt[5]{-1} \left(1+\sqrt[5]{-1}\right) \log \left(x+\sqrt[5]{-1}\right)-\frac{1}{5} (-1)^{4/5} \left(1-(-1)^{4/5}\right) \log \left(x-(-1)^{4/5}\right)",1,"RootSum[1 + #1 + #1^2 + #1^3 + #1^4 & , Log[x - #1]/(1 + 2*#1 + 3*#1^2 + 4*#1^3) & ]","C",1
223,1,187,208,0.1056111,"\int \frac{x^{11} \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{60 a^3 \log \left(a+b x^3\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)+b x^3 \left(-60 a^5 f+30 a^4 b \left(2 e+f x^3\right)-10 a^3 b^2 \left(6 d+3 e x^3+2 f x^6\right)+5 a^2 b^3 \left(12 c+6 d x^3+4 e x^6+3 f x^9\right)-a b^4 x^3 \left(30 c+20 d x^3+15 e x^6+12 f x^9\right)+b^5 x^6 \left(20 c+15 d x^3+12 e x^6+10 f x^9\right)\right)}{180 b^7}","\frac{x^{12} \left(a^2 f-a b e+b^2 d\right)}{12 b^3}-\frac{a^3 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^7}+\frac{a^2 x^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^6}-\frac{a x^6 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^5}+\frac{x^9 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{9 b^4}+\frac{x^{15} (b e-a f)}{15 b^2}+\frac{f x^{18}}{18 b}",1,"(b*x^3*(-60*a^5*f + 30*a^4*b*(2*e + f*x^3) - 10*a^3*b^2*(6*d + 3*e*x^3 + 2*f*x^6) + 5*a^2*b^3*(12*c + 6*d*x^3 + 4*e*x^6 + 3*f*x^9) + b^5*x^6*(20*c + 15*d*x^3 + 12*e*x^6 + 10*f*x^9) - a*b^4*x^3*(30*c + 20*d*x^3 + 15*e*x^6 + 12*f*x^9)) + 60*a^3*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(180*b^7)","A",1
224,1,154,170,0.0930652,"\int \frac{x^8 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{b x^3 \left(60 a^4 f-30 a^3 b \left(2 e+f x^3\right)+10 a^2 b^2 \left(6 d+3 e x^3+2 f x^6\right)-5 a b^3 \left(12 c+6 d x^3+4 e x^6+3 f x^9\right)+b^4 x^3 \left(30 c+20 d x^3+15 e x^6+12 f x^9\right)\right)-60 a^2 \log \left(a+b x^3\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{180 b^6}","\frac{x^9 \left(a^2 f-a b e+b^2 d\right)}{9 b^3}+\frac{a^2 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^6}-\frac{a x^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5}+\frac{x^6 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^4}+\frac{x^{12} (b e-a f)}{12 b^2}+\frac{f x^{15}}{15 b}",1,"(b*x^3*(60*a^4*f - 30*a^3*b*(2*e + f*x^3) + 10*a^2*b^2*(6*d + 3*e*x^3 + 2*f*x^6) - 5*a*b^3*(12*c + 6*d*x^3 + 4*e*x^6 + 3*f*x^9) + b^4*x^3*(30*c + 20*d*x^3 + 15*e*x^6 + 12*f*x^9)) - 60*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(180*b^6)","A",1
225,1,119,132,0.0732447,"\int \frac{x^5 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{12 a \log \left(a+b x^3\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)+b x^3 \left(-12 a^3 f+6 a^2 b \left(2 e+f x^3\right)-2 a b^2 \left(6 d+3 e x^3+2 f x^6\right)+b^3 \left(12 c+6 d x^3+4 e x^6+3 f x^9\right)\right)}{36 b^5}","\frac{x^6 \left(a^2 f-a b e+b^2 d\right)}{6 b^3}-\frac{a \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5}+\frac{x^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^4}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{12}}{12 b}",1,"(b*x^3*(-12*a^3*f + 6*a^2*b*(2*e + f*x^3) - 2*a*b^2*(6*d + 3*e*x^3 + 2*f*x^6) + b^3*(12*c + 6*d*x^3 + 4*e*x^6 + 3*f*x^9)) + 12*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(36*b^5)","A",1
226,1,88,96,0.0505837,"\int \frac{x^2 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{b x^3 \left(6 a^2 f-3 a b \left(2 e+f x^3\right)+b^2 \left(6 d+3 e x^3+2 f x^6\right)\right)+6 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{18 b^4}","\frac{x^3 \left(a^2 f-a b e+b^2 d\right)}{3 b^3}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^4}+\frac{x^6 (b e-a f)}{6 b^2}+\frac{f x^9}{9 b}",1,"(b*x^3*(6*a^2*f - 3*a*b*(2*e + f*x^3) + b^2*(6*d + 3*e*x^3 + 2*f*x^6)) + 6*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(18*b^4)","A",1
227,1,75,80,0.0372876,"\int \frac{c+d x^3+e x^6+f x^9}{x \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)),x]","\frac{-2 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+a b x^3 \left(-2 a f+2 b e+b f x^3\right)+6 b^3 c \log (x)}{6 a b^3}","-\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a b^3}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{c \log (x)}{a}+\frac{f x^6}{6 b}",1,"(a*b*x^3*(2*b*e - 2*a*f + b*f*x^3) + 6*b^3*c*Log[x] - 2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(6*a*b^3)","A",1
228,1,77,81,0.0475581,"\int \frac{c+d x^3+e x^6+f x^9}{x^4 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)),x]","\frac{1}{3} \left(\frac{3 \log (x) (a d-b c)}{a^2}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^2 b^2}-\frac{c}{a x^3}+\frac{f x^3}{b}\right)","-\frac{\log (x) (b c-a d)}{a^2}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^2 b^2}-\frac{c}{3 a x^3}+\frac{f x^3}{3 b}",1,"(-(c/(a*x^3)) + (f*x^3)/b + (3*(-(b*c) + a*d)*Log[x])/a^2 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(a^2*b^2))/3","A",1
229,1,88,95,0.0753455,"\int \frac{c+d x^3+e x^6+f x^9}{x^7 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)),x]","\frac{6 \log (x) \left(a^2 e-a b d+b^2 c\right)+\log \left(a+b x^3\right) \left(\frac{2 a^3 f}{b}-2 a^2 e+2 a b d-2 b^2 c\right)-\frac{a \left(a c+2 a d x^3-2 b c x^3\right)}{x^6}}{6 a^3}","\frac{b c-a d}{3 a^2 x^3}+\frac{\log (x) \left(a^2 e-a b d+b^2 c\right)}{a^3}-\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^3 b}-\frac{c}{6 a x^6}",1,"(-((a*(a*c - 2*b*c*x^3 + 2*a*d*x^3))/x^6) + 6*(b^2*c - a*b*d + a^2*e)*Log[x] + (-2*b^2*c + 2*a*b*d - 2*a^2*e + (2*a^3*f)/b)*Log[a + b*x^3])/(6*a^3)","A",1
230,1,128,128,0.093202,"\int \frac{c+d x^3+e x^6+f x^9}{x^{10} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x]","\frac{b c-a d}{6 a^2 x^6}+\frac{a^2 (-e)+a b d-b^2 c}{3 a^3 x^3}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^4}+\frac{\log (x) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^4}-\frac{c}{9 a x^9}","\frac{b c-a d}{6 a^2 x^6}-\frac{a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^4}-\frac{\log (x) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^4}-\frac{c}{9 a x^9}",1,"-1/9*c/(a*x^9) + (b*c - a*d)/(6*a^2*x^6) + (-(b^2*c) + a*b*d - a^2*e)/(3*a^3*x^3) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[x])/a^4 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(3*a^4)","A",1
231,1,164,164,0.0874987,"\int \frac{c+d x^3+e x^6+f x^9}{x^{13} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]","\frac{-a^4 \left(3 c+4 d x^3+6 e x^6+12 f x^9\right)+2 a^3 b x^3 \left(2 c+3 d x^3+6 e x^6\right)-6 a^2 b^2 x^6 \left(c+2 d x^3\right)+36 b x^{12} \log (x) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)-12 b x^{12} \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+12 a b^3 c x^9}{36 a^5 x^{12}}","\frac{b c-a d}{9 a^2 x^9}-\frac{a^2 e-a b d+b^2 c}{6 a^3 x^6}-\frac{b \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5}+\frac{b \log (x) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^5}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{c}{12 a x^{12}}",1,"(12*a*b^3*c*x^9 - 6*a^2*b^2*x^6*(c + 2*d*x^3) + 2*a^3*b*x^3*(2*c + 3*d*x^3 + 6*e*x^6) - a^4*(3*c + 4*d*x^3 + 6*e*x^6 + 12*f*x^9) + 36*b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^12*Log[x] - 12*b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^12*Log[a + b*x^3])/(36*a^5*x^12)","A",1
232,1,194,205,0.2371615,"\int \frac{c+d x^3+e x^6+f x^9}{x^{16} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^16*(a + b*x^3)),x]","-\frac{-60 b^2 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+180 b^2 \log (x) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+\frac{a \left(a^4 \left(12 c+15 d x^3+20 e x^6+30 f x^9\right)-5 a^3 b x^3 \left(3 c+4 d x^3+6 e x^6+12 f x^9\right)+10 a^2 b^2 x^6 \left(2 c+3 d x^3+6 e x^6\right)-30 a b^3 x^9 \left(c+2 d x^3\right)+60 b^4 c x^{12}\right)}{x^{15}}}{180 a^6}","\frac{b c-a d}{12 a^2 x^{12}}-\frac{a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac{b^2 \log \left(a+b x^3\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^6}-\frac{b^2 \log (x) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^6}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5 x^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 x^6}-\frac{c}{15 a x^{15}}",1,"-1/180*((a*(60*b^4*c*x^12 - 30*a*b^3*x^9*(c + 2*d*x^3) + 10*a^2*b^2*x^6*(2*c + 3*d*x^3 + 6*e*x^6) - 5*a^3*b*x^3*(3*c + 4*d*x^3 + 6*e*x^6 + 12*f*x^9) + a^4*(12*c + 15*d*x^3 + 20*e*x^6 + 30*f*x^9)))/x^15 + 180*b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x] - 60*b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/a^6","A",1
233,1,351,348,0.088438,"\int \frac{x^9 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{x^{10} \left(a^2 f-a b e+b^2 d\right)}{10 b^3}-\frac{a^2 x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^6}+\frac{a x^4 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{4 b^5}+\frac{x^7 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{7 b^4}-\frac{a^{7/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 b^{19/3}}+\frac{a^{7/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt{3} b^{19/3}}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b}","\frac{x^{10} \left(a^2 f-a b e+b^2 d\right)}{10 b^3}+\frac{a^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^6}-\frac{a x^4 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 b^5}+\frac{x^7 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{7 b^4}+\frac{a^{7/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^{19/3}}-\frac{a^{7/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} b^{19/3}}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b}",1,"-((a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^6) + (a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^4)/(4*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^7)/(7*b^4) + ((b^2*d - a*b*e + a^2*f)*x^10)/(10*b^3) + ((b*e - a*f)*x^13)/(13*b^2) + (f*x^16)/(16*b) + (a^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(19/3)) + (a^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(19/3)) - (a^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(19/3))","A",1
234,1,311,316,0.1070152,"\int \frac{x^7 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{x^8 \left(a^2 f-a b e+b^2 d\right)}{8 b^3}+\frac{a x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{2 b^5}+\frac{x^5 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{5 b^4}-\frac{a^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 b^{17/3}}+\frac{a^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 b^{17/3}}+\frac{a^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt{3} b^{17/3}}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b}","\frac{x^8 \left(a^2 f-a b e+b^2 d\right)}{8 b^3}-\frac{a x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 b^5}+\frac{x^5 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{5 b^4}+\frac{a^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^{17/3}}-\frac{a^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^{17/3}}-\frac{a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} b^{17/3}}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b}",1,"(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(2*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^8)/(8*b^3) + ((b*e - a*f)*x^11)/(11*b^2) + (f*x^14)/(14*b) + (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(17/3)) + (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(17/3)) - (a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(17/3))","A",1
235,1,306,312,0.1127477,"\int \frac{x^6 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{x^7 \left(a^2 f-a b e+b^2 d\right)}{7 b^3}+\frac{a x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^5}+\frac{x^4 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 b^4}+\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 b^{16/3}}-\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 b^{16/3}}+\frac{a^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt{3} b^{16/3}}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b}","\frac{x^7 \left(a^2 f-a b e+b^2 d\right)}{7 b^3}-\frac{a x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^5}+\frac{x^4 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 b^4}-\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^{16/3}}+\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^{16/3}}-\frac{a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} b^{16/3}}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b}",1,"(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^5 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^4)/(4*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10)/(10*b^2) + (f*x^13)/(13*b) + (a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(16/3)) - (a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) + (a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(16/3))","A",1
236,1,266,279,0.1165381,"\int \frac{x^4 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{264 b^{5/3} x^5 \left(a^2 f-a b e+b^2 d\right)+660 b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)-440 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)-440 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)+220 a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)+165 b^{8/3} x^8 (b e-a f)+120 b^{11/3} f x^{11}}{1320 b^{14/3}}","\frac{x^5 \left(a^2 f-a b e+b^2 d\right)}{5 b^3}+\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 b^4}-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^{14/3}}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^{14/3}}+\frac{a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} b^{14/3}}+\frac{x^8 (b e-a f)}{8 b^2}+\frac{f x^{11}}{11 b}",1,"(660*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2 + 264*b^(5/3)*(b^2*d - a*b*e + a^2*f)*x^5 + 165*b^(8/3)*(b*e - a*f)*x^8 + 120*b^(11/3)*f*x^11 - 440*Sqrt[3]*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 440*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1320*b^(14/3))","A",1
237,1,264,274,0.108835,"\int \frac{x^3 \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{105 b^{4/3} x^4 \left(a^2 f-a b e+b^2 d\right)+420 \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+140 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)-140 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)-70 \sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)+60 b^{7/3} x^7 (b e-a f)+42 b^{10/3} f x^{10}}{420 b^{13/3}}","\frac{x^4 \left(a^2 f-a b e+b^2 d\right)}{4 b^3}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^{13/3}}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} b^{13/3}}+\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^4}+\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^{13/3}}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b}",1,"(420*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x + 105*b^(4/3)*(b^2*d - a*b*e + a^2*f)*x^4 + 60*b^(7/3)*(b*e - a*f)*x^7 + 42*b^(10/3)*f*x^10 - 140*Sqrt[3]*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*b^(13/3))","A",1
238,1,231,245,0.1789937,"\int \frac{x \left(c+d x^3+e x^6+f x^9\right)}{a+b x^3} \, dx","Integrate[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]","\frac{60 b^{2/3} x^2 \left(a^2 f-a b e+b^2 d\right)+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt[3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt[3]{a}}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt[3]{a}}+24 b^{5/3} x^5 (b e-a f)+15 b^{8/3} f x^8}{120 b^{11/3}}","\frac{x^2 \left(a^2 f-a b e+b^2 d\right)}{2 b^3}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 \sqrt[3]{a} b^{11/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} \sqrt[3]{a} b^{11/3}}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 \sqrt[3]{a} b^{11/3}}+\frac{x^5 (b e-a f)}{5 b^2}+\frac{f x^8}{8 b}",1,"(60*b^(2/3)*(b^2*d - a*b*e + a^2*f)*x^2 + 24*b^(5/3)*(b*e - a*f)*x^5 + 15*b^(8/3)*f*x^8 + (40*Sqrt[3]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(120*b^(11/3))","A",1
239,1,229,240,0.1683794,"\int \frac{c+d x^3+e x^6+f x^9}{a+b x^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(a + b*x^3),x]","\frac{84 \sqrt[3]{b} x \left(a^2 f-a b e+b^2 d\right)+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{2/3}}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{2/3}}+21 b^{4/3} x^4 (b e-a f)+12 b^{7/3} f x^7}{84 b^{10/3}}","\frac{x \left(a^2 f-a b e+b^2 d\right)}{b^3}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{2/3} b^{10/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{2/3} b^{10/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{x^4 (b e-a f)}{4 b^2}+\frac{f x^7}{7 b}",1,"(84*b^(1/3)*(b^2*d - a*b*e + a^2*f)*x + 21*b^(4/3)*(b*e - a*f)*x^4 + 12*b^(7/3)*f*x^7 + (28*Sqrt[3]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(84*b^(10/3))","A",1
240,1,224,227,0.2052939,"\int \frac{c+d x^3+e x^6+f x^9}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^2*(a + b*x^3)),x]","\frac{15 a^{4/3} b^{2/3} x^3 (b e-a f)+6 a^{4/3} b^{5/3} f x^6+10 x \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+10 \sqrt{3} x \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)-5 x \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)-30 \sqrt[3]{a} b^{8/3} c}{30 a^{4/3} b^{8/3} x}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{4/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{4/3} b^{8/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{4/3} b^{8/3}}+\frac{x^2 (b e-a f)}{2 b^2}-\frac{c}{a x}+\frac{f x^5}{5 b}",1,"(-30*a^(1/3)*b^(8/3)*c + 15*a^(4/3)*b^(2/3)*(b*e - a*f)*x^3 + 6*a^(4/3)*b^(5/3)*f*x^6 + 10*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*Log[a^(1/3) + b^(1/3)*x] - 5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(30*a^(4/3)*b^(8/3)*x)","A",1
241,1,218,224,0.1606995,"\int \frac{c+d x^3+e x^6+f x^9}{x^3 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)),x]","\frac{1}{12} \left(\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{5/3} b^{7/3}}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{5/3} b^{7/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{5/3} b^{7/3}}+\frac{12 x (b e-a f)}{b^2}-\frac{6 c}{a x^2}+\frac{3 f x^4}{b}\right)","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{5/3} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{5/3} b^{7/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b e-a f)}{b^2}-\frac{c}{2 a x^2}+\frac{f x^4}{4 b}",1,"((-6*c)/(a*x^2) + (12*(b*e - a*f)*x)/b^2 + (3*f*x^4)/b + (4*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(5/3)*b^(7/3)) + (4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(5/3)*b^(7/3)) + (2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(5/3)*b^(7/3)))/12","A",1
242,1,220,227,0.1565025,"\int \frac{c+d x^3+e x^6+f x^9}{x^5 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)),x]","\frac{1}{12} \left(\frac{12 (b c-a d)}{a^2 x}+\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^{7/3} b^{5/3}}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{7/3} b^{5/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^{7/3} b^{5/3}}-\frac{3 c}{a x^4}+\frac{6 f x^2}{b}\right)","\frac{b c-a d}{a^2 x}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{7/3} b^{5/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{7/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{7/3} b^{5/3}}-\frac{c}{4 a x^4}+\frac{f x^2}{2 b}",1,"((-3*c)/(a*x^4) + (12*(b*c - a*d))/(a^2*x) + (6*f*x^2)/b + (4*Sqrt[3]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(7/3)*b^(5/3)) + (4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(7/3)*b^(5/3)) + (2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(7/3)*b^(5/3)))/12","A",1
243,1,220,225,0.1218331,"\int \frac{c+d x^3+e x^6+f x^9}{x^6 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)),x]","\frac{b c-a d}{2 a^2 x^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 a^{8/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{8/3} b^{4/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt{3} a^{8/3} b^{4/3}}-\frac{c}{5 a x^5}+\frac{f x}{b}","\frac{b c-a d}{2 a^2 x^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{8/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{8/3} b^{4/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{8/3} b^{4/3}}-\frac{c}{5 a x^5}+\frac{f x}{b}",1,"-1/5*c/(a*x^5) + (b*c - a*d)/(2*a^2*x^2) + (f*x)/b + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(8/3)*b^(4/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(8/3)*b^(4/3)) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(8/3)*b^(4/3))","A",1
244,1,231,242,0.1451622,"\int \frac{c+d x^3+e x^6+f x^9}{x^8 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)),x]","\frac{\frac{21 a^{4/3} (b c-a d)}{x^4}-\frac{12 a^{7/3} c}{x^7}-\frac{84 \sqrt[3]{a} \left(a^2 e-a b d+b^2 c\right)}{x}+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b^{2/3}}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^{2/3}}}{84 a^{10/3}}","\frac{b c-a d}{4 a^2 x^4}-\frac{a^2 e-a b d+b^2 c}{a^3 x}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{10/3} b^{2/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{10/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{10/3} b^{2/3}}-\frac{c}{7 a x^7}",1,"((-12*a^(7/3)*c)/x^7 + (21*a^(4/3)*(b*c - a*d))/x^4 - (84*a^(1/3)*(b^2*c - a*b*d + a^2*e))/x + (28*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(84*a^(10/3))","A",1
245,1,231,244,0.2450657,"\int \frac{c+d x^3+e x^6+f x^9}{x^9 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)),x]","\frac{\frac{24 a^{5/3} (b c-a d)}{x^5}-\frac{15 a^{8/3} c}{x^8}-\frac{60 a^{2/3} \left(a^2 e-a b d+b^2 c\right)}{x^2}+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt[3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt[3]{b}}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt[3]{b}}}{120 a^{11/3}}","\frac{b c-a d}{5 a^2 x^5}-\frac{a^2 e-a b d+b^2 c}{2 a^3 x^2}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{11/3} \sqrt[3]{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{11/3} \sqrt[3]{b}}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{11/3} \sqrt[3]{b}}-\frac{c}{8 a x^8}",1,"((-15*a^(8/3)*c)/x^8 + (24*a^(5/3)*(b*c - a*d))/x^5 - (60*a^(2/3)*(b^2*c - a*b*d + a^2*e))/x^2 + (40*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(120*a^(11/3))","A",1
246,1,266,277,0.1346347,"\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]","\frac{\frac{60 a^{7/3} (b c-a d)}{x^7}-\frac{42 a^{10/3} c}{x^{10}}-\frac{105 a^{4/3} \left(a^2 e-a b d+b^2 c\right)}{x^4}+\frac{420 \sqrt[3]{a} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{x}+140 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)-140 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+70 \sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{420 a^{13/3}}","\frac{b c-a d}{7 a^2 x^7}-\frac{a^2 e-a b d+b^2 c}{4 a^3 x^4}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{13/3}}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{13/3}}+\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{13/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac{c}{10 a x^{10}}",1,"((-42*a^(10/3)*c)/x^10 + (60*a^(7/3)*(b*c - a*d))/x^7 - (105*a^(4/3)*(b^2*c - a*b*d + a^2*e))/x^4 + (420*a^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/x - 140*Sqrt[3]*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(420*a^(13/3))","A",1
247,1,266,280,0.1630396,"\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)),x]","\frac{\frac{165 a^{8/3} (b c-a d)}{x^8}-\frac{120 a^{11/3} c}{x^{11}}-\frac{264 a^{5/3} \left(a^2 e-a b d+b^2 c\right)}{x^5}+440 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)-440 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)+\frac{660 a^{2/3} \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{x^2}+220 b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{1320 a^{14/3}}","\frac{b c-a d}{8 a^2 x^8}-\frac{a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{14/3}}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{14/3}}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{14/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^4 x^2}-\frac{c}{11 a x^{11}}",1,"((-120*a^(11/3)*c)/x^11 + (165*a^(8/3)*(b*c - a*d))/x^8 - (264*a^(5/3)*(b^2*c - a*b*d + a^2*e))/x^5 + (660*a^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/x^2 - 440*Sqrt[3]*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 440*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*b^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1320*a^(14/3))","A",1
248,1,308,313,0.1345723,"\int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)),x]","\frac{b c-a d}{10 a^2 x^{10}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 a^{16/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{16/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{16/3}}+\frac{b \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{4 a^4 x^4}-\frac{c}{13 a x^{13}}","\frac{b c-a d}{10 a^2 x^{10}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}-\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{16/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{16/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{16/3}}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^5 x}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{4 a^4 x^4}-\frac{c}{13 a x^{13}}",1,"-1/13*c/(a*x^13) + (b*c - a*d)/(10*a^2*x^10) - (b^2*c - a*b*d + a^2*e)/(7*a^3*x^7) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(4*a^4*x^4) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a^5*x) + (b^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(16/3)) + (b^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(16/3)) + (b^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(16/3))","A",1
249,1,311,315,0.1529358,"\int \frac{c+d x^3+e x^6+f x^9}{x^{15} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^15*(a + b*x^3)),x]","\frac{b c-a d}{11 a^2 x^{11}}-\frac{a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac{b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{17/3}}+\frac{b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 a^{17/3}}+\frac{b^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{17/3}}+\frac{b \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{2 a^5 x^2}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{14 a x^{14}}","\frac{b c-a d}{11 a^2 x^{11}}-\frac{a^2 e-a b d+b^2 c}{8 a^3 x^8}+\frac{b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{17/3}}-\frac{b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{17/3}}+\frac{b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{17/3}}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{2 a^5 x^2}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{5 a^4 x^5}-\frac{c}{14 a x^{14}}",1,"-1/14*c/(a*x^14) + (b*c - a*d)/(11*a^2*x^11) - (b^2*c - a*b*d + a^2*e)/(8*a^3*x^8) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(5*a^4*x^5) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(2*a^5*x^2) + (b^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(17/3)) + (b^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(17/3)) + (b^(5/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(17/3))","A",1
250,1,346,351,0.1455427,"\int \frac{c+d x^3+e x^6+f x^9}{x^{17} \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^17*(a + b*x^3)),x]","\frac{b c-a d}{13 a^2 x^{13}}-\frac{a^2 e-a b d+b^2 c}{10 a^3 x^{10}}+\frac{b^{7/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{19/3}}+\frac{b^{7/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 a^{19/3}}+\frac{b^{7/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\sqrt{3} a^{19/3}}+\frac{b^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^6 x}+\frac{b \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{4 a^5 x^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac{c}{16 a x^{16}}","\frac{b c-a d}{13 a^2 x^{13}}-\frac{a^2 e-a b d+b^2 c}{10 a^3 x^{10}}+\frac{b^{7/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^{19/3}}-\frac{b^{7/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^{19/3}}-\frac{b^{7/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\sqrt{3} a^{19/3}}+\frac{b^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a^6 x}-\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{4 a^5 x^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{7 a^4 x^7}-\frac{c}{16 a x^{16}}",1,"-1/16*c/(a*x^16) + (b*c - a*d)/(13*a^2*x^13) - (b^2*c - a*b*d + a^2*e)/(10*a^3*x^10) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(7*a^4*x^7) + (b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(4*a^5*x^4) + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a^6*x) + (b^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(19/3)) + (b^(7/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(19/3)) + (b^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(19/3))","A",1
251,1,205,220,0.2088924,"\int \frac{x^{11} \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{20 b^3 x^9 \left(3 a^2 f-2 a b e+b^2 d\right)+30 b^2 x^6 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)+60 a b x^3 \left(5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right)-\frac{60 a^3 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+60 a^2 \log \left(a+b x^3\right) \left(-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right)+15 b^4 x^{12} (b e-2 a f)+12 b^5 f x^{15}}{180 b^7}","\frac{x^9 \left(3 a^2 f-2 a b e+b^2 d\right)}{9 b^4}+\frac{a^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^7 \left(a+b x^3\right)}+\frac{a^2 \log \left(a+b x^3\right) \left(-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right)}{3 b^7}-\frac{a x^3 \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{3 b^6}+\frac{x^6 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{6 b^5}+\frac{x^{12} (b e-2 a f)}{12 b^3}+\frac{f x^{15}}{15 b^2}",1,"(60*a*b*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x^3 + 30*b^2*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^6 + 20*b^3*(b^2*d - 2*a*b*e + 3*a^2*f)*x^9 + 15*b^4*(b*e - 2*a*f)*x^12 + 12*b^5*f*x^15 - (60*a^3*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 60*a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f)*Log[a + b*x^3])/(180*b^7)","A",1
252,1,167,180,0.1424638,"\int \frac{x^8 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{6 b^2 x^6 \left(3 a^2 f-2 a b e+b^2 d\right)+12 b x^3 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)+\frac{12 a^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+12 a \log \left(a+b x^3\right) \left(5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right)+4 b^3 x^9 (b e-2 a f)+3 b^4 f x^{12}}{36 b^6}","\frac{x^6 \left(3 a^2 f-2 a b e+b^2 d\right)}{6 b^4}-\frac{a^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^6 \left(a+b x^3\right)}-\frac{a \log \left(a+b x^3\right) \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{3 b^6}+\frac{x^3 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{3 b^5}+\frac{x^9 (b e-2 a f)}{9 b^3}+\frac{f x^{12}}{12 b^2}",1,"(12*b*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3 + 6*b^2*(b^2*d - 2*a*b*e + 3*a^2*f)*x^6 + 4*b^3*(b*e - 2*a*f)*x^9 + 3*b^4*f*x^12 + (12*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 12*a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*Log[a + b*x^3])/(36*b^6)","A",1
253,1,129,140,0.1243437,"\int \frac{x^5 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{6 b x^3 \left(3 a^2 f-2 a b e+b^2 d\right)+\frac{6 a \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a+b x^3}+6 \log \left(a+b x^3\right) \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)+3 b^2 x^6 (b e-2 a f)+2 b^3 f x^9}{18 b^5}","\frac{x^3 \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}+\frac{a \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5 \left(a+b x^3\right)}+\frac{\log \left(a+b x^3\right) \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{3 b^5}+\frac{x^6 (b e-2 a f)}{6 b^3}+\frac{f x^9}{9 b^2}",1,"(6*b*(b^2*d - 2*a*b*e + 3*a^2*f)*x^3 + 3*b^2*(b*e - 2*a*f)*x^6 + 2*b^3*f*x^9 + (6*a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a + b*x^3) + 6*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*Log[a + b*x^3])/(18*b^5)","A",1
254,1,93,103,0.0680948,"\int \frac{x^2 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{2 \log \left(a+b x^3\right) \left(3 a^2 f-2 a b e+b^2 d\right)+\frac{2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+2 b x^3 (b e-2 a f)+b^2 f x^6}{6 b^4}","\frac{\log \left(a+b x^3\right) \left(3 a^2 f-2 a b e+b^2 d\right)}{3 b^4}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 b^4 \left(a+b x^3\right)}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^6}{6 b^2}",1,"(2*b*(b*e - 2*a*f)*x^3 + b^2*f*x^6 + (2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 2*(b^2*d - 2*a*b*e + 3*a^2*f)*Log[a + b*x^3])/(6*b^4)","A",1
255,1,95,100,0.1808482,"\int \frac{c+d x^3+e x^6+f x^9}{x \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x]","\frac{\frac{\log \left(a+b x^3\right) \left(-2 a^3 f+a^2 b e-b^3 c\right)+\frac{a \left(a^3 (-f)+a^2 b \left(e+f x^3\right)+a b^2 \left(f x^6-d\right)+b^3 c\right)}{a+b x^3}}{b^3}+3 c \log (x)}{3 a^2}","\frac{c \log (x)}{a^2}-\frac{\log \left(a+b x^3\right) \left(2 a^3 f-a^2 b e+b^3 c\right)}{3 a^2 b^3}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a b^3 \left(a+b x^3\right)}+\frac{f x^3}{3 b^2}",1,"(3*c*Log[x] + ((a*(b^3*c - a^3*f + a^2*b*(e + f*x^3) + a*b^2*(-d + f*x^6)))/(a + b*x^3) + (-(b^3*c) + a^2*b*e - 2*a^3*f)*Log[a + b*x^3])/b^3)/(3*a^2)","A",1
256,1,97,109,0.1511959,"\int \frac{c+d x^3+e x^6+f x^9}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)^2),x]","\frac{\frac{\log \left(a+b x^3\right) \left(a^3 f-a b^2 d+2 b^3 c\right)}{b^2}+\frac{a \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^2 \left(a+b x^3\right)}+3 \log (x) (a d-2 b c)-\frac{a c}{x^3}}{3 a^3}","\frac{\log \left(a+b x^3\right) \left(a^3 f-a b^2 d+2 b^3 c\right)}{3 a^3 b^2}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{c}{3 a^2 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^2 b^2 \left(a+b x^3\right)}",1,"(-((a*c)/x^3) + (a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(b^2*(a + b*x^3)) + 3*(-2*b*c + a*d)*Log[x] + ((2*b^3*c - a*b^2*d + a^3*f)*Log[a + b*x^3])/b^2)/(3*a^3)","A",1
257,1,118,130,0.1371806,"\int \frac{c+d x^3+e x^6+f x^9}{x^7 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^2),x]","-\frac{2 \log \left(a+b x^3\right) \left(a^2 e-2 a b d+3 b^2 c\right)-6 \log (x) \left(a^2 e-2 a b d+3 b^2 c\right)+\frac{a^2 c}{x^6}+\frac{2 a \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b \left(a+b x^3\right)}+\frac{2 a (a d-2 b c)}{x^3}}{6 a^4}","\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{6 a^2 x^6}-\frac{\log \left(a+b x^3\right) \left(a^2 e-2 a b d+3 b^2 c\right)}{3 a^4}+\frac{\log (x) \left(a^2 e-2 a b d+3 b^2 c\right)}{a^4}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^3 b \left(a+b x^3\right)}",1,"-1/6*((a^2*c)/x^6 + (2*a*(-2*b*c + a*d))/x^3 + (2*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(b*(a + b*x^3)) - 6*(3*b^2*c - 2*a*b*d + a^2*e)*Log[x] + 2*(3*b^2*c - 2*a*b*d + a^2*e)*Log[a + b*x^3])/a^4","A",1
258,1,160,175,0.1387202,"\int \frac{c+d x^3+e x^6+f x^9}{x^{10} \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^2),x]","\frac{-\frac{2 a^3 c}{x^9}-\frac{6 a \left(a^2 e-2 a b d+3 b^2 c\right)}{x^3}-\frac{3 a^2 (a d-2 b c)}{x^6}+\frac{6 a \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+6 \log \left(a+b x^3\right) \left(a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right)+18 \log (x) \left(a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right)}{18 a^5}","\frac{2 b c-a d}{6 a^3 x^6}-\frac{c}{9 a^2 x^9}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right)}{3 a^5}-\frac{\log (x) \left(a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c\right)}{a^5}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 \left(a+b x^3\right)}",1,"((-2*a^3*c)/x^9 - (3*a^2*(-2*b*c + a*d))/x^6 - (6*a*(3*b^2*c - 2*a*b*d + a^2*e))/x^3 + (6*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 18*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f)*Log[x] + 6*(4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)*Log[a + b*x^3])/(18*a^5)","A",1
259,1,198,214,0.2637142,"\int \frac{c+d x^3+e x^6+f x^9}{x^{13} \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)^2),x]","-\frac{\frac{3 a^4 c}{x^{12}}+\frac{4 a^3 (a d-2 b c)}{x^9}+\frac{6 a^2 \left(a^2 e-2 a b d+3 b^2 c\right)}{x^6}+\frac{12 a b \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+\frac{12 a \left(a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right)}{x^3}+12 b \log \left(a+b x^3\right) \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)-36 b \log (x) \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{36 a^6}","\frac{2 b c-a d}{9 a^3 x^9}-\frac{c}{12 a^2 x^{12}}-\frac{a^2 e-2 a b d+3 b^2 c}{6 a^4 x^6}-\frac{b \log \left(a+b x^3\right) \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{3 a^6}+\frac{b \log (x) \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{a^6}+\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5 \left(a+b x^3\right)}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3}",1,"-1/36*((3*a^4*c)/x^12 + (4*a^3*(-2*b*c + a*d))/x^9 + (6*a^2*(3*b^2*c - 2*a*b*d + a^2*e))/x^6 + (12*a*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/x^3 + (12*a*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) - 36*b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f)*Log[x] + 12*b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f)*Log[a + b*x^3])/a^6","A",1
260,1,364,369,0.4353815,"\int \frac{x^9 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{x^7 \left(3 a^2 f-2 a b e+b^2 d\right)}{7 b^4}+\frac{a^2 x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 b^6 \left(a+b x^3\right)}+\frac{a x \left(5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right)}{b^6}+\frac{x^4 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{4 b^5}+\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right)}{18 b^{19/3}}-\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right)}{9 b^{19/3}}+\frac{a^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(16 a^3 f-13 a^2 b e+10 a b^2 d-7 b^3 c\right)}{3 \sqrt{3} b^{19/3}}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2}","\frac{x^7 \left(3 a^2 f-2 a b e+b^2 d\right)}{7 b^4}-\frac{a^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^6 \left(a+b x^3\right)}-\frac{a x \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{b^6}+\frac{x^4 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{4 b^5}-\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right)}{18 b^{19/3}}+\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right)}{9 b^{19/3}}-\frac{a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right)}{3 \sqrt{3} b^{19/3}}+\frac{x^{10} (b e-2 a f)}{10 b^3}+\frac{f x^{13}}{13 b^2}",1,"(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^4)/(4*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^7)/(7*b^4) + ((b*e - 2*a*f)*x^10)/(10*b^3) + (f*x^13)/(13*b^2) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(3*b^6*(a + b*x^3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(19/3)) - (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(19/3)) + (a^(4/3)*(-7*b^3*c + 10*a*b^2*d - 13*a^2*b*e + 16*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(19/3))","A",1
261,1,319,335,0.1962431,"\int \frac{x^7 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{792 b^{5/3} x^5 \left(3 a^2 f-2 a b e+b^2 d\right)+1980 b^{2/3} x^2 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)+\frac{1320 a b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a+b x^3}-440 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right)-440 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right)+220 a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right)+495 b^{8/3} x^8 (b e-2 a f)+360 b^{11/3} f x^{11}}{3960 b^{17/3}}","\frac{x^5 \left(3 a^2 f-2 a b e+b^2 d\right)}{5 b^4}+\frac{x^2 \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{2 b^5}+\frac{a x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5 \left(a+b x^3\right)}-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right)}{18 b^{17/3}}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right)}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right)}{3 \sqrt{3} b^{17/3}}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2}",1,"(1980*b^(2/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^2 + 792*b^(5/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^5 + 495*b^(8/3)*(b*e - 2*a*f)*x^8 + 360*b^(11/3)*f*x^11 + (1320*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3) - 440*Sqrt[3]*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 440*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*a^(2/3)*(-5*b^3*c + 8*a*b^2*d - 11*a^2*b*e + 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*b^(17/3))","A",1
262,1,315,328,0.3039149,"\int \frac{x^6 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{315 b^{4/3} x^4 \left(3 a^2 f-2 a b e+b^2 d\right)+\frac{420 a \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a+b x^3}+1260 \sqrt[3]{b} x \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)+140 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right)-140 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right)-70 \sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(13 a^3 f-10 a^2 b e+7 a b^2 d-4 b^3 c\right)+180 b^{7/3} x^7 (b e-2 a f)+126 b^{10/3} f x^{10}}{1260 b^{16/3}}","\frac{x^4 \left(3 a^2 f-2 a b e+b^2 d\right)}{4 b^4}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right)}{9 b^{16/3}}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right)}{3 \sqrt{3} b^{16/3}}+\frac{a x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^5 \left(a+b x^3\right)}+\frac{x \left(-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right)}{b^5}+\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right)}{18 b^{16/3}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^{10}}{10 b^2}",1,"(1260*b^(1/3)*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x + 315*b^(4/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^4 + 180*b^(7/3)*(b*e - 2*a*f)*x^7 + 126*b^(10/3)*f*x^10 + (420*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3) - 140*Sqrt[3]*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-4*b^3*c + 7*a*b^2*d - 10*a^2*b*e + 13*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*b^(16/3))","A",1
263,1,282,298,0.2992827,"\int \frac{x^4 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{180 b^{2/3} x^2 \left(3 a^2 f-2 a b e+b^2 d\right)+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right)}{\sqrt[3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right)}{\sqrt[3]{a}}-\frac{120 b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a+b x^3}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right)}{\sqrt[3]{a}}+72 b^{5/3} x^5 (b e-2 a f)+45 b^{8/3} f x^8}{360 b^{14/3}}","\frac{x^2 \left(3 a^2 f-2 a b e+b^2 d\right)}{2 b^4}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right)}{9 \sqrt[3]{a} b^{14/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right)}{3 \sqrt{3} \sqrt[3]{a} b^{14/3}}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^4 \left(a+b x^3\right)}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right)}{18 \sqrt[3]{a} b^{14/3}}+\frac{x^5 (b e-2 a f)}{5 b^3}+\frac{f x^8}{8 b^2}",1,"(180*b^(2/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x^2 + 72*b^(5/3)*(b*e - 2*a*f)*x^5 + 45*b^(8/3)*f*x^8 - (120*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3) + (40*Sqrt[3]*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-2*b^3*c + 5*a*b^2*d - 8*a^2*b*e + 11*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(2*b^3*c - 5*a*b^2*d + 8*a^2*b*e - 11*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(360*b^(14/3))","A",1
264,1,277,288,0.1926155,"\int \frac{x^3 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{252 \sqrt[3]{b} x \left(3 a^2 f-2 a b e+b^2 d\right)-\frac{84 \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a+b x^3}+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right)}{a^{2/3}}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right)}{a^{2/3}}+63 b^{4/3} x^4 (b e-2 a f)+36 b^{7/3} f x^7}{252 b^{13/3}}","\frac{x \left(3 a^2 f-2 a b e+b^2 d\right)}{b^4}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 b^4 \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{18 a^{2/3} b^{13/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2}",1,"(252*b^(1/3)*(b^2*d - 2*a*b*e + 3*a^2*f)*x + 63*b^(4/3)*(b*e - 2*a*f)*x^4 + 36*b^(7/3)*f*x^7 - (84*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(b^3*c) + 4*a*b^2*d - 7*a^2*b*e + 10*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(252*b^(13/3))","A",1
265,1,255,271,0.1872003,"\int \frac{x \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]","\frac{\frac{30 b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)}-\frac{10 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{4/3}}+\frac{5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{4/3}}+45 b^{2/3} x^2 (b e-2 a f)+18 b^{5/3} f x^5}{90 b^{11/3}}","\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a b^3 \left(a+b x^3\right)}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{18 a^{4/3} b^{11/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{9 a^{4/3} b^{11/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(8 a^3 f-5 a^2 b e+2 a b^2 d+b^3 c\right)}{3 \sqrt{3} a^{4/3} b^{11/3}}+\frac{x^2 (b e-2 a f)}{2 b^3}+\frac{f x^5}{5 b^2}",1,"(45*b^(2/3)*(b*e - 2*a*f)*x^2 + 18*b^(5/3)*f*x^5 + (30*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (10*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 2*a*b^2*d - 5*a^2*b*e + 8*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(90*b^(11/3))","A",1
266,1,251,264,0.2002124,"\int \frac{c+d x^3+e x^6+f x^9}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(a + b*x^3)^2,x]","\frac{\frac{12 \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{5/3}}-\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{5/3}}+36 \sqrt[3]{b} x (b e-2 a f)+9 b^{4/3} f x^4}{36 b^{10/3}}","\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a b^3 \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{18 a^{5/3} b^{10/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{9 a^{5/3} b^{10/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^4}{4 b^2}",1,"(36*b^(1/3)*(b*e - 2*a*f)*x + 9*b^(4/3)*f*x^4 + (12*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(36*b^(10/3))","A",1
267,1,255,265,0.2116453,"\int \frac{c+d x^3+e x^6+f x^9}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^2*(a + b*x^3)^2),x]","\frac{1}{18} \left(-\frac{18 c}{a^2 x}+\frac{6 x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^2 b^2 \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{a^{7/3} b^{8/3}}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{a^{7/3} b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{a^{7/3} b^{8/3}}+\frac{9 f x^2}{b^2}\right)","-\frac{c}{a^2 x}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^2 b^2 \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{18 a^{7/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{9 a^{7/3} b^{8/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(5 a^3 f-2 a^2 b e-a b^2 d+4 b^3 c\right)}{3 \sqrt{3} a^{7/3} b^{8/3}}+\frac{f x^2}{2 b^2}",1,"((-18*c)/(a^2*x) + (9*f*x^2)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a^2*b^2*(a + b*x^3)) + (2*Sqrt[3]*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(7/3)*b^(8/3)) + (2*(4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(7/3)*b^(8/3)) - ((4*b^3*c - a*b^2*d - 2*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(7/3)*b^(8/3)))/18","A",1
268,1,250,260,0.2192356,"\int \frac{c+d x^3+e x^6+f x^9}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^2),x]","\frac{1}{18} \left(-\frac{9 c}{a^2 x^2}+\frac{6 x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a^2 b^2 \left(a+b x^3\right)}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{a^{8/3} b^{7/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{a^{8/3} b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{a^{8/3} b^{7/3}}+\frac{18 f x}{b^2}\right)","-\frac{c}{2 a^2 x^2}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^2 b^2 \left(a+b x^3\right)}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{18 a^{8/3} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{9 a^{8/3} b^{7/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{3 \sqrt{3} a^{8/3} b^{7/3}}+\frac{f x}{b^2}",1,"((-9*c)/(a^2*x^2) + (18*f*x)/b^2 + (6*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a^2*b^2*(a + b*x^3)) + (2*Sqrt[3]*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(8/3)*b^(7/3)) - (2*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(8/3)*b^(7/3)) + ((5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(8/3)*b^(7/3)))/18","A",1
269,1,255,269,0.2739542,"\int \frac{c+d x^3+e x^6+f x^9}{x^5 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^2),x]","\frac{-\frac{9 a^{4/3} c}{x^4}-\frac{12 \sqrt[3]{a} x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b \left(a+b x^3\right)}-\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{b^{5/3}}+\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{b^{5/3}}-\frac{36 \sqrt[3]{a} (a d-2 b c)}{x}}{36 a^{10/3}}","\frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4}+\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^3 b \left(a+b x^3\right)}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{18 a^{10/3} b^{5/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right)}{3 \sqrt{3} a^{10/3} b^{5/3}}",1,"((-9*a^(4/3)*c)/x^4 - (36*a^(1/3)*(-2*b*c + a*d))/x - (12*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)) - (4*Sqrt[3]*(7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) - (4*(7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*(7*b^3*c - 4*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3))/(36*a^(10/3))","A",1
270,1,253,270,0.1916506,"\int \frac{c+d x^3+e x^6+f x^9}{x^6 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^2),x]","\frac{-\frac{45 a^{2/3} (a d-2 b c)}{x^2}-\frac{18 a^{5/3} c}{x^5}+\frac{10 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{b^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{b^{4/3}}-\frac{30 a^{2/3} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b \left(a+b x^3\right)}-\frac{5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{b^{4/3}}}{90 a^{11/3}}","\frac{2 b c-a d}{2 a^3 x^2}-\frac{c}{5 a^2 x^5}+\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^3 b \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{18 a^{11/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{9 a^{11/3} b^{4/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{3 \sqrt{3} a^{11/3} b^{4/3}}",1,"((-18*a^(5/3)*c)/x^5 - (45*a^(2/3)*(-2*b*c + a*d))/x^2 - (30*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) + (10*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) - (5*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3))/(90*a^(11/3))","A",1
271,1,281,297,0.2577335,"\int \frac{c+d x^3+e x^6+f x^9}{x^8 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^2),x]","\frac{-\frac{63 a^{4/3} (a d-2 b c)}{x^4}-\frac{36 a^{7/3} c}{x^7}-\frac{252 \sqrt[3]{a} \left(a^2 e-2 a b d+3 b^2 c\right)}{x}+\frac{84 \sqrt[3]{a} x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right)}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right)}{b^{2/3}}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f-4 a^2 b e+7 a b^2 d-10 b^3 c\right)}{b^{2/3}}}{252 a^{13/3}}","\frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right)}{18 a^{13/3} b^{2/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right)}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 (-f)+4 a^2 b e-7 a b^2 d+10 b^3 c\right)}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^4 \left(a+b x^3\right)}",1,"((-36*a^(7/3)*c)/x^7 - (63*a^(4/3)*(-2*b*c + a*d))/x^4 - (252*a^(1/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x + (84*a^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3) + (28*Sqrt[3]*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(10*b^3*c - 7*a*b^2*d + 4*a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-10*b^3*c + 7*a*b^2*d - 4*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(252*a^(13/3))","A",1
272,1,280,297,0.2402017,"\int \frac{c+d x^3+e x^6+f x^9}{x^9 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^2),x]","\frac{-\frac{72 a^{5/3} (a d-2 b c)}{x^5}-\frac{45 a^{8/3} c}{x^8}-\frac{180 a^{2/3} \left(a^2 e-2 a b d+3 b^2 c\right)}{x^2}+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^3 f-5 a^2 b e+8 a b^2 d-11 b^3 c\right)}{\sqrt[3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{\sqrt[3]{b}}+\frac{120 a^{2/3} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{\sqrt[3]{b}}}{360 a^{14/3}}","\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{9 a^{14/3} \sqrt[3]{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{3 \sqrt{3} a^{14/3} \sqrt[3]{b}}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{18 a^{14/3} \sqrt[3]{b}}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^4 \left(a+b x^3\right)}",1,"((-45*a^(8/3)*c)/x^8 - (72*a^(5/3)*(-2*b*c + a*d))/x^5 - (180*a^(2/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^2 + (120*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-11*b^3*c + 8*a*b^2*d - 5*a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(11*b^3*c - 8*a*b^2*d + 5*a^2*b*e - 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(360*a^(14/3))","A",1
273,1,319,334,0.2056489,"\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^2),x]","\frac{-\frac{180 a^{7/3} (a d-2 b c)}{x^7}-\frac{126 a^{10/3} c}{x^{10}}-\frac{315 a^{4/3} \left(a^2 e-2 a b d+3 b^2 c\right)}{x^4}-\frac{420 \sqrt[3]{a} b x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}-\frac{1260 \sqrt[3]{a} \left(a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right)}{x}+140 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(4 a^3 f-7 a^2 b e+10 a b^2 d-13 b^3 c\right)-140 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right)+70 \sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right)}{1260 a^{16/3}}","\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{10 a^2 x^{10}}-\frac{a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right)}{9 a^{16/3}}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right)}{3 \sqrt{3} a^{16/3}}+\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-4 a^3 f+7 a^2 b e-10 a b^2 d+13 b^3 c\right)}{18 a^{16/3}}+\frac{b x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5 \left(a+b x^3\right)}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x}",1,"((-126*a^(10/3)*c)/x^10 - (180*a^(7/3)*(-2*b*c + a*d))/x^7 - (315*a^(4/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^4 - (1260*a^(1/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/x - (420*a^(1/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3) - 140*Sqrt[3]*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-13*b^3*c + 10*a*b^2*d - 7*a^2*b*e + 4*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(13*b^3*c - 10*a*b^2*d + 7*a^2*b*e - 4*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1260*a^(16/3))","A",1
274,1,317,335,0.3254369,"\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^2),x]","\frac{-\frac{495 a^{8/3} (a d-2 b c)}{x^8}-\frac{360 a^{11/3} c}{x^{11}}-\frac{792 a^{5/3} \left(a^2 e-2 a b d+3 b^2 c\right)}{x^5}+440 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)-440 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)-\frac{1320 a^{2/3} b x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{a+b x^3}-\frac{1980 a^{2/3} \left(a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right)}{x^2}+220 b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right)}{3960 a^{17/3}}","\frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)}{18 a^{17/3}}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)}{3 \sqrt{3} a^{17/3}}+\frac{b x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^5 \left(a+b x^3\right)}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{2 a^5 x^2}",1,"((-360*a^(11/3)*c)/x^11 - (495*a^(8/3)*(-2*b*c + a*d))/x^8 - (792*a^(5/3)*(3*b^2*c - 2*a*b*d + a^2*e))/x^5 - (1980*a^(2/3)*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/x^2 - (1320*a^(2/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3) - 440*Sqrt[3]*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 440*b^(2/3)*(14*b^3*c - 11*a*b^2*d + 8*a^2*b*e - 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 220*b^(2/3)*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3960*a^(17/3))","A",1
275,1,370,375,0.4131433,"\int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x]","\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(7 a^3 f-10 a^2 b e+13 a b^2 d-16 b^3 c\right)}{18 a^{19/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right)}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right)}{3 \sqrt{3} a^{19/3}}+\frac{b^2 x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{3 a^6 \left(a+b x^3\right)}+\frac{b \left(2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right)}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4}","\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}-\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right)}{18 a^{19/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right)}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right)}{3 \sqrt{3} a^{19/3}}-\frac{b^2 x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{3 a^6 \left(a+b x^3\right)}-\frac{b \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4}",1,"-1/13*c/(a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*a^4*x^7) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) + (b*(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*a^(19/3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(19/3)) + (b^(4/3)*(-16*b^3*c + 13*a*b^2*d - 10*a^2*b*e + 7*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(19/3))","A",1
276,1,246,266,0.190321,"\int \frac{x^{14} \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^14*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{20 b^3 x^9 \left(6 a^2 f-3 a b e+b^2 d\right)+30 b^2 x^6 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)+60 a b x^3 \left(15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right)-\frac{60 a^3 \left(7 a^3 f-6 a^2 b e+5 a b^2 d-4 b^3 c\right)}{a+b x^3}+60 a^2 \log \left(a+b x^3\right) \left(-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right)+\frac{30 a^4 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+15 b^4 x^{12} (b e-3 a f)+12 b^5 f x^{15}}{180 b^8}","\frac{x^9 \left(6 a^2 f-3 a b e+b^2 d\right)}{9 b^5}+\frac{a^3 \left(-7 a^3 f+6 a^2 b e-5 a b^2 d+4 b^3 c\right)}{3 b^8 \left(a+b x^3\right)}+\frac{a^2 \log \left(a+b x^3\right) \left(-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right)}{3 b^8}-\frac{a x^3 \left(-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right)}{3 b^7}+\frac{x^6 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{6 b^6}-\frac{a^4 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^8 \left(a+b x^3\right)^2}+\frac{x^{12} (b e-3 a f)}{12 b^4}+\frac{f x^{15}}{15 b^3}",1,"(60*a*b*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x^3 + 30*b^2*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^6 + 20*b^3*(b^2*d - 3*a*b*e + 6*a^2*f)*x^9 + 15*b^4*(b*e - 3*a*f)*x^12 + 12*b^5*f*x^15 + (30*a^4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 - (60*a^3*(-4*b^3*c + 5*a*b^2*d - 6*a^2*b*e + 7*a^3*f))/(a + b*x^3) + 60*a^2*(6*b^3*c - 10*a*b^2*d + 15*a^2*b*e - 21*a^3*f)*Log[a + b*x^3])/(180*b^8)","A",1
277,1,208,226,0.2021952,"\int \frac{x^{11} \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{6 b^2 x^6 \left(6 a^2 f-3 a b e+b^2 d\right)+12 b x^3 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)+\frac{12 a^2 \left(6 a^3 f-5 a^2 b e+4 a b^2 d-3 b^3 c\right)}{a+b x^3}+\frac{6 a^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\left(a+b x^3\right)^2}+12 a \log \left(a+b x^3\right) \left(15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right)+4 b^3 x^9 (b e-3 a f)+3 b^4 f x^{12}}{36 b^7}","\frac{x^6 \left(6 a^2 f-3 a b e+b^2 d\right)}{6 b^5}-\frac{a^2 \left(-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right)}{3 b^7 \left(a+b x^3\right)}+\frac{a^3 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^7 \left(a+b x^3\right)^2}-\frac{a \log \left(a+b x^3\right) \left(-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right)}{3 b^7}+\frac{x^3 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{3 b^6}+\frac{x^9 (b e-3 a f)}{9 b^4}+\frac{f x^{12}}{12 b^3}",1,"(12*b*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3 + 6*b^2*(b^2*d - 3*a*b*e + 6*a^2*f)*x^6 + 4*b^3*(b*e - 3*a*f)*x^9 + 3*b^4*f*x^12 + (6*a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a + b*x^3)^2 + (12*a^2*(-3*b^3*c + 4*a*b^2*d - 5*a^2*b*e + 6*a^3*f))/(a + b*x^3) + 12*a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*Log[a + b*x^3])/(36*b^7)","A",1
278,1,170,186,0.1654626,"\int \frac{x^8 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{6 b x^3 \left(6 a^2 f-3 a b e+b^2 d\right)-\frac{6 a \left(5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right)}{a+b x^3}+\frac{3 a^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+6 \log \left(a+b x^3\right) \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)+3 b^2 x^6 (b e-3 a f)+2 b^3 f x^9}{18 b^6}","\frac{x^3 \left(6 a^2 f-3 a b e+b^2 d\right)}{3 b^5}+\frac{a \left(-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right)}{3 b^6 \left(a+b x^3\right)}-\frac{a^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^6 \left(a+b x^3\right)^2}+\frac{\log \left(a+b x^3\right) \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{3 b^6}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3}",1,"(6*b*(b^2*d - 3*a*b*e + 6*a^2*f)*x^3 + 3*b^2*(b*e - 3*a*f)*x^6 + 2*b^3*f*x^9 + (3*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 - (6*a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f))/(a + b*x^3) + 6*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*Log[a + b*x^3])/(18*b^6)","A",1
279,1,145,146,0.102708,"\int \frac{x^5 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{7 a^4 f+a^3 b \left(2 f x^3-5 e\right)+2 \left(a+b x^3\right)^2 \log \left(a+b x^3\right) \left(6 a^2 f-3 a b e+b^2 d\right)+a^2 b^2 \left(3 d-4 e x^3-11 f x^6\right)-a b^3 \left(c-4 x^3 \left(d+e x^3-f x^6\right)\right)+b^4 x^3 \left(-2 c+2 e x^6+f x^9\right)}{6 b^5 \left(a+b x^3\right)^2}","\frac{\log \left(a+b x^3\right) \left(6 a^2 f-3 a b e+b^2 d\right)}{3 b^5}-\frac{-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c}{3 b^5 \left(a+b x^3\right)}+\frac{a \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^5 \left(a+b x^3\right)^2}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3}",1,"(7*a^4*f + a^3*b*(-5*e + 2*f*x^3) + a^2*b^2*(3*d - 4*e*x^3 - 11*f*x^6) + b^4*x^3*(-2*c + 2*e*x^6 + f*x^9) - a*b^3*(c - 4*x^3*(d + e*x^3 - f*x^6)) + 2*(b^2*d - 3*a*b*e + 6*a^2*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^5*(a + b*x^3)^2)","A",1
280,1,105,109,0.0611496,"\int \frac{x^2 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{-5 a^3 f+a^2 b \left(3 e-4 f x^3\right)+a b^2 \left(-d+4 e x^3+4 f x^6\right)+2 \left(a+b x^3\right)^2 (b e-3 a f) \log \left(a+b x^3\right)-b^3 \left(c+2 d x^3-2 f x^9\right)}{6 b^4 \left(a+b x^3\right)^2}","-\frac{3 a^2 f-2 a b e+b^2 d}{3 b^4 \left(a+b x^3\right)}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 b^4 \left(a+b x^3\right)^2}+\frac{(b e-3 a f) \log \left(a+b x^3\right)}{3 b^4}+\frac{f x^3}{3 b^3}",1,"(-5*a^3*f + a^2*b*(3*e - 4*f*x^3) + a*b^2*(-d + 4*e*x^3 + 4*f*x^6) - b^3*(c + 2*d*x^3 - 2*f*x^9) + 2*(b*e - 3*a*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^4*(a + b*x^3)^2)","A",1
281,1,104,114,0.1263251,"\int \frac{c+d x^3+e x^6+f x^9}{x \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^3),x]","\frac{\frac{2 \left(a^3 f-b^3 c\right) \log \left(a+b x^3\right)+\frac{a \left(3 a^4 f-a^3 b \left(e-4 f x^3\right)-a^2 b^2 \left(d+2 e x^3\right)+3 a b^3 c+2 b^4 c x^3\right)}{\left(a+b x^3\right)^2}}{b^3}+6 c \log (x)}{6 a^3}","-\frac{1}{3} \left(\frac{c}{a^3}-\frac{f}{b^3}\right) \log \left(a+b x^3\right)+\frac{c \log (x)}{a^3}+\frac{2 a^3 f-a^2 b e+b^3 c}{3 a^2 b^3 \left(a+b x^3\right)}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a b^3 \left(a+b x^3\right)^2}",1,"(6*c*Log[x] + ((a*(3*a*b^3*c + 3*a^4*f + 2*b^4*c*x^3 - a^2*b^2*(d + 2*e*x^3) - a^3*b*(e - 4*f*x^3)))/(a + b*x^3)^2 + 2*(-(b^3*c) + a^3*f)*Log[a + b*x^3])/b^3)/(6*a^3)","A",1
282,1,121,134,0.1086812,"\int \frac{c+d x^3+e x^6+f x^9}{x^4 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)^3),x]","\frac{-\frac{2 a \left(a^3 f-a b^2 d+2 b^3 c\right)}{b^2 \left(a+b x^3\right)}+\frac{a^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^2 \left(a+b x^3\right)^2}+2 (3 b c-a d) \log \left(a+b x^3\right)+6 \log (x) (a d-3 b c)-\frac{2 a c}{x^3}}{6 a^4}","\frac{(3 b c-a d) \log \left(a+b x^3\right)}{3 a^4}-\frac{\log (x) (3 b c-a d)}{a^4}-\frac{a^3 f-a b^2 d+2 b^3 c}{3 a^3 b^2 \left(a+b x^3\right)}-\frac{c}{3 a^3 x^3}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^2 b^2 \left(a+b x^3\right)^2}",1,"((-2*a*c)/x^3 + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(b^2*(a + b*x^3)^2) - (2*a*(2*b^3*c - a*b^2*d + a^3*f))/(b^2*(a + b*x^3)) + 6*(-3*b*c + a*d)*Log[x] + 2*(3*b*c - a*d)*Log[a + b*x^3])/(6*a^4)","A",1
283,1,149,163,0.1325968,"\int \frac{c+d x^3+e x^6+f x^9}{x^7 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^3),x]","\frac{\frac{2 a \left(a^2 e-2 a b d+3 b^2 c\right)}{a+b x^3}-2 \log \left(a+b x^3\right) \left(a^2 e-3 a b d+6 b^2 c\right)+6 \log (x) \left(a^2 e-3 a b d+6 b^2 c\right)-\frac{a^2 c}{x^6}+\frac{a^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{b \left(a+b x^3\right)^2}-\frac{2 a (a d-3 b c)}{x^3}}{6 a^5}","\frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6}-\frac{\log \left(a+b x^3\right) \left(a^2 e-3 a b d+6 b^2 c\right)}{3 a^5}+\frac{\log (x) \left(a^2 e-3 a b d+6 b^2 c\right)}{a^5}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left(a+b x^3\right)}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^3 b \left(a+b x^3\right)^2}",1,"(-((a^2*c)/x^6) - (2*a*(-3*b*c + a*d))/x^3 + (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(b*(a + b*x^3)^2) + (2*a*(3*b^2*c - 2*a*b*d + a^2*e))/(a + b*x^3) + 6*(6*b^2*c - 3*a*b*d + a^2*e)*Log[x] - 2*(6*b^2*c - 3*a*b*d + a^2*e)*Log[a + b*x^3])/(6*a^5)","A",1
284,1,200,218,0.1659447,"\int \frac{c+d x^3+e x^6+f x^9}{x^{10} \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^3),x]","\frac{-\frac{2 a^3 c}{x^9}-\frac{6 a \left(a^2 e-3 a b d+6 b^2 c\right)}{x^3}-\frac{3 a^2 (a d-3 b c)}{x^6}+\frac{3 a^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{6 a \left(a^3 f-2 a^2 b e+3 a b^2 d-4 b^3 c\right)}{a+b x^3}+6 \log \left(a+b x^3\right) \left(a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right)+18 \log (x) \left(a^3 f-3 a^2 b e+6 a b^2 d-10 b^3 c\right)}{18 a^6}","\frac{3 b c-a d}{6 a^4 x^6}-\frac{c}{9 a^3 x^9}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{\log \left(a+b x^3\right) \left(a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right)}{3 a^6}-\frac{\log (x) \left(a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right)}{a^6}-\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 \left(a+b x^3\right)}-\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 \left(a+b x^3\right)^2}",1,"((-2*a^3*c)/x^9 - (3*a^2*(-3*b*c + a*d))/x^6 - (6*a*(6*b^2*c - 3*a*b*d + a^2*e))/x^3 + (3*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 + (6*a*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/(a + b*x^3) + 18*(-10*b^3*c + 6*a*b^2*d - 3*a^2*b*e + a^3*f)*Log[x] + 6*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])/(18*a^6)","A",1
285,1,238,258,0.2671515,"\int \frac{c+d x^3+e x^6+f x^9}{x^{13} \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)^3),x]","\frac{12 b \log \left(a+b x^3\right) \left(3 a^3 f-6 a^2 b e+10 a b^2 d-15 b^3 c\right)+36 b \log (x) \left(-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right)-\frac{a \left(a^5 \left(3 c+4 d x^3+6 e x^6+12 f x^9\right)-2 a^4 b x^3 \left(3 c+5 d x^3+12 e x^6-27 f x^9\right)+a^3 b^2 x^6 \left(15 c+40 d x^3-108 e x^6+36 f x^9\right)-12 a^2 b^3 x^9 \left(5 c-15 d x^3+6 e x^6\right)+30 a b^4 x^{12} \left(4 d x^3-9 c\right)-180 b^5 c x^{15}\right)}{x^{12} \left(a+b x^3\right)^2}}{36 a^7}","\frac{3 b c-a d}{9 a^4 x^9}-\frac{c}{12 a^3 x^{12}}-\frac{a^2 e-3 a b d+6 b^2 c}{6 a^5 x^6}-\frac{b \log \left(a+b x^3\right) \left(-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right)}{3 a^7}+\frac{b \log (x) \left(-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right)}{a^7}+\frac{b \left(-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right)}{3 a^6 \left(a+b x^3\right)}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}+\frac{b \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^5 \left(a+b x^3\right)^2}",1,"(-((a*(-180*b^5*c*x^15 + 30*a*b^4*x^12*(-9*c + 4*d*x^3) - 12*a^2*b^3*x^9*(5*c - 15*d*x^3 + 6*e*x^6) - 2*a^4*b*x^3*(3*c + 5*d*x^3 + 12*e*x^6 - 27*f*x^9) + a^5*(3*c + 4*d*x^3 + 6*e*x^6 + 12*f*x^9) + a^3*b^2*x^6*(15*c + 40*d*x^3 - 108*e*x^6 + 36*f*x^9)))/(x^12*(a + b*x^3)^2)) + 36*b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*Log[x] + 12*b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f)*Log[a + b*x^3])/(36*a^7)","A",1
286,1,411,416,0.6877377,"\int \frac{x^{12} \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^12*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{x^7 \left(6 a^2 f-3 a b e+b^2 d\right)}{7 b^5}+\frac{a^2 x \left(37 a^3 f-31 a^2 b e+25 a b^2 d-19 b^3 c\right)}{18 b^7 \left(a+b x^3\right)}+\frac{a^3 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^7 \left(a+b x^3\right)^2}+\frac{a x \left(15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right)}{b^7}+\frac{x^4 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{4 b^6}+\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right)}{54 b^{22/3}}-\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right)}{27 b^{22/3}}+\frac{a^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(152 a^3 f-104 a^2 b e+65 a b^2 d-35 b^3 c\right)}{9 \sqrt{3} b^{22/3}}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3}","\frac{x^7 \left(6 a^2 f-3 a b e+b^2 d\right)}{7 b^5}-\frac{a^2 x \left(-37 a^3 f+31 a^2 b e-25 a b^2 d+19 b^3 c\right)}{18 b^7 \left(a+b x^3\right)}+\frac{a^3 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^7 \left(a+b x^3\right)^2}-\frac{a x \left(-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right)}{b^7}+\frac{x^4 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{4 b^6}-\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right)}{54 b^{22/3}}+\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right)}{27 b^{22/3}}-\frac{a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-152 a^3 f+104 a^2 b e-65 a b^2 d+35 b^3 c\right)}{9 \sqrt{3} b^{22/3}}+\frac{x^{10} (b e-3 a f)}{10 b^4}+\frac{f x^{13}}{13 b^3}",1,"(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^4)/(4*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^7)/(7*b^5) + ((b*e - 3*a*f)*x^10)/(10*b^4) + (f*x^13)/(13*b^3) + (a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*b^7*(a + b*x^3)^2) + (a^2*(-19*b^3*c + 25*a*b^2*d - 31*a^2*b*e + 37*a^3*f)*x)/(18*b^7*(a + b*x^3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(22/3)) - (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(22/3)) + (a^(4/3)*(-35*b^3*c + 65*a*b^2*d - 104*a^2*b*e + 152*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(22/3))","A",1
287,1,380,384,0.5540693,"\int \frac{x^{10} \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^10*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{x^5 \left(6 a^2 f-3 a b e+b^2 d\right)}{5 b^5}+\frac{x^2 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{2 b^6}+\frac{a x^2 \left(-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right)}{9 b^6 \left(a+b x^3\right)}+\frac{a^2 x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 b^6 \left(a+b x^3\right)^2}+\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right)}{54 b^{20/3}}-\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right)}{27 b^{20/3}}-\frac{a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right)}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3}","\frac{x^5 \left(6 a^2 f-3 a b e+b^2 d\right)}{5 b^5}+\frac{x^2 \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{2 b^6}+\frac{a x^2 \left(-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right)}{9 b^6 \left(a+b x^3\right)}-\frac{a^2 x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^6 \left(a+b x^3\right)^2}-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right)}{54 b^{20/3}}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right)}{27 b^{20/3}}+\frac{a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right)}{9 \sqrt{3} b^{20/3}}+\frac{x^8 (b e-3 a f)}{8 b^4}+\frac{f x^{11}}{11 b^3}",1,"((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^2)/(2*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^8)/(8*b^4) + (f*x^11)/(11*b^3) + (a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*b^6*(a + b*x^3)^2) + (a*(7*b^3*c - 10*a*b^2*d + 13*a^2*b*e - 16*a^3*f)*x^2)/(9*b^6*(a + b*x^3)) - (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(20/3)) - (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(20/3)) + (a^(2/3)*(-20*b^3*c + 44*a*b^2*d - 77*a^2*b*e + 119*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(20/3))","A",1
288,1,362,375,0.4680425,"\int \frac{x^9 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^9*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{945 b^{4/3} x^4 \left(6 a^2 f-3 a b e+b^2 d\right)+\frac{210 a \sqrt[3]{b} x \left(-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right)}{a+b x^3}+\frac{630 a^2 \sqrt[3]{b} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+3780 \sqrt[3]{b} x \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)+140 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right)-140 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right)-70 \sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(104 a^3 f-65 a^2 b e+35 a b^2 d-14 b^3 c\right)+540 b^{7/3} x^7 (b e-3 a f)+378 b^{10/3} f x^{10}}{3780 b^{19/3}}","\frac{x^4 \left(6 a^2 f-3 a b e+b^2 d\right)}{4 b^5}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right)}{27 b^{19/3}}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right)}{9 \sqrt{3} b^{19/3}}+\frac{a x \left(-31 a^3 f+25 a^2 b e-19 a b^2 d+13 b^3 c\right)}{18 b^6 \left(a+b x^3\right)}-\frac{a^2 x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^6 \left(a+b x^3\right)^2}+\frac{x \left(-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right)}{b^6}+\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-104 a^3 f+65 a^2 b e-35 a b^2 d+14 b^3 c\right)}{54 b^{19/3}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3}",1,"(3780*b^(1/3)*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x + 945*b^(4/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^4 + 540*b^(7/3)*(b*e - 3*a*f)*x^7 + 378*b^(10/3)*f*x^10 + (630*a^2*b^(1/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3)^2 + (210*a*b^(1/3)*(13*b^3*c - 19*a*b^2*d + 25*a^2*b*e - 31*a^3*f)*x)/(a + b*x^3) - 140*Sqrt[3]*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(1/3) + b^(1/3)*x] - 70*a^(1/3)*(-14*b^3*c + 35*a*b^2*d - 65*a^2*b*e + 104*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*b^(19/3))","A",1
289,1,329,345,0.365139,"\int \frac{x^7 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{540 b^{2/3} x^2 \left(6 a^2 f-3 a b e+b^2 d\right)+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(77 a^3 f-44 a^2 b e+20 a b^2 d-5 b^3 c\right)}{\sqrt[3]{a}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(77 a^3 f-44 a^2 b e+20 a b^2 d-5 b^3 c\right)}{\sqrt[3]{a}}-\frac{120 b^{2/3} x^2 \left(-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right)}{a+b x^3}+\frac{180 a b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right)}{\sqrt[3]{a}}+216 b^{5/3} x^5 (b e-3 a f)+135 b^{8/3} f x^8}{1080 b^{17/3}}","\frac{x^2 \left(6 a^2 f-3 a b e+b^2 d\right)}{2 b^5}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right)}{27 \sqrt[3]{a} b^{17/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right)}{9 \sqrt{3} \sqrt[3]{a} b^{17/3}}-\frac{x^2 \left(-13 a^3 f+10 a^2 b e-7 a b^2 d+4 b^3 c\right)}{9 b^5 \left(a+b x^3\right)}+\frac{a x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^5 \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-77 a^3 f+44 a^2 b e-20 a b^2 d+5 b^3 c\right)}{54 \sqrt[3]{a} b^{17/3}}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^8}{8 b^3}",1,"(540*b^(2/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x^2 + 216*b^(5/3)*(b*e - 3*a*f)*x^5 + 135*b^(8/3)*f*x^8 + (180*a*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3)^2 - (120*b^(2/3)*(4*b^3*c - 7*a*b^2*d + 10*a^2*b*e - 13*a^3*f)*x^2)/(a + b*x^3) + (40*Sqrt[3]*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (40*(-5*b^3*c + 20*a*b^2*d - 44*a^2*b*e + 77*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (20*(5*b^3*c - 20*a*b^2*d + 44*a^2*b*e - 77*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(1080*b^(17/3))","A",1
290,1,323,336,0.3675668,"\int \frac{x^6 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{756 \sqrt[3]{b} x \left(6 a^2 f-3 a b e+b^2 d\right)-\frac{42 \sqrt[3]{b} x \left(-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right)}{a+b x^3}+\frac{126 a \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right)}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right)}{a^{2/3}}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(65 a^3 f-35 a^2 b e+14 a b^2 d-2 b^3 c\right)}{a^{2/3}}+189 b^{4/3} x^4 (b e-3 a f)+108 b^{7/3} f x^7}{756 b^{16/3}}","\frac{x \left(6 a^2 f-3 a b e+b^2 d\right)}{b^5}-\frac{x \left(-25 a^3 f+19 a^2 b e-13 a b^2 d+7 b^3 c\right)}{18 b^5 \left(a+b x^3\right)}+\frac{a x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^5 \left(a+b x^3\right)^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right)}{54 a^{2/3} b^{16/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right)}{27 a^{2/3} b^{16/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-65 a^3 f+35 a^2 b e-14 a b^2 d+2 b^3 c\right)}{9 \sqrt{3} a^{2/3} b^{16/3}}+\frac{x^4 (b e-3 a f)}{4 b^4}+\frac{f x^7}{7 b^3}",1,"(756*b^(1/3)*(b^2*d - 3*a*b*e + 6*a^2*f)*x + 189*b^(4/3)*(b*e - 3*a*f)*x^4 + 108*b^(7/3)*f*x^7 + (126*a*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 - (42*b^(1/3)*(7*b^3*c - 13*a*b^2*d + 19*a^2*b*e - 25*a^3*f)*x)/(a + b*x^3) + (28*Sqrt[3]*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (28*(2*b^3*c - 14*a*b^2*d + 35*a^2*b*e - 65*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-2*b^3*c + 14*a*b^2*d - 35*a^2*b*e + 65*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(756*b^(16/3))","A",1
291,1,300,316,0.2916706,"\int \frac{x^4 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{\frac{30 b^{2/3} x^2 \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)}-\frac{45 b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\left(a+b x^3\right)^2}-\frac{10 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{a^{4/3}}+\frac{5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{a^{4/3}}+135 b^{2/3} x^2 (b e-3 a f)+54 b^{5/3} f x^5}{270 b^{14/3}}","\frac{x^2 \left(-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right)}{9 a b^4 \left(a+b x^3\right)}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^4 \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{54 a^{4/3} b^{14/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{27 a^{4/3} b^{14/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right)}{9 \sqrt{3} a^{4/3} b^{14/3}}+\frac{x^2 (b e-3 a f)}{2 b^4}+\frac{f x^5}{5 b^3}",1,"(135*b^(2/3)*(b*e - 3*a*f)*x^2 + 54*b^(5/3)*f*x^5 - (45*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a + b*x^3)^2 + (30*b^(2/3)*(b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (10*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(270*b^(14/3))","A",1
292,1,294,307,0.3076296,"\int \frac{x^3 \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{\frac{6 \sqrt[3]{b} x \left(-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)}-\frac{18 \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{5/3}}-\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{a^{5/3}}+108 \sqrt[3]{b} x (b e-3 a f)+27 b^{4/3} f x^4}{108 b^{13/3}}","\frac{x \left(-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right)}{18 a b^4 \left(a+b x^3\right)}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 b^4 \left(a+b x^3\right)^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{54 a^{5/3} b^{13/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{27 a^{5/3} b^{13/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right)}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^4}{4 b^3}",1,"(108*b^(1/3)*(b*e - 3*a*f)*x + 27*b^(4/3)*f*x^4 - (18*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 + (6*b^(1/3)*(b^3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(108*b^(13/3))","A",1
293,1,284,301,0.3214837,"\int \frac{x \left(c+d x^3+e x^6+f x^9\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]","\frac{\frac{6 b^{2/3} x^2 \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{a^2 \left(a+b x^3\right)}+\frac{9 b^{2/3} x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)^2}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{7/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{7/3}}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{a^{7/3}}+27 b^{2/3} f x^2}{54 b^{11/3}}","\frac{x^2 \left(7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right)}{9 a^2 b^3 \left(a+b x^3\right)}+\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a b^3 \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{54 a^{7/3} b^{11/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right)}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3}",1,"(27*b^(2/3)*f*x^2 + (9*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a + b*x^3)^2) + (6*b^(2/3)*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(a^2*(a + b*x^3)) - (2*Sqrt[3]*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (2*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) + ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3))/(54*b^(11/3))","A",1
294,1,279,292,0.2354098,"\int \frac{c+d x^3+e x^6+f x^9}{\left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(a + b*x^3)^3,x]","\frac{\frac{3 \sqrt[3]{b} x \left(13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right)}{a^2 \left(a+b x^3\right)}+\frac{9 \sqrt[3]{b} x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{a \left(a+b x^3\right)^2}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{a^{8/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{a^{8/3}}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{a^{8/3}}+54 \sqrt[3]{b} f x}{54 b^{10/3}}","\frac{x \left(13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right)}{18 a^2 b^3 \left(a+b x^3\right)}+\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a b^3 \left(a+b x^3\right)^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{54 a^{8/3} b^{10/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{27 a^{8/3} b^{10/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right)}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{f x}{b^3}",1,"(54*b^(1/3)*f*x + (9*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^3)^2) + (3*b^(1/3)*(5*b^3*c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(a^2*(a + b*x^3)) - (2*Sqrt[3]*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3))/(54*b^(10/3))","A",1
295,1,286,303,0.293005,"\int \frac{c+d x^3+e x^6+f x^9}{x^2 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^2*(a + b*x^3)^3),x]","\frac{-\frac{6 \sqrt[3]{a} x^2 \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{b^2 \left(a+b x^3\right)}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right)}{b^{8/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right)}{b^{8/3}}+\frac{9 a^{4/3} x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^2 \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^3 f+a^2 b e+2 a b^2 d-14 b^3 c\right)}{b^{8/3}}-\frac{54 \sqrt[3]{a} c}{x}}{54 a^{10/3}}","-\frac{c}{a^3 x}-\frac{x^2 \left(4 a^3 f-a^2 b e-2 a b^2 d+5 b^3 c\right)}{9 a^3 b^2 \left(a+b x^3\right)}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^2 b^2 \left(a+b x^3\right)^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right)}{54 a^{10/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right)}{27 a^{10/3} b^{8/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^3 f-a^2 b e-2 a b^2 d+14 b^3 c\right)}{9 \sqrt{3} a^{10/3} b^{8/3}}",1,"((-54*a^(1/3)*c)/x + (9*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b^2*(a + b*x^3)^2) - (6*a^(1/3)*(5*b^3*c - 2*a*b^2*d - a^2*b*e + 4*a^3*f)*x^2)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(14*b^3*c - 2*a*b^2*d - a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(8/3) - (2*(-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(8/3) + ((-14*b^3*c + 2*a*b^2*d + a^2*b*e + 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(8/3))/(54*a^(10/3))","A",1
296,1,283,301,0.295042,"\int \frac{c+d x^3+e x^6+f x^9}{x^3 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^3*(a + b*x^3)^3),x]","\frac{-\frac{27 a^{2/3} c}{x^2}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^3 f+a^2 b e+5 a b^2 d-20 b^3 c\right)}{b^{7/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right)}{b^{7/3}}+\frac{9 a^{5/3} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b^2 \left(a+b x^3\right)^2}-\frac{3 a^{2/3} x \left(7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right)}{b^2 \left(a+b x^3\right)}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^3 f+a^2 b e+5 a b^2 d-20 b^3 c\right)}{b^{7/3}}}{54 a^{11/3}}","-\frac{c}{2 a^3 x^2}-\frac{x \left(7 a^3 f-a^2 b e-5 a b^2 d+11 b^3 c\right)}{18 a^3 b^2 \left(a+b x^3\right)}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^2 b^2 \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right)}{54 a^{11/3} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right)}{27 a^{11/3} b^{7/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^3 f-a^2 b e-5 a b^2 d+20 b^3 c\right)}{9 \sqrt{3} a^{11/3} b^{7/3}}",1,"((-27*a^(2/3)*c)/x^2 + (9*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b^2*(a + b*x^3)^2) - (3*a^(2/3)*(11*b^3*c - 5*a*b^2*d - a^2*b*e + 7*a^3*f)*x)/(b^2*(a + b*x^3)) + (2*Sqrt[3]*(20*b^3*c - 5*a*b^2*d - a^2*b*e - 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(7/3) + (2*(-20*b^3*c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(7/3) - ((-20*b^3*c + 5*a*b^2*d + a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(7/3))/(54*a^(11/3))","A",1
297,1,303,317,0.3202476,"\int \frac{c+d x^3+e x^6+f x^9}{x^5 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^5*(a + b*x^3)^3),x]","\frac{-\frac{27 a^{4/3} c}{x^4}+\frac{12 \sqrt[3]{a} x^2 \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{b \left(a+b x^3\right)}-\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{b^{5/3}}-\frac{18 a^{4/3} x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b \left(a+b x^3\right)^2}+\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{b^{5/3}}-\frac{108 \sqrt[3]{a} (a d-3 b c)}{x}}{108 a^{13/3}}","\frac{3 b c-a d}{a^4 x}-\frac{c}{4 a^3 x^4}+\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^3 b \left(a+b x^3\right)^2}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{54 a^{13/3} b^{5/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{27 a^{13/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 f+2 a^2 b e-14 a b^2 d+35 b^3 c\right)}{9 \sqrt{3} a^{13/3} b^{5/3}}+\frac{x^2 \left(a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right)}{9 a^4 b \left(a+b x^3\right)}",1,"((-27*a^(4/3)*c)/x^4 - (108*a^(1/3)*(-3*b*c + a*d))/x - (18*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)^2) + (12*a^(1/3)*(8*b^3*c - 5*a*b^2*d + 2*a^2*b*e + a^3*f)*x^2)/(b*(a + b*x^3)) - (4*Sqrt[3]*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) - (4*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*(35*b^3*c - 14*a*b^2*d + 2*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3))/(108*a^(13/3))","A",1
298,1,299,316,0.2754709,"\int \frac{c+d x^3+e x^6+f x^9}{x^6 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^3),x]","\frac{-\frac{135 a^{2/3} (a d-3 b c)}{x^2}-\frac{54 a^{5/3} c}{x^5}+\frac{10 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{b^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{b^{4/3}}-\frac{45 a^{5/3} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{b \left(a+b x^3\right)^2}+\frac{15 a^{2/3} x \left(a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right)}{b \left(a+b x^3\right)}-\frac{5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{b^{4/3}}}{270 a^{14/3}}","\frac{3 b c-a d}{2 a^4 x^2}-\frac{c}{5 a^3 x^5}+\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^3 b \left(a+b x^3\right)^2}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{54 a^{14/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{27 a^{14/3} b^{4/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right)}{9 \sqrt{3} a^{14/3} b^{4/3}}+\frac{x \left(a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right)}{18 a^4 b \left(a+b x^3\right)}",1,"((-54*a^(5/3)*c)/x^5 - (135*a^(2/3)*(-3*b*c + a*d))/x^2 - (45*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)^2) + (15*a^(2/3)*(17*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) + (10*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) - (5*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3))/(270*a^(14/3))","A",1
299,1,328,343,0.3116678,"\int \frac{c+d x^3+e x^6+f x^9}{x^8 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^8*(a + b*x^3)^3),x]","\frac{-\frac{189 a^{4/3} (a d-3 b c)}{x^4}-\frac{108 a^{7/3} c}{x^7}-\frac{756 \sqrt[3]{a} \left(a^2 e-3 a b d+6 b^2 c\right)}{x}+\frac{84 \sqrt[3]{a} x^2 \left(2 a^3 f-5 a^2 b e+8 a b^2 d-11 b^3 c\right)}{a+b x^3}+\frac{28 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right)}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right)}{b^{2/3}}+\frac{126 a^{4/3} x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{14 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^3 f-14 a^2 b e+35 a b^2 d-65 b^3 c\right)}{b^{2/3}}}{756 a^{16/3}}","\frac{3 b c-a d}{4 a^4 x^4}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right)}{54 a^{16/3} b^{2/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right)}{27 a^{16/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^3 f+14 a^2 b e-35 a b^2 d+65 b^3 c\right)}{9 \sqrt{3} a^{16/3} b^{2/3}}-\frac{x^2 \left(-2 a^3 f+5 a^2 b e-8 a b^2 d+11 b^3 c\right)}{9 a^5 \left(a+b x^3\right)}-\frac{x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^4 \left(a+b x^3\right)^2}",1,"((-108*a^(7/3)*c)/x^7 - (189*a^(4/3)*(-3*b*c + a*d))/x^4 - (756*a^(1/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x + (126*a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3)^2 + (84*a^(1/3)*(-11*b^3*c + 8*a*b^2*d - 5*a^2*b*e + 2*a^3*f)*x^2)/(a + b*x^3) + (28*Sqrt[3]*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(65*b^3*c - 35*a*b^2*d + 14*a^2*b*e - 2*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-65*b^3*c + 35*a*b^2*d - 14*a^2*b*e + 2*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(756*a^(16/3))","A",1
300,1,324,341,0.3455973,"\int \frac{c+d x^3+e x^6+f x^9}{x^9 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^9*(a + b*x^3)^3),x]","\frac{-\frac{216 a^{5/3} (a d-3 b c)}{x^5}-\frac{135 a^{8/3} c}{x^8}-\frac{540 a^{2/3} \left(a^2 e-3 a b d+6 b^2 c\right)}{x^2}+\frac{40 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^3 f-20 a^2 b e+44 a b^2 d-77 b^3 c\right)}{\sqrt[3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right)}{\sqrt[3]{b}}+\frac{180 a^{5/3} x \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+\frac{60 a^{2/3} x \left(5 a^3 f-11 a^2 b e+17 a b^2 d-23 b^3 c\right)}{a+b x^3}+\frac{20 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right)}{\sqrt[3]{b}}}{1080 a^{17/3}}","\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{8 a^3 x^8}-\frac{a^2 e-3 a b d+6 b^2 c}{2 a^5 x^2}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right)}{27 a^{17/3} \sqrt[3]{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right)}{9 \sqrt{3} a^{17/3} \sqrt[3]{b}}+\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-5 a^3 f+20 a^2 b e-44 a b^2 d+77 b^3 c\right)}{54 a^{17/3} \sqrt[3]{b}}-\frac{x \left(-5 a^3 f+11 a^2 b e-17 a b^2 d+23 b^3 c\right)}{18 a^5 \left(a+b x^3\right)}-\frac{x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^4 \left(a+b x^3\right)^2}",1,"((-135*a^(8/3)*c)/x^8 - (216*a^(5/3)*(-3*b*c + a*d))/x^5 - (540*a^(2/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x^2 + (180*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(a + b*x^3)^2 + (60*a^(2/3)*(-23*b^3*c + 17*a*b^2*d - 11*a^2*b*e + 5*a^3*f)*x)/(a + b*x^3) + (40*Sqrt[3]*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-77*b^3*c + 44*a*b^2*d - 20*a^2*b*e + 5*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(77*b^3*c - 44*a*b^2*d + 20*a^2*b*e - 5*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(1080*a^(17/3))","A",1
301,1,366,381,0.5882519,"\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)^3),x]","\frac{-\frac{540 a^{7/3} (a d-3 b c)}{x^7}-\frac{378 a^{10/3} c}{x^{10}}-\frac{945 a^{4/3} \left(a^2 e-3 a b d+6 b^2 c\right)}{x^4}-\frac{420 \sqrt[3]{a} b x^2 \left(5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right)}{a+b x^3}-\frac{3780 \sqrt[3]{a} \left(a^3 f-3 a^2 b e+6 a b^2 d-10 b^3 c\right)}{x}+140 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(14 a^3 f-35 a^2 b e+65 a b^2 d-104 b^3 c\right)-140 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right)-\frac{630 a^{4/3} b x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{\left(a+b x^3\right)^2}+70 \sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right)}{3780 a^{19/3}}","\frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{10 a^3 x^{10}}-\frac{a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right)}{27 a^{19/3}}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right)}{9 \sqrt{3} a^{19/3}}+\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right)}{54 a^{19/3}}+\frac{b x^2 \left(-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right)}{9 a^6 \left(a+b x^3\right)}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^5 \left(a+b x^3\right)^2}",1,"((-378*a^(10/3)*c)/x^10 - (540*a^(7/3)*(-3*b*c + a*d))/x^7 - (945*a^(4/3)*(6*b^2*c - 3*a*b*d + a^2*e))/x^4 - (3780*a^(1/3)*(-10*b^3*c + 6*a*b^2*d - 3*a^2*b*e + a^3*f))/x - (630*a^(4/3)*b*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(a + b*x^3)^2 - (420*a^(1/3)*b*(-14*b^3*c + 11*a*b^2*d - 8*a^2*b*e + 5*a^3*f)*x^2)/(a + b*x^3) - 140*Sqrt[3]*b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 140*b^(1/3)*(-104*b^3*c + 65*a*b^2*d - 35*a^2*b*e + 14*a^3*f)*Log[a^(1/3) + b^(1/3)*x] + 70*b^(1/3)*(104*b^3*c - 65*a*b^2*d + 35*a^2*b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3780*a^(19/3))","A",1
302,1,376,380,0.5765833,"\int \frac{c+d x^3+e x^6+f x^9}{x^{12} \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^12*(a + b*x^3)^3),x]","\frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right)}{54 a^{20/3}}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right)}{27 a^{20/3}}+\frac{b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right)}{9 \sqrt{3} a^{20/3}}+\frac{b x \left(-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right)}{18 a^6 \left(a+b x^3\right)}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^5 \left(a+b x^3\right)^2}","\frac{3 b c-a d}{8 a^4 x^8}-\frac{c}{11 a^3 x^{11}}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right)}{54 a^{20/3}}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right)}{27 a^{20/3}}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right)}{9 \sqrt{3} a^{20/3}}+\frac{b x \left(-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right)}{18 a^6 \left(a+b x^3\right)}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac{b x \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^5 \left(a+b x^3\right)^2}",1,"-1/11*c/(a^3*x^11) + (3*b*c - a*d)/(8*a^4*x^8) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(2*a^6*x^2) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^5*(a + b*x^3)^2) + (b*(29*b^3*c - 23*a*b^2*d + 17*a^2*b*e - 11*a^3*f)*x)/(18*a^6*(a + b*x^3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d - 44*a^2*b*e + 20*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(20/3)) + (b^(2/3)*(119*b^3*c - 77*a*b^2*d + 44*a^2*b*e - 20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(20/3)) + (b^(2/3)*(-119*b^3*c + 77*a*b^2*d - 44*a^2*b*e + 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(20/3))","A",1
303,1,419,424,0.6663264,"\int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^3),x]","\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(35 a^3 f-65 a^2 b e+104 a b^2 d-152 b^3 c\right)}{54 a^{22/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right)}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right)}{9 \sqrt{3} a^{22/3}}+\frac{b^2 x^2 \left(8 a^3 f-11 a^2 b e+14 a b^2 d-17 b^3 c\right)}{9 a^7 \left(a+b x^3\right)}+\frac{b \left(3 a^3 f-6 a^2 b e+10 a b^2 d-15 b^3 c\right)}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}+\frac{b^2 x^2 \left(a^3 f-a^2 b e+a b^2 d-b^3 c\right)}{6 a^6 \left(a+b x^3\right)^2}","\frac{3 b c-a d}{10 a^4 x^{10}}-\frac{c}{13 a^3 x^{13}}-\frac{a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}-\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right)}{54 a^{22/3}}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right)}{27 a^{22/3}}+\frac{b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right)}{9 \sqrt{3} a^{22/3}}-\frac{b^2 x^2 \left(-8 a^3 f+11 a^2 b e-14 a b^2 d+17 b^3 c\right)}{9 a^7 \left(a+b x^3\right)}-\frac{b \left(-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right)}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac{b^2 x^2 \left(a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right)}{6 a^6 \left(a+b x^3\right)^2}",1,"-1/13*c/(a^3*x^13) + (3*b*c - a*d)/(10*a^4*x^10) - (6*b^2*c - 3*a*b*d + a^2*e)/(7*a^5*x^7) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(4*a^6*x^4) + (b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(6*a^6*(a + b*x^3)^2) + (b^2*(-17*b^3*c + 14*a*b^2*d - 11*a^2*b*e + 8*a^3*f)*x^2)/(9*a^7*(a + b*x^3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(9*Sqrt[3]*a^(22/3)) + (b^(4/3)*(152*b^3*c - 104*a*b^2*d + 65*a^2*b*e - 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(22/3)) + (b^(4/3)*(-152*b^3*c + 104*a*b^2*d - 65*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(22/3))","A",1
304,1,59,54,0.015224,"\int \frac{(1-x) x^4}{1+x^3} \, dx","Integrate[((1 - x)*x^4)/(1 + x^3),x]","\frac{1}{6} \left(-2 x^3+2 \log \left(x^3+1\right)+3 x^2-\log \left(x^2-x+1\right)+2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{x^3}{3}+\frac{x^2}{2}+\frac{1}{6} \log \left(x^2-x+1\right)+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(3*x^2 - 2*x^3 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - Log[1 - x + x^2] + 2*Log[1 + x^3])/6","A",1
305,1,30,30,0.0072962,"\int \frac{(1-x) x^3}{1+x^3} \, dx","Integrate[((1 - x)*x^3)/(1 + x^3),x]","-\frac{x^2}{2}+\frac{1}{3} \log \left(x^2-x+1\right)+x-\frac{2}{3} \log (x+1)","-\frac{x^2}{2}+\frac{1}{3} \log \left(x^2-x+1\right)+x-\frac{2}{3} \log (x+1)",1,"x - x^2/2 - (2*Log[1 + x])/3 + Log[1 - x + x^2]/3","A",1
306,1,53,44,0.0105313,"\int \frac{(1-x) x^2}{1+x^3} \, dx","Integrate[((1 - x)*x^2)/(1 + x^3),x]","\frac{1}{3} \log \left(x^3+1\right)-\frac{1}{6} \log \left(x^2-x+1\right)-x+\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{6} \log \left(x^2-x+1\right)-x+\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-x + ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3] + Log[1 + x]/3 - Log[1 - x + x^2]/6 + Log[1 + x^3]/3","A",1
307,1,50,41,0.0088994,"\int \frac{(1-x) x}{1+x^3} \, dx","Integrate[((1 - x)*x)/(1 + x^3),x]","-\frac{1}{3} \log \left(x^3+1\right)+\frac{1}{6} \log \left(x^2-x+1\right)-\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{6} \log \left(x^2-x+1\right)-\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3] - Log[1 + x]/3 + Log[1 - x + x^2]/6 - Log[1 + x^3]/3","A",1
308,1,53,42,0.0090263,"\int \frac{1-x}{x \left(1+x^3\right)} \, dx","Integrate[(1 - x)/(x*(1 + x^3)),x]","-\frac{1}{3} \log \left(x^3+1\right)+\frac{1}{6} \log \left(x^2-x+1\right)+\log (x)-\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{6} \log \left(x^2-x+1\right)+\log (x)-\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-(ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[x] - Log[1 + x]/3 + Log[1 - x + x^2]/6 - Log[1 + x^3]/3","A",1
309,1,60,49,0.0182563,"\int \frac{1-x}{x^2 \left(1+x^3\right)} \, dx","Integrate[(1 - x)/(x^2*(1 + x^3)),x]","\frac{1}{3} \log \left(x^3+1\right)-\frac{1}{6} \log \left(x^2-x+1\right)-\frac{1}{x}-\log (x)+\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{6} \log \left(x^2-x+1\right)-\frac{1}{x}-\log (x)+\frac{2}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-x^(-1) - ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3] - Log[x] + Log[1 + x]/3 - Log[1 - x + x^2]/6 + Log[1 + x^3]/3","A",1
310,1,32,32,0.0054034,"\int \frac{1-x}{x^3 \left(1+x^3\right)} \, dx","Integrate[(1 - x)/(x^3*(1 + x^3)),x]","-\frac{1}{2 x^2}+\frac{1}{3} \log \left(x^2-x+1\right)+\frac{1}{x}-\frac{2}{3} \log (x+1)","-\frac{1}{2 x^2}+\frac{1}{3} \log \left(x^2-x+1\right)+\frac{1}{x}-\frac{2}{3} \log (x+1)",1,"-1/2*1/x^2 + x^(-1) - (2*Log[1 + x])/3 + Log[1 - x + x^2]/3","A",1
311,1,47,41,0.034673,"\int \frac{x (1+2 x)}{1+x^3} \, dx","Integrate[(x*(1 + 2*x))/(1 + x^3),x]","\frac{1}{6} \left(4 \log \left(x^3+1\right)+\log \left(x^2-x+1\right)-2 \log (x+1)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{5}{6} \log \left(x^2-x+1\right)+\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{\sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + Log[1 - x + x^2] + 4*Log[1 + x^3])/6","A",1
312,1,53,39,0.0203803,"\int \frac{x (1+2 x)}{1-x^3} \, dx","Integrate[(x*(1 + 2*x))/(1 - x^3),x]","-\frac{2}{3} \log \left(1-x^3\right)+\frac{1}{6} \log \left(x^2+x+1\right)-\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{2} \log \left(x^2+x+1\right)-\log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-(ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3]) - Log[1 - x]/3 + Log[1 + x + x^2]/6 - (2*Log[1 - x^3])/3","A",1
313,1,55,55,0.0253823,"\int x^2 \left(c+d x+e x^2\right) \left(a+b x^3\right) \, dx","Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3),x]","\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8","\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8",1,"(a*c*x^3)/3 + (a*d*x^4)/4 + (a*e*x^5)/5 + (b*c*x^6)/6 + (b*d*x^7)/7 + (b*e*x^8)/8","A",1
314,1,55,55,0.0027379,"\int x \left(c+d x+e x^2\right) \left(a+b x^3\right) \, dx","Integrate[x*(c + d*x + e*x^2)*(a + b*x^3),x]","\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{5} b c x^5+\frac{1}{6} b d x^6+\frac{1}{7} b e x^7","\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{5} b c x^5+\frac{1}{6} b d x^6+\frac{1}{7} b e x^7",1,"(a*c*x^2)/2 + (a*d*x^3)/3 + (a*e*x^4)/4 + (b*c*x^5)/5 + (b*d*x^6)/6 + (b*e*x^7)/7","A",1
315,1,50,50,0.0021437,"\int \left(c+d x+e x^2\right) \left(a+b x^3\right) \, dx","Integrate[(c + d*x + e*x^2)*(a + b*x^3),x]","a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6","a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6",1,"a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + (b*c*x^4)/4 + (b*d*x^5)/5 + (b*e*x^6)/6","A",1
316,1,46,46,0.0050063,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)}{x} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3))/x,x]","a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4+\frac{1}{5} b e x^5","a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4+\frac{1}{5} b e x^5",1,"a*d*x + (a*e*x^2)/2 + (b*c*x^3)/3 + (b*d*x^4)/4 + (b*e*x^5)/5 + a*c*Log[x]","A",1
317,1,44,44,0.0059469,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)}{x^2} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3))/x^2,x]","-\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4","-\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4",1,"-((a*c)/x) + a*e*x + (b*c*x^2)/2 + (b*d*x^3)/3 + (b*e*x^4)/4 + a*d*Log[x]","A",1
318,1,44,44,0.0107526,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)}{x^3} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3))/x^3,x]","-\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3","-\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3",1,"-1/2*(a*c)/x^2 - (a*d)/x + b*c*x + (b*d*x^2)/2 + (b*e*x^3)/3 + a*e*Log[x]","A",1
319,1,97,82,0.0043346,"\int x^2 \left(c+d x+e x^2\right) \left(a+b x^3\right)^2 \, dx","Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^2,x]","\frac{1}{3} a^2 c x^3+\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{3} a b c x^6+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}","\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{c \left(a+b x^3\right)^3}{9 b}+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}",1,"(a^2*c*x^3)/3 + (a^2*d*x^4)/4 + (a^2*e*x^5)/5 + (a*b*c*x^6)/3 + (2*a*b*d*x^7)/7 + (a*b*e*x^8)/4 + (b^2*c*x^9)/9 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11","A",1
320,1,97,82,0.0034608,"\int x \left(c+d x+e x^2\right) \left(a+b x^3\right)^2 \, dx","Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^2,x]","\frac{1}{2} a^2 c x^2+\frac{1}{3} a^2 d x^3+\frac{1}{4} a^2 e x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{8} b^2 c x^8+\frac{1}{9} b^2 d x^9+\frac{1}{10} b^2 e x^{10}","\frac{1}{2} a^2 c x^2+\frac{1}{4} a^2 e x^4+\frac{2}{5} a b c x^5+\frac{d \left(a+b x^3\right)^3}{9 b}+\frac{2}{7} a b e x^7+\frac{1}{8} b^2 c x^8+\frac{1}{10} b^2 e x^{10}",1,"(a^2*c*x^2)/2 + (a^2*d*x^3)/3 + (a^2*e*x^4)/4 + (2*a*b*c*x^5)/5 + (a*b*d*x^6)/3 + (2*a*b*e*x^7)/7 + (b^2*c*x^8)/8 + (b^2*d*x^9)/9 + (b^2*e*x^10)/10","A",1
321,1,92,77,0.0040876,"\int \left(c+d x+e x^2\right) \left(a+b x^3\right)^2 \, dx","Integrate[(c + d*x + e*x^2)*(a + b*x^3)^2,x]","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{3} a b e x^6+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8+\frac{1}{9} b^2 e x^9","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{e \left(a+b x^3\right)^3}{9 b}+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8",1,"a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (a*b*e*x^6)/3 + (b^2*c*x^7)/7 + (b^2*d*x^8)/8 + (b^2*e*x^9)/9","A",1
322,1,88,88,0.009696,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^2}{x} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^2)/x,x]","a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{2} a b d x^4+\frac{2}{5} a b e x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b^2 d x^7+\frac{1}{8} b^2 e x^8","a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{2} a b d x^4+\frac{2}{5} a b e x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b^2 d x^7+\frac{1}{8} b^2 e x^8",1,"a^2*d*x + (a^2*e*x^2)/2 + (2*a*b*c*x^3)/3 + (a*b*d*x^4)/2 + (2*a*b*e*x^5)/5 + (b^2*c*x^6)/6 + (b^2*d*x^7)/7 + (b^2*e*x^8)/8 + a^2*c*Log[x]","A",1
323,1,83,83,0.0138854,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^2}{x^2} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^2)/x^2,x]","-\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+a b c x^2+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{1}{5} b^2 c x^5+\frac{1}{6} b^2 d x^6+\frac{1}{7} b^2 e x^7","-\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+a b c x^2+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{1}{5} b^2 c x^5+\frac{1}{6} b^2 d x^6+\frac{1}{7} b^2 e x^7",1,"-((a^2*c)/x) + a^2*e*x + a*b*c*x^2 + (2*a*b*d*x^3)/3 + (a*b*e*x^4)/2 + (b^2*c*x^5)/5 + (b^2*d*x^6)/6 + (b^2*e*x^7)/7 + a^2*d*Log[x]","A",1
324,1,84,84,0.0087788,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^2}{x^3} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^2)/x^3,x]","-\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+2 a b c x+a b d x^2+\frac{2}{3} a b e x^3+\frac{1}{4} b^2 c x^4+\frac{1}{5} b^2 d x^5+\frac{1}{6} b^2 e x^6","-\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+2 a b c x+a b d x^2+\frac{2}{3} a b e x^3+\frac{1}{4} b^2 c x^4+\frac{1}{5} b^2 d x^5+\frac{1}{6} b^2 e x^6",1,"-1/2*(a^2*c)/x^2 - (a^2*d)/x + 2*a*b*c*x + a*b*d*x^2 + (2*a*b*e*x^3)/3 + (b^2*c*x^4)/4 + (b^2*d*x^5)/5 + (b^2*e*x^6)/6 + a^2*e*Log[x]","A",1
325,1,139,110,0.0049483,"\int x^2 \left(c+d x+e x^2\right) \left(a+b x^3\right)^3 \, dx","Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^3,x]","\frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{2} a^2 b c x^6+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{12} b^3 c x^{12}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14}","\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{c \left(a+b x^3\right)^4}{12 b}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14}",1,"(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b*d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14","A",1
326,1,139,110,0.0048201,"\int x \left(c+d x+e x^2\right) \left(a+b x^3\right)^3 \, dx","Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^3,x]","\frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a b^2 c x^8+\frac{1}{3} a b^2 d x^9+\frac{3}{10} a b^2 e x^{10}+\frac{1}{11} b^3 c x^{11}+\frac{1}{12} b^3 d x^{12}+\frac{1}{13} b^3 e x^{13}","\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{3}{5} a^2 b c x^5+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a b^2 c x^8+\frac{3}{10} a b^2 e x^{10}+\frac{d \left(a+b x^3\right)^4}{12 b}+\frac{1}{11} b^3 c x^{11}+\frac{1}{13} b^3 e x^{13}",1,"(a^3*c*x^2)/2 + (a^3*d*x^3)/3 + (a^3*e*x^4)/4 + (3*a^2*b*c*x^5)/5 + (a^2*b*d*x^6)/2 + (3*a^2*b*e*x^7)/7 + (3*a*b^2*c*x^8)/8 + (a*b^2*d*x^9)/3 + (3*a*b^2*e*x^10)/10 + (b^3*c*x^11)/11 + (b^3*d*x^12)/12 + (b^3*e*x^13)/13","A",1
327,1,134,105,0.0369936,"\int \left(c+d x+e x^2\right) \left(a+b x^3\right)^3 \, dx","Integrate[(c + d*x + e*x^2)*(a + b*x^3)^3,x]","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{1}{2} a^2 b e x^6+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{3} a b^2 e x^9+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}+\frac{1}{12} b^3 e x^{12}","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{e \left(a+b x^3\right)^4}{12 b}+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}",1,"a^3*c*x + (a^3*d*x^2)/2 + (a^3*e*x^3)/3 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (a^2*b*e*x^6)/2 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (a*b^2*e*x^9)/3 + (b^3*c*x^10)/10 + (b^3*d*x^11)/11 + (b^3*e*x^12)/12","A",1
328,1,127,127,0.0122241,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^3}{x} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^3)/x,x]","a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{3}{4} a^2 b d x^4+\frac{3}{5} a^2 b e x^5+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b^2 d x^7+\frac{3}{8} a b^2 e x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^3 d x^{10}+\frac{1}{11} b^3 e x^{11}","a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{3}{4} a^2 b d x^4+\frac{3}{5} a^2 b e x^5+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b^2 d x^7+\frac{3}{8} a b^2 e x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^3 d x^{10}+\frac{1}{11} b^3 e x^{11}",1,"a^3*d*x + (a^3*e*x^2)/2 + a^2*b*c*x^3 + (3*a^2*b*d*x^4)/4 + (3*a^2*b*e*x^5)/5 + (a*b^2*c*x^6)/2 + (3*a*b^2*d*x^7)/7 + (3*a*b^2*e*x^8)/8 + (b^3*c*x^9)/9 + (b^3*d*x^10)/10 + (b^3*e*x^11)/11 + a^3*c*Log[x]","A",1
329,1,125,125,0.0157986,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^3}{x^2} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^3)/x^2,x]","-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10}","-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10}",1,"-((a^3*c)/x) + a^3*e*x + (3*a^2*b*c*x^2)/2 + a^2*b*d*x^3 + (3*a^2*b*e*x^4)/4 + (3*a*b^2*c*x^5)/5 + (a*b^2*d*x^6)/2 + (3*a*b^2*e*x^7)/7 + (b^3*c*x^8)/8 + (b^3*d*x^9)/9 + (b^3*e*x^10)/10 + a^3*d*Log[x]","A",1
330,1,126,126,0.0100045,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^3}{x^3} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^3)/x^3,x]","-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9","-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9",1,"-1/2*(a^3*c)/x^2 - (a^3*d)/x + 3*a^2*b*c*x + (3*a^2*b*d*x^2)/2 + a^2*b*e*x^3 + (3*a*b^2*c*x^4)/4 + (3*a*b^2*d*x^5)/5 + (a*b^2*e*x^6)/2 + (b^3*c*x^7)/7 + (b^3*d*x^8)/8 + (b^3*e*x^9)/9 + a^3*e*Log[x]","A",1
331,1,181,138,0.0059748,"\int x^2 \left(c+d x+e x^2\right) \left(a+b x^3\right)^4 \, dx","Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^4,x]","\frac{1}{3} a^4 c x^3+\frac{1}{4} a^4 d x^4+\frac{1}{5} a^4 e x^5+\frac{2}{3} a^3 b c x^6+\frac{4}{7} a^3 b d x^7+\frac{1}{2} a^3 b e x^8+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{3} a b^3 c x^{12}+\frac{4}{13} a b^3 d x^{13}+\frac{2}{7} a b^3 e x^{14}+\frac{1}{15} b^4 c x^{15}+\frac{1}{16} b^4 d x^{16}+\frac{1}{17} b^4 e x^{17}","\frac{1}{4} a^4 d x^4+\frac{1}{5} a^4 e x^5+\frac{4}{7} a^3 b d x^7+\frac{1}{2} a^3 b e x^8+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{4}{13} a b^3 d x^{13}+\frac{2}{7} a b^3 e x^{14}+\frac{c \left(a+b x^3\right)^5}{15 b}+\frac{1}{16} b^4 d x^{16}+\frac{1}{17} b^4 e x^{17}",1,"(a^4*c*x^3)/3 + (a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (2*a^3*b*c*x^6)/3 + (4*a^3*b*d*x^7)/7 + (a^3*b*e*x^8)/2 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (a*b^3*c*x^12)/3 + (4*a*b^3*d*x^13)/13 + (2*a*b^3*e*x^14)/7 + (b^4*c*x^15)/15 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17","A",1
332,1,181,138,0.0053071,"\int x \left(c+d x+e x^2\right) \left(a+b x^3\right)^4 \, dx","Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^4,x]","\frac{1}{2} a^4 c x^2+\frac{1}{3} a^4 d x^3+\frac{1}{4} a^4 e x^4+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{3}{4} a^2 b^2 c x^8+\frac{2}{3} a^2 b^2 d x^9+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{11} a b^3 c x^{11}+\frac{1}{3} a b^3 d x^{12}+\frac{4}{13} a b^3 e x^{13}+\frac{1}{14} b^4 c x^{14}+\frac{1}{15} b^4 d x^{15}+\frac{1}{16} b^4 e x^{16}","\frac{1}{2} a^4 c x^2+\frac{1}{4} a^4 e x^4+\frac{4}{5} a^3 b c x^5+\frac{4}{7} a^3 b e x^7+\frac{3}{4} a^2 b^2 c x^8+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{11} a b^3 c x^{11}+\frac{4}{13} a b^3 e x^{13}+\frac{d \left(a+b x^3\right)^5}{15 b}+\frac{1}{14} b^4 c x^{14}+\frac{1}{16} b^4 e x^{16}",1,"(a^4*c*x^2)/2 + (a^4*d*x^3)/3 + (a^4*e*x^4)/4 + (4*a^3*b*c*x^5)/5 + (2*a^3*b*d*x^6)/3 + (4*a^3*b*e*x^7)/7 + (3*a^2*b^2*c*x^8)/4 + (2*a^2*b^2*d*x^9)/3 + (3*a^2*b^2*e*x^10)/5 + (4*a*b^3*c*x^11)/11 + (a*b^3*d*x^12)/3 + (4*a*b^3*e*x^13)/13 + (b^4*c*x^14)/14 + (b^4*d*x^15)/15 + (b^4*e*x^16)/16","A",1
333,1,173,130,0.0051015,"\int \left(c+d x+e x^2\right) \left(a+b x^3\right)^4 \, dx","Integrate[(c + d*x + e*x^2)*(a + b*x^3)^4,x]","a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{2}{3} a^3 b e x^6+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{3} a^2 b^2 e x^9+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{3} a b^3 e x^{12}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}+\frac{1}{15} b^4 e x^{15}","a^4 c x+\frac{1}{2} a^4 d x^2+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{e \left(a+b x^3\right)^5}{15 b}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}",1,"a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (2*a^3*b*e*x^6)/3 + (6*a^2*b^2*c*x^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a^2*b^2*e*x^9)/3 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (a*b^3*e*x^12)/3 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + (b^4*e*x^15)/15","A",1
334,1,166,166,0.0097622,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^4}{x} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^4)/x,x]","a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14}","a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14}",1,"a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5)/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3*c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 + (b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*c*Log[x]","A",1
335,1,162,162,0.0119273,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^4}{x^2} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^4)/x^2,x]","-\frac{a^4 c}{x}+a^4 d \log (x)+a^4 e x+2 a^3 b c x^2+\frac{4}{3} a^3 b d x^3+a^3 b e x^4+\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{6}{7} a^2 b^2 e x^7+\frac{1}{2} a b^3 c x^8+\frac{4}{9} a b^3 d x^9+\frac{2}{5} a b^3 e x^{10}+\frac{1}{11} b^4 c x^{11}+\frac{1}{12} b^4 d x^{12}+\frac{1}{13} b^4 e x^{13}","-\frac{a^4 c}{x}+a^4 d \log (x)+a^4 e x+2 a^3 b c x^2+\frac{4}{3} a^3 b d x^3+a^3 b e x^4+\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{6}{7} a^2 b^2 e x^7+\frac{1}{2} a b^3 c x^8+\frac{4}{9} a b^3 d x^9+\frac{2}{5} a b^3 e x^{10}+\frac{1}{11} b^4 c x^{11}+\frac{1}{12} b^4 d x^{12}+\frac{1}{13} b^4 e x^{13}",1,"-((a^4*c)/x) + a^4*e*x + 2*a^3*b*c*x^2 + (4*a^3*b*d*x^3)/3 + a^3*b*e*x^4 + (6*a^2*b^2*c*x^5)/5 + a^2*b^2*d*x^6 + (6*a^2*b^2*e*x^7)/7 + (a*b^3*c*x^8)/2 + (4*a*b^3*d*x^9)/9 + (2*a*b^3*e*x^10)/5 + (b^4*c*x^11)/11 + (b^4*d*x^12)/12 + (b^4*e*x^13)/13 + a^4*d*Log[x]","A",1
336,1,166,166,0.0104678,"\int \frac{\left(c+d x+e x^2\right) \left(a+b x^3\right)^4}{x^3} \, dx","Integrate[((c + d*x + e*x^2)*(a + b*x^3)^4)/x^3,x]","-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3+\frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12}","-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3+\frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12}",1,"-1/2*(a^4*c)/x^2 - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^4*d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]","A",1
337,1,191,205,0.1107782,"\int \frac{x^3 \left(c+d x+e x^2\right)}{a+b x^3} \, dx","Integrate[(x^3*(c + d*x + e*x^2))/(a + b*x^3),x]","\frac{\sqrt[3]{b} \left(\sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(a^{2/3} d-\sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-2 a e \log \left(a+b x^3\right)+6 b c x+3 b d x^2+2 b e x^3}{6 b^2}","\frac{\sqrt[3]{a} \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{4/3}}-\frac{\sqrt[3]{a} \left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{5/3}}+\frac{\sqrt[3]{a} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{5/3}}-\frac{a e \log \left(a+b x^3\right)}{3 b^2}+\frac{c x}{b}+\frac{d x^2}{2 b}+\frac{e x^3}{3 b}",1,"(6*b*c*x + 3*b*d*x^2 + 2*b*e*x^3 + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(-(a^(1/3)*b^(1/3)*c) + a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*e*Log[a + b*x^3])/(6*b^2)","A",1
338,1,184,193,0.095817,"\int \frac{x^2 \left(c+d x+e x^2\right)}{a+b x^3} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/(a + b*x^3),x]","\frac{-\left(a^{2/3} e-\sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \left(a^{2/3} e-\sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 b^{2/3} c \log \left(a+b x^3\right)+2 \sqrt{3} \sqrt[3]{a} \left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+6 b^{2/3} d x+3 b^{2/3} e x^2}{6 b^{5/3}}","\frac{\sqrt[3]{a} \left(d-\frac{\sqrt[3]{a} e}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{4/3}}-\frac{\sqrt[3]{a} \left(\sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{5/3}}+\frac{\sqrt[3]{a} \left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{5/3}}+\frac{c \log \left(a+b x^3\right)}{3 b}+\frac{d x}{b}+\frac{e x^2}{2 b}",1,"(6*b^(2/3)*d*x + 3*b^(2/3)*e*x^2 + 2*Sqrt[3]*a^(1/3)*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] - (-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*b^(2/3)*c*Log[a + b*x^3])/(6*b^(5/3))","A",1
339,1,200,183,0.060081,"\int \frac{x \left(c+d x+e x^2\right)}{a+b x^3} \, dx","Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3),x]","-\frac{\left(a^{4/3} \left(-\sqrt[3]{b}\right) e-a^{2/3} b c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a b^{5/3}}+\frac{\left(a^{4/3} \left(-\sqrt[3]{b}\right) e-a^{2/3} b c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a b^{5/3}}+\frac{\left(a^{2/3} b c-a^{4/3} \sqrt[3]{b} e\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a b^{5/3}}+\frac{d \log \left(a+b x^3\right)}{3 b}+\frac{e x}{b}","\frac{\left(a^{2/3} e+b^{2/3} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} b^{4/3}}-\frac{\left(a^{2/3} e+b^{2/3} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} b^{4/3}}-\frac{\left(b^{2/3} c-a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{4/3}}+\frac{d \log \left(a+b x^3\right)}{3 b}+\frac{e x}{b}",1,"(e*x)/b + ((a^(2/3)*b*c - a^(4/3)*b^(1/3)*e)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a*b^(5/3)) + ((-(a^(2/3)*b*c) - a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*a*b^(5/3)) - ((-(a^(2/3)*b*c) - a^(4/3)*b^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a*b^(5/3)) + (d*Log[a + b*x^3])/(3*b)","A",1
340,1,176,177,0.1048319,"\int \frac{c+d x+e x^2}{a+b x^3} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^3),x]","\frac{-\sqrt[3]{b} \left(\sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(\sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+2 a e \log \left(a+b x^3\right)}{6 a b}","-\frac{\left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{2/3}}+\frac{e \log \left(a+b x^3\right)}{3 b}",1,"(-2*Sqrt[3]*a^(1/3)*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] - b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*e*Log[a + b*x^3])/(6*a*b)","A",1
341,1,176,184,0.0987594,"\int \frac{c+d x+e x^2}{x \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)),x]","\frac{\left(a^{2/3} e-\sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \left(\sqrt[3]{a} \sqrt[3]{b} d-a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 b^{2/3} c \log \left(a+b x^3\right)-2 \sqrt{3} \sqrt[3]{a} \left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+6 b^{2/3} c \log (x)}{6 a b^{2/3}}","-\frac{\left(d-\frac{\sqrt[3]{a} e}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\left(\sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{2/3}}-\frac{c \log \left(a+b x^3\right)}{3 a}+\frac{c \log (x)}{a}",1,"(-2*Sqrt[3]*a^(1/3)*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 6*b^(2/3)*c*Log[x] + 2*(a^(1/3)*b^(1/3)*d - a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] + (-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*b^(2/3)*c*Log[a + b*x^3])/(6*a*b^(2/3))","A",1
342,1,184,192,0.2538388,"\int \frac{c+d x+e x^2}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)),x]","-\frac{\frac{\left(a^{2/3} b^{2/3} c+a^{4/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}-\frac{2 \left(a^{2/3} b^{2/3} c+a^{4/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{2 \sqrt{3} a^{2/3} \left(a^{2/3} e-b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}+2 a d \log \left(a+b x^3\right)+\frac{6 a c}{x}-6 a d \log (x)}{6 a^2}","-\frac{\left(a^{2/3} e+b^{2/3} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{4/3} \sqrt[3]{b}}+\frac{\left(a^{2/3} e+b^{2/3} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{4/3} \sqrt[3]{b}}+\frac{\left(b^{2/3} c-a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} \sqrt[3]{b}}-\frac{d \log \left(a+b x^3\right)}{3 a}-\frac{c}{a x}+\frac{d \log (x)}{a}",1,"-1/6*((6*a*c)/x + (2*Sqrt[3]*a^(2/3)*(-(b^(2/3)*c) + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - 6*a*d*Log[x] - (2*(a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) + 2*a*d*Log[a + b*x^3])/a^2","A",1
343,1,192,203,0.2312624,"\int \frac{c+d x+e x^2}{x^3 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)),x]","\frac{\sqrt[3]{b} \left(\sqrt[3]{a} \sqrt[3]{b} c-a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(a^{2/3} d-\sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-2 a e \log \left(a+b x^3\right)-\frac{3 a c}{x^2}-\frac{6 a d}{x}+6 a e \log (x)}{6 a^2}","\frac{b^{2/3} \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{5/3}}-\frac{\sqrt[3]{b} \left(\sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{5/3}}+\frac{\sqrt[3]{b} \left(\sqrt[3]{a} d+\sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}-\frac{e \log \left(a+b x^3\right)}{3 a}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a}",1,"((-3*a*c)/x^2 - (6*a*d)/x + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 6*a*e*Log[x] + 2*b^(1/3)*(-(a^(1/3)*b^(1/3)*c) + a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*e*Log[a + b*x^3])/(6*a^2)","A",1
344,1,174,190,0.2025471,"\int \frac{x^2 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/(a + b*x^3)^2,x]","\frac{\frac{\left(2 \sqrt[3]{a} e-\sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}+\frac{2 \left(\sqrt[3]{b} d-2 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}-\frac{2 \sqrt{3} \left(2 \sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{2/3}}-\frac{6 b^{2/3} (c+x (d+e x))}{a+b x^3}}{18 b^{5/3}}","-\frac{\left(d-\frac{2 \sqrt[3]{a} e}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{2/3} b^{4/3}}+\frac{\left(\sqrt[3]{b} d-2 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{2/3} b^{5/3}}-\frac{\left(2 \sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{2/3} b^{5/3}}-\frac{c+d x+e x^2}{3 b \left(a+b x^3\right)}",1,"((-6*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*(b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(b^(1/3)*d - 2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-(b^(1/3)*d) + 2*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^(5/3))","A",1
345,1,186,200,0.2036135,"\int \frac{x \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^2,x]","\frac{-\left(a^{4/3} \sqrt[3]{b} e-a^{2/3} b c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \left(a^{4/3} \sqrt[3]{b} e-a^{2/3} b c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \left(a^{2/3} b c+a^{4/3} \sqrt[3]{b} e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-\frac{6 a b^{2/3} \left(a (d+e x)-b c x^2\right)}{a+b x^3}}{18 a^2 b^{5/3}}","\frac{\left(b^{2/3} c-a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{4/3} b^{4/3}}-\frac{\left(b^{2/3} c-a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{4/3} b^{4/3}}-\frac{\left(a^{2/3} e+b^{2/3} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{4/3} b^{4/3}}-\frac{x \left(a e-b c x-b d x^2\right)}{3 a b \left(a+b x^3\right)}",1,"((-6*a*b^(2/3)*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3) - 2*Sqrt[3]*(a^(2/3)*b*c + a^(4/3)*b^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-(a^(2/3)*b*c) + a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x] - (-(a^(2/3)*b*c) + a^(4/3)*b^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^2*b^(5/3))","A",1
346,1,189,199,0.2816335,"\int \frac{c+d x+e x^2}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^3)^2,x]","\frac{\sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a} d-2 \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\left(4 \sqrt[3]{a} b^{2/3} c-2 a^{2/3} \sqrt[3]{b} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{6 a (b x (c+d x)-a e)}{a+b x^3}-2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{18 a^2 b}","-\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{2/3}}+\frac{\left(2 \sqrt[3]{b} c-\sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} d+2 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{2/3}}-\frac{a e-b x (c+d x)}{3 a b \left(a+b x^3\right)}",1,"((6*a*(-(a*e) + b*x*(c + d*x)))/(a + b*x^3) - 2*Sqrt[3]*a^(1/3)*b^(1/3)*(2*b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (4*a^(1/3)*b^(2/3)*c - 2*a^(2/3)*b^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x] + a^(1/3)*b^(1/3)*(-2*b^(1/3)*c + a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^2*b)","A",1
347,1,199,222,0.2360574,"\int \frac{c+d x+e x^2}{x \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^2),x]","\frac{\frac{\left(a^{2/3} e-2 \sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 \left(2 \sqrt[3]{a} \sqrt[3]{b} d-a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \left(\sqrt[3]{a} e+2 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 a (c+x (d+e x))}{a+b x^3}-6 c \log \left(a+b x^3\right)+18 c \log (x)}{18 a^2}","-\frac{\left(2 \sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{2/3}}+\frac{\left(2 \sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{2/3}}-\frac{\left(\sqrt[3]{a} e+2 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x \left(a d+a e x-b c x^2\right)}{3 a^2 \left(a+b x^3\right)}-\frac{c \log \left(a+b x^3\right)}{3 a^2}+\frac{c \log (x)}{a^2}",1,"((6*a*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 18*c*Log[x] + (2*(2*a^(1/3)*b^(1/3)*d - a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-2*a^(1/3)*b^(1/3)*d + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 6*c*Log[a + b*x^3])/(18*a^2)","A",1
348,1,213,231,0.3362466,"\int \frac{c+d x+e x^2}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^2),x]","-\frac{\frac{\left(2 a^{2/3} b^{2/3} c+a^{4/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}-\frac{2 \left(2 a^{2/3} b^{2/3} c+a^{4/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{2 \sqrt{3} a^{2/3} \left(a^{2/3} e-2 b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}-\frac{3 a \left(a (d+e x)-b c x^2\right)}{a+b x^3}+3 a d \log \left(a+b x^3\right)+\frac{9 a c}{x}-9 a d \log (x)}{9 a^3}","-\frac{\left(a^{2/3} e+2 b^{2/3} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{9 a^{7/3} \sqrt[3]{b}}+\frac{2 \left(a^{2/3} e+2 b^{2/3} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{7/3} \sqrt[3]{b}}+\frac{2 \left(2 b^{2/3} c-a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{7/3} \sqrt[3]{b}}+\frac{x \left(a e-b c x-b d x^2\right)}{3 a^2 \left(a+b x^3\right)}-\frac{d \log \left(a+b x^3\right)}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2}",1,"-1/9*((9*a*c)/x - (3*a*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3) + (2*Sqrt[3]*a^(2/3)*(-2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - 9*a*d*Log[x] - (2*(2*a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((2*a^(2/3)*b^(2/3)*c + a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) + 3*a*d*Log[a + b*x^3])/a^3","A",1
349,1,221,242,0.2158754,"\int \frac{c+d x+e x^2}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)^2),x]","\frac{\sqrt[3]{b} \left(5 \sqrt[3]{a} \sqrt[3]{b} c-4 a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(4 a^{2/3} d-5 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{6 a (a e-b x (c+d x))}{a+b x^3}+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(4 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-6 a e \log \left(a+b x^3\right)-\frac{9 a c}{x^2}-\frac{18 a d}{x}+18 a e \log (x)}{18 a^3}","\frac{\sqrt[3]{b} \left(5 \sqrt[3]{b} c-4 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{8/3}}-\frac{\sqrt[3]{b} \left(5 \sqrt[3]{b} c-4 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{8/3}}+\frac{\sqrt[3]{b} \left(4 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{8/3}}-\frac{x \left(b c+b d x+b e x^2\right)}{3 a^2 \left(a+b x^3\right)}-\frac{e \log \left(a+b x^3\right)}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2}",1,"((-9*a*c)/x^2 - (18*a*d)/x + (6*a*(a*e - b*x*(c + d*x)))/(a + b*x^3) + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 18*a*e*Log[x] + 2*b^(1/3)*(-5*a^(1/3)*b^(1/3)*c + 4*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*c - 4*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 6*a*e*Log[a + b*x^3])/(18*a^3)","A",1
350,1,225,262,0.2983628,"\int \frac{c+d x+e x^2}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^2),x]","\frac{\sqrt[3]{b} \left(5 \sqrt[3]{a} \sqrt[3]{b} d-4 a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(4 a^{2/3} e-5 \sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-\frac{6 a b (c+x (d+e x))}{a+b x^3}+12 b c \log \left(a+b x^3\right)+2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(4 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}-36 b c \log (x)}{18 a^3}","\frac{\sqrt[3]{b} \left(5 \sqrt[3]{b} d-4 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{8/3}}-\frac{\sqrt[3]{b} \left(5 \sqrt[3]{b} d-4 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{8/3}}+\frac{\sqrt[3]{b} \left(4 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{8/3}}+\frac{2 b c \log \left(a+b x^3\right)}{3 a^3}-\frac{2 b c \log (x)}{a^3}-\frac{x \left(-\frac{b^2 c x^2}{a}+b d+b e x\right)}{3 a^2 \left(a+b x^3\right)}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}",1,"((-6*a*c)/x^3 - (9*a*d)/x^2 - (18*a*e)/x - (6*a*b*(c + x*(d + e*x)))/(a + b*x^3) + 2*Sqrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 36*b*c*Log[x] + 2*b^(1/3)*(-5*a^(1/3)*b^(1/3)*d + 4*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*d - 4*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 12*b*c*Log[a + b*x^3])/(18*a^3)","A",1
351,1,198,215,0.2084512,"\int \frac{x^2 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/(a + b*x^3)^3,x]","\frac{\frac{\left(\sqrt[3]{a} e-\sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{2 \left(\sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}-\frac{2 \sqrt{3} \left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{5/3}}-\frac{9 b^{2/3} (c+x (d+e x))}{\left(a+b x^3\right)^2}+\frac{3 b^{2/3} x (d+2 e x)}{a \left(a+b x^3\right)}}{54 b^{5/3}}","-\frac{\left(d-\frac{\sqrt[3]{a} e}{\sqrt[3]{b}}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{5/3} b^{4/3}}+\frac{\left(\sqrt[3]{b} d-\sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{5/3} b^{5/3}}-\frac{\left(\sqrt[3]{a} e+\sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{5/3} b^{5/3}}-\frac{c+d x+e x^2}{6 b \left(a+b x^3\right)^2}+\frac{x (d+2 e x)}{18 a b \left(a+b x^3\right)}",1,"((3*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)) - (9*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3)^2 - (2*Sqrt[3]*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + ((-(b^(1/3)*d) + a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^(5/3))","A",1
352,1,214,239,0.4042028,"\int \frac{x \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^3,x]","\frac{\left(2 a^{2/3} b c-a^{4/3} \sqrt[3]{b} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt{3} a^{2/3} \sqrt[3]{b} \left(a^{2/3} e+2 b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+2 \left(a^{4/3} \sqrt[3]{b} e-2 a^{2/3} b c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{3 a b^{2/3} \left(-a^2 (3 d+2 e x)+a b x^2 \left(7 c+e x^2\right)+4 b^2 c x^5\right)}{\left(a+b x^3\right)^2}}{54 a^3 b^{5/3}}","\frac{\left(2 b^{2/3} c-a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{7/3} b^{4/3}}-\frac{\left(2 b^{2/3} c-a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{7/3} b^{4/3}}-\frac{\left(a^{2/3} e+2 b^{2/3} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{7/3} b^{4/3}}-\frac{3 a d-x (a e+4 b c x)}{18 a^2 b \left(a+b x^3\right)}-\frac{x \left(a e-b c x-b d x^2\right)}{6 a b \left(a+b x^3\right)^2}",1,"((3*a*b^(2/3)*(4*b^2*c*x^5 - a^2*(3*d + 2*e*x) + a*b*x^2*(7*c + e*x^2)))/(a + b*x^3)^2 - 2*Sqrt[3]*a^(2/3)*b^(1/3)*(2*b^(2/3)*c + a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2*a^(2/3)*b*c + a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x] + (2*a^(2/3)*b*c - a^(4/3)*b^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^3*b^(5/3))","A",1
353,1,213,225,0.3745003,"\int \frac{c+d x+e x^2}{\left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^3)^3,x]","\frac{\sqrt[3]{a} \sqrt[3]{b} \left(2 \sqrt[3]{a} d-5 \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \sqrt[3]{b} \left(5 \sqrt[3]{a} \sqrt[3]{b} c-2 a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{3 a \left(-3 a^2 e+a b x (8 c+7 d x)+b^2 x^4 (5 c+4 d x)\right)}{\left(a+b x^3\right)^2}-2 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(2 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^3 b}","-\frac{\left(5 \sqrt[3]{b} c-2 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{8/3} b^{2/3}}+\frac{\left(5 \sqrt[3]{b} c-2 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{8/3} b^{2/3}}-\frac{\left(2 \sqrt[3]{a} d+5 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left(a+b x^3\right)}-\frac{a e-b x (c+d x)}{6 a b \left(a+b x^3\right)^2}",1,"((3*a*(-3*a^2*e + b^2*x^4*(5*c + 4*d*x) + a*b*x*(8*c + 7*d*x)))/(a + b*x^3)^2 - 2*Sqrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*c + 2*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(5*a^(1/3)*b^(1/3)*c - 2*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + a^(1/3)*b^(1/3)*(-5*b^(1/3)*c + 2*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^3*b)","A",1
354,1,229,257,0.2055508,"\int \frac{c+d x+e x^2}{x \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^3),x]","\frac{\frac{\left(2 a^{2/3} e-5 \sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 \left(5 \sqrt[3]{a} \sqrt[3]{b} d-2 a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{9 a^2 (c+x (d+e x))}{\left(a+b x^3\right)^2}-\frac{2 \sqrt{3} \sqrt[3]{a} \left(2 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{3 a (6 c+x (5 d+4 e x))}{a+b x^3}-18 c \log \left(a+b x^3\right)+54 c \log (x)}{54 a^3}","-\frac{\left(5 \sqrt[3]{b} d-2 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{8/3} b^{2/3}}+\frac{\left(5 \sqrt[3]{b} d-2 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{8/3} b^{2/3}}-\frac{\left(2 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x \left(5 a d+4 a e x-9 b c x^2\right)}{18 a^3 \left(a+b x^3\right)}-\frac{c \log \left(a+b x^3\right)}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left(a d+a e x-b c x^2\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"((9*a^2*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*(6*c + x*(5*d + 4*e*x)))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(5*b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 54*c*Log[x] + (2*(5*a^(1/3)*b^(1/3)*d - 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-5*a^(1/3)*b^(1/3)*d + 2*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 18*c*Log[a + b*x^3])/(54*a^3)","A",1
355,1,248,267,0.3301798,"\int \frac{c+d x+e x^2}{x^2 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^3),x]","\frac{-\frac{\left(14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}+\frac{2 \left(14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}-\frac{2 \sqrt{3} a^{2/3} \left(5 a^{2/3} e-14 b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}+\frac{9 a^2 \left(a (d+e x)-b c x^2\right)}{\left(a+b x^3\right)^2}+\frac{3 a \left(6 a d+5 a e x-10 b c x^2\right)}{a+b x^3}-18 a d \log \left(a+b x^3\right)-\frac{54 a c}{x}+54 a d \log (x)}{54 a^4}","-\frac{\left(5 a^{2/3} e+14 b^{2/3} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{10/3} \sqrt[3]{b}}+\frac{\left(5 a^{2/3} e+14 b^{2/3} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{10/3} \sqrt[3]{b}}+\frac{\left(14 b^{2/3} c-5 a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{10/3} \sqrt[3]{b}}+\frac{x \left(5 a e-10 b c x-9 b d x^2\right)}{18 a^3 \left(a+b x^3\right)}-\frac{d \log \left(a+b x^3\right)}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left(a e-b c x-b d x^2\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"((-54*a*c)/x + (3*a*(6*a*d + 5*a*e*x - 10*b*c*x^2))/(a + b*x^3) + (9*a^2*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3)^2 - (2*Sqrt[3]*a^(2/3)*(-14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 54*a*d*Log[x] + (2*(14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - ((14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 18*a*d*Log[a + b*x^3])/(54*a^4)","A",1
356,1,253,276,0.3336831,"\int \frac{c+d x+e x^2}{x^3 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)^3),x]","\frac{2 \sqrt[3]{b} \left(10 \sqrt[3]{a} \sqrt[3]{b} c-7 a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+4 \sqrt[3]{b} \left(7 a^{2/3} d-10 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{9 a^2 (a e-b x (c+d x))}{\left(a+b x^3\right)^2}+\frac{3 a (6 a e-b x (11 c+10 d x))}{a+b x^3}+4 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(7 \sqrt[3]{a} d+10 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-18 a e \log \left(a+b x^3\right)-\frac{27 a c}{x^2}-\frac{54 a d}{x}+54 a e \log (x)}{54 a^4}","\frac{\sqrt[3]{b} \left(10 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{11/3}}-\frac{2 \sqrt[3]{b} \left(10 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{11/3}}+\frac{2 \sqrt[3]{b} \left(7 \sqrt[3]{a} d+10 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{11/3}}-\frac{x \left(11 b c+10 b d x+9 b e x^2\right)}{18 a^3 \left(a+b x^3\right)}-\frac{e \log \left(a+b x^3\right)}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left(b c+b d x+b e x^2\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"((-27*a*c)/x^2 - (54*a*d)/x + (9*a^2*(a*e - b*x*(c + d*x)))/(a + b*x^3)^2 + (3*a*(6*a*e - b*x*(11*c + 10*d*x)))/(a + b*x^3) + 4*Sqrt[3]*a^(1/3)*b^(1/3)*(10*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 54*a*e*Log[x] + 4*b^(1/3)*(-10*a^(1/3)*b^(1/3)*c + 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + 2*b^(1/3)*(10*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 18*a*e*Log[a + b*x^3])/(54*a^4)","A",1
357,1,255,298,0.4596949,"\int \frac{c+d x+e x^2}{x^4 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]","-\frac{-2 \sqrt[3]{b} \left(10 \sqrt[3]{a} \sqrt[3]{b} d-7 a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+4 \sqrt[3]{b} \left(10 \sqrt[3]{a} \sqrt[3]{b} d-7 a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{9 a^2 b (c+x (d+e x))}{\left(a+b x^3\right)^2}+\frac{3 a b (12 c+x (11 d+10 e x))}{a+b x^3}-54 b c \log \left(a+b x^3\right)-4 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(7 \sqrt[3]{a} e+10 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+\frac{18 a c}{x^3}+\frac{27 a d}{x^2}+\frac{54 a e}{x}+162 b c \log (x)}{54 a^4}","\frac{\sqrt[3]{b} \left(10 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{11/3}}-\frac{2 \sqrt[3]{b} \left(10 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{11/3}}+\frac{2 \sqrt[3]{b} \left(7 \sqrt[3]{a} e+10 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{11/3}}+\frac{b c \log \left(a+b x^3\right)}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{x \left(-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right)}{18 a^3 \left(a+b x^3\right)}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left(-\frac{b^2 c x^2}{a}+b d+b e x\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"-1/54*((18*a*c)/x^3 + (27*a*d)/x^2 + (54*a*e)/x + (9*a^2*b*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*b*(12*c + x*(11*d + 10*e*x)))/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 162*b*c*Log[x] + 4*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] - 2*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 54*b*c*Log[a + b*x^3])/a^4","A",1
358,1,230,248,0.2879585,"\int \frac{x^2 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^4} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/(a + b*x^3)^4,x]","\frac{\frac{\left(4 \sqrt[3]{a} e-5 \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{8/3}}+\frac{2 \left(5 \sqrt[3]{b} d-4 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{8/3}}-\frac{2 \sqrt{3} \left(4 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{8/3}}+\frac{3 b^{2/3} x (5 d+8 e x)}{a^2 \left(a+b x^3\right)}-\frac{54 b^{2/3} (c+x (d+e x))}{\left(a+b x^3\right)^3}+\frac{9 b^{2/3} x (d+2 e x)}{a \left(a+b x^3\right)^2}}{486 b^{5/3}}","-\frac{\left(5 \sqrt[3]{b} d-4 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{486 a^{8/3} b^{5/3}}+\frac{\left(5 \sqrt[3]{b} d-4 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{8/3} b^{5/3}}-\frac{\left(4 \sqrt[3]{a} e+5 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x (5 d+8 e x)}{162 a^2 b \left(a+b x^3\right)}-\frac{c+d x+e x^2}{9 b \left(a+b x^3\right)^3}+\frac{x (d+2 e x)}{54 a b \left(a+b x^3\right)^2}",1,"((9*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)^2) + (3*b^(2/3)*x*(5*d + 8*e*x))/(a^2*(a + b*x^3)) - (54*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3)^3 - (2*Sqrt[3]*(5*b^(1/3)*d + 4*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(5*b^(1/3)*d - 4*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) + ((-5*b^(1/3)*d + 4*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3))/(486*b^(5/3))","A",1
359,1,241,270,0.4266985,"\int \frac{x \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^4} \, dx","Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^4,x]","\frac{a^{2/3} \sqrt[3]{b} \left(14 b^{2/3} c-5 a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt{3} a^{2/3} \sqrt[3]{b} \left(5 a^{2/3} e+14 b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+2 \left(5 a^{4/3} \sqrt[3]{b} e-14 a^{2/3} b c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{3 a b^{2/3} \left(-2 a^3 (9 d+5 e x)+a^2 b x^2 \left(67 c+13 e x^2\right)+a b^2 x^5 \left(77 c+5 e x^2\right)+28 b^3 c x^8\right)}{\left(a+b x^3\right)^3}}{486 a^4 b^{5/3}}","\frac{\left(14 b^{2/3} c-5 a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{486 a^{10/3} b^{4/3}}-\frac{\left(14 b^{2/3} c-5 a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{10/3} b^{4/3}}-\frac{\left(5 a^{2/3} e+14 b^{2/3} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x (5 a e+28 b c x)}{162 a^3 b \left(a+b x^3\right)}-\frac{6 a d-x (a e+7 b c x)}{54 a^2 b \left(a+b x^3\right)^2}-\frac{x \left(a e-b c x-b d x^2\right)}{9 a b \left(a+b x^3\right)^3}",1,"((3*a*b^(2/3)*(28*b^3*c*x^8 - 2*a^3*(9*d + 5*e*x) + a*b^2*x^5*(77*c + 5*e*x^2) + a^2*b*x^2*(67*c + 13*e*x^2)))/(a + b*x^3)^3 - 2*Sqrt[3]*a^(2/3)*b^(1/3)*(14*b^(2/3)*c + 5*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-14*a^(2/3)*b*c + 5*a^(4/3)*b^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x] + a^(2/3)*b^(1/3)*(14*b^(2/3)*c - 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(486*a^4*b^(5/3))","A",1
360,1,239,250,0.2788233,"\int \frac{c+d x+e x^2}{\left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^3)^4,x]","\frac{\frac{2 \left(7 a^{2/3} d-20 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{4 \left(20 \sqrt[3]{a} \sqrt[3]{b} c-7 a^{2/3} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}-\frac{54 a^3 (a e-b x (c+d x))}{b \left(a+b x^3\right)^3}+\frac{9 a^2 x (8 c+7 d x)}{\left(a+b x^3\right)^2}-\frac{4 \sqrt{3} \sqrt[3]{a} \left(7 \sqrt[3]{a} d+20 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4}","-\frac{\left(20 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{11/3} b^{2/3}}+\frac{2 \left(20 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{11/3} b^{2/3}}-\frac{2 \left(7 \sqrt[3]{a} d+20 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left(a+b x^3\right)}+\frac{x (8 c+7 d x)}{54 a^2 \left(a+b x^3\right)^2}-\frac{a e-b x (c+d x)}{9 a b \left(a+b x^3\right)^3}",1,"((9*a^2*x*(8*c + 7*d*x))/(a + b*x^3)^2 + (12*a*x*(10*c + 7*d*x))/(a + b*x^3) - (54*a^3*(a*e - b*x*(c + d*x)))/(b*(a + b*x^3)^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(20*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/3)*c + 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(486*a^4)","A",1
361,1,259,291,0.3095673,"\int \frac{c+d x+e x^2}{x \left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^4),x]","\frac{\frac{2 \left(7 a^{2/3} e-20 \sqrt[3]{a} \sqrt[3]{b} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{4 \left(20 \sqrt[3]{a} \sqrt[3]{b} d-7 a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{54 a^3 (c+x (d+e x))}{\left(a+b x^3\right)^3}+\frac{9 a^2 (9 c+x (8 d+7 e x))}{\left(a+b x^3\right)^2}-\frac{4 \sqrt{3} \sqrt[3]{a} \left(7 \sqrt[3]{a} e+20 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 a (27 c+2 x (10 d+7 e x))}{a+b x^3}-162 c \log \left(a+b x^3\right)+486 c \log (x)}{486 a^4}","-\frac{\left(20 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{11/3} b^{2/3}}+\frac{2 \left(20 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{11/3} b^{2/3}}-\frac{2 \left(7 \sqrt[3]{a} e+20 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x \left(40 a d+28 a e x-99 b c x^2\right)}{162 a^4 \left(a+b x^3\right)}-\frac{c \log \left(a+b x^3\right)}{3 a^4}+\frac{c \log (x)}{a^4}+\frac{x \left(8 a d+7 a e x-15 b c x^2\right)}{54 a^3 \left(a+b x^3\right)^2}+\frac{x \left(a d+a e x-b c x^2\right)}{9 a^2 \left(a+b x^3\right)^3}",1,"((54*a^3*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*(9*c + x*(8*d + 7*e*x)))/(a + b*x^3)^2 + (6*a*(27*c + 2*x*(10*d + 7*e*x)))/(a + b*x^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 486*c*Log[x] + (4*(20*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/3)*d + 7*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 162*c*Log[a + b*x^3])/(486*a^4)","A",1
362,1,279,301,0.313238,"\int \frac{c+d x+e x^2}{x^2 \left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^4),x]","\frac{-\frac{20 \left(7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}+\frac{40 \left(7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}-\frac{40 \sqrt{3} a^{2/3} \left(2 a^{2/3} e-7 b^{2/3} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}+\frac{54 a^3 \left(a (d+e x)-b c x^2\right)}{\left(a+b x^3\right)^3}+\frac{9 a^2 \left(9 a d+8 a e x-16 b c x^2\right)}{\left(a+b x^3\right)^2}+\frac{6 a \left(27 a d+20 a e x-59 b c x^2\right)}{a+b x^3}-162 a d \log \left(a+b x^3\right)-\frac{486 a c}{x}+486 a d \log (x)}{486 a^5}","-\frac{10 \left(2 a^{2/3} e+7 b^{2/3} c\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{13/3} \sqrt[3]{b}}+\frac{20 \left(2 a^{2/3} e+7 b^{2/3} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{13/3} \sqrt[3]{b}}+\frac{20 \left(7 b^{2/3} c-2 a^{2/3} e\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{13/3} \sqrt[3]{b}}+\frac{x \left(40 a e-118 b c x-99 b d x^2\right)}{162 a^4 \left(a+b x^3\right)}-\frac{d \log \left(a+b x^3\right)}{3 a^4}-\frac{c}{a^4 x}+\frac{d \log (x)}{a^4}+\frac{x \left(8 a e-16 b c x-15 b d x^2\right)}{54 a^3 \left(a+b x^3\right)^2}+\frac{x \left(a e-b c x-b d x^2\right)}{9 a^2 \left(a+b x^3\right)^3}",1,"((-486*a*c)/x + (9*a^2*(9*a*d + 8*a*e*x - 16*b*c*x^2))/(a + b*x^3)^2 + (6*a*(27*a*d + 20*a*e*x - 59*b*c*x^2))/(a + b*x^3) + (54*a^3*(-(b*c*x^2) + a*(d + e*x)))/(a + b*x^3)^3 - (40*Sqrt[3]*a^(2/3)*(-7*b^(2/3)*c + 2*a^(2/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 486*a*d*Log[x] + (40*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (20*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 162*a*d*Log[a + b*x^3])/(486*a^5)","A",1
363,1,284,310,0.3149895,"\int \frac{c+d x+e x^2}{x^3 \left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(x^3*(a + b*x^3)^4),x]","\frac{20 \sqrt[3]{b} \left(11 \sqrt[3]{a} \sqrt[3]{b} c-7 a^{2/3} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+40 \sqrt[3]{b} \left(7 a^{2/3} d-11 \sqrt[3]{a} \sqrt[3]{b} c\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{54 a^3 (a e-b x (c+d x))}{\left(a+b x^3\right)^3}+\frac{9 a^2 (9 a e-b x (17 c+16 d x))}{\left(a+b x^3\right)^2}+\frac{3 a (54 a e-b x (139 c+118 d x))}{a+b x^3}+40 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(7 \sqrt[3]{a} d+11 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)-162 a e \log \left(a+b x^3\right)-\frac{243 a c}{x^2}-\frac{486 a d}{x}+486 a e \log (x)}{486 a^5}","\frac{10 \sqrt[3]{b} \left(11 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{14/3}}-\frac{20 \sqrt[3]{b} \left(11 \sqrt[3]{b} c-7 \sqrt[3]{a} d\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{14/3}}+\frac{20 \sqrt[3]{b} \left(7 \sqrt[3]{a} d+11 \sqrt[3]{b} c\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{14/3}}-\frac{x \left(139 b c+118 b d x+99 b e x^2\right)}{162 a^4 \left(a+b x^3\right)}-\frac{e \log \left(a+b x^3\right)}{3 a^4}-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}+\frac{e \log (x)}{a^4}-\frac{x \left(17 b c+16 b d x+15 b e x^2\right)}{54 a^3 \left(a+b x^3\right)^2}-\frac{x \left(b c+b d x+b e x^2\right)}{9 a^2 \left(a+b x^3\right)^3}",1,"((-243*a*c)/x^2 - (486*a*d)/x + (54*a^3*(a*e - b*x*(c + d*x)))/(a + b*x^3)^3 + (9*a^2*(9*a*e - b*x*(17*c + 16*d*x)))/(a + b*x^3)^2 + (3*a*(54*a*e - b*x*(139*c + 118*d*x)))/(a + b*x^3) + 40*Sqrt[3]*a^(1/3)*b^(1/3)*(11*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 486*a*e*Log[x] + 40*b^(1/3)*(-11*a^(1/3)*b^(1/3)*c + 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + 20*b^(1/3)*(11*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 162*a*e*Log[a + b*x^3])/(486*a^5)","A",1
364,1,284,340,0.6015601,"\int \frac{c+d x+e x^2}{x^4 \left(a+b x^3\right)^4} \, dx","Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^4),x]","-\frac{-20 \sqrt[3]{b} \left(11 \sqrt[3]{a} \sqrt[3]{b} d-7 a^{2/3} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+40 \sqrt[3]{b} \left(11 \sqrt[3]{a} \sqrt[3]{b} d-7 a^{2/3} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+\frac{54 a^3 b (c+x (d+e x))}{\left(a+b x^3\right)^3}+\frac{9 a^2 b (18 c+x (17 d+16 e x))}{\left(a+b x^3\right)^2}+\frac{3 a b (162 c+x (139 d+118 e x))}{a+b x^3}-648 b c \log \left(a+b x^3\right)-40 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \left(7 \sqrt[3]{a} e+11 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+\frac{162 a c}{x^3}+\frac{243 a d}{x^2}+\frac{486 a e}{x}+1944 b c \log (x)}{486 a^5}","\frac{10 \sqrt[3]{b} \left(11 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{243 a^{14/3}}-\frac{20 \sqrt[3]{b} \left(11 \sqrt[3]{b} d-7 \sqrt[3]{a} e\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{243 a^{14/3}}+\frac{20 \sqrt[3]{b} \left(7 \sqrt[3]{a} e+11 \sqrt[3]{b} d\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{81 \sqrt{3} a^{14/3}}+\frac{4 b c \log \left(a+b x^3\right)}{3 a^5}-\frac{4 b c \log (x)}{a^5}-\frac{x \left(-\frac{234 b^2 c x^2}{a}+139 b d+118 b e x\right)}{162 a^4 \left(a+b x^3\right)}-\frac{c}{3 a^4 x^3}-\frac{d}{2 a^4 x^2}-\frac{e}{a^4 x}-\frac{x \left(-\frac{24 b^2 c x^2}{a}+17 b d+16 b e x\right)}{54 a^3 \left(a+b x^3\right)^2}-\frac{x \left(-\frac{b^2 c x^2}{a}+b d+b e x\right)}{9 a^2 \left(a+b x^3\right)^3}",1,"-1/486*((162*a*c)/x^3 + (243*a*d)/x^2 + (486*a*e)/x + (54*a^3*b*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*b*(18*c + x*(17*d + 16*e*x)))/(a + b*x^3)^2 + (3*a*b*(162*c + x*(139*d + 118*e*x)))/(a + b*x^3) - 40*Sqrt[3]*a^(1/3)*b^(1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 1944*b*c*Log[x] + 40*b^(1/3)*(11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x] - 20*b^(1/3)*(11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 648*b*c*Log[a + b*x^3])/a^5","A",1
365,1,57,29,0.0185367,"\int \frac{2 a x-x^2}{a^3+x^3} \, dx","Integrate[(2*a*x - x^2)/(a^3 + x^3),x]","\frac{1}{3} \left(-\log \left(a^3+x^3\right)+\log \left(a^2-a x+x^2\right)-2 \log (a+x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-a}{\sqrt{3} a}\right)\right)","-\log (a+x)-\frac{2 \tan ^{-1}\left(\frac{a-2 x}{\sqrt{3} a}\right)}{\sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-a + 2*x)/(Sqrt[3]*a)] - 2*Log[a + x] + Log[a^2 - a*x + x^2] - Log[a^3 + x^3])/3","A",1
366,1,57,29,0.0067644,"\int \frac{(2 a-x) x}{a^3+x^3} \, dx","Integrate[((2*a - x)*x)/(a^3 + x^3),x]","\frac{1}{3} \left(-\log \left(a^3+x^3\right)+\log \left(a^2-a x+x^2\right)-2 \log (a+x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-a}{\sqrt{3} a}\right)\right)","-\log (a+x)-\frac{2 \tan ^{-1}\left(\frac{a-2 x}{\sqrt{3} a}\right)}{\sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-a + 2*x)/(Sqrt[3]*a)] - 2*Log[a + x] + Log[a^2 - a*x + x^2] - Log[a^3 + x^3])/3","A",1
367,1,58,31,0.0146005,"\int \frac{2 a x+x^2}{a^3-x^3} \, dx","Integrate[(2*a*x + x^2)/(a^3 - x^3),x]","\frac{1}{3} \left(-\log \left(x^3-a^3\right)+\log \left(a^2+a x+x^2\right)-2 \log (x-a)-2 \sqrt{3} \tan ^{-1}\left(\frac{a+2 x}{\sqrt{3} a}\right)\right)","-\log (a-x)-\frac{2 \tan ^{-1}\left(\frac{a+2 x}{\sqrt{3} a}\right)}{\sqrt{3}}",1,"(-2*Sqrt[3]*ArcTan[(a + 2*x)/(Sqrt[3]*a)] - 2*Log[-a + x] + Log[a^2 + a*x + x^2] - Log[-a^3 + x^3])/3","A",1
368,1,58,31,0.0072046,"\int \frac{x (2 a+x)}{a^3-x^3} \, dx","Integrate[(x*(2*a + x))/(a^3 - x^3),x]","\frac{1}{3} \left(-\log \left(x^3-a^3\right)+\log \left(a^2+a x+x^2\right)-2 \log (x-a)-2 \sqrt{3} \tan ^{-1}\left(\frac{a+2 x}{\sqrt{3} a}\right)\right)","-\log (a-x)-\frac{2 \tan ^{-1}\left(\frac{a+2 x}{\sqrt{3} a}\right)}{\sqrt{3}}",1,"(-2*Sqrt[3]*ArcTan[(a + 2*x)/(Sqrt[3]*a)] - 2*Log[-a + x] + Log[a^2 + a*x + x^2] - Log[-a^3 + x^3])/3","A",1
369,1,146,50,0.0444015,"\int \frac{x \left(-2 \sqrt[3]{\frac{a}{b}} C+C x\right)}{a+b x^3} \, dx","Integrate[(x*(-2*(a/b)^(1/3)*C + C*x))/(a + b*x^3),x]","\frac{C \left(-\sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\sqrt[3]{a} \log \left(a+b x^3\right)+2 \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{a} b}","\frac{C \log \left(\sqrt[3]{\frac{a}{b}}+x\right)}{b}+\frac{2 C \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"(C*(2*Sqrt[3]*(a/b)^(1/3)*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(a/b)^(1/3)*b^(1/3)*Log[a^(1/3) + b^(1/3)*x] - (a/b)^(1/3)*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(1/3)*Log[a + b*x^3]))/(3*a^(1/3)*b)","B",1
370,1,149,53,0.0835018,"\int \frac{x \left(-2 \sqrt[3]{-\frac{a}{b}} C+C x\right)}{a-b x^3} \, dx","Integrate[(x*(-2*(-(a/b))^(1/3)*C + C*x))/(a - b*x^3),x]","-\frac{C \left(\sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\sqrt[3]{a} \log \left(a-b x^3\right)-2 \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)-2 \sqrt{3} \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{a} b}","-\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}+x\right)}{b}-\frac{2 C \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"-1/3*(C*(-2*Sqrt[3]*(-(a/b))^(1/3)*b^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(-(a/b))^(1/3)*b^(1/3)*Log[a^(1/3) - b^(1/3)*x] + (-(a/b))^(1/3)*b^(1/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(1/3)*Log[a - b*x^3]))/(a^(1/3)*b)","B",1
371,1,148,54,0.0505539,"\int \frac{x \left(2 \sqrt[3]{-\frac{a}{b}} C+C x\right)}{a+b x^3} \, dx","Integrate[(x*(2*(-(a/b))^(1/3)*C + C*x))/(a + b*x^3),x]","\frac{C \left(\sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\sqrt[3]{a} \log \left(a+b x^3\right)-2 \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \sqrt[3]{b} \sqrt[3]{-\frac{a}{b}} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{a} b}","\frac{C \log \left(\sqrt[3]{-\frac{a}{b}}-x\right)}{b}+\frac{2 C \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"(C*(-2*Sqrt[3]*(-(a/b))^(1/3)*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(-(a/b))^(1/3)*b^(1/3)*Log[a^(1/3) + b^(1/3)*x] + (-(a/b))^(1/3)*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(1/3)*Log[a + b*x^3]))/(3*a^(1/3)*b)","B",1
372,1,147,53,0.0633688,"\int \frac{x \left(2 \sqrt[3]{\frac{a}{b}} C+C x\right)}{a-b x^3} \, dx","Integrate[(x*(2*(a/b)^(1/3)*C + C*x))/(a - b*x^3),x]","-\frac{C \left(-\sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\sqrt[3]{a} \log \left(a-b x^3\right)+2 \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \log \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{b} \sqrt[3]{\frac{a}{b}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{3 \sqrt[3]{a} b}","-\frac{C \log \left(\sqrt[3]{\frac{a}{b}}-x\right)}{b}-\frac{2 C \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{\frac{a}{b}}}+1}{\sqrt{3}}\right)}{\sqrt{3} b}",1,"-1/3*(C*(2*Sqrt[3]*(a/b)^(1/3)*b^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(a/b)^(1/3)*b^(1/3)*Log[a^(1/3) - b^(1/3)*x] - (a/b)^(1/3)*b^(1/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(1/3)*Log[a - b*x^3]))/(a^(1/3)*b)","B",1
373,1,97,97,0.043407,"\int x^4 \left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^4*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{8} x^8 (a f+b c)+\frac{1}{9} x^9 (a g+b d)+\frac{1}{10} x^{10} (a h+b e)+\frac{1}{5} a c x^5+\frac{1}{6} a d x^6+\frac{1}{7} a e x^7+\frac{1}{11} b f x^{11}+\frac{1}{12} b g x^{12}+\frac{1}{13} b h x^{13}","\frac{1}{8} x^8 (a f+b c)+\frac{1}{9} x^9 (a g+b d)+\frac{1}{10} x^{10} (a h+b e)+\frac{1}{5} a c x^5+\frac{1}{6} a d x^6+\frac{1}{7} a e x^7+\frac{1}{11} b f x^{11}+\frac{1}{12} b g x^{12}+\frac{1}{13} b h x^{13}",1,"(a*c*x^5)/5 + (a*d*x^6)/6 + (a*e*x^7)/7 + ((b*c + a*f)*x^8)/8 + ((b*d + a*g)*x^9)/9 + ((b*e + a*h)*x^10)/10 + (b*f*x^11)/11 + (b*g*x^12)/12 + (b*h*x^13)/13","A",1
374,1,97,97,0.0356905,"\int x^3 \left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^3*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{7} x^7 (a f+b c)+\frac{1}{8} x^8 (a g+b d)+\frac{1}{9} x^9 (a h+b e)+\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{10} b f x^{10}+\frac{1}{11} b g x^{11}+\frac{1}{12} b h x^{12}","\frac{1}{7} x^7 (a f+b c)+\frac{1}{8} x^8 (a g+b d)+\frac{1}{9} x^9 (a h+b e)+\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{10} b f x^{10}+\frac{1}{11} b g x^{11}+\frac{1}{12} b h x^{12}",1,"(a*c*x^4)/4 + (a*d*x^5)/5 + (a*e*x^6)/6 + ((b*c + a*f)*x^7)/7 + ((b*d + a*g)*x^8)/8 + ((b*e + a*h)*x^9)/9 + (b*f*x^10)/10 + (b*g*x^11)/11 + (b*h*x^12)/12","A",1
375,1,97,97,0.0316489,"\int x^2 \left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^2*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{6} x^6 (a f+b c)+\frac{1}{7} x^7 (a g+b d)+\frac{1}{8} x^8 (a h+b e)+\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{9} b f x^9+\frac{1}{10} b g x^{10}+\frac{1}{11} b h x^{11}","\frac{1}{6} x^6 (a f+b c)+\frac{1}{7} x^7 (a g+b d)+\frac{1}{8} x^8 (a h+b e)+\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{9} b f x^9+\frac{1}{10} b g x^{10}+\frac{1}{11} b h x^{11}",1,"(a*c*x^3)/3 + (a*d*x^4)/4 + (a*e*x^5)/5 + ((b*c + a*f)*x^6)/6 + ((b*d + a*g)*x^7)/7 + ((b*e + a*h)*x^8)/8 + (b*f*x^9)/9 + (b*g*x^10)/10 + (b*h*x^11)/11","A",1
376,1,97,97,0.0268759,"\int x \left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{5} x^5 (a f+b c)+\frac{1}{6} x^6 (a g+b d)+\frac{1}{7} x^7 (a h+b e)+\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{8} b f x^8+\frac{1}{9} b g x^9+\frac{1}{10} b h x^{10}","\frac{1}{5} x^5 (a f+b c)+\frac{1}{6} x^6 (a g+b d)+\frac{1}{7} x^7 (a h+b e)+\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{8} b f x^8+\frac{1}{9} b g x^9+\frac{1}{10} b h x^{10}",1,"(a*c*x^2)/2 + (a*d*x^3)/3 + (a*e*x^4)/4 + ((b*c + a*f)*x^5)/5 + ((b*d + a*g)*x^6)/6 + ((b*e + a*h)*x^7)/7 + (b*f*x^8)/8 + (b*g*x^9)/9 + (b*h*x^10)/10","A",1
377,1,92,92,0.0150315,"\int \left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{4} x^4 (a f+b c)+\frac{1}{5} x^5 (a g+b d)+\frac{1}{6} x^6 (a h+b e)+a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9","\frac{1}{4} x^4 (a f+b c)+\frac{1}{5} x^5 (a g+b d)+\frac{1}{6} x^6 (a h+b e)+a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9",1,"a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + ((b*c + a*f)*x^4)/4 + ((b*d + a*g)*x^5)/5 + ((b*e + a*h)*x^6)/6 + (b*f*x^7)/7 + (b*g*x^8)/8 + (b*h*x^9)/9","A",1
378,1,88,88,0.0654218,"\int \frac{\left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x} \, dx","Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x,x]","\frac{1}{3} x^3 (a f+b c)+\frac{1}{4} x^4 (a g+b d)+\frac{1}{5} x^5 (a h+b e)+a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8","\frac{1}{3} x^3 (a f+b c)+\frac{1}{4} x^4 (a g+b d)+\frac{1}{5} x^5 (a h+b e)+a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} b f x^6+\frac{1}{7} b g x^7+\frac{1}{8} b h x^8",1,"a*d*x + (a*e*x^2)/2 + ((b*c + a*f)*x^3)/3 + ((b*d + a*g)*x^4)/4 + ((b*e + a*h)*x^5)/5 + (b*f*x^6)/6 + (b*g*x^7)/7 + (b*h*x^8)/8 + a*c*Log[x]","A",1
379,1,86,86,0.0730086,"\int \frac{\left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^2} \, dx","Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^2,x]","\frac{1}{2} x^2 (a f+b c)+\frac{1}{3} x^3 (a g+b d)+\frac{1}{4} x^4 (a h+b e)-\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{5} b f x^5+\frac{1}{6} b g x^6+\frac{1}{7} b h x^7","\frac{1}{2} x^2 (a f+b c)+\frac{1}{3} x^3 (a g+b d)+\frac{1}{4} x^4 (a h+b e)-\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{5} b f x^5+\frac{1}{6} b g x^6+\frac{1}{7} b h x^7",1,"-((a*c)/x) + a*e*x + ((b*c + a*f)*x^2)/2 + ((b*d + a*g)*x^3)/3 + ((b*e + a*h)*x^4)/4 + (b*f*x^5)/5 + (b*g*x^6)/6 + (b*h*x^7)/7 + a*d*Log[x]","A",1
380,1,78,86,0.0724917,"\int \frac{\left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^3} \, dx","Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]","\frac{a \left(-3 c-6 d x+6 f x^3+3 g x^4+2 h x^5\right)}{6 x^2}+a e \log (x)+b c x+\frac{1}{60} b x^2 \left(30 d+x \left(20 e+15 f x+12 g x^2+10 h x^3\right)\right)","x (a f+b c)+\frac{1}{2} x^2 (a g+b d)+\frac{1}{3} x^3 (a h+b e)-\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+\frac{1}{4} b f x^4+\frac{1}{5} b g x^5+\frac{1}{6} b h x^6",1,"b*c*x + (a*(-3*c - 6*d*x + 6*f*x^3 + 3*g*x^4 + 2*h*x^5))/(6*x^2) + (b*x^2*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3)))/60 + a*e*Log[x]","A",1
381,1,76,86,0.0694741,"\int \frac{\left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^4} \, dx","Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^4,x]","\log (x) (a f+b c)-\frac{a \left(2 c+3 x \left(d+2 e x-\left(x^3 (2 g+h x)\right)\right)\right)}{6 x^3}+\frac{1}{60} b x \left(60 d+x \left(30 e+x \left(20 f+15 g x+12 h x^2\right)\right)\right)","\log (x) (a f+b c)+x (a g+b d)+\frac{1}{2} x^2 (a h+b e)-\frac{a c}{3 x^3}-\frac{a d}{2 x^2}-\frac{a e}{x}+\frac{1}{3} b f x^3+\frac{1}{4} b g x^4+\frac{1}{5} b h x^5",1,"-1/6*(a*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/x^3 + (b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*x^2))))/60 + (b*c + a*f)*Log[x]","A",1
382,1,77,86,0.0763892,"\int \frac{\left(a+b x^3\right) \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^5} \, dx","Integrate[((a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^5,x]","\log (x) (a g+b d)-\frac{a \left(3 c+4 d x+6 x^2 \left(e+2 f x-2 h x^3\right)\right)}{12 x^4}+b \left(-\frac{c}{x}+e x+\frac{1}{12} x^2 \left(6 f+4 g x+3 h x^2\right)\right)","-\frac{a f+b c}{x}+\log (x) (a g+b d)+x (a h+b e)-\frac{a c}{4 x^4}-\frac{a d}{3 x^3}-\frac{a e}{2 x^2}+\frac{1}{2} b f x^2+\frac{1}{3} b g x^3+\frac{1}{4} b h x^4",1,"b*(-(c/x) + e*x + (x^2*(6*f + 4*g*x + 3*h*x^2))/12) - (a*(3*c + 4*d*x + 6*x^2*(e + 2*f*x - 2*h*x^3)))/(12*x^4) + (b*d + a*g)*Log[x]","A",1
383,1,163,163,0.0502168,"\int x^4 \left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^4*(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{5} a^2 c x^5+\frac{1}{6} a^2 d x^6+\frac{1}{7} a^2 e x^7+\frac{1}{11} b x^{11} (2 a f+b c)+\frac{1}{8} a x^8 (a f+2 b c)+\frac{1}{12} b x^{12} (2 a g+b d)+\frac{1}{9} a x^9 (a g+2 b d)+\frac{1}{13} b x^{13} (2 a h+b e)+\frac{1}{10} a x^{10} (a h+2 b e)+\frac{1}{14} b^2 f x^{14}+\frac{1}{15} b^2 g x^{15}+\frac{1}{16} b^2 h x^{16}","\frac{1}{5} a^2 c x^5+\frac{1}{6} a^2 d x^6+\frac{1}{7} a^2 e x^7+\frac{1}{11} b x^{11} (2 a f+b c)+\frac{1}{8} a x^8 (a f+2 b c)+\frac{1}{12} b x^{12} (2 a g+b d)+\frac{1}{9} a x^9 (a g+2 b d)+\frac{1}{13} b x^{13} (2 a h+b e)+\frac{1}{10} a x^{10} (a h+2 b e)+\frac{1}{14} b^2 f x^{14}+\frac{1}{15} b^2 g x^{15}+\frac{1}{16} b^2 h x^{16}",1,"(a^2*c*x^5)/5 + (a^2*d*x^6)/6 + (a^2*e*x^7)/7 + (a*(2*b*c + a*f)*x^8)/8 + (a*(2*b*d + a*g)*x^9)/9 + (a*(2*b*e + a*h)*x^10)/10 + (b*(b*c + 2*a*f)*x^11)/11 + (b*(b*d + 2*a*g)*x^12)/12 + (b*(b*e + 2*a*h)*x^13)/13 + (b^2*f*x^14)/14 + (b^2*g*x^15)/15 + (b^2*h*x^16)/16","A",1
384,1,163,163,0.0318673,"\int x^3 \left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^3*(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15}","\frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15}",1,"(a^2*c*x^4)/4 + (a^2*d*x^5)/5 + (a^2*e*x^6)/6 + (a*(2*b*c + a*f)*x^7)/7 + (a*(2*b*d + a*g)*x^8)/8 + (a*(2*b*e + a*h)*x^9)/9 + (b*(b*c + 2*a*f)*x^10)/10 + (b*(b*d + 2*a*g)*x^11)/11 + (b*(b*e + 2*a*h)*x^12)/12 + (b^2*f*x^13)/13 + (b^2*g*x^14)/14 + (b^2*h*x^15)/15","A",1
385,1,150,158,0.0823145,"\int x^2 \left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^2*(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","a^2 \left(\frac{c x^3}{3}+\frac{d x^4}{4}+\frac{e x^5}{5}+\frac{f x^6}{6}+\frac{g x^7}{7}+\frac{h x^8}{8}\right)+a b \left(\frac{c x^6}{3}+\frac{2 d x^7}{7}+\frac{e x^8}{4}+\frac{2 f x^9}{9}+\frac{g x^{10}}{5}+\frac{2 h x^{11}}{11}\right)+\frac{b^2 x^9 \left(20020 c+3 x \left(6006 d+5460 e x+55 x^2 \left(91 f+84 g x+78 h x^2\right)\right)\right)}{180180}","\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{6} a^2 f x^6+\frac{c \left(a+b x^3\right)^3}{9 b}+\frac{1}{10} b x^{10} (2 a g+b d)+\frac{1}{7} a x^7 (a g+2 b d)+\frac{1}{11} b x^{11} (2 a h+b e)+\frac{1}{8} a x^8 (a h+2 b e)+\frac{2}{9} a b f x^9+\frac{1}{12} b^2 f x^{12}+\frac{1}{13} b^2 g x^{13}+\frac{1}{14} b^2 h x^{14}",1,"a^2*((c*x^3)/3 + (d*x^4)/4 + (e*x^5)/5 + (f*x^6)/6 + (g*x^7)/7 + (h*x^8)/8) + a*b*((c*x^6)/3 + (2*d*x^7)/7 + (e*x^8)/4 + (2*f*x^9)/9 + (g*x^10)/5 + (2*h*x^11)/11) + (b^2*x^9*(20020*c + 3*x*(6006*d + 5460*e*x + 55*x^2*(91*f + 84*g*x + 78*h*x^2))))/180180","A",1
386,1,163,158,0.0263152,"\int x \left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x*(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{2} a^2 c x^2+\frac{1}{3} a^2 d x^3+\frac{1}{4} a^2 e x^4+\frac{1}{8} b x^8 (2 a f+b c)+\frac{1}{5} a x^5 (a f+2 b c)+\frac{1}{9} b x^9 (2 a g+b d)+\frac{1}{6} a x^6 (a g+2 b d)+\frac{1}{10} b x^{10} (2 a h+b e)+\frac{1}{7} a x^7 (a h+2 b e)+\frac{1}{11} b^2 f x^{11}+\frac{1}{12} b^2 g x^{12}+\frac{1}{13} b^2 h x^{13}","\frac{1}{2} a^2 c x^2+\frac{1}{4} a^2 e x^4+\frac{1}{6} a^2 g x^6+\frac{1}{8} b x^8 (2 a f+b c)+\frac{1}{5} a x^5 (a f+2 b c)+\frac{d \left(a+b x^3\right)^3}{9 b}+\frac{1}{10} b x^{10} (2 a h+b e)+\frac{1}{7} a x^7 (a h+2 b e)+\frac{2}{9} a b g x^9+\frac{1}{11} b^2 f x^{11}+\frac{1}{12} b^2 g x^{12}+\frac{1}{13} b^2 h x^{13}",1,"(a^2*c*x^2)/2 + (a^2*d*x^3)/3 + (a^2*e*x^4)/4 + (a*(2*b*c + a*f)*x^5)/5 + (a*(2*b*d + a*g)*x^6)/6 + (a*(2*b*e + a*h)*x^7)/7 + (b*(b*c + 2*a*f)*x^8)/8 + (b*(b*d + 2*a*g)*x^9)/9 + (b*(b*e + 2*a*h)*x^10)/10 + (b^2*f*x^11)/11 + (b^2*g*x^12)/12 + (b^2*h*x^13)/13","A",1
387,1,125,153,0.0870191,"\int \left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{462 a^2 x \left(60 c+x \left(30 d+x \left(20 e+15 f x+12 g x^2+10 h x^3\right)\right)\right)+22 a b x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+b^2 x^7 \left(3960 c+7 x \left(495 d+440 e x+6 x^2 \left(66 f+60 g x+55 h x^2\right)\right)\right)}{27720}","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{6} a^2 h x^6+\frac{1}{7} b x^7 (2 a f+b c)+\frac{1}{4} a x^4 (a f+2 b c)+\frac{1}{8} b x^8 (2 a g+b d)+\frac{1}{5} a x^5 (a g+2 b d)+\frac{e \left(a+b x^3\right)^3}{9 b}+\frac{2}{9} a b h x^9+\frac{1}{10} b^2 f x^{10}+\frac{1}{11} b^2 g x^{11}+\frac{1}{12} b^2 h x^{12}",1,"(b^2*x^7*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2))) + 462*a^2*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))) + 22*a*b*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/27720","A",1
388,1,154,149,0.0462183,"\int \frac{\left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x} \, dx","Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x,x]","a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{6} b x^6 (2 a f+b c)+\frac{1}{3} a x^3 (a f+2 b c)+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{1}{9} b^2 f x^9+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11}","a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{f \left(a+b x^3\right)^3}{9 b}+\frac{1}{6} b^2 c x^6+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11}",1,"a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*c + a*f)*x^3)/3 + (a*(2*b*d + a*g)*x^4)/4 + (a*(2*b*e + a*h)*x^5)/5 + (b*(b*c + 2*a*f)*x^6)/6 + (b*(b*d + 2*a*g)*x^7)/7 + (b*(b*e + 2*a*h)*x^8)/8 + (b^2*f*x^9)/9 + (b^2*g*x^10)/10 + (b^2*h*x^11)/11 + a^2*c*Log[x]","A",1
389,1,152,147,0.0704055,"\int \frac{\left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^2} \, dx","Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^2,x]","-\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+\frac{1}{5} b x^5 (2 a f+b c)+\frac{1}{2} a x^2 (a f+2 b c)+\frac{1}{6} b x^6 (2 a g+b d)+\frac{1}{3} a x^3 (a g+2 b d)+\frac{1}{7} b x^7 (2 a h+b e)+\frac{1}{4} a x^4 (a h+2 b e)+\frac{1}{8} b^2 f x^8+\frac{1}{9} b^2 g x^9+\frac{1}{10} b^2 h x^{10}","-\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+\frac{1}{5} b x^5 (2 a f+b c)+\frac{1}{2} a x^2 (a f+2 b c)+\frac{2}{3} a b d x^3+\frac{1}{7} b x^7 (2 a h+b e)+\frac{1}{4} a x^4 (a h+2 b e)+\frac{g \left(a+b x^3\right)^3}{9 b}+\frac{1}{6} b^2 d x^6+\frac{1}{8} b^2 f x^8+\frac{1}{10} b^2 h x^{10}",1,"-((a^2*c)/x) + a^2*e*x + (a*(2*b*c + a*f)*x^2)/2 + (a*(2*b*d + a*g)*x^3)/3 + (a*(2*b*e + a*h)*x^4)/4 + (b*(b*c + 2*a*f)*x^5)/5 + (b*(b*d + 2*a*g)*x^6)/6 + (b*(b*e + 2*a*h)*x^7)/7 + (b^2*f*x^8)/8 + (b^2*g*x^9)/9 + (b^2*h*x^10)/10 + a^2*d*Log[x]","A",1
390,1,127,147,0.0951185,"\int \frac{\left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^3} \, dx","Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]","\frac{a^2 \left(-3 c-6 d x+x^3 \left(6 f+3 g x+2 h x^2\right)\right)}{6 x^2}+a^2 e \log (x)+\frac{1}{30} a b x \left(60 c+x \left(30 d+x \left(20 e+15 f x+12 g x^2+10 h x^3\right)\right)\right)+\frac{b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))}{2520}","-\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+\frac{1}{4} b x^4 (2 a f+b c)+a x (a f+2 b c)+\frac{1}{5} b x^5 (2 a g+b d)+\frac{1}{2} a x^2 (a g+2 b d)+\frac{2}{3} a b e x^3+\frac{h \left(a+b x^3\right)^3}{9 b}+\frac{1}{6} b^2 e x^6+\frac{1}{7} b^2 f x^7+\frac{1}{8} b^2 g x^8",1,"(a^2*(-3*c - 6*d*x + x^3*(6*f + 3*g*x + 2*h*x^2)))/(6*x^2) + (a*b*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))))/30 + (b^2*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/2520 + a^2*e*Log[x]","A",1
391,1,123,152,0.100511,"\int \frac{\left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^4} \, dx","Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^4,x]","-\frac{a^2 \left(2 c+3 x \left(d+2 e x-\left(x^3 (2 g+h x)\right)\right)\right)}{6 x^3}+a \log (x) (a f+2 b c)+\frac{1}{30} a b x \left(60 d+x \left(30 e+x \left(20 f+15 g x+12 h x^2\right)\right)\right)+\frac{1}{840} b^2 x^3 \left(280 c+x \left(210 d+x \left(168 e+140 f x+120 g x^2+105 h x^3\right)\right)\right)","-\frac{a^2 c}{3 x^3}-\frac{a^2 d}{2 x^2}-\frac{a^2 e}{x}+\frac{1}{3} b x^3 (2 a f+b c)+a \log (x) (a f+2 b c)+\frac{1}{4} b x^4 (2 a g+b d)+a x (a g+2 b d)+\frac{1}{5} b x^5 (2 a h+b e)+\frac{1}{2} a x^2 (a h+2 b e)+\frac{1}{6} b^2 f x^6+\frac{1}{7} b^2 g x^7+\frac{1}{8} b^2 h x^8",1,"-1/6*(a^2*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/x^3 + (a*b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*x^2))))/30 + (b^2*x^3*(280*c + x*(210*d + x*(168*e + 140*f*x + 120*g*x^2 + 105*h*x^3))))/840 + a*(2*b*c + a*f)*Log[x]","A",1
392,1,125,152,0.1151289,"\int \frac{\left(a+b x^3\right)^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^5} \, dx","Integrate[((a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^5,x]","-\frac{a^2 \left(3 c+4 d x+6 x^2 \left(e+2 f x-2 h x^3\right)\right)}{12 x^4}-\frac{2 a b c}{x}+a \log (x) (a g+2 b d)+\frac{1}{6} a b x (12 e+x (6 f+x (4 g+3 h x)))+\frac{1}{420} b^2 x^2 \left(210 c+x \left(140 d+x \left(105 e+84 f x+70 g x^2+60 h x^3\right)\right)\right)","-\frac{a^2 c}{4 x^4}-\frac{a^2 d}{3 x^3}-\frac{a^2 e}{2 x^2}+\frac{1}{2} b x^2 (2 a f+b c)-\frac{a (a f+2 b c)}{x}+\frac{1}{3} b x^3 (2 a g+b d)+a \log (x) (a g+2 b d)+\frac{1}{4} b x^4 (2 a h+b e)+a x (a h+2 b e)+\frac{1}{5} b^2 f x^5+\frac{1}{6} b^2 g x^6+\frac{1}{7} b^2 h x^7",1,"(-2*a*b*c)/x - (a^2*(3*c + 4*d*x + 6*x^2*(e + 2*f*x - 2*h*x^3)))/(12*x^4) + (a*b*x*(12*e + x*(6*f + x*(4*g + 3*h*x))))/6 + (b^2*x^2*(210*c + x*(140*d + x*(105*e + 84*f*x + 70*g*x^2 + 60*h*x^3))))/420 + a*(2*b*d + a*g)*Log[x]","A",1
393,1,223,223,0.0618809,"\int x^4 \left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^4*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{5} a^3 c x^5+\frac{1}{6} a^3 d x^6+\frac{1}{7} a^3 e x^7+\frac{1}{8} a^2 x^8 (a f+3 b c)+\frac{1}{9} a^2 x^9 (a g+3 b d)+\frac{1}{10} a^2 x^{10} (a h+3 b e)+\frac{1}{14} b^2 x^{14} (3 a f+b c)+\frac{1}{15} b^2 x^{15} (3 a g+b d)+\frac{1}{16} b^2 x^{16} (3 a h+b e)+\frac{3}{11} a b x^{11} (a f+b c)+\frac{1}{4} a b x^{12} (a g+b d)+\frac{3}{13} a b x^{13} (a h+b e)+\frac{1}{17} b^3 f x^{17}+\frac{1}{18} b^3 g x^{18}+\frac{1}{19} b^3 h x^{19}","\frac{1}{5} a^3 c x^5+\frac{1}{6} a^3 d x^6+\frac{1}{7} a^3 e x^7+\frac{1}{8} a^2 x^8 (a f+3 b c)+\frac{1}{9} a^2 x^9 (a g+3 b d)+\frac{1}{10} a^2 x^{10} (a h+3 b e)+\frac{1}{14} b^2 x^{14} (3 a f+b c)+\frac{1}{15} b^2 x^{15} (3 a g+b d)+\frac{1}{16} b^2 x^{16} (3 a h+b e)+\frac{3}{11} a b x^{11} (a f+b c)+\frac{1}{4} a b x^{12} (a g+b d)+\frac{3}{13} a b x^{13} (a h+b e)+\frac{1}{17} b^3 f x^{17}+\frac{1}{18} b^3 g x^{18}+\frac{1}{19} b^3 h x^{19}",1,"(a^3*c*x^5)/5 + (a^3*d*x^6)/6 + (a^3*e*x^7)/7 + (a^2*(3*b*c + a*f)*x^8)/8 + (a^2*(3*b*d + a*g)*x^9)/9 + (a^2*(3*b*e + a*h)*x^10)/10 + (3*a*b*(b*c + a*f)*x^11)/11 + (a*b*(b*d + a*g)*x^12)/4 + (3*a*b*(b*e + a*h)*x^13)/13 + (b^2*(b*c + 3*a*f)*x^14)/14 + (b^2*(b*d + 3*a*g)*x^15)/15 + (b^2*(b*e + 3*a*h)*x^16)/16 + (b^3*f*x^17)/17 + (b^3*g*x^18)/18 + (b^3*h*x^19)/19","A",1
394,1,223,223,0.0546927,"\int x^3 \left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^3*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^2 x^7 (a f+3 b c)+\frac{1}{8} a^2 x^8 (a g+3 b d)+\frac{1}{9} a^2 x^9 (a h+3 b e)+\frac{1}{13} b^2 x^{13} (3 a f+b c)+\frac{1}{14} b^2 x^{14} (3 a g+b d)+\frac{1}{15} b^2 x^{15} (3 a h+b e)+\frac{3}{10} a b x^{10} (a f+b c)+\frac{3}{11} a b x^{11} (a g+b d)+\frac{1}{4} a b x^{12} (a h+b e)+\frac{1}{16} b^3 f x^{16}+\frac{1}{17} b^3 g x^{17}+\frac{1}{18} b^3 h x^{18}","\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^2 x^7 (a f+3 b c)+\frac{1}{8} a^2 x^8 (a g+3 b d)+\frac{1}{9} a^2 x^9 (a h+3 b e)+\frac{1}{13} b^2 x^{13} (3 a f+b c)+\frac{1}{14} b^2 x^{14} (3 a g+b d)+\frac{1}{15} b^2 x^{15} (3 a h+b e)+\frac{3}{10} a b x^{10} (a f+b c)+\frac{3}{11} a b x^{11} (a g+b d)+\frac{1}{4} a b x^{12} (a h+b e)+\frac{1}{16} b^3 f x^{16}+\frac{1}{17} b^3 g x^{17}+\frac{1}{18} b^3 h x^{18}",1,"(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^2*(3*b*c + a*f)*x^7)/7 + (a^2*(3*b*d + a*g)*x^8)/8 + (a^2*(3*b*e + a*h)*x^9)/9 + (3*a*b*(b*c + a*f)*x^10)/10 + (3*a*b*(b*d + a*g)*x^11)/11 + (a*b*(b*e + a*h)*x^12)/4 + (b^2*(b*c + 3*a*f)*x^13)/13 + (b^2*(b*d + 3*a*g)*x^14)/14 + (b^2*(b*e + 3*a*h)*x^15)/15 + (b^3*f*x^16)/16 + (b^3*g*x^17)/17 + (b^3*h*x^18)/18","A",1
395,1,223,212,0.0628648,"\int x^2 \left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x^2*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{6} a^2 x^6 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a g+3 b d)+\frac{1}{8} a^2 x^8 (a h+3 b e)+\frac{1}{12} b^2 x^{12} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a g+b d)+\frac{1}{14} b^2 x^{14} (3 a h+b e)+\frac{1}{3} a b x^9 (a f+b c)+\frac{3}{10} a b x^{10} (a g+b d)+\frac{3}{11} a b x^{11} (a h+b e)+\frac{1}{15} b^3 f x^{15}+\frac{1}{16} b^3 g x^{16}+\frac{1}{17} b^3 h x^{17}","\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{6} a^3 f x^6+\frac{1}{7} a^2 x^7 (a g+3 b d)+\frac{1}{8} a^2 x^8 (a h+3 b e)+\frac{1}{3} a^2 b f x^9+\frac{1}{13} b^2 x^{13} (3 a g+b d)+\frac{1}{14} b^2 x^{14} (3 a h+b e)+\frac{1}{4} a b^2 f x^{12}+\frac{c \left(a+b x^3\right)^4}{12 b}+\frac{3}{10} a b x^{10} (a g+b d)+\frac{3}{11} a b x^{11} (a h+b e)+\frac{1}{15} b^3 f x^{15}+\frac{1}{16} b^3 g x^{16}+\frac{1}{17} b^3 h x^{17}",1,"(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*(3*b*c + a*f)*x^6)/6 + (a^2*(3*b*d + a*g)*x^7)/7 + (a^2*(3*b*e + a*h)*x^8)/8 + (a*b*(b*c + a*f)*x^9)/3 + (3*a*b*(b*d + a*g)*x^10)/10 + (3*a*b*(b*e + a*h)*x^11)/11 + (b^2*(b*c + 3*a*f)*x^12)/12 + (b^2*(b*d + 3*a*g)*x^13)/13 + (b^2*(b*e + 3*a*h)*x^14)/14 + (b^3*f*x^15)/15 + (b^3*g*x^16)/16 + (b^3*h*x^17)/17","A",1
396,1,223,212,0.037355,"\int x \left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[x*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{6} a^2 x^6 (a g+3 b d)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{12} b^2 x^{12} (3 a g+b d)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{3}{8} a b x^8 (a f+b c)+\frac{1}{3} a b x^9 (a g+b d)+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16}","\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{6} a^3 g x^6+\frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{3} a^2 b g x^9+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{1}{4} a b^2 g x^{12}+\frac{3}{8} a b x^8 (a f+b c)+\frac{d \left(a+b x^3\right)^4}{12 b}+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16}",1,"(a^3*c*x^2)/2 + (a^3*d*x^3)/3 + (a^3*e*x^4)/4 + (a^2*(3*b*c + a*f)*x^5)/5 + (a^2*(3*b*d + a*g)*x^6)/6 + (a^2*(3*b*e + a*h)*x^7)/7 + (3*a*b*(b*c + a*f)*x^8)/8 + (a*b*(b*d + a*g)*x^9)/3 + (3*a*b*(b*e + a*h)*x^10)/10 + (b^2*(b*c + 3*a*f)*x^11)/11 + (b^2*(b*d + 3*a*g)*x^12)/12 + (b^2*(b*e + 3*a*h)*x^13)/13 + (b^3*f*x^14)/14 + (b^3*g*x^15)/15 + (b^3*h*x^16)/16","A",1
397,1,170,207,0.1086005,"\int \left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right) \, dx","Integrate[(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]","\frac{x \left(2002 a^3 \left(60 c+x \left(30 d+x \left(20 e+15 f x+12 g x^2+10 h x^3\right)\right)\right)+143 a^2 b x^3 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+13 a b^2 x^6 \left(3960 c+7 x \left(495 d+440 e x+6 x^2 \left(66 f+60 g x+55 h x^2\right)\right)\right)+2 b^3 x^9 \left(6006 c+x \left(5460 d+11 x \left(455 e+420 f x+390 g x^2+364 h x^3\right)\right)\right)\right)}{120120}","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{6} a^3 h x^6+\frac{1}{4} a^2 x^4 (a f+3 b c)+\frac{1}{5} a^2 x^5 (a g+3 b d)+\frac{1}{3} a^2 b h x^9+\frac{1}{10} b^2 x^{10} (3 a f+b c)+\frac{1}{11} b^2 x^{11} (3 a g+b d)+\frac{1}{4} a b^2 h x^{12}+\frac{3}{7} a b x^7 (a f+b c)+\frac{3}{8} a b x^8 (a g+b d)+\frac{e \left(a+b x^3\right)^4}{12 b}+\frac{1}{13} b^3 f x^{13}+\frac{1}{14} b^3 g x^{14}+\frac{1}{15} b^3 h x^{15}",1,"(x*(13*a*b^2*x^6*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2))) + 2002*a^3*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))) + 2*b^3*x^9*(6006*c + x*(5460*d + 11*x*(455*e + 420*f*x + 390*g*x^2 + 364*h*x^3))) + 143*a^2*b*x^3*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x)))))))/120120","A",1
398,1,214,200,0.1345584,"\int \frac{\left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x} \, dx","Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x,x]","a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{3} a^2 x^3 (a f+3 b c)+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+\frac{1}{9} b^2 x^9 (3 a f+b c)+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{1}{2} a b x^6 (a f+b c)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{1}{12} b^3 f x^{12}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14}","a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left(a+b x^3\right)^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14}",1,"a^3*d*x + (a^3*e*x^2)/2 + (a^2*(3*b*c + a*f)*x^3)/3 + (a^2*(3*b*d + a*g)*x^4)/4 + (a^2*(3*b*e + a*h)*x^5)/5 + (a*b*(b*c + a*f)*x^6)/2 + (3*a*b*(b*d + a*g)*x^7)/7 + (3*a*b*(b*e + a*h)*x^8)/8 + (b^2*(b*c + 3*a*f)*x^9)/9 + (b^2*(b*d + 3*a*g)*x^10)/10 + (b^2*(b*e + 3*a*h)*x^11)/11 + (b^3*f*x^12)/12 + (b^3*g*x^13)/13 + (b^3*h*x^14)/14 + a^3*c*Log[x]","A",1
399,1,172,198,0.2075213,"\int \frac{\left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^2} \, dx","Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^2,x]","a^3 \left(-\frac{c}{x}+e x+\frac{1}{12} x^2 \left(6 f+4 g x+3 h x^2\right)\right)+a^3 d \log (x)+\frac{1}{140} a^2 b x^2 \left(210 c+x \left(140 d+x \left(105 e+84 f x+70 g x^2+60 h x^3\right)\right)\right)+\frac{1}{840} a b^2 x^5 \left(504 c+x \left(420 d+x \left(360 e+315 f x+280 g x^2+252 h x^3\right)\right)\right)+\frac{b^3 x^8 \left(6435 c+5720 d x+6 x^2 \left(858 e+780 f x+715 g x^2+660 h x^3\right)\right)}{51480}","-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{1}{2} a^2 x^2 (a f+3 b c)+a^2 b d x^3+\frac{1}{4} a^2 x^4 (a h+3 b e)+\frac{1}{8} b^2 x^8 (3 a f+b c)+\frac{1}{2} a b^2 d x^6+\frac{1}{10} b^2 x^{10} (3 a h+b e)+\frac{3}{5} a b x^5 (a f+b c)+\frac{3}{7} a b x^7 (a h+b e)+\frac{g \left(a+b x^3\right)^4}{12 b}+\frac{1}{9} b^3 d x^9+\frac{1}{11} b^3 f x^{11}+\frac{1}{13} b^3 h x^{13}",1,"a^3*(-(c/x) + e*x + (x^2*(6*f + 4*g*x + 3*h*x^2))/12) + (b^3*x^8*(6435*c + 5720*d*x + 6*x^2*(858*e + 780*f*x + 715*g*x^2 + 660*h*x^3)))/51480 + (a^2*b*x^2*(210*c + x*(140*d + x*(105*e + 84*f*x + 70*g*x^2 + 60*h*x^3))))/140 + (a*b^2*x^5*(504*c + x*(420*d + x*(360*e + 315*f*x + 280*g*x^2 + 252*h*x^3))))/840 + a^3*d*Log[x]","A",1
400,1,174,198,0.1484133,"\int \frac{\left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^3} \, dx","Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^3,x]","\frac{a^3 \left(-3 c-6 d x+x^3 \left(6 f+3 g x+2 h x^2\right)\right)}{6 x^2}+a^3 e \log (x)+\frac{1}{20} a^2 b x \left(60 c+x \left(30 d+x \left(20 e+15 f x+12 g x^2+10 h x^3\right)\right)\right)+\frac{1}{840} a b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+\frac{b^3 x^7 \left(3960 c+7 x \left(495 d+440 e x+6 x^2 \left(66 f+60 g x+55 h x^2\right)\right)\right)}{27720}","-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+a^2 x (a f+3 b c)+\frac{1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3+\frac{1}{7} b^2 x^7 (3 a f+b c)+\frac{1}{8} b^2 x^8 (3 a g+b d)+\frac{1}{2} a b^2 e x^6+\frac{3}{4} a b x^4 (a f+b c)+\frac{3}{5} a b x^5 (a g+b d)+\frac{h \left(a+b x^3\right)^4}{12 b}+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11}",1,"(a^3*(-3*c - 6*d*x + x^3*(6*f + 3*g*x + 2*h*x^2)))/(6*x^2) + (b^3*x^7*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2))))/27720 + (a^2*b*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))))/20 + (a*b^2*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/840 + a^3*e*Log[x]","A",1
401,1,172,209,0.1461326,"\int \frac{\left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^4} \, dx","Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^4,x]","-\frac{a^3 \left(2 c+3 x \left(d+2 e x-\left(x^3 (2 g+h x)\right)\right)\right)}{6 x^3}+a^2 \log (x) (a f+3 b c)+\frac{1}{20} a^2 b x \left(60 d+x \left(30 e+x \left(20 f+15 g x+12 h x^2\right)\right)\right)+\frac{1}{280} a b^2 x^3 \left(280 c+x \left(210 d+x \left(168 e+140 f x+120 g x^2+105 h x^3\right)\right)\right)+\frac{b^3 x^6 \left(4620 c+x \left(3960 d+7 x \left(495 e+4 x \left(110 f+99 g x+90 h x^2\right)\right)\right)\right)}{27720}","-\frac{a^3 c}{3 x^3}-\frac{a^3 d}{2 x^2}-\frac{a^3 e}{x}+a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac{1}{2} a^2 x^2 (a h+3 b e)+\frac{1}{6} b^2 x^6 (3 a f+b c)+\frac{1}{7} b^2 x^7 (3 a g+b d)+\frac{1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac{3}{4} a b x^4 (a g+b d)+\frac{3}{5} a b x^5 (a h+b e)+\frac{1}{9} b^3 f x^9+\frac{1}{10} b^3 g x^{10}+\frac{1}{11} b^3 h x^{11}",1,"-1/6*(a^3*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/x^3 + (a^2*b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*x^2))))/20 + (a*b^2*x^3*(280*c + x*(210*d + x*(168*e + 140*f*x + 120*g*x^2 + 105*h*x^3))))/280 + (b^3*x^6*(4620*c + x*(3960*d + 7*x*(495*e + 4*x*(110*f + 99*g*x + 90*h*x^2)))))/27720 + a^2*(3*b*c + a*f)*Log[x]","A",1
402,1,170,209,0.1598166,"\int \frac{\left(a+b x^3\right)^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{x^5} \, dx","Integrate[((a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/x^5,x]","a^2 \log (x) (a g+3 b d)+\frac{-210 a^3 \left(3 c+4 d x+6 x^2 \left(e+2 f x-2 h x^3\right)\right)+630 a^2 b x^3 \left(x^2 \left(12 e+6 f x+4 g x^2+3 h x^3\right)-12 c\right)+18 a b^2 x^6 \left(210 c+x \left(140 d+105 e x+84 f x^2+70 g x^3+60 h x^4\right)\right)+b^3 x^9 \left(504 c+x \left(420 d+360 e x+315 f x^2+280 g x^3+252 h x^4\right)\right)}{2520 x^4}","-\frac{a^3 c}{4 x^4}-\frac{a^3 d}{3 x^3}-\frac{a^3 e}{2 x^2}-\frac{a^2 (a f+3 b c)}{x}+a^2 \log (x) (a g+3 b d)+a^2 x (a h+3 b e)+\frac{1}{5} b^2 x^5 (3 a f+b c)+\frac{1}{6} b^2 x^6 (3 a g+b d)+\frac{1}{7} b^2 x^7 (3 a h+b e)+\frac{3}{2} a b x^2 (a f+b c)+a b x^3 (a g+b d)+\frac{3}{4} a b x^4 (a h+b e)+\frac{1}{8} b^3 f x^8+\frac{1}{9} b^3 g x^9+\frac{1}{10} b^3 h x^{10}",1,"(-210*a^3*(3*c + 4*d*x + 6*x^2*(e + 2*f*x - 2*h*x^3)) + 630*a^2*b*x^3*(-12*c + x^2*(12*e + 6*f*x + 4*g*x^2 + 3*h*x^3)) + 18*a*b^2*x^6*(210*c + x*(140*d + 105*e*x + 84*f*x^2 + 70*g*x^3 + 60*h*x^4)) + b^3*x^9*(504*c + x*(420*d + 360*e*x + 315*f*x^2 + 280*g*x^3 + 252*h*x^4)))/(2520*x^4) + a^2*(3*b*d + a*g)*Log[x]","A",1
403,1,334,331,0.5582773,"\int \frac{x^4 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{a+b x^3} \, dx","Integrate[(x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]","\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right)}{6 b^{10/3}}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right)}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{\sqrt{3} b^{10/3}}+\frac{a (a g-b d) \log \left(a+b x^3\right)}{3 b^3}+\frac{a x (a h-b e)}{b^3}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}","-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{6 b^{10/3}}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{\sqrt{3} b^{10/3}}-\frac{a (b d-a g) \log \left(a+b x^3\right)}{3 b^3}-\frac{a x (b e-a h)}{b^3}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}",1,"(a*(-(b*e) + a*h)*x)/b^3 + ((b*c - a*f)*x^2)/(2*b^2) + ((b*d - a*g)*x^3)/(3*b^2) + ((b*e - a*h)*x^4)/(4*b^2) + (f*x^5)/(5*b) + (g*x^6)/(6*b) + (h*x^7)/(7*b) + (a^(2/3)*(b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(10/3)) + (a^(2/3)*(b^(5/3)*c + a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/3)) + (a^(2/3)*(-(b^(5/3)*c) - a^(2/3)*b*e + a*b^(2/3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3)) + (a*(-(b*d) + a*g)*Log[a + b*x^3])/(3*b^3)","A",1
404,1,299,313,0.2920525,"\int \frac{x^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{a+b x^3} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]","\frac{10 \sqrt[3]{a} \sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)-20 \sqrt[3]{a} \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)-20 \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} g-\sqrt[3]{a} b d+a \sqrt[3]{b} f-b^{4/3} c\right)+60 b x (b c-a f)+30 b x^2 (b d-a g)+20 b x^3 (b e-a h)+20 a (a h-b e) \log \left(a+b x^3\right)+15 b^2 f x^4+12 b^2 g x^5+10 b^2 h x^6}{60 b^3}","\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{6 b^{8/3}}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-g)+\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{\sqrt{3} b^{8/3}}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{3 b^{8/3}}-\frac{a (b e-a h) \log \left(a+b x^3\right)}{3 b^3}+\frac{x (b c-a f)}{b^2}+\frac{x^2 (b d-a g)}{2 b^2}+\frac{x^3 (b e-a h)}{3 b^2}+\frac{f x^4}{4 b}+\frac{g x^5}{5 b}+\frac{h x^6}{6 b}",1,"(60*b*(b*c - a*f)*x + 30*b*(b*d - a*g)*x^2 + 20*b*(b*e - a*h)*x^3 + 15*b^2*f*x^4 + 12*b^2*g*x^5 + 10*b^2*h*x^6 - 20*Sqrt[3]*a^(1/3)*b^(1/3)*(-(b^(4/3)*c) - a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 20*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + 10*a^(1/3)*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*a*(-(b*e) + a*h)*Log[a + b*x^3])/(60*b^3)","A",1
405,1,290,294,0.3128849,"\int \frac{x^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{a+b x^3} \, dx","Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]","\frac{10 \sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} h-\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)+20 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} (-h)+\sqrt[3]{a} b e+a \sqrt[3]{b} g-b^{4/3} d\right)-20 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} h-\sqrt[3]{a} b e+a \sqrt[3]{b} g-b^{4/3} d\right)+20 b^{2/3} (b c-a f) \log \left(a+b x^3\right)+60 b^{2/3} x (b d-a g)+30 b^{2/3} x^2 (b e-a h)+20 b^{5/3} f x^3+15 b^{5/3} g x^4+12 b^{5/3} h x^5}{60 b^{8/3}}","\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{6 b^{8/3}}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-h)+\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)}{\sqrt{3} b^{8/3}}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{3 b^{8/3}}+\frac{(b c-a f) \log \left(a+b x^3\right)}{3 b^2}+\frac{x (b d-a g)}{b^2}+\frac{x^2 (b e-a h)}{2 b^2}+\frac{f x^3}{3 b}+\frac{g x^4}{4 b}+\frac{h x^5}{5 b}",1,"(60*b^(2/3)*(b*d - a*g)*x + 30*b^(2/3)*(b*e - a*h)*x^2 + 20*b^(5/3)*f*x^3 + 15*b^(5/3)*g*x^4 + 12*b^(5/3)*h*x^5 - 20*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 20*a^(1/3)*(-(b^(4/3)*d) + a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x] + 10*a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*b^(2/3)*(b*c - a*f)*Log[a + b*x^3])/(60*b^(8/3))","A",1
406,1,272,275,0.4798703,"\int \frac{x \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{a+b x^3} \, dx","Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3),x]","\frac{\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right)}{\sqrt[3]{a}}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right)}{\sqrt[3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{\sqrt[3]{a}}+4 \sqrt[3]{b} (b d-a g) \log \left(a+b x^3\right)+12 \sqrt[3]{b} x (b e-a h)+6 b^{4/3} f x^2+4 b^{4/3} g x^3+3 b^{4/3} h x^4}{12 b^{7/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{6 \sqrt[3]{a} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{3 \sqrt[3]{a} b^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{\sqrt{3} \sqrt[3]{a} b^{7/3}}+\frac{(b d-a g) \log \left(a+b x^3\right)}{3 b^2}+\frac{x (b e-a h)}{b^2}+\frac{f x^2}{2 b}+\frac{g x^3}{3 b}+\frac{h x^4}{4 b}",1,"(12*b^(1/3)*(b*e - a*h)*x + 6*b^(4/3)*f*x^2 + 4*b^(4/3)*g*x^3 + 3*b^(4/3)*h*x^4 - (4*Sqrt[3]*(b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-(b^(5/3)*c) - a^(2/3)*b*e + a*b^(2/3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(b^(5/3)*c + a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 4*b^(1/3)*(b*d - a*g)*Log[a + b*x^3])/(12*b^(7/3))","A",1
407,1,254,259,0.3883275,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3),x]","\frac{-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{2/3}}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} g-\sqrt[3]{a} b d+a \sqrt[3]{b} f-b^{4/3} c\right)}{a^{2/3}}+\frac{2 (b e-a h) \log \left(a+b x^3\right)}{\sqrt[3]{b}}+6 b^{2/3} f x+3 b^{2/3} g x^2+2 b^{2/3} h x^3}{6 b^{5/3}}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{6 a^{2/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-g)+\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{\sqrt{3} a^{2/3} b^{5/3}}+\frac{(b e-a h) \log \left(a+b x^3\right)}{3 b^2}+\frac{f x}{b}+\frac{g x^2}{2 b}+\frac{h x^3}{3 b}",1,"(6*b^(2/3)*f*x + 3*b^(2/3)*g*x^2 + 2*b^(2/3)*h*x^3 + (2*Sqrt[3]*(-(b^(4/3)*c) - a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - ((b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) + (2*(b*e - a*h)*Log[a + b*x^3])/b^(1/3))/(6*b^(5/3))","A",1
408,1,258,258,0.3144758,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)),x]","\frac{-\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} h-\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)+2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} h-\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)+2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} h-\sqrt[3]{a} b e+a \sqrt[3]{b} g-b^{4/3} d\right)-2 b^{2/3} (b c-a f) \log \left(a+b x^3\right)+6 a b^{2/3} g x+3 a b^{2/3} h x^2+6 b^{5/3} c \log (x)}{6 a b^{5/3}}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{6 a^{2/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{3 a^{2/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-h)+\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)}{\sqrt{3} a^{2/3} b^{5/3}}-\frac{(b c-a f) \log \left(a+b x^3\right)}{3 a b}+\frac{c \log (x)}{a}+\frac{g x}{b}+\frac{h x^2}{2 b}",1,"(6*a*b^(2/3)*g*x + 3*a*b^(2/3)*h*x^2 + 2*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 6*b^(5/3)*c*Log[x] + 2*a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x] - a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*b^(2/3)*(b*c - a*f)*Log[a + b*x^3])/(6*a*b^(5/3))","A",1
409,1,257,253,0.321421,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^2*(a + b*x^3)),x]","\frac{1}{6} \left(\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right)}{a^{4/3} b^{4/3}}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right)}{a^{4/3} b^{4/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{a^{4/3} b^{4/3}}+\frac{2 (a g-b d) \log \left(a+b x^3\right)}{a b}-\frac{6 c}{a x}+\frac{6 d \log (x)}{a}+\frac{6 h x}{b}\right)","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{6 a^{4/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right)}{3 a^{4/3} b^{4/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right)}{\sqrt{3} a^{4/3} b^{4/3}}-\frac{(b d-a g) \log \left(a+b x^3\right)}{3 a b}-\frac{c}{a x}+\frac{d \log (x)}{a}+\frac{h x}{b}",1,"((-6*c)/(a*x) + (6*h*x)/b + (2*Sqrt[3]*(b^(5/3)*c - a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(4/3)*b^(4/3)) + (6*d*Log[x])/a + (2*(b^(5/3)*c + a^(2/3)*b*e - a*b^(2/3)*f - a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/(a^(4/3)*b^(4/3)) + ((-(b^(5/3)*c) - a^(2/3)*b*e + a*b^(2/3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(4/3)*b^(4/3)) + (2*(-(b*d) + a*g)*Log[a + b*x^3])/(a*b))/6","A",1
410,1,257,260,0.4635685,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)),x]","\frac{\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{b^{2/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} g-\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{b^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} (-g)+\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{b^{2/3}}+\frac{2 a^{2/3} (a h-b e) \log \left(a+b x^3\right)}{b}-\frac{3 a^{2/3} c}{x^2}-\frac{6 a^{2/3} d}{x}+6 a^{2/3} e \log (x)}{6 a^{5/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{6 a^{5/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b c-a f)-\sqrt[3]{a} (b d-a g)\right)}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-g)+\sqrt[3]{a} b d-a \sqrt[3]{b} f+b^{4/3} c\right)}{\sqrt{3} a^{5/3} b^{2/3}}-\frac{(b e-a h) \log \left(a+b x^3\right)}{3 a b}-\frac{c}{2 a x^2}-\frac{d}{a x}+\frac{e \log (x)}{a}",1,"((-3*a^(2/3)*c)/x^2 - (6*a^(2/3)*d)/x + (2*Sqrt[3]*(b^(4/3)*c + a^(1/3)*b*d - a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 6*a^(2/3)*e*Log[x] - (2*(b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((b^(4/3)*c - a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + (2*a^(2/3)*(-(b*e) + a*h)*Log[a + b*x^3])/b)/(6*a^(5/3))","A",1
411,1,264,276,0.5473425,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^4*(a + b*x^3)),x]","-\frac{-\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} h-\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)}{b^{2/3}}+\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} h-\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} h-\sqrt[3]{a} b e+a \sqrt[3]{b} g-b^{4/3} d\right)}{b^{2/3}}-2 (b c-a f) \log \left(a+b x^3\right)+6 \log (x) (b c-a f)+\frac{2 a c}{x^3}+\frac{3 a d}{x^2}+\frac{6 a e}{x}}{6 a^2}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{6 a^{5/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b d-a g)-\sqrt[3]{a} (b e-a h)\right)}{3 a^{5/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-h)+\sqrt[3]{a} b e-a \sqrt[3]{b} g+b^{4/3} d\right)}{\sqrt{3} a^{5/3} b^{2/3}}+\frac{(b c-a f) \log \left(a+b x^3\right)}{3 a^2}-\frac{\log (x) (b c-a f)}{a^2}-\frac{c}{3 a x^3}-\frac{d}{2 a x^2}-\frac{e}{a x}",1,"-1/6*((2*a*c)/x^3 + (3*a*d)/x^2 + (6*a*e)/x + (2*Sqrt[3]*a^(1/3)*(-(b^(4/3)*d) - a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 6*(b*c - a*f)*Log[x] + (2*a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(b^(4/3)*d - a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 2*(b*c - a*f)*Log[a + b*x^3])/a^2","A",1
412,1,334,337,0.5737965,"\int \frac{x^4 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]","\frac{\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(4 a^{2/3} b^{4/3} e-7 a^{5/3} \sqrt[3]{b} h-5 a b f+2 b^2 c\right)}{\sqrt[3]{a}}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt[3]{b} h+5 a b f-2 b^2 c\right)}{\sqrt[3]{a}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-4 a^{2/3} b^{4/3} e+7 a^{5/3} \sqrt[3]{b} h-5 a b f+2 b^2 c\right)}{\sqrt[3]{a}}-\frac{12 b^{2/3} \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{a+b x^3}+12 b^{2/3} (b d-2 a g) \log \left(a+b x^3\right)+36 b^{2/3} x (b e-2 a h)+18 b^{5/3} f x^2+12 b^{5/3} g x^3+9 b^{5/3} h x^4}{36 b^{11/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right)}{18 \sqrt[3]{a} b^{10/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right)}{9 \sqrt[3]{a} b^{10/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right)}{3 \sqrt{3} \sqrt[3]{a} b^{10/3}}+\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{3 b^3 \left(a+b x^3\right)}+\frac{(b d-2 a g) \log \left(a+b x^3\right)}{3 b^3}+\frac{x (b e-2 a h)}{b^3}+\frac{f x^2}{2 b^2}+\frac{g x^3}{3 b^2}+\frac{h x^4}{4 b^2}",1,"(36*b^(2/3)*(b*e - 2*a*h)*x + 18*b^(5/3)*f*x^2 + 12*b^(5/3)*g*x^3 + 9*b^(5/3)*h*x^4 - (12*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3) - (4*Sqrt[3]*(2*b^2*c - 4*a^(2/3)*b^(4/3)*e - 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (4*(-2*b^2*c - 4*a^(2/3)*b^(4/3)*e + 5*a*b*f + 7*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (2*(2*b^2*c + 4*a^(2/3)*b^(4/3)*e - 5*a*b*f - 7*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + 12*b^(2/3)*(b*d - 2*a*g)*Log[a + b*x^3])/(36*b^(11/3))","A",1
413,1,294,311,0.2217735,"\int \frac{x^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]","\frac{-\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^{4/3} g-2 \sqrt[3]{a} b d-4 a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{2/3}}+\frac{2 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^{4/3} g-2 \sqrt[3]{a} b d-4 a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{2/3}}+\frac{2 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^{4/3} g-2 \sqrt[3]{a} b d+4 a \sqrt[3]{b} f-b^{4/3} c\right)}{a^{2/3}}-\frac{6 \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{a+b x^3}+6 (b e-2 a h) \log \left(a+b x^3\right)+18 b f x+9 b g x^2+6 b h x^3}{18 b^3}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b c-4 a f)-\sqrt[3]{a} (2 b d-5 a g)\right)}{18 a^{2/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b c-4 a f)-\sqrt[3]{a} (2 b d-5 a g)\right)}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^{4/3} g+2 \sqrt[3]{a} b d-4 a \sqrt[3]{b} f+b^{4/3} c\right)}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{(b e-2 a h) \log \left(a+b x^3\right)}{3 b^3}-\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{3 b^2 \left(a+b x^3\right)}+\frac{f x}{b^2}+\frac{g x^2}{2 b^2}+\frac{h x^3}{3 b^2}",1,"(18*b*f*x + 9*b*g*x^2 + 6*b*h*x^3 - (6*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3) + (2*Sqrt[3]*b^(1/3)*(-(b^(4/3)*c) - 2*a^(1/3)*b*d + 4*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - (b^(1/3)*(b^(4/3)*c - 2*a^(1/3)*b*d - 4*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) + 6*(b*e - 2*a*h)*Log[a + b*x^3])/(18*b^3)","A",1
414,1,280,290,0.2514376,"\int \frac{x^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]","\frac{-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^{4/3} h-2 \sqrt[3]{a} b e-4 a \sqrt[3]{b} g+b^{4/3} d\right)}{a^{2/3}}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^{4/3} h-2 \sqrt[3]{a} b e-4 a \sqrt[3]{b} g+b^{4/3} d\right)}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^{4/3} h-2 \sqrt[3]{a} b e+4 a \sqrt[3]{b} g-b^{4/3} d\right)}{a^{2/3}}-\frac{6 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{a+b x^3}+6 b^{2/3} f \log \left(a+b x^3\right)+18 b^{2/3} g x+9 b^{2/3} h x^2}{18 b^{8/3}}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (b d-4 a g)-\sqrt[3]{a} (2 b e-5 a h)\right)}{18 a^{2/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (b d-4 a g)-\sqrt[3]{a} (2 b e-5 a h)\right)}{9 a^{2/3} b^{8/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^{4/3} h+2 \sqrt[3]{a} b e-4 a \sqrt[3]{b} g+b^{4/3} d\right)}{3 \sqrt{3} a^{2/3} b^{8/3}}+\frac{f \log \left(a+b x^3\right)}{3 b^2}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left(a+b x^3\right)}+\frac{4 g x}{3 b^2}+\frac{5 h x^2}{6 b^2}",1,"(18*b^(2/3)*g*x + 9*b^(2/3)*h*x^2 - (6*b^(2/3)*(b*(c + x*(d + e*x)) - a*(f + x*(g + h*x))))/(a + b*x^3) + (2*Sqrt[3]*(-(b^(4/3)*d) - 2*a^(1/3)*b*e + 4*a*b^(1/3)*g + 5*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(b^(4/3)*d - 2*a^(1/3)*b*e - 4*a*b^(1/3)*g + 5*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - ((b^(4/3)*d - 2*a^(1/3)*b*e - 4*a*b^(1/3)*g + 5*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) + 6*b^(2/3)*f*Log[a + b*x^3])/(18*b^(8/3))","A",1
415,1,285,289,0.2600782,"\int \frac{x \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]","\frac{\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt[3]{b} h+2 a b f+b^2 c\right)}{a^{4/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt[3]{b} h+2 a b f+b^2 c\right)}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{2/3} b^{4/3} e-4 a^{5/3} \sqrt[3]{b} h+2 a b f+b^2 c\right)}{a^{4/3}}+\frac{6 b^{2/3} \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{a \left(a+b x^3\right)}+6 b^{2/3} g \log \left(a+b x^3\right)+18 b^{2/3} h x}{18 b^{8/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right)}{18 a^{4/3} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right)}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right)}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{3 a b^2 \left(a+b x^3\right)}+\frac{g \log \left(a+b x^3\right)}{3 b^2}+\frac{h x}{b^2}",1,"(18*b^(2/3)*h*x + (6*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + a^(2/3)*b^(4/3)*e + 2*a*b*f - 4*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*(b^2*c - a^(2/3)*b^(4/3)*e + 2*a*b*f + 4*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2*c - a^(2/3)*b^(4/3)*e + 2*a*b*f + 4*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) + 6*b^(2/3)*g*Log[a + b*x^3])/(18*b^(8/3))","A",1
416,1,268,276,0.2089213,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3)^2,x]","\frac{\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^{4/3} g+\sqrt[3]{a} b d-a \sqrt[3]{b} f-2 b^{4/3} c\right)}{a^{5/3}}+\frac{2 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^{4/3} g-\sqrt[3]{a} b d+a \sqrt[3]{b} f+2 b^{4/3} c\right)}{a^{5/3}}-\frac{2 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^{4/3} g+\sqrt[3]{a} b d+a \sqrt[3]{b} f+2 b^{4/3} c\right)}{a^{5/3}}+\frac{6 \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{a \left(a+b x^3\right)}+6 h \log \left(a+b x^3\right)}{18 b^2}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (a f+2 b c)-\sqrt[3]{a} (2 a g+b d)\right)}{18 a^{5/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (a f+2 b c)-\sqrt[3]{a} (2 a g+b d)\right)}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(2 a^{4/3} g+\sqrt[3]{a} b d+a \sqrt[3]{b} f+2 b^{4/3} c\right)}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{h \log \left(a+b x^3\right)}{3 b^2}+\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{3 a b \left(a+b x^3\right)}",1,"((6*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*b^(1/3)*(2*b^(4/3)*c + a^(1/3)*b*d + a*b^(1/3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(2*b^(4/3)*c - a^(1/3)*b*d + a*b^(1/3)*f - 2*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-2*b^(4/3)*c + a^(1/3)*b*d - a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 6*h*Log[a + b*x^3])/(18*b^2)","A",1
417,1,269,289,0.2229887,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^2),x]","\frac{\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^{4/3} h+\sqrt[3]{a} b e-a \sqrt[3]{b} g-2 b^{4/3} d\right)}{b^{5/3}}+\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^{4/3} h-\sqrt[3]{a} b e+a \sqrt[3]{b} g+2 b^{4/3} d\right)}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^{4/3} h+\sqrt[3]{a} b e+a \sqrt[3]{b} g+2 b^{4/3} d\right)}{b^{5/3}}-\frac{6 a (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left(a+b x^3\right)}-6 c \log \left(a+b x^3\right)+18 c \log (x)}{18 a^2}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (a g+2 b d)-\sqrt[3]{a} (2 a h+b e)\right)}{18 a^{5/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (a g+2 b d)-\sqrt[3]{a} (2 a h+b e)\right)}{9 a^{5/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(2 a^{4/3} h+\sqrt[3]{a} b e+a \sqrt[3]{b} g+2 b^{4/3} d\right)}{3 \sqrt{3} a^{5/3} b^{5/3}}+\frac{x \left(-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right)}{3 a^2 b \left(a+b x^3\right)}-\frac{c \log \left(a+b x^3\right)}{3 a^2}+\frac{c \log (x)}{a^2}",1,"((-6*a*(-(b*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(b*(a + b*x^3)) - (2*Sqrt[3]*a^(1/3)*(2*b^(4/3)*d + a^(1/3)*b*e + a*b^(1/3)*g + 2*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + 18*c*Log[x] + (2*a^(1/3)*(2*b^(4/3)*d - a^(1/3)*b*e + a*b^(1/3)*g - 2*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (a^(1/3)*(-2*b^(4/3)*d + a^(1/3)*b*e - a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - 6*c*Log[a + b*x^3])/(18*a^2)","A",1
418,1,285,301,0.4022693,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^2*(a + b*x^3)^2),x]","-\frac{\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right)}{b^{4/3}}-\frac{2 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^{2/3} b e+a^{5/3} h-a b^{2/3} f+4 b^{5/3} c\right)}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^{2/3} b e+a^{5/3} h+a b^{2/3} f-4 b^{5/3} c\right)}{b^{4/3}}+\frac{6 a \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{b \left(a+b x^3\right)}+6 a d \log \left(a+b x^3\right)+\frac{18 a c}{x}-18 a d \log (x)}{18 a^3}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right)}{18 a^{7/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (a h+2 b e)+b^{2/3} (4 b c-a f)\right)}{9 a^{7/3} b^{4/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+4 b^{5/3} c\right)}{3 \sqrt{3} a^{7/3} b^{4/3}}+\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{3 a^2 b \left(a+b x^3\right)}-\frac{d \log \left(a+b x^3\right)}{3 a^2}-\frac{c}{a^2 x}+\frac{d \log (x)}{a^2}",1,"-1/18*((18*a*c)/x + (6*a*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(b*(a + b*x^3)) + (2*Sqrt[3]*a^(2/3)*(-4*b^(5/3)*c + 2*a^(2/3)*b*e + a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) - 18*a*d*Log[x] - (2*a^(2/3)*(4*b^(5/3)*c + 2*a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) + (a^(2/3)*(4*b^(5/3)*c + 2*a^(2/3)*b*e - a*b^(2/3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3) + 6*a*d*Log[a + b*x^3])/a^3","A",1
419,1,292,306,0.5388138,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^2),x]","-\frac{-\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} g-4 \sqrt[3]{a} b d-2 a \sqrt[3]{b} f+5 b^{4/3} c\right)}{b^{2/3}}+\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} g-4 \sqrt[3]{a} b d-2 a \sqrt[3]{b} f+5 b^{4/3} c\right)}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} g-4 \sqrt[3]{a} b d+2 a \sqrt[3]{b} f-5 b^{4/3} c\right)}{b^{2/3}}+\frac{6 a \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{b \left(a+b x^3\right)}+6 a e \log \left(a+b x^3\right)+\frac{9 a c}{x^2}+\frac{18 a d}{x}-18 a e \log (x)}{18 a^3}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (5 b c-2 a f)-\sqrt[3]{a} (4 b d-a g)\right)}{18 a^{8/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (5 b c-2 a f)-\sqrt[3]{a} (4 b d-a g)\right)}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-g)+4 \sqrt[3]{a} b d-2 a \sqrt[3]{b} f+5 b^{4/3} c\right)}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{3 a^2 \left(a+b x^3\right)}-\frac{e \log \left(a+b x^3\right)}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2}",1,"-1/18*((9*a*c)/x^2 + (18*a*d)/x + (6*a*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)) + (2*Sqrt[3]*a^(1/3)*(-5*b^(4/3)*c - 4*a^(1/3)*b*d + 2*a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - 18*a*e*Log[x] + (2*a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 6*a*e*Log[a + b*x^3])/a^3","A",1
420,1,303,338,0.622306,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^4*(a + b*x^3)^2),x]","\frac{\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} h-4 \sqrt[3]{a} b e-2 a \sqrt[3]{b} g+5 b^{4/3} d\right)}{b^{2/3}}-\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} h-4 \sqrt[3]{a} b e-2 a \sqrt[3]{b} g+5 b^{4/3} d\right)}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} h-4 \sqrt[3]{a} b e+2 a \sqrt[3]{b} g-5 b^{4/3} d\right)}{b^{2/3}}+\frac{a (6 a (f+x (g+h x))-6 b (c+x (d+e x)))}{a+b x^3}+6 (2 b c-a f) \log \left(a+b x^3\right)+18 \log (x) (a f-2 b c)-\frac{6 a c}{x^3}-\frac{9 a d}{x^2}-\frac{18 a e}{x}}{18 a^3}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (5 b d-2 a g)-\sqrt[3]{a} (4 b e-a h)\right)}{18 a^{8/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (5 b d-2 a g)-\sqrt[3]{a} (4 b e-a h)\right)}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} (-h)+4 \sqrt[3]{a} b e-2 a \sqrt[3]{b} g+5 b^{4/3} d\right)}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{(2 b c-a f) \log \left(a+b x^3\right)}{3 a^3}-\frac{\log (x) (2 b c-a f)}{a^3}-\frac{x \left(-b x^2 \left(\frac{b c}{a}-f\right)+x (b e-a h)-a g+b d\right)}{3 a^2 \left(a+b x^3\right)}-\frac{c}{3 a^2 x^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}",1,"((-6*a*c)/x^3 - (9*a*d)/x^2 - (18*a*e)/x + (a*(-6*b*(c + x*(d + e*x)) + 6*a*(f + x*(g + h*x))))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(-5*b^(4/3)*d - 4*a^(1/3)*b*e + 2*a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 18*(-2*b*c + a*f)*Log[x] - (2*a^(1/3)*(5*b^(4/3)*d - 4*a^(1/3)*b*e - 2*a*b^(1/3)*g + a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (a^(1/3)*(5*b^(4/3)*d - 4*a^(1/3)*b*e - 2*a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 6*(2*b*c - a*f)*Log[a + b*x^3])/(18*a^3)","A",1
421,1,342,345,0.3758948,"\int \frac{x^4 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^4*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]","\frac{\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt[3]{b} h+5 a b f+b^2 c\right)}{a^{4/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-2 a^{2/3} b^{4/3} e+14 a^{5/3} \sqrt[3]{b} h+5 a b f+b^2 c\right)}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^{2/3} b^{4/3} e-14 a^{5/3} \sqrt[3]{b} h+5 a b f+b^2 c\right)}{a^{4/3}}-\frac{9 b^{2/3} \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{\left(a+b x^3\right)^2}+\frac{3 b^{2/3} \left(a^2 (12 g+13 h x)-a b (6 d+x (7 e+8 f x))+2 b^2 c x^2\right)}{a \left(a+b x^3\right)}+18 b^{2/3} g \log \left(a+b x^3\right)+54 b^{2/3} h x}{54 b^{11/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right)}{54 a^{4/3} b^{10/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(b^{2/3} (5 a f+b c)-2 a^{2/3} (b e-7 a h)\right)}{27 a^{4/3} b^{10/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(2 a^{2/3} b e-14 a^{5/3} h+5 a b^{2/3} f+b^{5/3} c\right)}{9 \sqrt{3} a^{4/3} b^{10/3}}-\frac{x \left(-2 b x (b c-4 a f)-3 b x^2 (b d-3 a g)+a (7 b e-13 a h)\right)}{18 a b^3 \left(a+b x^3\right)}+\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{6 b^3 \left(a+b x^3\right)^2}+\frac{g \log \left(a+b x^3\right)}{3 b^3}+\frac{h x}{b^3}",1,"(54*b^(2/3)*h*x - (9*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(2*b^2*c*x^2 + a^2*(12*g + 13*h*x) - a*b*(6*d + x*(7*e + 8*f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c + 2*a^(2/3)*b^(4/3)*e + 5*a*b*f - 14*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*(b^2*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2*c - 2*a^(2/3)*b^(4/3)*e + 5*a*b*f + 14*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) + 18*b^(2/3)*g*Log[a + b*x^3])/(54*b^(11/3))","A",1
422,1,315,325,0.3378631,"\int \frac{x^3 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]","\frac{\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^{4/3} g+\sqrt[3]{a} b d-2 a \sqrt[3]{b} f-b^{4/3} c\right)}{a^{5/3}}+\frac{2 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^{4/3} g-\sqrt[3]{a} b d+2 a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{5/3}}-\frac{2 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^{4/3} g+\sqrt[3]{a} b d+2 a \sqrt[3]{b} f+b^{4/3} c\right)}{a^{5/3}}-\frac{9 \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{\left(a+b x^3\right)^2}+\frac{36 a^2 h-3 a b (6 e+x (7 f+8 g x))+3 b^2 x (c+2 d x)}{a \left(a+b x^3\right)}+18 h \log \left(a+b x^3\right)}{54 b^3}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (2 a f+b c)-\sqrt[3]{a} (5 a g+b d)\right)}{54 a^{5/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (2 a f+b c)-\sqrt[3]{a} (5 a g+b d)\right)}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(5 a^{4/3} g+\sqrt[3]{a} b d+2 a \sqrt[3]{b} f+b^{4/3} c\right)}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{h \log \left(a+b x^3\right)}{3 b^3}+\frac{x \left(2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right)}{18 a b^2 \left(a+b x^3\right)}-\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{6 b^2 \left(a+b x^3\right)^2}",1,"((-9*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 + (36*a^2*h + 3*b^2*x*(c + 2*d*x) - 3*a*b*(6*e + x*(7*f + 8*g*x)))/(a*(a + b*x^3)) - (2*Sqrt[3]*b^(1/3)*(b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d + 2*a*b^(1/3)*f - 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-(b^(4/3)*c) + a^(1/3)*b*d - 2*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 18*h*Log[a + b*x^3])/(54*b^3)","A",1
423,1,287,297,0.301875,"\int \frac{x^2 \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]","\frac{\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^{4/3} h+\sqrt[3]{a} b e-2 a \sqrt[3]{b} g-b^{4/3} d\right)}{a^{5/3}}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(-5 a^{4/3} h-\sqrt[3]{a} b e+2 a \sqrt[3]{b} g+b^{4/3} d\right)}{a^{5/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^{4/3} h+\sqrt[3]{a} b e+2 a \sqrt[3]{b} g+b^{4/3} d\right)}{a^{5/3}}-\frac{9 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{\left(a+b x^3\right)^2}+\frac{3 b^{2/3} (b x (d+2 e x)-a (6 f+x (7 g+8 h x)))}{a \left(a+b x^3\right)}}{54 b^{8/3}}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (2 a g+b d)-\sqrt[3]{a} (5 a h+b e)\right)}{54 a^{5/3} b^{8/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (2 a g+b d)-\sqrt[3]{a} (5 a h+b e)\right)}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(5 a^{4/3} h+\sqrt[3]{a} b e+2 a \sqrt[3]{b} g+b^{4/3} d\right)}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{x \left(x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right)}{18 a b^2 \left(a+b x^3\right)}-\frac{c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left(a+b x^3\right)^2}",1,"((-9*b^(2/3)*(b*(c + x*(d + e*x)) - a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(b*x*(d + 2*e*x) - a*(6*f + x*(7*g + 8*h*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^(4/3)*d + a^(1/3)*b*e + 2*a*b^(1/3)*g + 5*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^(4/3)*d - a^(1/3)*b*e + 2*a*b^(1/3)*g - 5*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + ((-(b^(4/3)*d) + a^(1/3)*b*e - 2*a*b^(1/3)*g + 5*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^(8/3))","A",1
424,1,297,323,0.3580828,"\int \frac{x \left(c+d x+e x^2+f x^3+g x^4+h x^5\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^3,x]","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(-a^{2/3} b e-2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right)+2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} b e+2 a^{5/3} h-a b^{2/3} f-2 b^{5/3} c\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right)-\frac{3 \sqrt[3]{a} \sqrt[3]{b} \left(a^2 (6 g+7 h x)-a b x (e+2 f x)-4 b^2 c x^2\right)}{a+b x^3}+\frac{9 a^{4/3} \sqrt[3]{b} \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{\left(a+b x^3\right)^2}}{54 a^{7/3} b^{7/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right)}{54 a^{7/3} b^{7/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right)}{27 a^{7/3} b^{7/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right)}{9 \sqrt{3} a^{7/3} b^{7/3}}+\frac{x \left(2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right)}{18 a^2 b^2 \left(a+b x^3\right)}-\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{6 a b^2 \left(a+b x^3\right)^2}",1,"((-3*a^(1/3)*b^(1/3)*(-4*b^2*c*x^2 - a*b*x*(e + 2*f*x) + a^2*(6*g + 7*h*x)))/(a + b*x^3) + (9*a^(4/3)*b^(1/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a + b*x^3)^2 - 2*Sqrt[3]*(2*b^(5/3)*c + a^(2/3)*b*e + a*b^(2/3)*f + 2*a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-2*b^(5/3)*c + a^(2/3)*b*e - a*b^(2/3)*f + 2*a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x] + (2*b^(5/3)*c - a^(2/3)*b*e + a*b^(2/3)*f - 2*a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(7/3))","A",1
425,1,295,313,0.2819533,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^3)^3,x]","\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} g+2 \sqrt[3]{a} b d-a \sqrt[3]{b} f-5 b^{4/3} c\right)+2 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} (-g)-2 \sqrt[3]{a} b d+a \sqrt[3]{b} f+5 b^{4/3} c\right)-2 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} g+2 \sqrt[3]{a} b d+a \sqrt[3]{b} f+5 b^{4/3} c\right)+\frac{9 a^{5/3} \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{\left(a+b x^3\right)^2}+\frac{3 a^{2/3} \left(-6 a^2 h+a b x (f+2 g x)+b^2 x (5 c+4 d x)\right)}{a+b x^3}}{54 a^{8/3} b^2}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (a f+5 b c)-\sqrt[3]{a} (a g+2 b d)\right)}{54 a^{8/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (a f+5 b c)-\sqrt[3]{a} (a g+2 b d)\right)}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} g+2 \sqrt[3]{a} b d+a \sqrt[3]{b} f+5 b^{4/3} c\right)}{9 \sqrt{3} a^{8/3} b^{5/3}}-\frac{3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left(a+b x^3\right)}+\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{6 a b \left(a+b x^3\right)^2}",1,"((3*a^(2/3)*(-6*a^2*h + b^2*x*(5*c + 4*d*x) + a*b*x*(f + 2*g*x)))/(a + b*x^3) + (9*a^(5/3)*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 - 2*Sqrt[3]*b^(1/3)*(5*b^(4/3)*c + 2*a^(1/3)*b*d + a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(5*b^(4/3)*c - 2*a^(1/3)*b*d + a*b^(1/3)*f - a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(-5*b^(4/3)*c + 2*a^(1/3)*b*d - a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^2)","A",1
426,1,311,347,0.349421,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x]","\frac{\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{4/3} h+2 \sqrt[3]{a} b e-a \sqrt[3]{b} g-5 b^{4/3} d\right)}{b^{5/3}}+\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{4/3} (-h)-2 \sqrt[3]{a} b e+a \sqrt[3]{b} g+5 b^{4/3} d\right)}{b^{5/3}}-\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(a^{4/3} h+2 \sqrt[3]{a} b e+a \sqrt[3]{b} g+5 b^{4/3} d\right)}{b^{5/3}}-\frac{9 a^2 (a (f+x (g+h x))-b (c+x (d+e x)))}{b \left(a+b x^3\right)^2}+\frac{3 a (a x (g+2 h x)+6 b c+b x (5 d+4 e x))}{b \left(a+b x^3\right)}-18 c \log \left(a+b x^3\right)+54 c \log (x)}{54 a^3}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(\sqrt[3]{b} (a g+5 b d)-\sqrt[3]{a} (a h+2 b e)\right)}{54 a^{8/3} b^{5/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\sqrt[3]{b} (a g+5 b d)-\sqrt[3]{a} (a h+2 b e)\right)}{27 a^{8/3} b^{5/3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(a^{4/3} h+2 \sqrt[3]{a} b e+a \sqrt[3]{b} g+5 b^{4/3} d\right)}{9 \sqrt{3} a^{8/3} b^{5/3}}+\frac{x \left(-3 b x^2 (3 b c-a f)+a (a g+5 b d)+2 a x (a h+2 b e)\right)}{18 a^3 b \left(a+b x^3\right)}-\frac{c \log \left(a+b x^3\right)}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left(-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right)}{6 a^2 b \left(a+b x^3\right)^2}",1,"((3*a*(6*b*c + b*x*(5*d + 4*e*x) + a*x*(g + 2*h*x)))/(b*(a + b*x^3)) - (9*a^2*(-(b*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(b*(a + b*x^3)^2) - (2*Sqrt[3]*a^(1/3)*(5*b^(4/3)*d + 2*a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + 54*c*Log[x] + (2*a^(1/3)*(5*b^(4/3)*d - 2*a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (a^(1/3)*(-5*b^(4/3)*d + 2*a^(1/3)*b*e - a*b^(1/3)*g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - 18*c*Log[a + b*x^3])/(54*a^3)","A",1
427,1,336,362,0.7793255,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^2*(a + b*x^3)^3),x]","-\frac{\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right)}{b^{4/3}}-\frac{2 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right)}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(5 a^{2/3} b e+a^{5/3} h+2 a b^{2/3} f-14 b^{5/3} c\right)}{b^{4/3}}+\frac{9 a^2 \left(a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right)}{b \left(a+b x^3\right)^2}-\frac{3 a \left(a^2 h x+a b (6 d+x (5 e+4 f x))-10 b^2 c x^2\right)}{b \left(a+b x^3\right)}+18 a d \log \left(a+b x^3\right)+\frac{54 a c}{x}-54 a d \log (x)}{54 a^4}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right)}{54 a^{10/3} b^{4/3}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right)}{27 a^{10/3} b^{4/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right)}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x \left(-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right)}{18 a^3 b \left(a+b x^3\right)}-\frac{d \log \left(a+b x^3\right)}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left(-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right)}{6 a^2 b \left(a+b x^3\right)^2}",1,"-1/54*((54*a*c)/x + (9*a^2*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(b*(a + b*x^3)^2) - (3*a*(a^2*h*x - 10*b^2*c*x^2 + a*b*(6*d + x*(5*e + 4*f*x))))/(b*(a + b*x^3)) + (2*Sqrt[3]*a^(2/3)*(-14*b^(5/3)*c + 5*a^(2/3)*b*e + 2*a*b^(2/3)*f + a^(5/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) - 54*a*d*Log[x] - (2*a^(2/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/3)*f + a^(5/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) + (a^(2/3)*(14*b^(5/3)*c + 5*a^(2/3)*b*e - 2*a*b^(2/3)*f + a^(5/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3) + 18*a*d*Log[a + b*x^3])/a^4","A",1
428,1,337,360,0.7133381,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]","-\frac{-\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^{4/3} g-14 \sqrt[3]{a} b d-5 a \sqrt[3]{b} f+20 b^{4/3} c\right)}{b^{2/3}}+\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^{4/3} g-14 \sqrt[3]{a} b d-5 a \sqrt[3]{b} f+20 b^{4/3} c\right)}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(2 a^{4/3} g-14 \sqrt[3]{a} b d+5 a \sqrt[3]{b} f-20 b^{4/3} c\right)}{b^{2/3}}+\frac{9 a^2 \left(a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right)}{b \left(a+b x^3\right)^2}-\frac{3 a (6 a e+a x (5 f+4 g x)-b x (11 c+10 d x))}{a+b x^3}+18 a e \log \left(a+b x^3\right)+\frac{27 a c}{x^2}+\frac{54 a d}{x}-54 a e \log (x)}{54 a^4}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 \sqrt[3]{b} (4 b c-a f)-2 \sqrt[3]{a} (7 b d-a g)\right)}{54 a^{11/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 \sqrt[3]{b} (4 b c-a f)-2 \sqrt[3]{a} (7 b d-a g)\right)}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^{4/3} g+14 \sqrt[3]{a} b d-5 a \sqrt[3]{b} f+20 b^{4/3} c\right)}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left(2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right)}{18 a^3 \left(a+b x^3\right)}-\frac{e \log \left(a+b x^3\right)}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left(x (b d-a g)+x^2 (b e-a h)-a f+b c\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"-1/54*((27*a*c)/x^2 + (54*a*d)/x - (3*a*(6*a*e - b*x*(11*c + 10*d*x) + a*x*(5*f + 4*g*x)))/(a + b*x^3) + (9*a^2*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)^2) + (2*Sqrt[3]*a^(1/3)*(-20*b^(4/3)*c - 14*a^(1/3)*b*d + 5*a*b^(1/3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - 54*a*e*Log[x] + (2*a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 18*a*e*Log[a + b*x^3])/a^4","A",1
429,1,352,395,0.7886417,"\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^4 \left(a+b x^3\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^4*(a + b*x^3)^3),x]","\frac{\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(2 a^{4/3} h-14 \sqrt[3]{a} b e-5 a \sqrt[3]{b} g+20 b^{4/3} d\right)}{b^{2/3}}-\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(2 a^{4/3} h-14 \sqrt[3]{a} b e-5 a \sqrt[3]{b} g+20 b^{4/3} d\right)}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right) \left(-2 a^{4/3} h+14 \sqrt[3]{a} b e-5 a \sqrt[3]{b} g+20 b^{4/3} d\right)}{b^{2/3}}+\frac{a^2 (9 a (f+x (g+h x))-9 b (c+x (d+e x)))}{\left(a+b x^3\right)^2}+\frac{3 a (6 a f+a x (5 g+4 h x)-12 b c-b x (11 d+10 e x))}{a+b x^3}+18 (3 b c-a f) \log \left(a+b x^3\right)+54 \log (x) (a f-3 b c)-\frac{18 a c}{x^3}-\frac{27 a d}{x^2}-\frac{54 a e}{x}}{54 a^4}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right) \left(5 \sqrt[3]{b} (4 b d-a g)-2 \sqrt[3]{a} (7 b e-a h)\right)}{54 a^{11/3} b^{2/3}}-\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \left(5 \sqrt[3]{b} (4 b d-a g)-2 \sqrt[3]{a} (7 b e-a h)\right)}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right) \left(-2 a^{4/3} h+14 \sqrt[3]{a} b e-5 a \sqrt[3]{b} g+20 b^{4/3} d\right)}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{(3 b c-a f) \log \left(a+b x^3\right)}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{x \left(-3 b x^2 \left(\frac{5 b c}{a}-3 f\right)+2 x (5 b e-2 a h)-5 a g+11 b d\right)}{18 a^3 \left(a+b x^3\right)}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left(-b x^2 \left(\frac{b c}{a}-f\right)+x (b e-a h)-a g+b d\right)}{6 a^2 \left(a+b x^3\right)^2}",1,"((-18*a*c)/x^3 - (27*a*d)/x^2 - (54*a*e)/x + (3*a*(-12*b*c + 6*a*f - b*x*(11*d + 10*e*x) + a*x*(5*g + 4*h*x)))/(a + b*x^3) + (a^2*(-9*b*(c + x*(d + e*x)) + 9*a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (2*Sqrt[3]*a^(1/3)*(20*b^(4/3)*d + 14*a^(1/3)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 54*(-3*b*c + a*f)*Log[x] - (2*a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 18*(3*b*c - a*f)*Log[a + b*x^3])/(54*a^4)","A",1
430,1,132,583,0.2088954,"\int \frac{x^3 \left(c+d x+e x^2\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^3*(c + d*x + e*x^2))/Sqrt[a + b*x^3],x]","\frac{-2 \left(a+b x^3\right) (70 a e-b x (63 c+5 x (9 d+7 e x)))-126 a b c x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-90 a b d x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{315 b^2 \sqrt{a+b x^3}}","-\frac{4 \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(7 \sqrt[3]{b} c-10 \left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{35 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}",1,"(-2*(a + b*x^3)*(70*a*e - b*x*(63*c + 5*x*(9*d + 7*e*x))) - 126*a*b*c*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] - 90*a*b*d*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(315*b^2*Sqrt[a + b*x^3])","C",1
431,1,121,560,0.1671184,"\int \frac{x^2 \left(c+d x+e x^2\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/Sqrt[a + b*x^3],x]","\frac{2 \left(a+b x^3\right) (35 c+3 x (7 d+5 e x))-42 a d x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-30 a e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{105 b \sqrt{a+b x^3}}","-\frac{4 \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(7 \sqrt[3]{b} d-10 \left(1-\sqrt{3}\right) \sqrt[3]{a} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{35 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{8 a e \sqrt{a+b x^3}}{7 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 c \sqrt{a+b x^3}}{3 b}+\frac{2 d x \sqrt{a+b x^3}}{5 b}+\frac{2 e x^2 \sqrt{a+b x^3}}{7 b}",1,"(2*(a + b*x^3)*(35*c + 3*x*(7*d + 5*e*x)) - 42*a*d*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] - 30*a*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(105*b*Sqrt[a + b*x^3])","C",1
432,1,114,537,0.072759,"\int \frac{x \left(c+d x+e x^2\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x*(c + d*x + e*x^2))/Sqrt[a + b*x^3],x]","\frac{15 b c x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 \left(a+b x^3\right) (5 d+3 e x)-12 a e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)}{30 b \sqrt{a+b x^3}}","-\frac{2 \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(2 a^{2/3} e+5 \left(1-\sqrt{3}\right) b^{2/3} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{5 \sqrt[4]{3} b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} c \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 c \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 d \sqrt{a+b x^3}}{3 b}+\frac{2 e x \sqrt{a+b x^3}}{5 b}",1,"(4*(5*d + 3*e*x)*(a + b*x^3) - 12*a*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 15*b*c*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(30*b*Sqrt[a + b*x^3])","C",1
433,1,107,509,0.0501965,"\int \frac{c+d x+e x^2}{\sqrt{a+b x^3}} \, dx","Integrate[(c + d*x + e*x^2)/Sqrt[a + b*x^3],x]","\frac{6 b c x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+3 b d x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 e \left(a+b x^3\right)}{6 b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} c-\left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 d \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 e \sqrt{a+b x^3}}{3 b}",1,"(4*e*(a + b*x^3) + 6*b*c*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*b*d*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(6*b*Sqrt[a + b*x^3])","C",1
434,1,128,518,0.2022856,"\int \frac{c+d x+e x^2}{x \sqrt{a+b x^3}} \, dx","Integrate[(c + d*x + e*x^2)/(x*Sqrt[a + b*x^3]),x]","-\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}+\frac{d x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{a+b x^3}}+\frac{e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} d-\left(1-\sqrt{3}\right) \sqrt[3]{a} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 e \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(-2*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a]) + (d*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/Sqrt[a + b*x^3] + (e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(2*Sqrt[a + b*x^3])","C",1
435,1,126,547,0.1408809,"\int \frac{c+d x+e x^2}{x^2 \sqrt{a+b x^3}} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*Sqrt[a + b*x^3]),x]","-\frac{c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a}\right)}{x \sqrt{a+b x^3}}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}+\frac{e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{a+b x^3}}","-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\left(1-\sqrt{3}\right) b^{2/3} c-2 a^{2/3} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} a^{2/3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} c \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a x}+\frac{\sqrt[3]{b} c \sqrt{a+b x^3}}{a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(-2*d*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a]) - (c*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 1/2, 2/3, -((b*x^3)/a)])/(x*Sqrt[a + b*x^3]) + (e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/Sqrt[a + b*x^3]","C",1
436,1,131,569,0.2182783,"\int \frac{c+d x+e x^2}{x^3 \sqrt{a+b x^3}} \, dx","Integrate[(c + d*x + e*x^2)/(x^3*Sqrt[a + b*x^3]),x]","-\frac{c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)}{2 x^2 \sqrt{a+b x^3}}-\frac{d \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a}\right)}{x \sqrt{a+b x^3}}-\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}","-\frac{\sqrt{2+\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(2 \left(1-\sqrt{3}\right) \sqrt[3]{a} d+\sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt[4]{3} a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{2 a x^2}-\frac{d \sqrt{a+b x^3}}{a x}+\frac{\sqrt[3]{b} d \sqrt{a+b x^3}}{a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(-2*e*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a]) - (c*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-2/3, 1/2, 1/3, -((b*x^3)/a)])/(2*x^2*Sqrt[a + b*x^3]) - (d*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 1/2, 2/3, -((b*x^3)/a)])/(x*Sqrt[a + b*x^3])","C",1
437,1,134,594,0.1281456,"\int \frac{x^5 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^5*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x]","\frac{2 \left(-56 a d x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+150 a e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+70 a c+56 a d x-150 a e x^2+35 b c x^3+21 b d x^4+15 b e x^5\right)}{105 b^2 \sqrt{a+b x^3}}","-\frac{16 \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(14 \sqrt[3]{b} d-25 \left(1-\sqrt{3}\right) \sqrt[3]{a} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{105 \sqrt[4]{3} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{40 \sqrt{2-\sqrt{3}} a^{4/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7\ 3^{3/4} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{80 a e \sqrt{a+b x^3}}{21 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x \left(a d+a e x-b c x^2\right)}{3 b^2 \sqrt{a+b x^3}}+\frac{4 c \sqrt{a+b x^3}}{3 b^2}+\frac{2 d x \sqrt{a+b x^3}}{5 b^2}+\frac{2 e x^2 \sqrt{a+b x^3}}{7 b^2}",1,"(2*(70*a*c + 56*a*d*x - 150*a*e*x^2 + 35*b*c*x^3 + 21*b*d*x^4 + 15*b*e*x^5 - 56*a*d*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 150*a*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(105*b^2*Sqrt[a + b*x^3])","C",1
438,1,127,574,0.1346758,"\int \frac{x^4 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^4*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x]","\frac{2 \left(-15 b c x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-8 a e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+10 a d+8 a e x+15 b c x^2+5 b d x^3+3 b e x^4\right)}{15 b^2 \sqrt{a+b x^3}}","-\frac{8 \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(4 a^{2/3} e+5 \left(1-\sqrt{3}\right) b^{2/3} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{15 \sqrt[4]{3} b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt[3]{a} c \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{8 c \sqrt{a+b x^3}}{3 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x \left(a e-b c x-b d x^2\right)}{3 b^2 \sqrt{a+b x^3}}+\frac{4 d \sqrt{a+b x^3}}{3 b^2}+\frac{2 e x \sqrt{a+b x^3}}{5 b^2}",1,"(2*(10*a*d + 8*a*e*x + 15*b*c*x^2 + 5*b*d*x^3 + 3*b*e*x^4 - 8*a*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] - 15*b*c*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(15*b^2*Sqrt[a + b*x^3])","C",1
439,1,118,542,0.1165694,"\int \frac{x^3 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^3*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x]","\frac{2 \left(b c x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-3 b d x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+2 a e-b c x+3 b d x^2+b e x^3\right)}{3 b^2 \sqrt{a+b x^3}}","\frac{4 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} c-2 \left(1-\sqrt{3}\right) \sqrt[3]{a} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt[3]{a} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{8 d \sqrt{a+b x^3}}{3 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{4 e \sqrt{a+b x^3}}{3 b^2}-\frac{2 x \left(c+d x+e x^2\right)}{3 b \sqrt{a+b x^3}}",1,"(2*(2*a*e - b*c*x + 3*b*d*x^2 + b*e*x^3 + b*c*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] - 3*b*d*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(3*b^2*Sqrt[a + b*x^3])","C",1
440,1,107,522,0.1032844,"\int \frac{x^2 \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^2*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x]","\frac{2 d x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-2 \left(3 e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+c+x (d-3 e x)\right)}{3 b \sqrt{a+b x^3}}","\frac{4 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} d-2 \left(1-\sqrt{3}\right) \sqrt[3]{a} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt[3]{a} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{8 e \sqrt{a+b x^3}}{3 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 \left(c+d x+e x^2\right)}{3 b \sqrt{a+b x^3}}",1,"(2*d*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] - 2*(c + x*(d - 3*e*x) + 3*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(3*b*Sqrt[a + b*x^3])","C",1
441,1,108,561,0.0762349,"\int \frac{x \left(c+d x+e x^2\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x*(c + d*x + e*x^2))/(a + b*x^3)^(3/2),x]","\frac{3 b c x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+4 a e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-4 a (d+e x)}{6 a b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(2 a^{2/3} e+b^{2/3} \left(c-\sqrt{3} c\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a^{2/3} b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} c \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 c \sqrt{a+b x^3}}{3 a b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 x \left(a e-b c x-b d x^2\right)}{3 a b \sqrt{a+b x^3}}-\frac{2 d \sqrt{a+b x^3}}{3 a b}",1,"(-4*a*(d + e*x) + 4*a*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*b*c*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)])/(6*a*b*Sqrt[a + b*x^3])","C",1
442,1,109,532,0.0759003,"\int \frac{c+d x+e x^2}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2)/(a + b*x^3)^(3/2),x]","\frac{2 b c x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+3 b d x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-4 a e+4 b c x}{6 a b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a} d+\sqrt[3]{b} c\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} d \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 d \sqrt{a+b x^3}}{3 a b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 (a e-b x (c+d x))}{3 a b \sqrt{a+b x^3}}",1,"(-4*a*e + 4*b*c*x + 2*b*c*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*b*d*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)])/(6*a*b*Sqrt[a + b*x^3])","C",1
443,1,119,579,0.117348,"\int \frac{c+d x+e x^2}{x \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^(3/2)),x]","\frac{4 c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right)+x \left(2 d \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+3 e x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+4 d\right)}{6 a \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a} e+\sqrt[3]{b} d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2}}+\frac{2 x \left(a d+a e x-b c x^2\right)}{3 a^2 \sqrt{a+b x^3}}+\frac{2 c \sqrt{a+b x^3}}{3 a^2}-\frac{2 e \sqrt{a+b x^3}}{3 a b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(4*c*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^3)/a] + x*(4*d + 2*d*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 3*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(6*a*Sqrt[a + b*x^3])","C",1
444,1,121,607,0.1129701,"\int \frac{c+d x+e x^2}{x^2 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^(3/2)),x]","\frac{-3 c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{1}{3},\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a}\right)+2 d x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right)+e x^2 \left(\sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+2\right)}{3 a x \sqrt{a+b x^3}}","-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(5 \left(1-\sqrt{3}\right) b^{2/3} c-2 a^{2/3} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a^{5/3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt[3]{b} c \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2\ 3^{3/4} a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2}}+\frac{2 x \left(a e-b c x-b d x^2\right)}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{5 \sqrt[3]{b} c \sqrt{a+b x^3}}{3 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 d \sqrt{a+b x^3}}{3 a^2}",1,"(2*d*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^3)/a] - 3*c*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 3/2, 2/3, -((b*x^3)/a)] + e*x^2*(2 + Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(3*a*x*Sqrt[a + b*x^3])","C",1
445,1,172,733,0.4649108,"\int x^3 \sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x^3*Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{2 \sqrt{a+b x^3} \left(-\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(a (92378 e+90 x (988 f+935 g x))-3 b x \left(62985 c+11 x \left(4845 d+13 x \left(323 e+285 f x+255 g x^2\right)\right)\right)\right)+11115 a x (8 a f-17 b c) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+8415 a x^2 (10 a g-19 b d) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2078505 b^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 b d-10 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{24 a^2 \sqrt{a+b x^3} (19 b d-10 a g)}{1729 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{4 a^2 e \sqrt{a+b x^3}}{45 b^2}-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(1729 \sqrt[3]{b} (17 b c-8 a f)-1870 \left(1-\sqrt{3}\right) \sqrt[3]{a} (19 b d-10 a g)\right)}{1616615 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a x \sqrt{a+b x^3} (17 b c-8 a f)}{935 b^2}+\frac{6 a x^2 \sqrt{a+b x^3} (19 b d-10 a g)}{1729 b^2}+\frac{2 x^3 \sqrt{a+b x^3} \left(62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right)}{692835}+\frac{2 a e x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a f x^4 \sqrt{a+b x^3}}{187 b}+\frac{6 a g x^5 \sqrt{a+b x^3}}{247 b}",1,"(2*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(a*(92378*e + 90*x*(988*f + 935*g*x)) - 3*b*x*(62985*c + 11*x*(4845*d + 13*x*(323*e + 285*f*x + 255*g*x^2))))) + 11115*a*(-17*b*c + 8*a*f)*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 8415*a*(-19*b*d + 10*a*g)*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2078505*b^2*Sqrt[1 + (b*x^3)/a])","C",1
446,1,158,681,0.2972519,"\int x^2 \sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x^2*Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{2 \sqrt{a+b x^3} \left(-\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(26 a (187 f+180 g x)-b \left(12155 c+9945 d x+33 x^2 (255 e+13 x (17 f+15 g x))\right)\right)+585 a x (8 a g-17 b d) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-8415 a b e x^2 \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{109395 b^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{24 a^2 e \sqrt{a+b x^3}}{91 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-1870 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-728 a g+1547 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{85085 b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 a \sqrt{a+b x^3} (5 b c-2 a f)}{45 b^2}+\frac{6 a x \sqrt{a+b x^3} (17 b d-8 a g)}{935 b^2}+\frac{2 x^2 \sqrt{a+b x^3} \left(12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right)}{109395}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}",1,"(2*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(26*a*(187*f + 180*g*x) - b*(12155*c + 9945*d*x + 33*x^2*(255*e + 13*x*(17*f + 15*g*x))))) + 585*a*(-17*b*d + 8*a*g)*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] - 8415*a*b*e*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(109395*b^2*Sqrt[1 + (b*x^3)/a])","C",1
447,1,143,667,0.3570815,"\int x \sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x*Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{\sqrt{a+b x^3} \left(495 b x^2 (13 b c-4 a f) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-4 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(286 a g-b \left(715 d+585 e x+495 f x^2+429 g x^3\right)\right)-2340 a b e x \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{12870 b^2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(182 a^{2/3} \sqrt[3]{b} e+55 \left(1-\sqrt{3}\right) (13 b c-4 a f)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{5005 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 b c-4 a f) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a \sqrt{a+b x^3} (13 b c-4 a f)}{91 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a \sqrt{a+b x^3} (5 b d-2 a g)}{45 b^2}+\frac{2 x \sqrt{a+b x^3} \left(6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right)}{45045}+\frac{6 a e x \sqrt{a+b x^3}}{55 b}+\frac{6 a f x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a g x^3 \sqrt{a+b x^3}}{45 b}",1,"(Sqrt[a + b*x^3]*(-4*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(286*a*g - b*(715*d + 585*e*x + 495*f*x^2 + 429*g*x^3)) - 2340*a*b*e*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 495*b*(13*b*c - 4*a*f)*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(12870*b^2*Sqrt[1 + (b*x^3)/a])","C",1
448,1,135,639,0.192474,"\int \sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{\sqrt{a+b x^3} \left(234 x (11 b c-2 a f) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+99 x^2 (13 b d-4 a g) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (143 e+9 x (13 f+11 g x))\right)}{2574 b \sqrt{\frac{b x^3}{a}+1}}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(91 \sqrt[3]{b} (11 b c-2 a f)-55 \left(1-\sqrt{3}\right) \sqrt[3]{a} (13 b d-4 a g)\right)}{5005 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 b d-4 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a \sqrt{a+b x^3} (13 b d-4 a g)}{91 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 \sqrt{a+b x^3} \left(9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right)}{45045}+\frac{2 a e \sqrt{a+b x^3}}{9 b}+\frac{6 a f x \sqrt{a+b x^3}}{55 b}+\frac{6 a g x^2 \sqrt{a+b x^3}}{91 b}",1,"(Sqrt[a + b*x^3]*(4*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(143*e + 9*x*(13*f + 11*g*x)) + 234*(11*b*c - 2*a*f)*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 99*(13*b*d - 4*a*g)*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2574*b*Sqrt[1 + (b*x^3)/a])","C",1
449,1,185,620,0.4329363,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x,x]","\frac{4 \sqrt{\frac{b x^3}{a}+1} \left(\sqrt{a+b x^3} \left(11 a f+9 a g x+33 b c+11 b f x^3+9 b g x^4\right)-33 \sqrt{a} b c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+18 x \sqrt{a+b x^3} (11 b d-2 a g) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+99 b e x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{198 b \sqrt{\frac{b x^3}{a}+1}}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-55 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-14 a g+77 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{385 b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a e \sqrt{a+b x^3}}{7 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 \sqrt{a+b x^3} \left(1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right)}{3465 x}-\frac{2}{3} \sqrt{a} c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{2 a f \sqrt{a+b x^3}}{9 b}+\frac{6 a g x \sqrt{a+b x^3}}{55 b}",1,"(4*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3]*(33*b*c + 11*a*f + 9*a*g*x + 11*b*f*x^3 + 9*b*g*x^4) - 33*Sqrt[a]*b*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 18*(11*b*d - 2*a*g)*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 99*b*e*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])/(198*b*Sqrt[1 + (b*x^3)/a])","C",1
450,1,211,638,0.2949988,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^2} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^2,x]","-\frac{c \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)}{x \sqrt{\frac{b x^3}{a}+1}}+\frac{2}{3} d \left(\sqrt{a+b x^3}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+\frac{e x \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}+\frac{f x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}+\frac{2 g \left(a+b x^3\right)^{3/2}}{9 b}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(14 a^{2/3} \sqrt[3]{b} e-5 \left(1-\sqrt{3}\right) (2 a f+7 b c)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{35 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a f+7 b c) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{14 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \sqrt{a+b x^3} (2 a f+7 b c)}{7 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 \sqrt{a+b x^3} \left(315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right)}{315 x^2}-\frac{3 c \sqrt{a+b x^3}}{x}-\frac{2}{3} \sqrt{a} d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{2 a g \sqrt{a+b x^3}}{9 b}",1,"(2*g*(a + b*x^3)^(3/2))/(9*b) + (2*d*(Sqrt[a + b*x^3] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]))/3 - (c*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)])/(x*Sqrt[1 + (b*x^3)/a]) + (e*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a] + (f*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])","C",1
451,1,218,640,0.5007384,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^3} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3,x]","\frac{x \left(x \left(4 e \sqrt{\frac{b x^3}{a}+1} \left(\sqrt{a+b x^3}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+6 f x \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+3 g x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)-6 d \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)\right)-3 c \sqrt{a+b x^3} \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)}{6 x^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(7 \sqrt[3]{b} (4 a f+5 b c)-10 \left(1-\sqrt{3}\right) \sqrt[3]{a} (2 a g+7 b d)\right)}{70 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a g+7 b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{14 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \sqrt{a+b x^3} (2 a g+7 b d)}{7 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 \sqrt{a+b x^3} \left(105 c x-105 d x^2-35 e x^3-21 f x^4-15 g x^5\right)}{105 x^3}+\frac{3 c \sqrt{a+b x^3}}{2 x^2}-\frac{3 d \sqrt{a+b x^3}}{x}-\frac{2}{3} \sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)",1,"(-3*c*Sqrt[a + b*x^3]*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)] + x*(-6*d*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)] + x*(4*e*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 6*f*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 3*g*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])))/(6*x^2*Sqrt[1 + (b*x^3)/a])","C",1
452,1,254,637,0.4356912,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^4} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]","-\frac{b c \left(\frac{a+b x^3}{b x^3}+\sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)\right)}{3 \sqrt{a+b x^3}}-\frac{d \sqrt{a+b x^3} \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)}{2 x^2 \sqrt{\frac{b x^3}{a}+1}}-\frac{e \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)}{x \sqrt{\frac{b x^3}{a}+1}}+\frac{2}{3} f \left(\sqrt{a+b x^3}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+\frac{g x \sqrt{a+b x^3} \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-10 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e+4 a g+5 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{10 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{b} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} \left(5 c x+15 d x^2-15 e x^3-5 f x^4-3 g x^5\right)}{15 x^4}-\frac{(2 a f+b c) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}+\frac{c \sqrt{a+b x^3}}{3 x^3}+\frac{3 d \sqrt{a+b x^3}}{2 x^2}-\frac{3 e \sqrt{a+b x^3}}{x}+\frac{3 \sqrt[3]{b} e \sqrt{a+b x^3}}{\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}",1,"(2*f*(Sqrt[a + b*x^3] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]))/3 - (b*c*((a + b*x^3)/(b*x^3) + Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]]))/(3*Sqrt[a + b*x^3]) - (d*Sqrt[a + b*x^3]*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)])/(2*x^2*Sqrt[1 + (b*x^3)/a]) - (e*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)])/(x*Sqrt[1 + (b*x^3)/a]) + (g*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]","C",1
453,1,253,694,0.4644134,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^5} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5,x]","\frac{-3 c \left(a+b x^3\right) \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)-4 d x \sqrt{\frac{b x^3}{a}+1} \left(b x^3 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+a+b x^3\right)-6 e x^2 \left(a+b x^3\right) \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)-12 f x^3 \left(a+b x^3\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)+8 g x^4 \sqrt{a+b x^3} \sqrt{\frac{b x^3}{a}+1} \left(\sqrt{a+b x^3}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)}{12 x^4 \sqrt{a+b x^3} \sqrt{\frac{b x^3}{a}+1}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(4 a^{2/3} \sqrt[3]{b} e-\left(1-\sqrt{3}\right) (8 a f+b c)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{8 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a f+b c) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} \left(3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right)}{15 x^5}-\frac{3 \sqrt{a+b x^3} (8 a f+b c)}{8 a x}+\frac{3 \sqrt[3]{b} \sqrt{a+b x^3} (8 a f+b c)}{8 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{3 c \sqrt{a+b x^3}}{20 x^4}-\frac{(2 a g+b d) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}+\frac{d \sqrt{a+b x^3}}{3 x^3}+\frac{3 e \sqrt{a+b x^3}}{2 x^2}",1,"(8*g*x^4*Sqrt[a + b*x^3]*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) - 4*d*x*Sqrt[1 + (b*x^3)/a]*(a + b*x^3 + b*x^3*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]]) - 3*c*(a + b*x^3)*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] - 6*e*x^2*(a + b*x^3)*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)] - 12*f*x^3*(a + b*x^3)*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)])/(12*x^4*Sqrt[a + b*x^3]*Sqrt[1 + (b*x^3)/a])","C",1
454,1,180,652,0.3471172,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^6} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^6,x]","-\frac{\sqrt{a+b x^3} \left(12 a c \, _2F_1\left(-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)+5 x \left(3 a d \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)+2 x \left(2 a e \sqrt{\frac{b x^3}{a}+1}+2 b e x^3 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+3 a f x \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)+6 a g x^2 \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)\right)\right)\right)}{60 a x^5 \sqrt{\frac{b x^3}{a}+1}}","-\frac{3^{3/4} \sqrt{2+\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(2 \sqrt[3]{b} (b c-10 a f)+5 \left(1-\sqrt{3}\right) \sqrt[3]{a} (8 a g+b d)\right)}{40 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a g+b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{1}{60} \sqrt{a+b x^3} \left(\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}+\frac{60 g}{x}\right)-\frac{3 b c \sqrt{a+b x^3}}{20 a x^2}+\frac{3 \sqrt[3]{b} \sqrt{a+b x^3} (8 a g+b d)}{8 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{3 b d \sqrt{a+b x^3}}{8 a x}-\frac{b e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"-1/60*(Sqrt[a + b*x^3]*(12*a*c*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)] + 5*x*(3*a*d*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 2*x*(2*a*e*Sqrt[1 + (b*x^3)/a] + 2*b*e*x^3*ArcTanh[Sqrt[1 + (b*x^3)/a]] + 3*a*f*x*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)] + 6*a*g*x^2*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)]))))/(a*x^5*Sqrt[1 + (b*x^3)/a])","C",1
455,1,211,659,0.5333888,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^7} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^7,x]","-\frac{\sqrt{a+b x^3} \left(36 a^3 d \, _2F_1\left(-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)+5 x \left(9 a^3 e \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)+2 x \left(9 a^3 g x \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)+6 a^2 f \left(a \sqrt{\frac{b x^3}{a}+1}+b x^3 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)\right)+4 b^2 c x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right)\right)\right)\right)}{180 a^3 x^5 \sqrt{\frac{b x^3}{a}+1}}","-\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(5 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-20 a g+2 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{40 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{b (b c-4 a f) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{12 a^{3/2}}+\frac{3 b^{4/3} e \sqrt{a+b x^3}}{8 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{1}{60} \sqrt{a+b x^3} \left(\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right)-\frac{b c \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b d \sqrt{a+b x^3}}{20 a x^2}-\frac{3 b e \sqrt{a+b x^3}}{8 a x}",1,"-1/180*(Sqrt[a + b*x^3]*(36*a^3*d*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)] + 5*x*(9*a^3*e*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 2*x*(6*a^2*f*(a*Sqrt[1 + (b*x^3)/a] + b*x^3*ArcTanh[Sqrt[1 + (b*x^3)/a]]) + 9*a^3*g*x*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)] + 4*b^2*c*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^3)/a]))))/(a^3*x^5*Sqrt[1 + (b*x^3)/a])","C",1
456,1,213,711,0.5171085,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^8} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^8,x]","-\frac{\sqrt{a+b x^3} \left(180 a^3 c \, _2F_1\left(-\frac{7}{3},-\frac{1}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)+7 x^2 \left(36 a^3 e \, _2F_1\left(-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)+5 x \left(9 a^3 f \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)+12 a^2 g x \left(a \sqrt{\frac{b x^3}{a}+1}+b x^3 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)\right)+8 b^2 d x^4 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right)\right)\right)\right)}{1260 a^3 x^7 \sqrt{\frac{b x^3}{a}+1}}","-\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(28 a^{2/3} \sqrt[3]{b} e-5 \left(1-\sqrt{3}\right) (5 b c-14 a f)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{560 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 b c-14 a f) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{b (b d-4 a g) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{12 a^{3/2}}-\frac{3 b^{4/3} \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{3 b \sqrt{a+b x^3} (5 b c-14 a f)}{112 a^2 x}-\frac{1}{420} \sqrt{a+b x^3} \left(\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right)-\frac{3 b c \sqrt{a+b x^3}}{56 a x^4}-\frac{b d \sqrt{a+b x^3}}{12 a x^3}-\frac{3 b e \sqrt{a+b x^3}}{20 a x^2}",1,"-1/1260*(Sqrt[a + b*x^3]*(180*a^3*c*Hypergeometric2F1[-7/3, -1/2, -4/3, -((b*x^3)/a)] + 7*x^2*(36*a^3*e*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)] + 5*x*(12*a^2*g*x*(a*Sqrt[1 + (b*x^3)/a] + b*x^3*ArcTanh[Sqrt[1 + (b*x^3)/a]]) + 9*a^3*f*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 8*b^2*d*x^4*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^3)/a]))))/(a^3*x^7*Sqrt[1 + (b*x^3)/a])","C",1
457,1,192,743,0.2503026,"\int \frac{\sqrt{a+b x^3} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^9} \, dx","Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^9,x]","-\frac{\sqrt{a+b x^3} \left(14 x^3 \left(36 a^3 f \, _2F_1\left(-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)+45 a^3 g x \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)+40 b^2 e x^5 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right)\right)+315 a^3 c \, _2F_1\left(-\frac{8}{3},-\frac{1}{2};-\frac{5}{3};-\frac{b x^3}{a}\right)+360 a^3 d x \, _2F_1\left(-\frac{7}{3},-\frac{1}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)\right)}{2520 a^3 x^8 \sqrt{\frac{b x^3}{a}+1}}","\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 b d-14 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{b^2 e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{12 a^{3/2}}-\frac{3 b^{4/3} \sqrt{a+b x^3} (5 b d-14 a g)}{112 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{3 b \sqrt{a+b x^3} (7 b c-16 a f)}{320 a^2 x^2}+\frac{3 b \sqrt{a+b x^3} (5 b d-14 a g)}{112 a^2 x}+\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(7 \sqrt[3]{b} (7 b c-16 a f)+20 \left(1-\sqrt{3}\right) \sqrt[3]{a} (5 b d-14 a g)\right)}{2240 a^2 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{1}{840} \sqrt{a+b x^3} \left(\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right)-\frac{3 b c \sqrt{a+b x^3}}{80 a x^5}-\frac{3 b d \sqrt{a+b x^3}}{56 a x^4}-\frac{b e \sqrt{a+b x^3}}{12 a x^3}",1,"-1/2520*(Sqrt[a + b*x^3]*(315*a^3*c*Hypergeometric2F1[-8/3, -1/2, -5/3, -((b*x^3)/a)] + 360*a^3*d*x*Hypergeometric2F1[-7/3, -1/2, -4/3, -((b*x^3)/a)] + 14*x^3*(36*a^3*f*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)] + 45*a^3*g*x*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 40*b^2*e*x^5*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^3)/a])))/(a^3*x^8*Sqrt[1 + (b*x^3)/a])","C",1
458,1,179,791,0.6542856,"\int x^3 \left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x^3*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{2 \sqrt{a+b x^3} \left(9975 a^2 x (8 a f-23 b c) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+41055 a^2 x^2 (2 a g-5 b d) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-\left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \left(10 a (7429 e+21 x (380 f+391 g x))-b x \left(229425 c+17 x \left(12075 d+19 x \left(575 e+525 f x+483 g x^2\right)\right)\right)\right)\right)}{3900225 b^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{108 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 b d-2 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{8645 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{216 a^3 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{4 a^3 e \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b c-8 a f)}{21505 b^2}+\frac{54 a^2 x^2 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(1729 \sqrt[3]{b} (23 b c-8 a f)-8602 \left(1-\sqrt{3}\right) \sqrt[3]{a} (5 b d-2 a g)\right)}{37182145 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x^3 \left(a+b x^3\right)^{3/2} \left(229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right)}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left(8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right)}{185910725}",1,"(2*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(10*a*(7429*e + 21*x*(380*f + 391*g*x)) - b*x*(229425*c + 17*x*(12075*d + 19*x*(575*e + 525*f*x + 483*g*x^2))))) + 9975*a^2*(-23*b*c + 8*a*f)*x*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 41055*a^2*(-5*b*d + 2*a*g)*x^2*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(3900225*b^2*Sqrt[1 + (b*x^3)/a])","C",1
459,1,162,742,0.4100588,"\int x^2 \left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x^2*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{2 \left(1995 a^3 x \sqrt{\frac{b x^3}{a}+1} (8 a g-23 b d) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-41055 a^3 b e x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+\left(a+b x^3\right)^3 (-38 a (391 f+420 g x)+52003 b c+5 b x (9177 d+17 x (483 e+19 x (23 f+21 g x))))\right)}{780045 b^2 \sqrt{a+b x^3}}","\frac{108 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{10/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{216 a^3 e \sqrt{a+b x^3}}{1729 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a^2 \sqrt{a+b x^3} (7 b c-2 a f)}{105 b^2}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b d-8 a g)}{21505 b^2}+\frac{54 a^2 e x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 f x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 g x^4 \sqrt{a+b x^3}}{4301 b}+\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(43010 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{37182145 b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x^2 \left(a+b x^3\right)^{3/2} \left(52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right)}{780045}+\frac{2 a x^2 \sqrt{a+b x^3} \left(7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right)}{111546435}",1,"(2*((a + b*x^3)^3*(52003*b*c - 38*a*(391*f + 420*g*x) + 5*b*x*(9177*d + 17*x*(483*e + 19*x*(23*f + 21*g*x)))) + 1995*a^3*(-23*b*d + 8*a*g)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] - 41055*a^3*b*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(780045*b^2*Sqrt[a + b*x^3])","C",1
460,1,148,723,0.3662108,"\int x \left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[x*(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","-\frac{\sqrt{a+b x^3} \left(7980 a^2 b e x \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+1785 a b x^2 (4 a f-19 b c) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} (646 a g-2261 b d-5 b x (399 e+17 x (21 f+19 g x)))\right)}{67830 b^2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(3458 a^{2/3} \sqrt[3]{b} e+935 \left(1-\sqrt{3}\right) (19 b c-4 a f)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1616615 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 b c-4 a f) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{54 a^2 \sqrt{a+b x^3} (19 b c-4 a f)}{1729 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a^2 \sqrt{a+b x^3} (7 b d-2 a g)}{105 b^2}+\frac{54 a^2 e x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 f x^2 \sqrt{a+b x^3}}{1729 b}+\frac{2 a^2 g x^3 \sqrt{a+b x^3}}{105 b}+\frac{2 x \left(a+b x^3\right)^{3/2} \left(33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right)}{440895}+\frac{2 a x \sqrt{a+b x^3} \left(479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right)}{4849845}",1,"-1/67830*(Sqrt[a + b*x^3]*(4*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(-2261*b*d + 646*a*g - 5*b*x*(399*e + 17*x*(21*f + 19*g*x))) + 7980*a^2*b*e*x*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 1785*a*b*(-19*b*c + 4*a*f)*x^2*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(b^2*Sqrt[1 + (b*x^3)/a])","C",1
461,1,139,694,0.2629809,"\int \left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right) \, dx","Integrate[(a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]","\frac{\sqrt{a+b x^3} \left(-570 a x (2 a f-17 b c) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-255 a x^2 (4 a g-19 b d) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} (323 e+15 x (19 f+17 g x))\right)}{9690 b \sqrt{\frac{b x^3}{a}+1}}","-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 b d-4 a g) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{54 a^2 \sqrt{a+b x^3} (19 b d-4 a g)}{1729 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a^2 e \sqrt{a+b x^3}}{15 b}+\frac{54 a^2 f x \sqrt{a+b x^3}}{935 b}+\frac{54 a^2 g x^2 \sqrt{a+b x^3}}{1729 b}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(1729 \sqrt[3]{b} (17 b c-2 a f)-935 \left(1-\sqrt{3}\right) \sqrt[3]{a} (19 b d-4 a g)\right)}{1616615 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 a \sqrt{a+b x^3} \left(793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right)}{4849845}+\frac{2 \left(a+b x^3\right)^{3/2} \left(62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right)}{692835}",1,"(Sqrt[a + b*x^3]*(4*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(323*e + 15*x*(19*f + 17*g*x)) - 570*a*(-17*b*c + 2*a*f)*x*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] - 255*a*(-19*b*d + 4*a*g)*x^2*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(9690*b*Sqrt[1 + (b*x^3)/a])","C",1
462,1,215,676,0.5839219,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x,x]","\frac{4 \sqrt{\frac{b x^3}{a}+1} \left(\sqrt{a+b x^3} \left(a^2 (51 f+45 g x)+2 a b \left(170 c+51 f x^3+45 g x^4\right)+b^2 x^3 \left(85 c+51 f x^3+45 g x^4\right)\right)-255 a^{3/2} b c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)-90 a x \sqrt{a+b x^3} (2 a g-17 b d) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+765 a b e x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{1530 b \sqrt{\frac{b x^3}{a}+1}}","-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} c \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{54 a^2 e \sqrt{a+b x^3}}{91 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a^2 f \sqrt{a+b x^3}}{15 b}+\frac{54 a^2 g x \sqrt{a+b x^3}}{935 b}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-935 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-182 a g+1547 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{85085 b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 a \sqrt{a+b x^3} \left(85085 c x+41769 d x^2+25245 e x^3+17017 f x^4+12285 g x^5\right)}{255255 x}+\frac{2 \left(a+b x^3\right)^{3/2} \left(12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right)}{109395 x}",1,"(4*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3]*(a^2*(51*f + 45*g*x) + b^2*x^3*(85*c + 51*f*x^3 + 45*g*x^4) + 2*a*b*(170*c + 51*f*x^3 + 45*g*x^4)) - 255*a^(3/2)*b*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) - 90*a*(-17*b*d + 2*a*g)*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 765*a*b*e*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/(1530*b*Sqrt[1 + (b*x^3)/a])","C",1
463,1,224,692,0.4529636,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^2} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^2,x]","\frac{2}{9} d \left(\sqrt{a+b x^3} \left(4 a+b x^3\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)-\frac{a c \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)}{x \sqrt{\frac{b x^3}{a}+1}}+\frac{a e x \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}+\frac{a f x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}+\frac{2 g \left(a+b x^3\right)^{5/2}}{15 b}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(182 a^{2/3} \sqrt[3]{b} e-55 \left(1-\sqrt{3}\right) (2 a f+13 b c)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{5005 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a f+13 b c) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{182 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{2 a^2 g \sqrt{a+b x^3}}{15 b}+\frac{27 a \sqrt{a+b x^3} (2 a f+13 b c)}{91 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 a \sqrt{a+b x^3} \left(19305 c x+5005 d x^2+2457 e x^3+1485 f x^4+1001 g x^5\right)}{15015 x^2}+\frac{2 \left(a+b x^3\right)^{3/2} \left(6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right)}{45045 x^2}-\frac{27 a c \sqrt{a+b x^3}}{7 x}",1,"(2*g*(a + b*x^3)^(5/2))/(15*b) + (2*d*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]))/9 - (a*c*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)])/(x*Sqrt[1 + (b*x^3)/a]) + (a*e*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a] + (a*f*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])","C",1
464,1,232,694,0.4280307,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^3} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^3,x]","\frac{4 e x^2 \sqrt{\frac{b x^3}{a}+1} \left(\sqrt{a+b x^3} \left(4 a+b x^3\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)-9 a c \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)-18 a d x \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)+18 a f x^3 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+9 a g x^4 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{18 x^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(91 \sqrt[3]{b} (4 a f+11 b c)-110 \left(1-\sqrt{3}\right) \sqrt[3]{a} (2 a g+13 b d)\right)}{10010 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a g+13 b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{182 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2}{3} a^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{27 a \sqrt{a+b x^3} (2 a g+13 b d)}{91 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 a \sqrt{a+b x^3} \left(27027 c x-19305 d x^2-5005 e x^3-2457 f x^4-1485 g x^5\right)}{15015 x^3}+\frac{2 \left(a+b x^3\right)^{3/2} \left(9009 c x+6435 d x^2+5005 e x^3+4095 f x^4+3465 g x^5\right)}{45045 x^3}+\frac{27 a c \sqrt{a+b x^3}}{10 x^2}-\frac{27 a d \sqrt{a+b x^3}}{7 x}",1,"(4*e*x^2*Sqrt[1 + (b*x^3)/a]*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) - 9*a*c*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] - 18*a*d*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 18*a*f*x^3*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 9*a*g*x^4*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/(18*x^2*Sqrt[1 + (b*x^3)/a])","C",1
465,1,243,692,0.7972222,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^4} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^4,x]","\frac{-45 a^3 d \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)-90 a^3 e x \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)+90 a^3 g x^3 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+4 x^2 \sqrt{\frac{b x^3}{a}+1} \left(5 a^2 f \left(\sqrt{a+b x^3} \left(4 a+b x^3\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+3 b c \left(a+b x^3\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)\right)}{90 a^2 x^2 \sqrt{\frac{b x^3}{a}+1}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-110 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e+28 a g+77 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{770 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \sqrt[3]{b} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{14 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \left(a+b x^3\right)^{3/2} \left(1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right)}{3465 x^4}-\frac{2 a \sqrt{a+b x^3} \left(1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right)}{1155 x^4}-\frac{1}{3} \sqrt{a} (2 a f+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}+\frac{27 a \sqrt[3]{b} e \sqrt{a+b x^3}}{7 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}",1,"(-45*a^3*d*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] - 90*a^3*e*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 90*a^3*g*x^3*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 4*x^2*Sqrt[1 + (b*x^3)/a]*(5*a^2*f*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 3*b*c*(a + b*x^3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(90*a^2*x^2*Sqrt[1 + (b*x^3)/a])","C",1
466,1,246,741,0.7632701,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^5} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5,x]","\frac{-45 a^3 c \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)-90 a^3 e x^2 \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)+4 x^3 \left(2 x \sqrt{\frac{b x^3}{a}+1} \left(5 a^2 g \left(\sqrt{a+b x^3} \left(4 a+b x^3\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)+3 b d \left(a+b x^3\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)\right)-45 a^3 f \sqrt{a+b x^3} \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)\right)}{180 a^2 x^4 \sqrt{\frac{b x^3}{a}+1}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(28 a^{2/3} \sqrt[3]{b} e-5 \left(1-\sqrt{3}\right) (8 a f+7 b c)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{280 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a f+7 b c) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{112 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \left(a+b x^3\right)^{3/2} \left(315 c x+105 d x^2+63 e x^3+45 f x^4+35 g x^5\right)}{315 x^5}-\frac{2 a \sqrt{a+b x^3} \left(189 c x+105 d x^2+189 e x^3-135 f x^4-35 g x^5\right)}{105 x^5}-\frac{27 \sqrt{a+b x^3} (8 a f+7 b c)}{56 x}+\frac{27 \sqrt[3]{b} \sqrt{a+b x^3} (8 a f+7 b c)}{56 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{27 a c \sqrt{a+b x^3}}{20 x^4}-\frac{1}{3} \sqrt{a} (2 a g+3 b d) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{a d \sqrt{a+b x^3}}{x^3}+\frac{27 a e \sqrt{a+b x^3}}{10 x^2}",1,"(-45*a^3*c*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)] - 90*a^3*e*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] + 4*x^3*(-45*a^3*f*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 2*x*Sqrt[1 + (b*x^3)/a]*(5*a^2*g*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]) + 3*b*d*(a + b*x^3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a])))/(180*a^2*x^4*Sqrt[1 + (b*x^3)/a])","C",1
467,1,191,689,0.2757485,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^6} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^6,x]","\frac{\sqrt{a+b x^3} \left(-12 a^3 c \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)-15 a^3 d x \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)-30 a^3 f x^3 \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)-60 a^3 g x^4 \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)+8 b e x^5 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)\right)}{60 a^2 x^5 \sqrt{\frac{b x^3}{a}+1}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(14 \sqrt[3]{b} (2 a f+b c)-5 \left(1-\sqrt{3}\right) \sqrt[3]{a} (8 a g+7 b d)\right)}{280 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a g+7 b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{112 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{1}{60} \left(a+b x^3\right)^{3/2} \left(\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}+\frac{60 g}{x}\right)-\frac{b \sqrt{a+b x^3} \left(252 c x-315 d x^2-140 e x^3-126 f x^4-180 g x^5\right)}{140 x^3}+\frac{27 b c \sqrt{a+b x^3}}{20 x^2}+\frac{27 \sqrt[3]{b} \sqrt{a+b x^3} (8 a g+7 b d)}{56 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{27 b d \sqrt{a+b x^3}}{8 x}-\sqrt{a} b e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)",1,"(Sqrt[a + b*x^3]*(-12*a^3*c*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)] - 15*a^3*d*x*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)] - 30*a^3*f*x^3*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] - 60*a^3*g*x^4*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 8*b*e*x^5*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(60*a^2*x^5*Sqrt[1 + (b*x^3)/a])","C",1
468,1,240,692,0.7315368,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^7} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^7,x]","\frac{-\frac{12 a^2 d \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)}{x^5}-\frac{15 a^2 e \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)}{x^4}+\frac{8 b f \left(a+b x^3\right)^3 \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)}{a^2}-\frac{30 a^2 g \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)}{x^2}-15 b^2 c \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)-\frac{15 b c \left(a+b x^3\right)}{x^3}-\frac{10 c \left(a+b x^3\right)^2}{x^6}}{60 \sqrt{a+b x^3}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(-5 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e+4 a g+2 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{40 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} b^{4/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} e \sqrt{a+b x^3}}{8 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{b \sqrt{a+b x^3} \left(10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right)}{20 x^4}-\frac{1}{60} \left(a+b x^3\right)^{3/2} \left(\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}+\frac{30 g}{x^2}\right)-\frac{b (4 a f+b c) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{4 \sqrt{a}}+\frac{b c \sqrt{a+b x^3}}{4 x^3}+\frac{27 b d \sqrt{a+b x^3}}{20 x^2}-\frac{27 b e \sqrt{a+b x^3}}{8 x}",1,"((-15*b*c*(a + b*x^3))/x^3 - (10*c*(a + b*x^3)^2)/x^6 - 15*b^2*c*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]] - (12*a^2*d*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)])/x^5 - (15*a^2*e*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)])/x^4 - (30*a^2*g*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)])/x^2 + (8*b*f*(a + b*x^3)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a])/a^2)/(60*Sqrt[a + b*x^3])","C",1
469,1,240,746,1.0727096,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^8} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^8,x]","\frac{-\frac{60 a^2 c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)}{x^7}-\frac{84 a^2 e \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)}{x^5}-\frac{105 a^2 f \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)}{x^4}+\frac{56 b g \left(a+b x^3\right)^3 \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)}{a^2}-105 b^2 d \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)-\frac{105 b d \left(a+b x^3\right)}{x^3}-\frac{70 d \left(a+b x^3\right)^2}{x^6}}{420 \sqrt{a+b x^3}}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(28 a^{2/3} \sqrt[3]{b} e-5 \left(1-\sqrt{3}\right) (14 a f+b c)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{560 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (14 a f+b c) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a f+b c)}{112 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{b \sqrt{a+b x^3} \left(36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right)}{140 x^5}-\frac{1}{420} \left(a+b x^3\right)^{3/2} \left(\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}+\frac{140 g}{x^3}\right)-\frac{27 b \sqrt{a+b x^3} (14 a f+b c)}{112 a x}+\frac{27 b c \sqrt{a+b x^3}}{280 x^4}-\frac{b (4 a g+b d) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{4 \sqrt{a}}+\frac{b d \sqrt{a+b x^3}}{4 x^3}+\frac{27 b e \sqrt{a+b x^3}}{20 x^2}",1,"((-105*b*d*(a + b*x^3))/x^3 - (70*d*(a + b*x^3)^2)/x^6 - 105*b^2*d*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]] - (60*a^2*c*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)])/x^7 - (84*a^2*e*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)])/x^5 - (105*a^2*f*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)])/x^4 + (56*b*g*(a + b*x^3)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a])/a^2)/(420*Sqrt[a + b*x^3])","C",1
470,1,202,705,0.6026608,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^9} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^9,x]","-\frac{\sqrt{a+b x^3} \left(2 x \left(7 x \left(5 \left(3 a^2 g x^2 \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)+3 b^2 e x^6 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+a e \left(2 a+5 b x^3\right) \sqrt{\frac{b x^3}{a}+1}\right)+12 a^2 f x \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)\right)+60 a^2 d \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)\right)+105 a^2 c \, _2F_1\left(-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{840 a x^8 \sqrt{\frac{b x^3}{a}+1}}","-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) \left(7 \sqrt[3]{b} (b c-16 a f)+20 \left(1-\sqrt{3}\right) \sqrt[3]{a} (14 a g+b d)\right)}{2240 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (14 a g+b d) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a g+b d)}{112 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{b^2 e \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{4 \sqrt{a}}-\frac{1}{560} b \sqrt{a+b x^3} \left(\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right)-\frac{1}{840} \left(a+b x^3\right)^{3/2} \left(\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right)",1,"-1/840*(Sqrt[a + b*x^3]*(105*a^2*c*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] + 2*x*(60*a^2*d*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] + 7*x*(12*a^2*f*x*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)] + 5*(a*e*(2*a + 5*b*x^3)*Sqrt[1 + (b*x^3)/a] + 3*b^2*e*x^6*ArcTanh[Sqrt[1 + (b*x^3)/a]] + 3*a^2*g*x^2*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)])))))/(a*x^8*Sqrt[1 + (b*x^3)/a])","C",1
471,1,226,714,0.8179031,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^{10}} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^10,x]","-\frac{\sqrt{a+b x^3} \left(105 a^5 d \, _2F_1\left(-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right)+2 x \left(60 a^5 e \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)+7 x \left(12 a^5 g x \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)+5 a^3 f \left(3 b^2 x^6 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+a \left(2 a+5 b x^3\right) \sqrt{\frac{b x^3}{a}+1}\right)-8 b^3 c x^6 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right)\right)\right)\right)}{840 a^4 x^8 \sqrt{\frac{b x^3}{a}+1}}","-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(20 \left(1-\sqrt{3}\right) \sqrt[3]{a} b^{2/3} e-112 a g+7 b d\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2240 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{7/3} e \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{b^2 (b c-6 a f) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{24 a^{3/2}}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{b \sqrt{a+b x^3} \left(\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right)}{1680}-\frac{\left(a+b x^3\right)^{3/2} \left(\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right)}{2520}",1,"-1/840*(Sqrt[a + b*x^3]*(105*a^5*d*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] + 2*x*(60*a^5*e*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] + 7*x*(5*a^3*f*(a*(2*a + 5*b*x^3)*Sqrt[1 + (b*x^3)/a] + 3*b^2*x^6*ArcTanh[Sqrt[1 + (b*x^3)/a]]) + 12*a^5*g*x*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)] - 8*b^3*c*x^6*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^3)/a]))))/(a^4*x^8*Sqrt[1 + (b*x^3)/a])","C",1
472,1,227,764,0.7213641,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^{11}} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^11,x]","-\frac{\sqrt{a+b x^3} \left(84 a^5 c \, _2F_1\left(-\frac{10}{3},-\frac{3}{2};-\frac{7}{3};-\frac{b x^3}{a}\right)+105 a^5 e x^2 \, _2F_1\left(-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right)+2 x^3 \left(60 a^5 f \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)+35 a^3 g x \left(3 b^2 x^6 \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+a \left(2 a+5 b x^3\right) \sqrt{\frac{b x^3}{a}+1}\right)-56 b^3 d x^7 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right)\right)\right)}{840 a^4 x^{10} \sqrt{\frac{b x^3}{a}+1}}","-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(7 a^{2/3} \sqrt[3]{b} e-5 \left(1-\sqrt{3}\right) (b c-4 a f)\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2240 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (b c-4 a f) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{896 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{b^2 (b d-6 a g) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{24 a^{3/2}}-\frac{27 b^{7/3} \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{27 b^2 \sqrt{a+b x^3} (b c-4 a f)}{448 a^2 x}-\frac{27 b^2 c \sqrt{a+b x^3}}{1120 a x^4}-\frac{b^2 d \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 e \sqrt{a+b x^3}}{320 a x^2}-\frac{b \sqrt{a+b x^3} \left(\frac{108 c}{x^7}+\frac{140 d}{x^6}+\frac{189 e}{x^5}+\frac{270 f}{x^4}+\frac{420 g}{x^3}\right)}{1680}-\frac{\left(a+b x^3\right)^{3/2} \left(\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}+\frac{420 g}{x^6}\right)}{2520}",1,"-1/840*(Sqrt[a + b*x^3]*(84*a^5*c*Hypergeometric2F1[-10/3, -3/2, -7/3, -((b*x^3)/a)] + 105*a^5*e*x^2*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] + 2*x^3*(35*a^3*g*x*(a*(2*a + 5*b*x^3)*Sqrt[1 + (b*x^3)/a] + 3*b^2*x^6*ArcTanh[Sqrt[1 + (b*x^3)/a]]) + 60*a^5*f*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] - 56*b^3*d*x^7*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^3)/a])))/(a^4*x^10*Sqrt[1 + (b*x^3)/a])","C",1
473,1,194,796,0.4734555,"\int \frac{\left(a+b x^3\right)^{3/2} \left(c+d x+e x^2+f x^3+g x^4\right)}{x^{12}} \, dx","Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x]","\frac{\sqrt{a+b x^3} \left(11 x^3 \left(-105 a^5 f \, _2F_1\left(-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right)-120 a^5 g x \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)+112 b^3 e x^8 \left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right)\right)-840 a^5 c \, _2F_1\left(-\frac{11}{3},-\frac{3}{2};-\frac{8}{3};-\frac{b x^3}{a}\right)-924 a^5 d x \, _2F_1\left(-\frac{10}{3},-\frac{3}{2};-\frac{7}{3};-\frac{b x^3}{a}\right)\right)}{9240 a^4 x^{11} \sqrt{\frac{b x^3}{a}+1}}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right) b^3}{24 a^{3/2}}+\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (b d-4 a g) \left(\sqrt[3]{b} x+\sqrt[3]{a}\right) \sqrt{\frac{b^{2/3} x^2-\sqrt[3]{a} \sqrt[3]{b} x+a^{2/3}}{\left(\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) b^{7/3}}{896 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{b} x+\sqrt[3]{a}\right)}{\left(\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}\right)^2}} \sqrt{b x^3+a}}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \left(7 \sqrt[3]{b} (7 b c-22 a f)+110 \left(1-\sqrt{3}\right) \sqrt[3]{a} (b d-4 a g)\right) \left(\sqrt[3]{b} x+\sqrt[3]{a}\right) \sqrt{\frac{b^{2/3} x^2-\sqrt[3]{a} \sqrt[3]{b} x+a^{2/3}}{\left(\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right) b^{7/3}}{49280 a^2 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{b} x+\sqrt[3]{a}\right)}{\left(\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}\right)^2}} \sqrt{b x^3+a}}-\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^{7/3}}{448 a^2 \left(\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}\right)}+\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^2}{448 a^2 x}+\frac{27 (7 b c-22 a f) \sqrt{b x^3+a} b^2}{7040 a^2 x^2}-\frac{e \sqrt{b x^3+a} b^2}{24 a x^3}-\frac{27 d \sqrt{b x^3+a} b^2}{1120 a x^4}-\frac{27 c \sqrt{b x^3+a} b^2}{1760 a x^5}-\frac{\left(\frac{945 c}{x^8}+\frac{2970 g}{x^4}+\frac{2079 f}{x^5}+\frac{1540 e}{x^6}+\frac{1188 d}{x^7}\right) \sqrt{b x^3+a} b}{18480}-\frac{\left(\frac{2520 c}{x^{11}}+\frac{3960 g}{x^7}+\frac{3465 f}{x^8}+\frac{3080 e}{x^9}+\frac{2772 d}{x^{10}}\right) \left(b x^3+a\right)^{3/2}}{27720}",1,"(Sqrt[a + b*x^3]*(-840*a^5*c*Hypergeometric2F1[-11/3, -3/2, -8/3, -((b*x^3)/a)] - 924*a^5*d*x*Hypergeometric2F1[-10/3, -3/2, -7/3, -((b*x^3)/a)] + 11*x^3*(-105*a^5*f*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] - 120*a^5*g*x*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] + 112*b^3*e*x^8*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^3)/a])))/(9240*a^4*x^11*Sqrt[1 + (b*x^3)/a])","C",1
474,1,114,102,0.0696871,"\int \left(c+d x+e x^2\right) \left(a+b x^3\right)^p \, dx","Integrate[(c + d*x + e*x^2)*(a + b*x^3)^p,x]","\frac{\left(a+b x^3\right)^p \left(\frac{b x^3}{a}+1\right)^{-p} \left(6 b c (p+1) x \, _2F_1\left(\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right)+3 b d (p+1) x^2 \, _2F_1\left(\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right)+2 e \left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^p\right)}{6 b (p+1)}","\frac{c x \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{a}+\frac{d x^2 \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 a}+\frac{e \left(a+b x^3\right)^{p+1}}{3 b (p+1)}",1,"((a + b*x^3)^p*(2*e*(a + b*x^3)*(1 + (b*x^3)/a)^p + 6*b*c*(1 + p)*x*Hypergeometric2F1[1/3, -p, 4/3, -((b*x^3)/a)] + 3*b*d*(1 + p)*x^2*Hypergeometric2F1[2/3, -p, 5/3, -((b*x^3)/a)]))/(6*b*(1 + p)*(1 + (b*x^3)/a)^p)","A",1
475,1,116,107,0.0966377,"\int x \left(c+d x+e x^2\right) \left(a+b x^3\right)^p \, dx","Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^p,x]","\frac{\left(a+b x^3\right)^p \left(\frac{b x^3}{a}+1\right)^{-p} \left(6 b c (p+1) x^2 \, _2F_1\left(\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right)+4 d \left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^p+3 b e (p+1) x^4 \, _2F_1\left(\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right)\right)}{12 b (p+1)}","\frac{c x^2 \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 a}+\frac{d \left(a+b x^3\right)^{p+1}}{3 b (p+1)}+\frac{e x^4 \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{4 a}",1,"((a + b*x^3)^p*(4*d*(a + b*x^3)*(1 + (b*x^3)/a)^p + 6*b*c*(1 + p)*x^2*Hypergeometric2F1[2/3, -p, 5/3, -((b*x^3)/a)] + 3*b*e*(1 + p)*x^4*Hypergeometric2F1[4/3, -p, 7/3, -((b*x^3)/a)]))/(12*b*(1 + p)*(1 + (b*x^3)/a)^p)","A",1
476,1,116,107,0.1159684,"\int x^2 \left(c+d x+e x^2\right) \left(a+b x^3\right)^p \, dx","Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^p,x]","\frac{\left(a+b x^3\right)^p \left(\frac{b x^3}{a}+1\right)^{-p} \left(20 c \left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^p+15 b d (p+1) x^4 \, _2F_1\left(\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right)+12 b e (p+1) x^5 \, _2F_1\left(\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right)\right)}{60 b (p+1)}","\frac{c \left(a+b x^3\right)^{p+1}}{3 b (p+1)}+\frac{d x^4 \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{4 a}+\frac{e x^5 \left(a+b x^3\right)^{p+1} \, _2F_1\left(1,p+\frac{8}{3};\frac{8}{3};-\frac{b x^3}{a}\right)}{5 a}",1,"((a + b*x^3)^p*(20*c*(a + b*x^3)*(1 + (b*x^3)/a)^p + 15*b*d*(1 + p)*x^4*Hypergeometric2F1[4/3, -p, 7/3, -((b*x^3)/a)] + 12*b*e*(1 + p)*x^5*Hypergeometric2F1[5/3, -p, 8/3, -((b*x^3)/a)]))/(60*b*(1 + p)*(1 + (b*x^3)/a)^p)","A",1
477,1,68,68,0.0062538,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right) \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4),x]","a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} a f x^4+\frac{1}{5} b c x^5+\frac{1}{6} b d x^6+\frac{1}{7} b e x^7+\frac{1}{8} b f x^8","a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} a f x^4+\frac{1}{5} b c x^5+\frac{1}{6} b d x^6+\frac{1}{7} b e x^7+\frac{1}{8} b f x^8",1,"a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + (a*f*x^4)/4 + (b*c*x^5)/5 + (b*d*x^6)/6 + (b*e*x^7)/7 + (b*f*x^8)/8","A",1
478,1,73,73,0.0036159,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right) \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4),x]","\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{7} a f x^7+\frac{1}{8} b c x^8+\frac{1}{9} b d x^9+\frac{1}{10} b e x^{10}+\frac{1}{11} b f x^{11}","\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{7} a f x^7+\frac{1}{8} b c x^8+\frac{1}{9} b d x^9+\frac{1}{10} b e x^{10}+\frac{1}{11} b f x^{11}",1,"(a*c*x^4)/4 + (a*d*x^5)/5 + (a*e*x^6)/6 + (a*f*x^7)/7 + (b*c*x^8)/8 + (b*d*x^9)/9 + (b*e*x^10)/10 + (b*f*x^11)/11","A",1
479,1,124,109,0.0041774,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^2 \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^2,x]","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{4} a^2 f x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{4} a b f x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{1}{12} b^2 f x^{12}","a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left(a+b x^4\right)^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}",1,"a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a^2*f*x^4)/4 + (2*a*b*c*x^5)/5 + (a*b*d*x^6)/3 + (2*a*b*e*x^7)/7 + (a*b*f*x^8)/4 + (b^2*c*x^9)/9 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11 + (b^2*f*x^12)/12","A",1
480,1,129,114,0.005034,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^2 \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^2,x]","\frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{1}{4} a b c x^8+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{12} b^2 c x^{12}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15}","\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{c \left(a+b x^4\right)^3}{12 b}+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15}",1,"(a^2*c*x^4)/4 + (a^2*d*x^5)/5 + (a^2*e*x^6)/6 + (a^2*f*x^7)/7 + (a*b*c*x^8)/4 + (2*a*b*d*x^9)/9 + (a*b*e*x^10)/5 + (2*a*b*f*x^11)/11 + (b^2*c*x^12)/12 + (b^2*d*x^13)/13 + (b^2*e*x^14)/14 + (b^2*f*x^15)/15","A",1
481,1,180,151,0.005188,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^3 \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^3,x]","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{1}{4} a^3 f x^4+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a^2 b f x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{4} a b^2 f x^{12}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}+\frac{1}{16} b^3 f x^{16}","a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{f \left(a+b x^4\right)^4}{16 b}+\frac{1}{13} b^3 c x^{13}+\frac{1}{14} b^3 d x^{14}+\frac{1}{15} b^3 e x^{15}",1,"a^3*c*x + (a^3*d*x^2)/2 + (a^3*e*x^3)/3 + (a^3*f*x^4)/4 + (3*a^2*b*c*x^5)/5 + (a^2*b*d*x^6)/2 + (3*a^2*b*e*x^7)/7 + (3*a^2*b*f*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)/11 + (a*b^2*f*x^12)/4 + (b^3*c*x^13)/13 + (b^3*d*x^14)/14 + (b^3*e*x^15)/15 + (b^3*f*x^16)/16","A",1
482,1,185,156,0.0176812,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^3 \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^3,x]","\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{3}{8} a^2 b c x^8+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{4} a b^2 c x^{12}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{1}{16} b^3 c x^{16}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19}","\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{c \left(a+b x^4\right)^4}{16 b}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19}",1,"(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (3*a^2*b*c*x^8)/8 + (a^2*b*d*x^9)/3 + (3*a^2*b*e*x^10)/10 + (3*a^2*b*f*x^11)/11 + (a*b^2*c*x^12)/4 + (3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^15)/5 + (b^3*c*x^16)/16 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19)/19","A",1
483,1,236,193,0.0069535,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^4 \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]","a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{1}{4} a^4 f x^4+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{1}{2} a^3 b f x^8+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{2} a^2 b^2 f x^{12}+\frac{4}{13} a b^3 c x^{13}+\frac{2}{7} a b^3 d x^{14}+\frac{4}{15} a b^3 e x^{15}+\frac{1}{4} a b^3 f x^{16}+\frac{1}{17} b^4 c x^{17}+\frac{1}{18} b^4 d x^{18}+\frac{1}{19} b^4 e x^{19}+\frac{1}{20} b^4 f x^{20}","a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{4}{13} a b^3 c x^{13}+\frac{2}{7} a b^3 d x^{14}+\frac{4}{15} a b^3 e x^{15}+\frac{f \left(a+b x^4\right)^5}{20 b}+\frac{1}{17} b^4 c x^{17}+\frac{1}{18} b^4 d x^{18}+\frac{1}{19} b^4 e x^{19}",1,"a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + (a^4*f*x^4)/4 + (4*a^3*b*c*x^5)/5 + (2*a^3*b*d*x^6)/3 + (4*a^3*b*e*x^7)/7 + (a^3*b*f*x^8)/2 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (a^2*b^2*f*x^12)/2 + (4*a*b^3*c*x^13)/13 + (2*a*b^3*d*x^14)/7 + (4*a*b^3*e*x^15)/15 + (a*b^3*f*x^16)/4 + (b^4*c*x^17)/17 + (b^4*d*x^18)/18 + (b^4*e*x^19)/19 + (b^4*f*x^20)/20","A",1
484,1,241,198,0.0073182,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^4 \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]","\frac{1}{4} a^4 c x^4+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{1}{2} a^3 b c x^8+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{2} a^2 b^2 c x^{12}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{1}{4} a b^3 c x^{16}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{20} b^4 c x^{20}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23}","\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left(a+b x^4\right)^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23}",1,"(a^4*c*x^4)/4 + (a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (a^3*b*c*x^8)/2 + (4*a^3*b*d*x^9)/9 + (2*a^3*b*e*x^10)/5 + (4*a^3*b*f*x^11)/11 + (a^2*b^2*c*x^12)/2 + (6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*b^2*f*x^15)/5 + (a*b^3*c*x^16)/4 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/9 + (4*a*b^3*f*x^19)/19 + (b^4*c*x^20)/20 + (b^4*d*x^21)/21 + (b^4*e*x^22)/22 + (b^4*f*x^23)/23","A",1
485,1,214,133,0.112502,"\int \frac{c+d x+e x^2+f x^3}{a-b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4),x]","-\frac{\log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{3/4} e+\sqrt[4]{a} \sqrt{b} c+\sqrt{a} \sqrt[4]{b} d\right)}{4 a b^{3/4}}-\frac{\log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(-a^{3/4} e-\sqrt[4]{a} \sqrt{b} c+\sqrt{a} \sqrt[4]{b} d\right)}{4 a b^{3/4}}+\frac{\left(\sqrt[4]{a} \sqrt{b} c-a^{3/4} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a b^{3/4}}+\frac{d \log \left(\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{a} \sqrt{b}}-\frac{f \log \left(a-b x^4\right)}{4 b}","\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left(a-b x^4\right)}{4 b}",1,"((a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a*b^(3/4)) - ((a^(1/4)*Sqrt[b]*c + Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/(4*a*b^(3/4)) - ((-(a^(1/4)*Sqrt[b]*c) + Sqrt[a]*b^(1/4)*d - a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/(4*a*b^(3/4)) + (d*Log[Sqrt[a] + Sqrt[b]*x^2])/(4*Sqrt[a]*Sqrt[b]) - (f*Log[a - b*x^4])/(4*b)","A",1
486,1,221,162,0.0949383,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{a-b x^4} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a - b*x^4),x]","\frac{-3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} x\right) \left(a^{3/4} f+\sqrt[4]{a} \sqrt{b} d+\sqrt{a} \sqrt[4]{b} e\right)+3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} x\right) \left(a^{3/4} f+\sqrt[4]{a} \sqrt{b} d-\sqrt{a} \sqrt[4]{b} e\right)+6 \left(\sqrt[4]{a} \sqrt{b} d-a^{3/4} f\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)-3 b^{3/4} c \log \left(a-b x^4\right)+3 \sqrt{a} \sqrt[4]{b} e \log \left(\sqrt{a}+\sqrt{b} x^2\right)-12 b^{3/4} d x-6 b^{3/4} e x^2-4 b^{3/4} f x^3}{12 b^{7/4}}","\frac{\sqrt[4]{a} \left(\sqrt{b} d-\sqrt{a} f\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 b^{7/4}}+\frac{\sqrt[4]{a} \left(\sqrt{a} f+\sqrt{b} d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 b^{3/2}}-\frac{c \log \left(a-b x^4\right)}{4 b}-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b}",1,"(-12*b^(3/4)*d*x - 6*b^(3/4)*e*x^2 - 4*b^(3/4)*f*x^3 + 6*(a^(1/4)*Sqrt[b]*d - a^(3/4)*f)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(a^(1/4)*Sqrt[b]*d + Sqrt[a]*b^(1/4)*e + a^(3/4)*f)*Log[a^(1/4) - b^(1/4)*x] + 3*(a^(1/4)*Sqrt[b]*d - Sqrt[a]*b^(1/4)*e + a^(3/4)*f)*Log[a^(1/4) + b^(1/4)*x] + 3*Sqrt[a]*b^(1/4)*e*Log[Sqrt[a] + Sqrt[b]*x^2] - 3*b^(3/4)*c*Log[a - b*x^4])/(12*b^(7/4))","A",1
487,1,296,293,0.2313805,"\int \frac{c+d x+e x^2+f x^3}{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4),x]","\frac{-\sqrt{2} \sqrt[4]{b} \left(\sqrt[4]{a} \sqrt{b} c-a^{3/4} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} \sqrt[4]{b} \left(\sqrt[4]{a} \sqrt{b} c-a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-2 \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+\sqrt{2} \sqrt{b} c\right)+2 \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-2 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+\sqrt{2} \sqrt{b} c\right)+2 a f \log \left(a+b x^4\right)}{8 a b}","-\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{b} c-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left(\sqrt{a} e+\sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b}}+\frac{f \log \left(a+b x^4\right)}{4 b}",1,"(-2*a^(1/4)*b^(1/4)*(Sqrt[2]*Sqrt[b]*c + 2*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*b^(1/4)*(Sqrt[2]*Sqrt[b]*c - 2*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*b^(1/4)*(a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 2*a*f*Log[a + b*x^4])/(8*a*b)","A",1
488,1,311,321,0.2392422,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{a+b x^4} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4),x]","\frac{-3 \sqrt{2} \left(a^{3/4} f-\sqrt[4]{a} \sqrt{b} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+3 \sqrt{2} \left(a^{3/4} f-\sqrt[4]{a} \sqrt{b} d\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+6 b^{3/4} c \log \left(a+b x^4\right)+6 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(2 \sqrt[4]{a} \sqrt[4]{b} e+\sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right)-6 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-2 \sqrt[4]{a} \sqrt[4]{b} e+\sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right)+24 b^{3/4} d x+12 b^{3/4} e x^2+8 b^{3/4} f x^3}{24 b^{7/4}}","\frac{\sqrt[4]{a} \left(\sqrt{b} d-\sqrt{a} f\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{7/4}}-\frac{\sqrt[4]{a} \left(\sqrt{b} d-\sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{7/4}}+\frac{\sqrt[4]{a} \left(\sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt[4]{a} \left(\sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} b^{7/4}}-\frac{\sqrt{a} e \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 b^{3/2}}+\frac{c \log \left(a+b x^4\right)}{4 b}+\frac{d x}{b}+\frac{e x^2}{2 b}+\frac{f x^3}{3 b}",1,"(24*b^(3/4)*d*x + 12*b^(3/4)*e*x^2 + 8*b^(3/4)*f*x^3 + 6*a^(1/4)*(Sqrt[2]*Sqrt[b]*d + 2*a^(1/4)*b^(1/4)*e + Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 6*a^(1/4)*(Sqrt[2]*Sqrt[b]*d - 2*a^(1/4)*b^(1/4)*e + Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*(-(a^(1/4)*Sqrt[b]*d) + a^(3/4)*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*(-(a^(1/4)*Sqrt[b]*d) + a^(3/4)*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 6*b^(3/4)*c*Log[a + b*x^4])/(24*b^(7/4))","A",1
489,1,315,318,0.4055762,"\int \frac{c+d x+e x^2+f x^3}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^2,x]","\frac{\sqrt{2} \sqrt[4]{b} \left(a^{3/4} e-3 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} \sqrt[4]{b} \left(3 \sqrt[4]{a} \sqrt{b} c-a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\frac{8 a (a f-b x (c+x (d+e x)))}{a+b x^4}-2 \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right)+2 \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 \sqrt[4]{a} \sqrt[4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right)}{32 a^2 b}","-\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left(3 \sqrt{b} c-\sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left(\sqrt{a} e+3 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 a^{3/2} \sqrt{b}}-\frac{a f-b x \left(c+d x+e x^2\right)}{4 a b \left(a+b x^4\right)}",1,"((-8*a*(a*f - b*x*(c + x*(d + e*x))))/(a + b*x^4) - 2*a^(1/4)*b^(1/4)*(3*Sqrt[2]*Sqrt[b]*c + 4*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*b^(1/4)*(3*Sqrt[2]*Sqrt[b]*c - 4*a^(1/4)*b^(1/4)*d + Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*b^(1/4)*(-3*a^(1/4)*Sqrt[b]*c + a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(1/4)*(3*a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^2*b)","A",1
490,1,294,310,0.3828458,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^2} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^2,x]","\frac{-\frac{2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(4 \sqrt[4]{a} \sqrt[4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right)}{a^{3/4}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-4 \sqrt[4]{a} \sqrt[4]{b} e+3 \sqrt{2} \sqrt{a} f+\sqrt{2} \sqrt{b} d\right)}{a^{3/4}}+\frac{\sqrt{2} \left(3 \sqrt{a} f-\sqrt{b} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{3/4}}+\frac{\sqrt{2} \left(\sqrt{b} d-3 \sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{3/4}}-\frac{8 b^{3/4} (c+x (d+x (e+f x)))}{a+b x^4}}{32 b^{7/4}}","-\frac{\left(\sqrt{b} d-3 \sqrt{a} f\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left(\sqrt{b} d-3 \sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\left(3 \sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\left(3 \sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{3/4} b^{7/4}}+\frac{e \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{4 \sqrt{a} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{4 b \left(a+b x^4\right)}",1,"((-8*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4) - (2*(Sqrt[2]*Sqrt[b]*d + 4*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (2*(Sqrt[2]*Sqrt[b]*d - 4*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (Sqrt[2]*(-(Sqrt[b]*d) + 3*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (Sqrt[2]*(Sqrt[b]*d - 3*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(32*b^(7/4))","A",1
491,1,347,351,0.4389803,"\int \frac{c+d x+e x^2+f x^3}{\left(a+b x^4\right)^3} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^3,x]","\frac{\frac{\sqrt{2} \left(5 a^{3/4} e-21 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{\sqrt{2} \left(21 \sqrt[4]{a} \sqrt{b} c-5 a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}-\frac{32 a^2 (a f-b x (c+x (d+e x)))}{b \left(a+b x^4\right)^2}-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(24 \sqrt[4]{a} \sqrt[4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-24 \sqrt[4]{a} \sqrt[4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3}","-\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left(21 \sqrt{b} c-5 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left(5 \sqrt{a} e+21 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{5/2} \sqrt{b}}+\frac{x \left(7 c+6 d x+5 e x^2\right)}{32 a^2 \left(a+b x^4\right)}-\frac{a f-b x \left(c+d x+e x^2\right)}{8 a b \left(a+b x^4\right)^2}",1,"((8*a*x*(7*c + x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^2*(a*f - b*x*(c + x*(d + e*x))))/(b*(a + b*x^4)^2) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c + 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[b]*c - 24*a^(1/4)*b^(1/4)*d + 5*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (Sqrt[2]*(-21*a^(1/4)*Sqrt[b]*c + 5*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[b]*c - 5*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(256*a^3)","A",1
492,1,329,340,0.3908617,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^3} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^3,x]","\frac{-\frac{2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(8 \sqrt[4]{a} \sqrt[4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right)}{a^{7/4}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-8 \sqrt[4]{a} \sqrt[4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right)}{a^{7/4}}+\frac{3 \sqrt{2} \left(\sqrt{a} f-\sqrt{b} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{3 \sqrt{2} \left(\sqrt{b} d-\sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}-\frac{32 b^{3/4} (c+x (d+x (e+f x)))}{\left(a+b x^4\right)^2}+\frac{8 b^{3/4} x (d+x (2 e+3 f x))}{a \left(a+b x^4\right)}}{256 b^{7/4}}","-\frac{3 \left(\sqrt{b} d-\sqrt{a} f\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left(\sqrt{b} d-\sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left(\sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left(\sqrt{a} f+\sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{e \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{16 a^{3/2} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{8 b \left(a+b x^4\right)^2}+\frac{x \left(d+2 e x+3 f x^2\right)}{32 a b \left(a+b x^4\right)}",1,"((8*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)) - (32*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4)^2 - (2*(3*Sqrt[2]*Sqrt[b]*d + 8*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (2*(3*Sqrt[2]*Sqrt[b]*d - 8*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (3*Sqrt[2]*(-(Sqrt[b]*d) + Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(256*b^(7/4))","A",1
493,1,379,382,0.4420632,"\int \frac{c+d x+e x^2+f x^3}{\left(a+b x^4\right)^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^4,x]","\frac{\frac{3 \sqrt{2} \left(15 a^{3/4} e-77 \sqrt[4]{a} \sqrt{b} c\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}+\frac{3 \sqrt{2} \left(77 \sqrt[4]{a} \sqrt{b} c-15 a^{3/4} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{3/4}}-\frac{256 a^3 (a f-b x (c+x (d+e x)))}{b \left(a+b x^4\right)^3}+\frac{32 a^2 x (11 c+x (10 d+9 e x))}{\left(a+b x^4\right)^2}-\frac{6 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(80 \sqrt[4]{a} \sqrt[4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{6 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-80 \sqrt[4]{a} \sqrt[4]{b} d+15 \sqrt{2} \sqrt{a} e+77 \sqrt{2} \sqrt{b} c\right)}{b^{3/4}}+\frac{8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}}{3072 a^4}","-\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left(77 \sqrt{b} c-15 \sqrt{a} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{15/4} b^{3/4}}-\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{\left(15 \sqrt{a} e+77 \sqrt{b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{256 \sqrt{2} a^{15/4} b^{3/4}}+\frac{5 d \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{7/2} \sqrt{b}}+\frac{x \left(77 c+60 d x+45 e x^2\right)}{384 a^3 \left(a+b x^4\right)}+\frac{x \left(11 c+10 d x+9 e x^2\right)}{96 a^2 \left(a+b x^4\right)^2}-\frac{a f-b x \left(c+d x+e x^2\right)}{12 a b \left(a+b x^4\right)^3}",1,"((8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (256*a^3*(a*f - b*x*(c + x*(d + e*x))))/(b*(a + b*x^4)^3) - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c - 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-77*a^(1/4)*Sqrt[b]*c + 15*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (3*Sqrt[2]*(77*a^(1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(3072*a^4)","A",1
494,1,366,380,0.4588347,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^4} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^4,x]","\frac{-\frac{6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right) \left(16 \sqrt[4]{a} \sqrt[4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right)}{a^{11/4}}+\frac{6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right) \left(-16 \sqrt[4]{a} \sqrt[4]{b} e+5 \sqrt{2} \sqrt{a} f+7 \sqrt{2} \sqrt{b} d\right)}{a^{11/4}}+\frac{3 \sqrt{2} \left(5 \sqrt{a} f-7 \sqrt{b} d\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{11/4}}+\frac{3 \sqrt{2} \left(7 \sqrt{b} d-5 \sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{11/4}}+\frac{8 b^{3/4} x (7 d+3 x (4 e+5 f x))}{a^2 \left(a+b x^4\right)}-\frac{256 b^{3/4} (c+x (d+x (e+f x)))}{\left(a+b x^4\right)^3}+\frac{32 b^{3/4} x (d+x (2 e+3 f x))}{a \left(a+b x^4\right)^2}}{3072 b^{7/4}}","-\frac{\left(7 \sqrt{b} d-5 \sqrt{a} f\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left(7 \sqrt{b} d-5 \sqrt{a} f\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{512 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\left(5 \sqrt{a} f+7 \sqrt{b} d\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\left(5 \sqrt{a} f+7 \sqrt{b} d\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{256 \sqrt{2} a^{11/4} b^{7/4}}+\frac{e \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{32 a^{5/2} b^{3/2}}+\frac{x \left(7 d+12 e x+15 f x^2\right)}{384 a^2 b \left(a+b x^4\right)}-\frac{c+d x+e x^2+f x^3}{12 b \left(a+b x^4\right)^3}+\frac{x \left(d+2 e x+3 f x^2\right)}{96 a b \left(a+b x^4\right)^2}",1,"((32*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)^2) + (8*b^(3/4)*x*(7*d + 3*x*(4*e + 5*f*x)))/(a^2*(a + b*x^4)) - (256*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4)^3 - (6*(7*Sqrt[2]*Sqrt[b]*d + 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*(7*Sqrt[2]*Sqrt[b]*d - 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (3*Sqrt[2]*(-7*Sqrt[b]*d + 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4) + (3*Sqrt[2]*(7*Sqrt[b]*d - 5*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4))/(3072*b^(7/4))","A",1
495,1,202,418,0.727332,"\int x^4 \left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4} \, dx","Integrate[x^4*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]","\frac{\sqrt{a+b x^4} \left(-\frac{315 a^{3/2} \sqrt{b} d \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{\frac{b x^4}{a}+1}}-\frac{720 a b c x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+720 b c x \left(a+b x^4\right)+315 b d x^2 \left(a+2 b x^4\right)-\frac{560 a b e x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+560 b e x^3 \left(a+b x^4\right)+168 f \left(a+b x^4\right) \left(3 b x^4-2 a\right)\right)}{5040 b^2}","-\frac{a^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(7 \sqrt{a} e+5 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{7/4} \sqrt{a+b x^4}}+\frac{2 a^{9/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 b^{3/2}}-\frac{2 a^2 e x \sqrt{a+b x^4}}{15 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{\left(a+b x^4\right)^{3/2} \left(8 a f-15 b d x^2\right)}{120 b^2}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left(9 c+7 e x^2\right)+\frac{2 a c x \sqrt{a+b x^4}}{21 b}-\frac{a d x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a e x^3 \sqrt{a+b x^4}}{45 b}+\frac{f x^4 \left(a+b x^4\right)^{3/2}}{10 b}",1,"(Sqrt[a + b*x^4]*(720*b*c*x*(a + b*x^4) + 560*b*e*x^3*(a + b*x^4) + 315*b*d*x^2*(a + 2*b*x^4) + 168*f*(a + b*x^4)*(-2*a + 3*b*x^4) - (315*a^(3/2)*Sqrt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a] - (720*a*b*c*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (560*a*b*e*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(5040*b^2)","C",1
496,1,215,394,0.6592995,"\int x^3 \left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4} \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]","\frac{\sqrt{a+b x^4} \left(63 e \left(\sqrt{b} x^2 \left(a+2 b x^4\right)-\frac{a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{\frac{b x^4}{a}+1}}\right)+168 \sqrt{b} c \left(a+b x^4\right)-\frac{144 a \sqrt{b} d x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+144 \sqrt{b} d x \left(a+b x^4\right)-\frac{112 a \sqrt{b} f x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+112 \sqrt{b} f x^3 \left(a+b x^4\right)\right)}{1008 b^{3/2}}","-\frac{a^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(7 \sqrt{a} f+5 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{7/4} \sqrt{a+b x^4}}+\frac{2 a^{9/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 b^{3/2}}-\frac{2 a^2 f x \sqrt{a+b x^4}}{15 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{\left(a+b x^4\right)^{3/2} \left(4 c+3 e x^2\right)}{24 b}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left(9 d+7 f x^2\right)+\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b}",1,"(Sqrt[a + b*x^4]*(168*Sqrt[b]*c*(a + b*x^4) + 144*Sqrt[b]*d*x*(a + b*x^4) + 112*Sqrt[b]*f*x^3*(a + b*x^4) + 63*e*(Sqrt[b]*x^2*(a + 2*b*x^4) - (a^(3/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a]) - (144*a*Sqrt[b]*d*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (112*a*Sqrt[b]*f*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(1008*b^(3/2))","C",1
497,1,182,369,0.7808345,"\int x^2 \left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4} \, dx","Integrate[x^2*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]","\frac{1}{336} \sqrt{a+b x^4} \left(-\frac{21 a^{3/2} f \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{b^{3/2} \sqrt{\frac{b x^4}{a}+1}}+\frac{112 c x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{56 d \left(a+b x^4\right)}{b}-\frac{48 a e x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{b \sqrt{\frac{b x^4}{a}+1}}+\frac{48 e x \left(a+b x^4\right)}{b}+\frac{21 f x^2 \left(a+2 b x^4\right)}{b}\right)","\frac{a^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(21 \sqrt{b} c-5 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^4}}-\frac{a^2 f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 b^{3/2}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left(7 c+5 e x^2\right)+\frac{2 a c x \sqrt{a+b x^4}}{5 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{\left(a+b x^4\right)^{3/2} \left(4 d+3 f x^2\right)}{24 b}+\frac{2 a e x \sqrt{a+b x^4}}{21 b}-\frac{a f x^2 \sqrt{a+b x^4}}{16 b}",1,"(Sqrt[a + b*x^4]*((56*d*(a + b*x^4))/b + (48*e*x*(a + b*x^4))/b + (21*f*x^2*(a + 2*b*x^4))/b - (21*a^(3/2)*f*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/(b^(3/2)*Sqrt[1 + (b*x^4)/a]) - (48*a*e*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/(b*Sqrt[1 + (b*x^4)/a]) + (112*c*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/336","C",1
498,1,211,354,0.2049359,"\int x \left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4} \, dx","Integrate[x*(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]","\frac{\sqrt{a+b x^4} \left(21 \sqrt{a} \sqrt{b} c \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)+21 b c x^2 \sqrt{\frac{b x^4}{a}+1}+28 b d x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+14 b e x^4 \sqrt{\frac{b x^4}{a}+1}+14 a e \sqrt{\frac{b x^4}{a}+1}-12 a f x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)+12 a f x \sqrt{\frac{b x^4}{a}+1}+12 b f x^5 \sqrt{\frac{b x^4}{a}+1}\right)}{84 b \sqrt{\frac{b x^4}{a}+1}}","\frac{a^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(21 \sqrt{b} d-5 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{5/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} c x^2 \sqrt{a+b x^4}+\frac{a c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 \sqrt{b}}+\frac{1}{35} x^3 \sqrt{a+b x^4} \left(7 d+5 f x^2\right)+\frac{2 a d x \sqrt{a+b x^4}}{5 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{e \left(a+b x^4\right)^{3/2}}{6 b}+\frac{2 a f x \sqrt{a+b x^4}}{21 b}",1,"(Sqrt[a + b*x^4]*(14*a*e*Sqrt[1 + (b*x^4)/a] + 12*a*f*x*Sqrt[1 + (b*x^4)/a] + 21*b*c*x^2*Sqrt[1 + (b*x^4)/a] + 14*b*e*x^4*Sqrt[1 + (b*x^4)/a] + 12*b*f*x^5*Sqrt[1 + (b*x^4)/a] + 21*Sqrt[a]*Sqrt[b]*c*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] - 12*a*f*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)] + 28*b*d*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)]))/(84*b*Sqrt[1 + (b*x^4)/a])","C",1
499,1,171,331,0.1385656,"\int \left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4],x]","\frac{\sqrt{a+b x^4} \left(12 b c x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)+3 \sqrt{a} \sqrt{b} d \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)+3 b d x^2 \sqrt{\frac{b x^4}{a}+1}+4 b e x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+2 b f x^4 \sqrt{\frac{b x^4}{a}+1}+2 a f \sqrt{\frac{b x^4}{a}+1}\right)}{12 b \sqrt{\frac{b x^4}{a}+1}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} e+5 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{15} x \sqrt{a+b x^4} \left(5 c+3 e x^2\right)+\frac{1}{4} d x^2 \sqrt{a+b x^4}+\frac{a d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 \sqrt{b}}+\frac{2 a e x \sqrt{a+b x^4}}{5 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{f \left(a+b x^4\right)^{3/2}}{6 b}",1,"(Sqrt[a + b*x^4]*(2*a*f*Sqrt[1 + (b*x^4)/a] + 3*b*d*x^2*Sqrt[1 + (b*x^4)/a] + 2*b*f*x^4*Sqrt[1 + (b*x^4)/a] + 3*Sqrt[a]*Sqrt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 12*b*c*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)] + 4*b*e*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)]))/(12*b*Sqrt[1 + (b*x^4)/a])","C",1
500,1,208,345,0.3866292,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x,x]","\frac{3 a^{3/2} e \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)+3 \sqrt{b} \left(\left(a+b x^4\right) \left(2 c+e x^2\right)-2 \sqrt{a} c \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)+12 a \sqrt{b} d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)+4 a \sqrt{b} f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{12 \sqrt{b} \sqrt{a+b x^4}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} f+5 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{2 a^{5/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left(2 c+e x^2\right)-\frac{1}{2} \sqrt{a} c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{1}{15} x \sqrt{a+b x^4} \left(5 d+3 f x^2\right)+\frac{a e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 \sqrt{b}}+\frac{2 a f x \sqrt{a+b x^4}}{5 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(3*a^(3/2)*e*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 3*Sqrt[b]*((2*c + e*x^2)*(a + b*x^4) - 2*Sqrt[a]*c*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) + 12*a*Sqrt[b]*d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)] + 4*a*Sqrt[b]*f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^4)/a)])/(12*Sqrt[b]*Sqrt[a + b*x^4])","C",1
501,1,208,341,0.4273639,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^2} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^2,x]","\frac{x \left(\sqrt{b} \sqrt{\frac{b x^4}{a}+1} \left(\sqrt{a+b x^4} \left(2 d+f x^2\right)-2 \sqrt{a} d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)+4 \sqrt{b} e x \sqrt{a+b x^4} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)+\sqrt{a} f \sqrt{a+b x^4} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right)-4 \sqrt{b} c \sqrt{a+b x^4} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)}{4 \sqrt{b} x \sqrt{\frac{b x^4}{a}+1}}","-\frac{\sqrt{a+b x^4} \left(3 c-e x^2\right)}{3 x}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} e+3 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{a+b x^4}}+\frac{2 \sqrt{b} c x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt[4]{a} \sqrt[4]{b} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b x^4}}+\frac{1}{4} \sqrt{a+b x^4} \left(2 d+f x^2\right)-\frac{1}{2} \sqrt{a} d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{a f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 \sqrt{b}}",1,"(-4*Sqrt[b]*c*Sqrt[a + b*x^4]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^4)/a)] + x*(Sqrt[a]*f*Sqrt[a + b*x^4]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + Sqrt[b]*Sqrt[1 + (b*x^4)/a]*((2*d + f*x^2)*Sqrt[a + b*x^4] - 2*Sqrt[a]*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) + 4*Sqrt[b]*e*x*Sqrt[a + b*x^4]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)]))/(4*Sqrt[b]*x*Sqrt[1 + (b*x^4)/a])","C",1
502,1,204,342,0.2585306,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^3} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^3,x]","\frac{\sqrt{a} \sqrt{b} c x^2 \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-2 a d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)-\sqrt{a} e x^2 \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+2 a f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)-a c+a e x^2-b c x^4+b e x^6}{2 x^2 \sqrt{a+b x^4}}","-\frac{\sqrt{a+b x^4} \left(c-e x^2\right)}{2 x^2}+\frac{1}{2} \sqrt{b} c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-\frac{\sqrt{a+b x^4} \left(3 d-f x^2\right)}{3 x}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} f+3 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{a+b x^4}}+\frac{2 \sqrt{b} d x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b x^4}}-\frac{1}{2} \sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)",1,"(-(a*c) + a*e*x^2 - b*c*x^4 + b*e*x^6 + Sqrt[a]*Sqrt[b]*c*x^2*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] - Sqrt[a]*e*x^2*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]] - 2*a*d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^4)/a)] + 2*a*f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^4)/a)])/(2*x^2*Sqrt[a + b*x^4])","C",1
503,1,205,357,0.3308102,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^4} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^4,x]","\frac{3 x \left(\sqrt{a} \sqrt{b} d x^2 \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-2 a e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)-\sqrt{a} f x^2 \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)-a d+a f x^2-b d x^4+b f x^6\right)-2 a c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)}{6 x^3 \sqrt{a+b x^4}}","\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} e+\sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{a} \sqrt{a+b x^4}}-\frac{\sqrt{a+b x^4} \left(c-3 e x^2\right)}{3 x^3}-\frac{\sqrt{a+b x^4} \left(d-f x^2\right)}{2 x^2}+\frac{1}{2} \sqrt{b} d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-\frac{2 e \sqrt{a+b x^4}}{x}+\frac{2 \sqrt{b} e x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt[4]{a} \sqrt[4]{b} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b x^4}}-\frac{1}{2} \sqrt{a} f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)",1,"(-2*a*c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^4)/a)] + 3*x*(-(a*d) + a*f*x^2 - b*d*x^4 + b*f*x^6 + Sqrt[a]*Sqrt[b]*d*x^2*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] - Sqrt[a]*f*x^2*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]] - 2*a*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^4)/a)]))/(6*x^3*Sqrt[a + b*x^4])","C",1
504,1,175,329,0.2721501,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^5} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^5,x]","-\frac{\sqrt{\frac{b x^4}{a}+1} \left(3 a c \sqrt{\frac{b x^4}{a}+1}+3 b c x^4 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+4 a d x \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)+6 a e x^2 \sqrt{\frac{b x^4}{a}+1}-6 \sqrt{a} \sqrt{b} e x^4 \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)+12 a f x^3 \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)\right)}{12 x^4 \sqrt{a+b x^4}}","-\frac{1}{12} \sqrt{a+b x^4} \left(\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right)-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 \sqrt{a}}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} f+\sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{a} \sqrt{a+b x^4}}+\frac{1}{2} \sqrt{b} e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{2 \sqrt{b} f x \sqrt{a+b x^4}}{\sqrt{a}+\sqrt{b} x^2}-\frac{2 \sqrt[4]{a} \sqrt[4]{b} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b x^4}}",1,"-1/12*(Sqrt[1 + (b*x^4)/a]*(3*a*c*Sqrt[1 + (b*x^4)/a] + 6*a*e*x^2*Sqrt[1 + (b*x^4)/a] - 6*Sqrt[a]*Sqrt[b]*e*x^4*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 3*b*c*x^4*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*d*x*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^4)/a)] + 12*a*f*x^3*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^4)/a)]))/(x^4*Sqrt[a + b*x^4])","C",1
505,1,179,360,0.2628634,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^6} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^6,x]","-\frac{\sqrt{a+b x^4} \left(12 a c \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right)+5 x \left(3 a d \sqrt{\frac{b x^4}{a}+1}+3 b d x^4 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+4 a e x \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)+6 a f x^2 \sqrt{\frac{b x^4}{a}+1}-6 \sqrt{a} \sqrt{b} f x^4 \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right)\right)}{60 a x^5 \sqrt{\frac{b x^4}{a}+1}}","\frac{b^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} e+3 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} c x \sqrt{a+b x^4}}{5 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{1}{60} \sqrt{a+b x^4} \left(\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right)-\frac{2 b c \sqrt{a+b x^4}}{5 a x}-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 \sqrt{a}}+\frac{1}{2} \sqrt{b} f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)",1,"-1/60*(Sqrt[a + b*x^4]*(12*a*c*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^4)/a)] + 5*x*(3*a*d*Sqrt[1 + (b*x^4)/a] + 6*a*f*x^2*Sqrt[1 + (b*x^4)/a] - 6*Sqrt[a]*Sqrt[b]*f*x^4*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 3*b*d*x^4*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*e*x*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^4)/a)])))/(a*x^5*Sqrt[1 + (b*x^4)/a])","C",1
506,1,145,352,0.2672618,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^7} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^7,x]","-\frac{\sqrt{a+b x^4} \left(5 \left(\sqrt{\frac{b x^4}{a}+1} \left(2 a c+3 a e x^2+2 b c x^4\right)+3 b e x^6 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+4 a f x^3 \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)\right)+12 a d x \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right)\right)}{60 a x^6 \sqrt{\frac{b x^4}{a}+1}}","\frac{b^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} f+3 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} d x \sqrt{a+b x^4}}{5 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{1}{60} \sqrt{a+b x^4} \left(\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right)-\frac{b c \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b d \sqrt{a+b x^4}}{5 a x}-\frac{b e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 \sqrt{a}}",1,"-1/60*(Sqrt[a + b*x^4]*(12*a*d*x*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^4)/a)] + 5*(Sqrt[1 + (b*x^4)/a]*(2*a*c + 3*a*e*x^2 + 2*b*c*x^4) + 3*b*e*x^6*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*f*x^3*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^4)/a)])))/(a*x^6*Sqrt[1 + (b*x^4)/a])","C",1
507,1,145,375,0.2649339,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^8} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^8,x]","-\frac{\sqrt{a+b x^4} \left(60 a c \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)+35 x \left(\sqrt{\frac{b x^4}{a}+1} \left(2 a d+3 a f x^2+2 b d x^4\right)+3 b f x^6 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)\right)+84 a e x^2 \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right)\right)}{420 a x^7 \sqrt{\frac{b x^4}{a}+1}}","-\frac{b^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{b} c-21 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 a^{5/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{3/2} e x \sqrt{a+b x^4}}{5 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{1}{420} \sqrt{a+b x^4} \left(\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}\right)-\frac{2 b c \sqrt{a+b x^4}}{21 a x^3}-\frac{b d \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b e \sqrt{a+b x^4}}{5 a x}-\frac{b f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 \sqrt{a}}",1,"-1/420*(Sqrt[a + b*x^4]*(35*x*(Sqrt[1 + (b*x^4)/a]*(2*a*d + 3*a*f*x^2 + 2*b*d*x^4) + 3*b*f*x^6*ArcTanh[Sqrt[1 + (b*x^4)/a]]) + 60*a*c*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^4)/a)] + 84*a*e*x^2*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^4)/a)]))/(a*x^7*Sqrt[1 + (b*x^4)/a])","C",1
508,1,146,400,0.1767914,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^9} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^9,x]","-\frac{\sqrt{a+b x^4} \left(30 a^3 d \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)+7 x \left(6 a^3 f x \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right)+5 \left(a+b x^4\right) \sqrt{\frac{b x^4}{a}+1} \left(a^2 e+b^2 c x^6 \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^4}{a}+1\right)\right)\right)\right)}{210 a^3 x^7 \sqrt{\frac{b x^4}{a}+1}}","-\frac{b^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{b} d-21 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 a^{5/4} \sqrt{a+b x^4}}-\frac{2 b^{5/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 a^{3/4} \sqrt{a+b x^4}}+\frac{b^2 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 a^{3/2}}+\frac{2 b^{3/2} f x \sqrt{a+b x^4}}{5 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{1}{840} \sqrt{a+b x^4} \left(\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}\right)-\frac{b c \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b d \sqrt{a+b x^4}}{21 a x^3}-\frac{b e \sqrt{a+b x^4}}{6 a x^2}-\frac{2 b f \sqrt{a+b x^4}}{5 a x}",1,"-1/210*(Sqrt[a + b*x^4]*(30*a^3*d*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^4)/a)] + 7*x*(6*a^3*f*x*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^4)/a)] + 5*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*(a^2*e + b^2*c*x^6*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^4)/a]))))/(a^3*x^7*Sqrt[1 + (b*x^4)/a])","C",1
509,1,148,425,0.1818108,"\int \frac{\left(c+d x+e x^2+f x^3\right) \sqrt{a+b x^4}}{x^{10}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^10,x]","-\frac{\sqrt{a+b x^4} \left(14 a^3 c \, _2F_1\left(-\frac{9}{4},-\frac{1}{2};-\frac{5}{4};-\frac{b x^4}{a}\right)+3 x^2 \left(6 a^3 e \, _2F_1\left(-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)+7 x \left(a+b x^4\right) \sqrt{\frac{b x^4}{a}+1} \left(a^2 f+b^2 d x^6 \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^4}{a}+1\right)\right)\right)\right)}{126 a^3 x^9 \sqrt{\frac{b x^4}{a}+1}}","-\frac{b^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} e+7 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 a^{7/4} \sqrt{a+b x^4}}+\frac{2 b^{9/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{7/4} \sqrt{a+b x^4}}+\frac{b^2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 a^{3/2}}-\frac{2 b^{5/2} c x \sqrt{a+b x^4}}{15 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{2 b^2 c \sqrt{a+b x^4}}{15 a^2 x}-\frac{1}{504} \sqrt{a+b x^4} \left(\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right)-\frac{2 b c \sqrt{a+b x^4}}{45 a x^5}-\frac{b d \sqrt{a+b x^4}}{16 a x^4}-\frac{2 b e \sqrt{a+b x^4}}{21 a x^3}-\frac{b f \sqrt{a+b x^4}}{6 a x^2}",1,"-1/126*(Sqrt[a + b*x^4]*(14*a^3*c*Hypergeometric2F1[-9/4, -1/2, -5/4, -((b*x^4)/a)] + 3*x^2*(6*a^3*e*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^4)/a)] + 7*x*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*(a^2*f + b^2*d*x^6*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^4)/a]))))/(a^3*x^9*Sqrt[1 + (b*x^4)/a])","C",1
510,1,225,476,0.9422405,"\int x^4 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2} \, dx","Integrate[x^4*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]","\frac{\sqrt{a+b x^4} \left(-\frac{15015 a^{5/2} \sqrt{b} d \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{\frac{b x^4}{a}+1}}+5005 b d x^2 \left(3 a^2+14 a b x^4+8 b^2 x^8\right)-\frac{43680 a^2 b c x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}-\frac{36960 a^2 b e x^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+43680 b c x \left(a+b x^4\right)^2+36960 b e x^3 \left(a+b x^4\right)^2+6864 f \left(a+b x^4\right)^2 \left(5 b x^4-2 a\right)\right)}{480480 b^2}","-\frac{2 a^{11/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(77 \sqrt{a} e+65 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{32 b^{3/2}}-\frac{4 a^3 e x \sqrt{a+b x^4}}{65 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{4 a^2 c x \sqrt{a+b x^4}}{77 b}-\frac{a^2 d x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 e x^3 \sqrt{a+b x^4}}{195 b}-\frac{\left(a+b x^4\right)^{5/2} \left(12 a f-35 b d x^2\right)}{420 b^2}+\frac{1}{143} x^5 \left(a+b x^4\right)^{3/2} \left(13 c+11 e x^2\right)+\frac{2 a x^5 \sqrt{a+b x^4} \left(117 c+77 e x^2\right)}{3003}-\frac{a d x^2 \left(a+b x^4\right)^{3/2}}{48 b}+\frac{f x^4 \left(a+b x^4\right)^{5/2}}{14 b}",1,"(Sqrt[a + b*x^4]*(43680*b*c*x*(a + b*x^4)^2 + 36960*b*e*x^3*(a + b*x^4)^2 + 6864*f*(a + b*x^4)^2*(-2*a + 5*b*x^4) + 5005*b*d*x^2*(3*a^2 + 14*a*b*x^4 + 8*b^2*x^8) - (15015*a^(5/2)*Sqrt[b]*d*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a] - (43680*a^2*b*c*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (36960*a^2*b*e*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(480480*b^2)","C",1
511,1,238,452,0.7716714,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2} \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]","\frac{\sqrt{a+b x^4} \left(-\frac{6240 a^2 \sqrt{b} d x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}-\frac{5280 a^2 \sqrt{b} f x^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+715 e \left(\sqrt{b} x^2 \left(3 a^2+14 a b x^4+8 b^2 x^8\right)-\frac{3 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{\frac{b x^4}{a}+1}}\right)+6864 \sqrt{b} c \left(a+b x^4\right)^2+6240 \sqrt{b} d x \left(a+b x^4\right)^2+5280 \sqrt{b} f x^3 \left(a+b x^4\right)^2\right)}{68640 b^{3/2}}","-\frac{2 a^{11/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(77 \sqrt{a} f+65 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5005 b^{7/4} \sqrt{a+b x^4}}+\frac{4 a^{13/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{65 b^{7/4} \sqrt{a+b x^4}}-\frac{a^3 e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{32 b^{3/2}}-\frac{4 a^3 f x \sqrt{a+b x^4}}{65 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{4 a^2 d x \sqrt{a+b x^4}}{77 b}-\frac{a^2 e x^2 \sqrt{a+b x^4}}{32 b}+\frac{4 a^2 f x^3 \sqrt{a+b x^4}}{195 b}+\frac{\left(a+b x^4\right)^{5/2} \left(6 c+5 e x^2\right)}{60 b}+\frac{1}{143} x^5 \left(a+b x^4\right)^{3/2} \left(13 d+11 f x^2\right)+\frac{2 a x^5 \sqrt{a+b x^4} \left(117 d+77 f x^2\right)}{3003}-\frac{a e x^2 \left(a+b x^4\right)^{3/2}}{48 b}",1,"(Sqrt[a + b*x^4]*(6864*Sqrt[b]*c*(a + b*x^4)^2 + 6240*Sqrt[b]*d*x*(a + b*x^4)^2 + 5280*Sqrt[b]*f*x^3*(a + b*x^4)^2 + 715*e*(Sqrt[b]*x^2*(3*a^2 + 14*a*b*x^4 + 8*b^2*x^8) - (3*a^(5/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a]) - (6240*a^2*Sqrt[b]*d*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] - (5280*a^2*Sqrt[b]*f*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/(68640*b^(3/2))","C",1
512,1,205,427,0.9354364,"\int x^2 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2} \, dx","Integrate[x^2*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]","\frac{\sqrt{a+b x^4} \left(-\frac{480 a^2 e x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{b \sqrt{\frac{b x^4}{a}+1}}+\frac{55 f \left(\sqrt{b} x^2 \left(3 a^2+14 a b x^4+8 b^2 x^8\right)-\frac{3 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{\frac{b x^4}{a}+1}}\right)}{b^{3/2}}+\frac{1760 a c x^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{528 d \left(a+b x^4\right)^2}{b}+\frac{480 e x \left(a+b x^4\right)^2}{b}\right)}{5280}","\frac{2 a^{9/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(77 \sqrt{b} c-15 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{a^3 f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{32 b^{3/2}}+\frac{4 a^2 c x \sqrt{a+b x^4}}{15 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{4 a^2 e x \sqrt{a+b x^4}}{77 b}-\frac{a^2 f x^2 \sqrt{a+b x^4}}{32 b}+\frac{2 a x^3 \sqrt{a+b x^4} \left(77 c+45 e x^2\right)}{1155}+\frac{1}{99} x^3 \left(a+b x^4\right)^{3/2} \left(11 c+9 e x^2\right)+\frac{\left(a+b x^4\right)^{5/2} \left(6 d+5 f x^2\right)}{60 b}-\frac{a f x^2 \left(a+b x^4\right)^{3/2}}{48 b}",1,"(Sqrt[a + b*x^4]*((528*d*(a + b*x^4)^2)/b + (480*e*x*(a + b*x^4)^2)/b + (55*f*(Sqrt[b]*x^2*(3*a^2 + 14*a*b*x^4 + 8*b^2*x^8) - (3*a^(5/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[1 + (b*x^4)/a]))/b^(3/2) - (480*a^2*e*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/(b*Sqrt[1 + (b*x^4)/a]) + (1760*a*c*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/5280","C",1
513,1,196,409,0.9779054,"\int x \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2} \, dx","Integrate[x*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]","\frac{\sqrt{a+b x^4} \left(165 c \left(\frac{3 a^{5/2} \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{b} \left(a+b x^4\right)}+5 a x^2+2 b x^6\right)-\frac{240 a^2 f x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{b \sqrt{\frac{b x^4}{a}+1}}+\frac{880 a d x^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{264 e \left(a+b x^4\right)^2}{b}+\frac{240 f x \left(a+b x^4\right)^2}{b}\right)}{2640}","\frac{2 a^{9/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(77 \sqrt{b} d-15 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{1155 b^{5/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 \sqrt{b}}+\frac{4 a^2 d x \sqrt{a+b x^4}}{15 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{4 a^2 f x \sqrt{a+b x^4}}{77 b}+\frac{1}{8} c x^2 \left(a+b x^4\right)^{3/2}+\frac{3}{16} a c x^2 \sqrt{a+b x^4}+\frac{1}{99} x^3 \left(a+b x^4\right)^{3/2} \left(11 d+9 f x^2\right)+\frac{2 a x^3 \sqrt{a+b x^4} \left(77 d+45 f x^2\right)}{1155}+\frac{e \left(a+b x^4\right)^{5/2}}{10 b}",1,"(Sqrt[a + b*x^4]*((264*e*(a + b*x^4)^2)/b + (240*f*x*(a + b*x^4)^2)/b + 165*c*(5*a*x^2 + 2*b*x^6 + (3*a^(5/2)*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*(a + b*x^4))) - (240*a^2*f*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/(b*Sqrt[1 + (b*x^4)/a]) + (880*a*d*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/2640","C",1
514,1,175,382,0.5603721,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2),x]","\frac{1}{240} \sqrt{a+b x^4} \left(15 d \left(\frac{3 a^{5/2} \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{b} \left(a+b x^4\right)}+5 a x^2+2 b x^6\right)+\frac{240 a c x \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{80 a e x^3 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{24 f \left(a+b x^4\right)^2}{b}\right)","\frac{2 a^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(7 \sqrt{a} e+15 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{3/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 \sqrt{b}}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{1}{63} x \left(a+b x^4\right)^{3/2} \left(9 c+7 e x^2\right)+\frac{2}{105} a x \sqrt{a+b x^4} \left(15 c+7 e x^2\right)+\frac{1}{8} d x^2 \left(a+b x^4\right)^{3/2}+\frac{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{f \left(a+b x^4\right)^{5/2}}{10 b}",1,"(Sqrt[a + b*x^4]*((24*f*(a + b*x^4)^2)/b + 15*d*(5*a*x^2 + 2*b*x^6 + (3*a^(5/2)*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*(a + b*x^4))) + (240*a*c*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] + (80*a*e*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]))/240","C",1
515,1,224,403,0.5711971,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x,x]","\frac{1}{6} c \left(\sqrt{a+b x^4} \left(4 a+b x^4\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)+\frac{1}{16} e \sqrt{a+b x^4} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^4}{a}+1}}+5 a x^2+2 b x^6\right)+\frac{a d x \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}+\frac{a f x^3 \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{3 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 a^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(7 \sqrt{a} f+15 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 b^{3/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{3 a^2 e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 \sqrt{b}}+\frac{4 a^2 f x \sqrt{a+b x^4}}{15 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{1}{16} a \sqrt{a+b x^4} \left(8 c+3 e x^2\right)+\frac{1}{24} \left(a+b x^4\right)^{3/2} \left(4 c+3 e x^2\right)+\frac{2}{105} a x \sqrt{a+b x^4} \left(15 d+7 f x^2\right)+\frac{1}{63} x \left(a+b x^4\right)^{3/2} \left(9 d+7 f x^2\right)",1,"(e*Sqrt[a + b*x^4]*(5*a*x^2 + 2*b*x^6 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^4)/a])))/16 + (c*(Sqrt[a + b*x^4]*(4*a + b*x^4) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]))/6 + (a*d*x*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a] + (a*f*x^3*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^4)/a)])/(3*Sqrt[1 + (b*x^4)/a])","C",1
516,1,222,404,0.5655284,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^2} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^2,x]","\frac{1}{6} d \left(\sqrt{a+b x^4} \left(4 a+b x^4\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)+\frac{1}{16} f \sqrt{a+b x^4} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^4}{a}+1}}+5 a x^2+2 b x^6\right)-\frac{a c \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)}{x \sqrt{\frac{b x^4}{a}+1}}+\frac{a e x \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{\frac{b x^4}{a}+1}}","\frac{2 a^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} e+21 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{35 \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt[4]{b} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{3 a^2 f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{16 \sqrt{b}}-\frac{\left(a+b x^4\right)^{3/2} \left(7 c-e x^2\right)}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left(5 a e+21 b c x^2\right)+\frac{12 a \sqrt{b} c x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{1}{16} a \sqrt{a+b x^4} \left(8 d+3 f x^2\right)+\frac{1}{24} \left(a+b x^4\right)^{3/2} \left(4 d+3 f x^2\right)",1,"(f*Sqrt[a + b*x^4]*(5*a*x^2 + 2*b*x^6 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^4)/a])))/16 + (d*(Sqrt[a + b*x^4]*(4*a + b*x^4) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]))/6 - (a*c*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^4)/a)])/(x*Sqrt[1 + (b*x^4)/a]) + (a*e*x*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)])/Sqrt[1 + (b*x^4)/a]","C",1
517,1,194,406,0.4024547,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^3} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^3,x]","\frac{x \left(e x \sqrt{\frac{b x^4}{a}+1} \left(\sqrt{a+b x^4} \left(4 a+b x^4\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)-6 a d \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)+6 a f x^2 \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right)\right)-3 a c \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^4}{a}\right)}{6 x^2 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 a^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} f+21 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{35 \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)-\frac{\left(a+b x^4\right)^{3/2} \left(3 c-e x^2\right)}{6 x^2}+\frac{1}{4} \sqrt{a+b x^4} \left(2 a e+3 b c x^2\right)+\frac{3}{4} a \sqrt{b} c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-\frac{\left(a+b x^4\right)^{3/2} \left(7 d-f x^2\right)}{7 x}+\frac{2}{35} x \sqrt{a+b x^4} \left(5 a f+21 b d x^2\right)+\frac{12 a \sqrt{b} d x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(-3*a*c*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^4)/a)] + x*(e*x*Sqrt[1 + (b*x^4)/a]*(Sqrt[a + b*x^4]*(4*a + b*x^4) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 6*a*d*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^4)/a)] + 6*a*f*x^2*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^4)/a)]))/(6*x^2*Sqrt[1 + (b*x^4)/a])","C",1
518,1,194,408,0.3902371,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^4} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^4,x]","\frac{x^2 \left(f x \sqrt{\frac{b x^4}{a}+1} \left(\sqrt{a+b x^4} \left(4 a+b x^4\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)\right)-6 a e \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)\right)-2 a c \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^4}{a}\right)-3 a d x \sqrt{a+b x^4} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^4}{a}\right)}{6 x^3 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 a^{3/4} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{a} e+5 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt[4]{b} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{2} a^{3/2} f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)-\frac{2 \sqrt{a+b x^4} \left(9 a e-5 b c x^2\right)}{15 x}-\frac{\left(a+b x^4\right)^{3/2} \left(5 c-3 e x^2\right)}{15 x^3}-\frac{\left(a+b x^4\right)^{3/2} \left(3 d-f x^2\right)}{6 x^2}+\frac{1}{4} \sqrt{a+b x^4} \left(2 a f+3 b d x^2\right)+\frac{3}{4} a \sqrt{b} d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{12 a \sqrt{b} e x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(-2*a*c*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] - 3*a*d*x*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^4)/a)] + x^2*(f*x*Sqrt[1 + (b*x^4)/a]*(Sqrt[a + b*x^4]*(4*a + b*x^4) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 6*a*e*Sqrt[a + b*x^4]*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^4)/a)]))/(6*x^3*Sqrt[1 + (b*x^4)/a])","C",1
519,1,163,386,0.2471023,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^5} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^5,x]","\frac{\sqrt{a+b x^4} \left(3 x \left(-5 a^3 e \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^4}{a}\right)-10 a^3 f x \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^4}{a}\right)+b c x^2 \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right)\right)-10 a^3 d \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^4}{a}\right)\right)}{30 a^2 x^3 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 a^{3/4} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{a} f+5 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 \sqrt{a+b x^4}}-\frac{12 a^{5/4} \sqrt[4]{b} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{12} \left(a+b x^4\right)^{3/2} \left(\frac{3 c}{x^4}+\frac{4 d}{x^3}+\frac{6 e}{x^2}+\frac{12 f}{x}\right)+\frac{3}{4} b \sqrt{a+b x^4} \left(c+e x^2\right)-\frac{3}{4} \sqrt{a} b c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{2}{15} b x \sqrt{a+b x^4} \left(5 d+9 f x^2\right)+\frac{3}{4} a \sqrt{b} e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{12 a \sqrt{b} f x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(Sqrt[a + b*x^4]*(-10*a^3*d*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] + 3*x*(-5*a^3*e*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^4)/a)] - 10*a^3*f*x*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^4)/a)] + b*c*x^2*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a])))/(30*a^2*x^3*Sqrt[1 + (b*x^4)/a])","C",1
520,1,165,387,0.2346931,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^6} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^6,x]","\frac{\sqrt{a+b x^4} \left(-6 a^3 c \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^4}{a}\right)-10 a^3 e x^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^4}{a}\right)-15 a^3 f x^3 \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^4}{a}\right)+3 b d x^5 \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right)\right)}{30 a^2 x^5 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 \sqrt[4]{a} b^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} e+9 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 \sqrt{a+b x^4}}+\frac{12 b^{3/2} c x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{12 \sqrt[4]{a} b^{5/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{60} \left(a+b x^4\right)^{3/2} \left(\frac{12 c}{x^5}+\frac{15 d}{x^4}+\frac{20 e}{x^3}+\frac{30 f}{x^2}\right)-\frac{2 b \sqrt{a+b x^4} \left(9 c-5 e x^2\right)}{15 x}+\frac{3}{4} b \sqrt{a+b x^4} \left(d+f x^2\right)-\frac{3}{4} \sqrt{a} b d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+\frac{3}{4} a \sqrt{b} f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)",1,"(Sqrt[a + b*x^4]*(-6*a^3*c*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)] - 10*a^3*e*x^2*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] - 15*a^3*f*x^3*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^4)/a)] + 3*b*d*x^5*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a]))/(30*a^2*x^5*Sqrt[1 + (b*x^4)/a])","C",1
521,1,163,392,0.2195257,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^7} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^7,x]","\frac{\sqrt{a+b x^4} \left(-5 a^3 c \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^4}{a}\right)-6 a^3 d x \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^4}{a}\right)-10 a^3 f x^3 \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^4}{a}\right)+3 b e x^6 \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right)\right)}{30 a^2 x^6 \sqrt{\frac{b x^4}{a}+1}}","\frac{1}{2} b^{3/2} c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{2 \sqrt[4]{a} b^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} f+9 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 \sqrt{a+b x^4}}+\frac{12 b^{3/2} d x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{12 \sqrt[4]{a} b^{5/4} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{60} \left(a+b x^4\right)^{3/2} \left(\frac{10 c}{x^6}+\frac{12 d}{x^5}+\frac{15 e}{x^4}+\frac{20 f}{x^3}\right)-\frac{b \sqrt{a+b x^4} \left(2 c-3 e x^2\right)}{4 x^2}-\frac{2 b \sqrt{a+b x^4} \left(9 d-5 f x^2\right)}{15 x}-\frac{3}{4} \sqrt{a} b e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)",1,"(Sqrt[a + b*x^4]*(-5*a^3*c*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a)] - 6*a^3*d*x*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)] - 10*a^3*f*x^3*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] + 3*b*e*x^6*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a]))/(30*a^2*x^6*Sqrt[1 + (b*x^4)/a])","C",1
522,1,164,412,0.2456005,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^8} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^8,x]","\frac{\sqrt{a+b x^4} \left(7 x \left(-5 a^3 d \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^4}{a}\right)-6 a^3 e x \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^4}{a}\right)+3 b f x^6 \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right)\right)-30 a^3 c \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)\right)}{210 a^2 x^7 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 b^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(21 \sqrt{a} e+5 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{35 \sqrt[4]{a} \sqrt{a+b x^4}}+\frac{1}{2} b^{3/2} d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{12 b^{3/2} e x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{12 \sqrt[4]{a} b^{5/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{1}{420} \left(a+b x^4\right)^{3/2} \left(\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}\right)-\frac{2 b \sqrt{a+b x^4} \left(5 c-21 e x^2\right)}{35 x^3}-\frac{b \sqrt{a+b x^4} \left(2 d-3 f x^2\right)}{4 x^2}-\frac{12 b e \sqrt{a+b x^4}}{5 x}-\frac{3}{4} \sqrt{a} b f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)",1,"(Sqrt[a + b*x^4]*(-30*a^3*c*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/a)] + 7*x*(-5*a^3*d*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a)] - 6*a^3*e*x*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)] + 3*b*f*x^6*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a])))/(210*a^2*x^7*Sqrt[1 + (b*x^4)/a])","C",1
523,1,174,377,0.3164246,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^9} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^9,x]","-\frac{\sqrt{a+b x^4} \left(7 \left(40 a^2 e x^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^4}{a}\right)+48 a^2 f x^3 \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^4}{a}\right)+15 c \left(3 b^2 x^8 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+a \left(2 a+5 b x^4\right) \sqrt{\frac{b x^4}{a}+1}\right)\right)+240 a^2 d x \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)\right)}{1680 a x^8 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 b^{5/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(21 \sqrt{a} f+5 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{35 \sqrt[4]{a} \sqrt{a+b x^4}}+\frac{1}{2} b^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+\frac{12 b^{3/2} f x \sqrt{a+b x^4}}{5 \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{12 \sqrt[4]{a} b^{5/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 \sqrt{a+b x^4}}-\frac{3 b^2 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{1}{560} b \sqrt{a+b x^4} \left(\frac{105 c}{x^4}+\frac{160 d}{x^3}+\frac{280 e}{x^2}+\frac{672 f}{x}\right)-\frac{1}{840} \left(a+b x^4\right)^{3/2} \left(\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}\right)",1,"-1/1680*(Sqrt[a + b*x^4]*(240*a^2*d*x*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/a)] + 7*(15*c*(a*(2*a + 5*b*x^4)*Sqrt[1 + (b*x^4)/a] + 3*b^2*x^8*ArcTanh[Sqrt[1 + (b*x^4)/a]]) + 40*a^2*e*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a)] + 48*a^2*f*x^3*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)])))/(a*x^8*Sqrt[1 + (b*x^4)/a])","C",1
524,1,174,405,0.3538757,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^{10}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^10,x]","-\frac{\sqrt{a+b x^4} \left(3 x \left(7 \left(8 a^2 f x^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^4}{a}\right)+9 b^2 d x^8 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+3 a d \left(2 a+5 b x^4\right) \sqrt{\frac{b x^4}{a}+1}\right)+48 a^2 e x \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)\right)+112 a^2 c \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^4}{a}\right)\right)}{1008 a x^9 \sqrt{\frac{b x^4}{a}+1}}","\frac{2 b^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(15 \sqrt{a} e+7 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 a^{3/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{4 b^{5/2} c x \sqrt{a+b x^4}}{15 a \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{1}{2} b^{3/2} f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-\frac{4 b^2 c \sqrt{a+b x^4}}{15 a x}-\frac{3 b^2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{b \sqrt{a+b x^4} \left(\frac{224 c}{x^5}+\frac{315 d}{x^4}+\frac{480 e}{x^3}+\frac{840 f}{x^2}\right)}{1680}-\frac{1}{504} \left(a+b x^4\right)^{3/2} \left(\frac{56 c}{x^9}+\frac{63 d}{x^8}+\frac{72 e}{x^7}+\frac{84 f}{x^6}\right)",1,"-1/1008*(Sqrt[a + b*x^4]*(112*a^2*c*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)] + 3*x*(48*a^2*e*x*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/a)] + 7*(3*a*d*(2*a + 5*b*x^4)*Sqrt[1 + (b*x^4)/a] + 9*b^2*d*x^8*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 8*a^2*f*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a)]))))/(a*x^9*Sqrt[1 + (b*x^4)/a])","C",1
525,1,171,399,0.3508985,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^{11}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^11,x]","-\frac{\sqrt{a+b x^4} \left(63 \sqrt{\frac{b x^4}{a}+1} \left(2 a^2 \left(4 c+5 e x^2\right)+a b x^4 \left(16 c+25 e x^2\right)+8 b^2 c x^8\right)+560 a^2 d x \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^4}{a}\right)+720 a^2 f x^3 \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right)+945 b^2 e x^{10} \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)\right)}{5040 a x^{10} \sqrt{\frac{b x^4}{a}+1}}","\frac{2 b^{7/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(15 \sqrt{a} f+7 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{105 a^{3/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}-\frac{3 b^2 e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{b \sqrt{a+b x^4} \left(\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right)}{1680}-\frac{\left(a+b x^4\right)^{3/2} \left(\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right)}{2520}",1,"-1/5040*(Sqrt[a + b*x^4]*(63*Sqrt[1 + (b*x^4)/a]*(8*b^2*c*x^8 + 2*a^2*(4*c + 5*e*x^2) + a*b*x^4*(16*c + 25*e*x^2)) + 945*b^2*e*x^10*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 560*a^2*d*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)] + 720*a^2*f*x^3*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/a)]))/(a*x^10*Sqrt[1 + (b*x^4)/a])","C",1
526,1,172,424,0.4249414,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^{12}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^12,x]","-\frac{\sqrt{a+b x^4} \left(11 x \left(9 \sqrt{\frac{b x^4}{a}+1} \left(2 a^2 \left(4 d+5 f x^2\right)+a b x^4 \left(16 d+25 f x^2\right)+8 b^2 d x^8\right)+80 a^2 e x \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^4}{a}\right)+135 b^2 f x^{10} \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)\right)+720 a^2 c \, _2F_1\left(-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};-\frac{b x^4}{a}\right)\right)}{7920 a x^{11} \sqrt{\frac{b x^4}{a}+1}}","-\frac{2 b^{9/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(15 \sqrt{b} c-77 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{1155 a^{5/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{4 b^{5/2} e x \sqrt{a+b x^4}}{15 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{4 b^2 c \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 d \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 e \sqrt{a+b x^4}}{15 a x}-\frac{3 b^2 f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{b \sqrt{a+b x^4} \left(\frac{1440 c}{x^7}+\frac{1848 d}{x^6}+\frac{2464 e}{x^5}+\frac{3465 f}{x^4}\right)}{18480}-\frac{\left(a+b x^4\right)^{3/2} \left(\frac{360 c}{x^{11}}+\frac{396 d}{x^{10}}+\frac{440 e}{x^9}+\frac{495 f}{x^8}\right)}{3960}",1,"-1/7920*(Sqrt[a + b*x^4]*(720*a^2*c*Hypergeometric2F1[-11/4, -3/2, -7/4, -((b*x^4)/a)] + 11*x*(9*Sqrt[1 + (b*x^4)/a]*(8*b^2*d*x^8 + 2*a^2*(4*d + 5*f*x^2) + a*b*x^4*(16*d + 25*f*x^2)) + 135*b^2*f*x^10*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 80*a^2*e*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)])))/(a*x^11*Sqrt[1 + (b*x^4)/a])","C",1
527,1,149,449,0.207757,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^{13}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]","-\frac{\sqrt{a+b x^4} \left(90 a^5 d \, _2F_1\left(-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};-\frac{b x^4}{a}\right)+11 x \left(10 a^5 f x \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^4}{a}\right)+9 \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \left(a^3 e-b^3 c x^{10} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^4}{a}+1\right)\right)\right)\right)}{990 a^4 x^{11} \sqrt{\frac{b x^4}{a}+1}}","-\frac{2 b^{9/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(15 \sqrt{b} d-77 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{1155 a^{5/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{b^3 c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{32 a^{3/2}}+\frac{4 b^{5/2} f x \sqrt{a+b x^4}}{15 a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{b^2 c \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 d \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 e \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 f \sqrt{a+b x^4}}{15 a x}-\frac{\left(a+b x^4\right)^{3/2} \left(\frac{165 c}{x^{12}}+\frac{180 d}{x^{11}}+\frac{198 e}{x^{10}}+\frac{220 f}{x^9}\right)}{1980}-\frac{b \sqrt{a+b x^4} \left(\frac{1155 c}{x^8}+\frac{1440 d}{x^7}+\frac{1848 e}{x^6}+\frac{2464 f}{x^5}\right)}{18480}",1,"-1/990*(Sqrt[a + b*x^4]*(90*a^5*d*Hypergeometric2F1[-11/4, -3/2, -7/4, -((b*x^4)/a)] + 11*x*(10*a^5*f*x*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)] + 9*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*(a^3*e - b^3*c*x^10*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^4)/a]))))/(a^4*x^11*Sqrt[1 + (b*x^4)/a])","C",1
528,1,151,474,0.2133307,"\int \frac{\left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^{3/2}}{x^{14}} \, dx","Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]","-\frac{\sqrt{a+b x^4} \left(110 a^5 c \, _2F_1\left(-\frac{13}{4},-\frac{3}{2};-\frac{9}{4};-\frac{b x^4}{a}\right)+13 x^2 \left(10 a^5 e \, _2F_1\left(-\frac{11}{4},-\frac{3}{2};-\frac{7}{4};-\frac{b x^4}{a}\right)+11 x \left(a+b x^4\right)^2 \sqrt{\frac{b x^4}{a}+1} \left(a^3 f-b^3 d x^{10} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^4}{a}+1\right)\right)\right)\right)}{1430 a^4 x^{13} \sqrt{\frac{b x^4}{a}+1}}","-\frac{2 b^{11/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(65 \sqrt{a} e+77 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5005 a^{7/4} \sqrt{a+b x^4}}+\frac{4 b^{13/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{65 a^{7/4} \sqrt{a+b x^4}}+\frac{b^3 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{32 a^{3/2}}-\frac{4 b^{7/2} c x \sqrt{a+b x^4}}{65 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{4 b^3 c \sqrt{a+b x^4}}{65 a^2 x}-\frac{4 b^2 c \sqrt{a+b x^4}}{195 a x^5}-\frac{b^2 d \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 e \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 f \sqrt{a+b x^4}}{10 a x^2}-\frac{\left(a+b x^4\right)^{3/2} \left(\frac{660 c}{x^{13}}+\frac{715 d}{x^{12}}+\frac{780 e}{x^{11}}+\frac{858 f}{x^{10}}\right)}{8580}-\frac{b \sqrt{a+b x^4} \left(\frac{12320 c}{x^9}+\frac{15015 d}{x^8}+\frac{18720 e}{x^7}+\frac{24024 f}{x^6}\right)}{240240}",1,"-1/1430*(Sqrt[a + b*x^4]*(110*a^5*c*Hypergeometric2F1[-13/4, -3/2, -9/4, -((b*x^4)/a)] + 13*x^2*(10*a^5*e*Hypergeometric2F1[-11/4, -3/2, -7/4, -((b*x^4)/a)] + 11*x*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*(a^3*f - b^3*d*x^10*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^4)/a]))))/(a^4*x^13*Sqrt[1 + (b*x^4)/a])","C",1
529,1,212,361,0.1444165,"\int \frac{x^4 \left(c+d x+e x^2+f x^3\right)}{\sqrt{a+b x^4}} \, dx","Integrate[(x^4*(c + d*x + e*x^2 + f*x^3))/Sqrt[a + b*x^4],x]","\frac{-20 a^2 f-20 a b c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+20 a b c x+15 a b d x^2-15 a \sqrt{b} d \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-12 a b e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+12 a b e x^3-10 a b f x^4+20 b^2 c x^5+15 b^2 d x^6+12 b^2 e x^7+10 b^2 f x^8}{60 b^2 \sqrt{a+b x^4}}","-\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{a} e+5 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{a d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 b^{3/2}}-\frac{3 a e x \sqrt{a+b x^4}}{5 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{\sqrt{a+b x^4} \left(4 a f-3 b d x^2\right)}{12 b^2}+\frac{c x \sqrt{a+b x^4}}{3 b}+\frac{e x^3 \sqrt{a+b x^4}}{5 b}+\frac{f x^4 \sqrt{a+b x^4}}{6 b}",1,"(-20*a^2*f + 20*a*b*c*x + 15*a*b*d*x^2 + 12*a*b*e*x^3 - 10*a*b*f*x^4 + 20*b^2*c*x^5 + 15*b^2*d*x^6 + 12*b^2*e*x^7 + 10*b^2*f*x^8 - 15*a*Sqrt[b]*d*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 20*a*b*c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 12*a*b*e*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(60*b^2*Sqrt[a + b*x^4])","C",1
530,1,212,336,0.1623986,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{\sqrt{a+b x^4}} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/Sqrt[a + b*x^4],x]","\frac{30 \sqrt{b} c \left(a+b x^4\right)-20 a \sqrt{b} d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+20 \sqrt{b} d x \left(a+b x^4\right)+15 \sqrt{b} e x^2 \left(a+b x^4\right)-15 a e \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)-12 a \sqrt{b} f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+12 \sqrt{b} f x^3 \left(a+b x^4\right)}{60 b^{3/2} \sqrt{a+b x^4}}","-\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{a} f+5 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{30 b^{7/4} \sqrt{a+b x^4}}+\frac{3 a^{5/4} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^4}}-\frac{a e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 b^{3/2}}-\frac{3 a f x \sqrt{a+b x^4}}{5 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{\sqrt{a+b x^4} \left(2 c+e x^2\right)}{4 b}+\frac{d x \sqrt{a+b x^4}}{3 b}+\frac{f x^3 \sqrt{a+b x^4}}{5 b}",1,"(30*Sqrt[b]*c*(a + b*x^4) + 20*Sqrt[b]*d*x*(a + b*x^4) + 15*Sqrt[b]*e*x^2*(a + b*x^4) + 12*Sqrt[b]*f*x^3*(a + b*x^4) - 15*a*e*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 20*a*Sqrt[b]*d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 12*a*Sqrt[b]*f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(60*b^(3/2)*Sqrt[a + b*x^4])","C",1
531,1,193,308,0.1828722,"\int \frac{x^2 \left(c+d x+e x^2+f x^3\right)}{\sqrt{a+b x^4}} \, dx","Integrate[(x^2*(c + d*x + e*x^2 + f*x^3))/Sqrt[a + b*x^4],x]","\frac{4 b^{3/2} c x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)+6 \sqrt{b} d \left(a+b x^4\right)-4 a \sqrt{b} e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+4 \sqrt{b} e x \left(a+b x^4\right)+3 \sqrt{b} f x^2 \left(a+b x^4\right)-3 a f \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{12 b^{3/2} \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{b} c-\sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 b^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{a} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^4}}-\frac{a f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 b^{3/2}}+\frac{c x \sqrt{a+b x^4}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{\sqrt{a+b x^4} \left(2 d+f x^2\right)}{4 b}+\frac{e x \sqrt{a+b x^4}}{3 b}",1,"(6*Sqrt[b]*d*(a + b*x^4) + 4*Sqrt[b]*e*x*(a + b*x^4) + 3*Sqrt[b]*f*x^2*(a + b*x^4) - 3*a*f*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 4*a*Sqrt[b]*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 4*b^(3/2)*c*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(12*b^(3/2)*Sqrt[a + b*x^4])","C",1
532,1,160,299,0.1036599,"\int \frac{x \left(c+d x+e x^2+f x^3\right)}{\sqrt{a+b x^4}} \, dx","Integrate[(x*(c + d*x + e*x^2 + f*x^3))/Sqrt[a + b*x^4],x]","\frac{3 \sqrt{b} c \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)+2 b d x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)-2 a f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+3 a e+2 a f x+3 b e x^4+2 b f x^5}{6 b \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{b} d-\sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 b^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{a} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^4}}+\frac{c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{d x \sqrt{a+b x^4}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{e \sqrt{a+b x^4}}{2 b}+\frac{f x \sqrt{a+b x^4}}{3 b}",1,"(3*a*e + 2*a*f*x + 3*b*e*x^4 + 2*b*f*x^5 + 3*Sqrt[b]*c*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - 2*a*f*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 2*b*d*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(6*b*Sqrt[a + b*x^4])","C",1
533,1,150,276,0.1054161,"\int \frac{c+d x+e x^2+f x^3}{\sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x^4],x]","\frac{c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{3 \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{2 b}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\frac{\sqrt{b} c}{\sqrt{a}}+e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{a} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{e x \sqrt{a+b x^4}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{f \sqrt{a+b x^4}}{2 b}",1,"(f*Sqrt[a + b*x^4])/(2*b) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) + (c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4] + (e*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(3*Sqrt[a + b*x^4])","C",1
534,1,159,285,0.2419652,"\int \frac{c+d x+e x^2+f x^3}{x \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x*Sqrt[a + b*x^4]),x]","-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{3 \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\frac{\sqrt{b} d}{\sqrt{a}}+f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{3/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{a} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^4}}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}+\frac{f x \sqrt{a+b x^4}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(e*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) - (c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) + (d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4] + (f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)])/(3*Sqrt[a + b*x^4])","C",1
535,1,157,309,0.252342,"\int \frac{c+d x+e x^2+f x^3}{x^2 \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^2*Sqrt[a + b*x^4]),x]","-\frac{c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right)}{x \sqrt{a+b x^4}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} e+\sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{\sqrt[4]{b} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} c x \sqrt{a+b x^4}}{a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 \sqrt{b}}",1,"(f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) - (d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^4)/a)])/(x*Sqrt[a + b*x^4]) + (e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4]","C",1
536,1,148,300,0.1522978,"\int \frac{c+d x+e x^2+f x^3}{x^3 \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^3*Sqrt[a + b*x^4]),x]","-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right)}{x \sqrt{a+b x^4}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}+\frac{f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{\sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} f+\sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{\sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a x^2}-\frac{d \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} d x \sqrt{a+b x^4}}{a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"-1/2*(c*Sqrt[a + b*x^4])/(a*x^2) - (e*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (d*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^4)/a)])/(x*Sqrt[a + b*x^4]) + (f*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/Sqrt[a + b*x^4]","C",1
537,1,149,323,0.173092,"\int \frac{c+d x+e x^2+f x^3}{x^4 \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^4*Sqrt[a + b*x^4]),x]","\frac{-2 a c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)-3 x \left(2 a e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right)+\sqrt{a} f x^2 \sqrt{a+b x^4} \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)+a d+b d x^4\right)}{6 a x^3 \sqrt{a+b x^4}}","-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} c-3 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{b} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{3 a x^3}-\frac{d \sqrt{a+b x^4}}{2 a x^2}-\frac{e \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} e x \sqrt{a+b x^4}}{a \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"(-2*a*c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^4)/a)] - 3*x*(a*d + b*d*x^4 + Sqrt[a]*f*x^2*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]] + 2*a*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^4)/a)]))/(6*a*x^3*Sqrt[a + b*x^4])","C",1
538,1,147,346,0.1558293,"\int \frac{c+d x+e x^2+f x^3}{x^5 \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^5*Sqrt[a + b*x^4]),x]","-\frac{\sqrt{a+b x^4} \left(3 a c \sqrt{\frac{b x^4}{a}+1}-3 b c x^4 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+4 a d x \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)+6 a e x^2 \sqrt{\frac{b x^4}{a}+1}+12 a f x^3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^4}{a}\right)\right)}{12 a^2 x^4 \sqrt{\frac{b x^4}{a}+1}}","\frac{b c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 a^{3/2}}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} d-3 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 a^{5/4} \sqrt{a+b x^4}}-\frac{\sqrt[4]{b} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{a^{3/4} \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{4 a x^4}-\frac{d \sqrt{a+b x^4}}{3 a x^3}-\frac{e \sqrt{a+b x^4}}{2 a x^2}-\frac{f \sqrt{a+b x^4}}{a x}+\frac{\sqrt{b} f x \sqrt{a+b x^4}}{a \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"-1/12*(Sqrt[a + b*x^4]*(3*a*c*Sqrt[1 + (b*x^4)/a] + 6*a*e*x^2*Sqrt[1 + (b*x^4)/a] - 3*b*c*x^4*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*d*x*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^4)/a)] + 12*a*f*x^3*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^4)/a)]))/(a^2*x^4*Sqrt[1 + (b*x^4)/a])","C",1
539,1,134,377,0.2321843,"\int \frac{c+d x+e x^2+f x^3}{x^6 \sqrt{a+b x^4}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]","-\frac{\sqrt{a+b x^4} \left(12 a c \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{b x^4}{a}\right)+5 x \left(3 a \sqrt{\frac{b x^4}{a}+1} \left(d+2 f x^2\right)-3 b d x^4 \tanh ^{-1}\left(\sqrt{\frac{b x^4}{a}+1}\right)+4 a e x \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^4}{a}\right)\right)\right)}{60 a^2 x^5 \sqrt{\frac{b x^4}{a}+1}}","-\frac{b^{3/4} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{a} e+9 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{30 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 a^{7/4} \sqrt{a+b x^4}}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{4 a^{3/2}}-\frac{3 b^{3/2} c x \sqrt{a+b x^4}}{5 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{3 b c \sqrt{a+b x^4}}{5 a^2 x}-\frac{c \sqrt{a+b x^4}}{5 a x^5}-\frac{d \sqrt{a+b x^4}}{4 a x^4}-\frac{e \sqrt{a+b x^4}}{3 a x^3}-\frac{f \sqrt{a+b x^4}}{2 a x^2}",1,"-1/60*(Sqrt[a + b*x^4]*(12*a*c*Hypergeometric2F1[-5/4, 1/2, -1/4, -((b*x^4)/a)] + 5*x*(3*a*(d + 2*f*x^2)*Sqrt[1 + (b*x^4)/a] - 3*b*d*x^4*ArcTanh[Sqrt[1 + (b*x^4)/a]] + 4*a*e*x*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^4)/a)])))/(a^2*x^5*Sqrt[1 + (b*x^4)/a])","C",1
540,1,220,365,0.1898567,"\int \frac{x^6 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x^6*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{-9 a^{3/2} f \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-12 b^{3/2} c x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)+12 a \sqrt{b} d-10 a \sqrt{b} e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+10 a \sqrt{b} e x+9 a \sqrt{b} f x^2+12 b^{3/2} c x^3+6 b^{3/2} d x^4+4 b^{3/2} e x^5+3 b^{3/2} f x^6}{12 b^{5/2} \sqrt{a+b x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{b} c-5 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{3 c x \sqrt{a+b x^4}}{2 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{3 \sqrt[4]{a} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{7/4} \sqrt{a+b x^4}}-\frac{3 a f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{4 b^{5/2}}+\frac{x \left(a e+a f x-b c x^2-b d x^3\right)}{2 b^2 \sqrt{a+b x^4}}+\frac{d \sqrt{a+b x^4}}{b^2}+\frac{e x \sqrt{a+b x^4}}{3 b^2}+\frac{f x^2 \sqrt{a+b x^4}}{4 b^2}",1,"(12*a*Sqrt[b]*d + 10*a*Sqrt[b]*e*x + 9*a*Sqrt[b]*f*x^2 + 12*b^(3/2)*c*x^3 + 6*b^(3/2)*d*x^4 + 4*b^(3/2)*e*x^5 + 3*b^(3/2)*f*x^6 - 9*a^(3/2)*f*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] - 10*a*Sqrt[b]*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 12*b^(3/2)*c*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(12*b^(5/2)*Sqrt[a + b*x^4])","C",1
541,1,176,343,0.1669309,"\int \frac{x^5 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x^5*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{3 \sqrt{a} \sqrt{b} c \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-6 b d x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)-5 a f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+6 a e+5 a f x-3 b c x^2+6 b d x^3+3 b e x^4+2 b f x^5}{6 b^2 \sqrt{a+b x^4}}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 b^{3/2}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(9 \sqrt{b} d-5 \sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{3 d x \sqrt{a+b x^4}}{2 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{3 \sqrt[4]{a} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{x \left(a f-b c x-b d x^2-b e x^3\right)}{2 b^2 \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2}",1,"(6*a*e + 5*a*f*x - 3*b*c*x^2 + 6*b*d*x^3 + 3*b*e*x^4 + 2*b*f*x^5 + 3*Sqrt[a]*Sqrt[b]*c*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] - 5*a*f*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 6*b*d*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(6*b^2*Sqrt[a + b*x^4])","C",1
542,1,166,314,0.1580645,"\int \frac{x^4 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x^4*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{b c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+\sqrt{a} \sqrt{b} d \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-2 b e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)+2 a f-b c x-b d x^2+2 b e x^3+b f x^4}{2 b^2 \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} e+\sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 b^{3/2}}+\frac{3 e x \sqrt{a+b x^4}}{2 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{3 \sqrt[4]{a} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{f \sqrt{a+b x^4}}{b^2}-\frac{x \left(c+d x+e x^2+f x^3\right)}{2 b \sqrt{a+b x^4}}",1,"(2*a*f - b*c*x - b*d*x^2 + 2*b*e*x^3 + b*f*x^4 + Sqrt[a]*Sqrt[b]*d*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + b*c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 2*b*e*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(2*b^2*Sqrt[a + b*x^4])","C",1
543,1,181,302,0.1405092,"\int \frac{x^3 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{\sqrt{b} d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+\sqrt{a} e \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)-2 \sqrt{b} f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)-\sqrt{b} c-\sqrt{b} d x-\sqrt{b} e x^2+2 \sqrt{b} f x^3}{2 b^{3/2} \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(3 \sqrt{a} f+\sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} b^{7/4} \sqrt{a+b x^4}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 b^{3/2}}+\frac{3 f x \sqrt{a+b x^4}}{2 b^{3/2} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{3 \sqrt[4]{a} f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{-c-d x-e x^2-f x^3}{2 b \sqrt{a+b x^4}}",1,"(-(Sqrt[b]*c) - Sqrt[b]*d*x - Sqrt[b]*e*x^2 + 2*Sqrt[b]*f*x^3 + Sqrt[a]*e*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + Sqrt[b]*d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] - 2*Sqrt[b]*f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(2*b^(3/2)*Sqrt[a + b*x^4])","C",1
544,1,165,333,0.2188311,"\int \frac{x^2 \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x^2*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{3 a^{3/2} f \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)+2 b^{3/2} c x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)-3 a \sqrt{b} (d+x (e+f x))+3 a \sqrt{b} e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)}{6 a b^{3/2} \sqrt{a+b x^4}}","-\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} c-\sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{3/4} b^{5/4} \sqrt{a+b x^4}}+\frac{c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right)}{2 b^{3/2}}-\frac{x \left(a e+a f x-b c x^2-b d x^3\right)}{2 a b \sqrt{a+b x^4}}-\frac{c x \sqrt{a+b x^4}}{2 a \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{d \sqrt{a+b x^4}}{2 a b}",1,"(-3*a*Sqrt[b]*(d + x*(e + f*x)) + 3*a^(3/2)*f*Sqrt[1 + (b*x^4)/a]*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 3*a*Sqrt[b]*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 2*b^(3/2)*c*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(6*a*b^(3/2)*Sqrt[a + b*x^4])","C",1
545,1,116,303,0.0816351,"\int \frac{x \left(c+d x+e x^2+f x^3\right)}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(x*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^(3/2),x]","\frac{2 b d x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)+3 a f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)-3 a e-3 a f x+3 b c x^2}{6 a b \sqrt{a+b x^4}}","-\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} d-\sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{3/4} b^{5/4} \sqrt{a+b x^4}}+\frac{d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}-\frac{x \left(a f-b c x-b d x^2-b e x^3\right)}{2 a b \sqrt{a+b x^4}}-\frac{d x \sqrt{a+b x^4}}{2 a \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}-\frac{e \sqrt{a+b x^4}}{2 a b}",1,"(-3*a*e - 3*a*f*x + 3*b*c*x^2 + 3*a*f*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 2*b*d*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(6*a*b*Sqrt[a + b*x^4])","C",1
546,1,116,275,0.0628853,"\int \frac{c+d x+e x^2+f x^3}{\left(a+b x^4\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^(3/2),x]","\frac{3 b c x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+2 b e x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)-3 a f+3 b c x+3 b d x^2}{6 a b \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} c-\sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} b^{3/4} \sqrt{a+b x^4}}+\frac{e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}-\frac{a f-b x \left(c+d x+e x^2\right)}{2 a b \sqrt{a+b x^4}}-\frac{e x \sqrt{a+b x^4}}{2 a \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(-3*a*f + 3*b*c*x + 3*b*d*x^2 + 3*b*c*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 2*b*e*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)])/(6*a*b*Sqrt[a + b*x^4])","C",1
547,1,125,323,0.1353184,"\int \frac{c+d x+e x^2+f x^3}{x \left(a+b x^4\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x*(a + b*x^4)^(3/2)),x]","\frac{3 c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right)+x \left(3 d \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+2 f x^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right)+3 d+3 e x\right)}{6 a \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{b} d-\sqrt{a} f\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} b^{3/4} \sqrt{a+b x^4}}+\frac{f \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{3/4} \sqrt{a+b x^4}}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{x \left(a d+a e x+a f x^2-b c x^3\right)}{2 a^2 \sqrt{a+b x^4}}+\frac{c \sqrt{a+b x^4}}{2 a^2}-\frac{f x \sqrt{a+b x^4}}{2 a \sqrt{b} \left(\sqrt{a}+\sqrt{b} x^2\right)}",1,"(3*c*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^4)/a] + x*(3*d + 3*e*x + 3*d*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)] + 2*f*x^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)]))/(6*a*Sqrt[a + b*x^4])","C",1
548,1,123,344,0.1264472,"\int \frac{c+d x+e x^2+f x^3}{x^2 \left(a+b x^4\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^2*(a + b*x^4)^(3/2)),x]","\frac{-2 c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^4}{a}\right)+d x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right)+x^2 \left(e \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)+e+f x\right)}{2 a x \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} e+3 \sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{7/4} \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{3 \sqrt[4]{b} c \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{x \left(a e+a f x-b c x^2-b d x^3\right)}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} c x \sqrt{a+b x^4}}{2 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{d \sqrt{a+b x^4}}{2 a^2}",1,"(d*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^4)/a] - 2*c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^4)/a)] + x^2*(e + f*x + e*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]))/(2*a*x*Sqrt[a + b*x^4])","C",1
549,1,140,367,0.1265183,"\int \frac{c+d x+e x^2+f x^3}{x^3 \left(a+b x^4\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^3*(a + b*x^4)^(3/2)),x]","\frac{-2 a d x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^4}{a}\right)+a e x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right)+a f x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right)-a c+a f x^3-2 b c x^4}{2 a^2 x^2 \sqrt{a+b x^4}}","\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} f+3 \sqrt{b} d\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{7/4} \sqrt[4]{b} \sqrt{a+b x^4}}-\frac{3 \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 a^{3/2}}+\frac{x \left(a f-b c x-b d x^2-b e x^3\right)}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{2 a^2 x^2}-\frac{d \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} d x \sqrt{a+b x^4}}{2 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{e \sqrt{a+b x^4}}{2 a^2}",1,"(-(a*c) + a*f*x^3 - 2*b*c*x^4 + a*e*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^4)/a] - 2*a*d*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^4)/a)] + a*f*x^3*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/(2*a^2*x^2*Sqrt[a + b*x^4])","C",1
550,1,136,387,0.1376893,"\int \frac{c+d x+e x^2+f x^3}{x^4 \left(a+b x^4\right)^{3/2}} \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)/(x^4*(a + b*x^4)^(3/2)),x]","\frac{-2 a c \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{b x^4}{a}\right)-3 x \left(2 a e x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^4}{a}\right)-a f x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^4}{a}+1\right)+a d+2 b d x^4\right)}{6 a^2 x^3 \sqrt{a+b x^4}}","-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(5 \sqrt{b} c-9 \sqrt{a} e\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{12 a^{9/4} \sqrt{a+b x^4}}-\frac{3 \sqrt[4]{b} e \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{f \tanh ^{-1}\left(\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{x \left(b c+b d x+b e x^2+b f x^3\right)}{2 a^2 \sqrt{a+b x^4}}-\frac{c \sqrt{a+b x^4}}{3 a^2 x^3}-\frac{d \sqrt{a+b x^4}}{2 a^2 x^2}-\frac{e \sqrt{a+b x^4}}{a^2 x}+\frac{3 \sqrt{b} e x \sqrt{a+b x^4}}{2 a^2 \left(\sqrt{a}+\sqrt{b} x^2\right)}+\frac{f \sqrt{a+b x^4}}{2 a^2}",1,"(-2*a*c*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-3/4, 3/2, 1/4, -((b*x^4)/a)] - 3*x*(a*d + 2*b*d*x^4 - a*f*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^4)/a] + 2*a*e*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^4)/a)]))/(6*a^2*x^3*Sqrt[a + b*x^4])","C",1
551,1,174,269,0.2403087,"\int (g x)^m \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^p \, dx","Integrate[(g*x)^m*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]","x (g x)^m \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(\frac{c \, _2F_1\left(\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right)}{m+1}+x \left(\frac{d \, _2F_1\left(\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right)}{m+2}+x \left(\frac{e \, _2F_1\left(\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right)}{m+3}+\frac{f x \, _2F_1\left(\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right)}{m+4}\right)\right)\right)","\frac{c (g x)^{m+1} \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right)}{g (m+1)}+\frac{d (g x)^{m+2} \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right)}{g^2 (m+2)}+\frac{e (g x)^{m+3} \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right)}{g^3 (m+3)}+\frac{f (g x)^{m+4} \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right)}{g^4 (m+4)}",1,"(x*(g*x)^m*(a + b*x^4)^p*((c*Hypergeometric2F1[(1 + m)/4, -p, (5 + m)/4, -((b*x^4)/a)])/(1 + m) + x*((d*Hypergeometric2F1[(2 + m)/4, -p, (6 + m)/4, -((b*x^4)/a)])/(2 + m) + x*((e*Hypergeometric2F1[(3 + m)/4, -p, (7 + m)/4, -((b*x^4)/a)])/(3 + m) + (f*x*Hypergeometric2F1[(4 + m)/4, -p, (8 + m)/4, -((b*x^4)/a)])/(4 + m)))))/(1 + (b*x^4)/a)^p","A",1
552,1,147,143,0.1249358,"\int \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^p \, dx","Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]","\frac{1}{12} \left(a+b x^4\right)^p \left(12 c x \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right)+6 d x^2 \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^4}{a}\right)+4 e x^3 \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right)+\frac{3 f \left(a+b x^4\right)}{b (p+1)}\right)","\frac{c x \left(a+b x^4\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{4};\frac{5}{4};-\frac{b x^4}{a}\right)}{a}+\frac{d x^2 \left(a+b x^4\right)^{p+1} \, _2F_1\left(1,p+\frac{3}{2};\frac{3}{2};-\frac{b x^4}{a}\right)}{2 a}+\frac{e x^3 \left(a+b x^4\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{4};\frac{7}{4};-\frac{b x^4}{a}\right)}{3 a}+\frac{f \left(a+b x^4\right)^{p+1}}{4 b (p+1)}",1,"((a + b*x^4)^p*((3*f*(a + b*x^4))/(b*(1 + p)) + (12*c*x*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p + (6*d*x^2*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p + (4*e*x^3*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p))/12","A",1
553,1,145,175,0.1385196,"\int x^3 \left(c+d x+e x^2+f x^3\right) \left(a+b x^4\right)^p \, dx","Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]","\frac{\left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(105 c \left(a+b x^4\right) \left(\frac{b x^4}{a}+1\right)^p+84 b d (p+1) x^5 \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right)+70 b e (p+1) x^6 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right)+60 b f (p+1) x^7 \, _2F_1\left(\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right)\right)}{420 b (p+1)}","\frac{c \left(a+b x^4\right)^{p+1}}{4 b (p+1)}+\frac{1}{5} d x^5 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right)+\frac{1}{6} e x^6 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right)+\frac{1}{7} f x^7 \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right)",1,"((a + b*x^4)^p*(105*c*(a + b*x^4)*(1 + (b*x^4)/a)^p + 84*b*d*(1 + p)*x^5*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^4)/a)] + 70*b*e*(1 + p)*x^6*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^4)/a)] + 60*b*f*(1 + p)*x^7*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^4)/a)]))/(420*b*(1 + p)*(1 + (b*x^4)/a)^p)","A",1
554,1,8,8,0.0012397,"\int \frac{1+x+x^2+x^3+x^4}{1-x^5} \, dx","Integrate[(1 + x + x^2 + x^3 + x^4)/(1 - x^5),x]","-\log (1-x)","-\log (1-x)",1,"-Log[1 - x]","A",1
555,1,10,10,0.0012399,"\int \frac{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{729-64 x^6} \, dx","Integrate[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6),x]","\frac{1}{2} \log (2 x+3)","\frac{1}{2} \log (2 x+3)",1,"Log[3 + 2*x]/2","A",1
556,1,10,10,0.0013135,"\int \frac{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx","Integrate[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6),x]","-\frac{1}{2} \log (3-2 x)","-\frac{1}{2} \log (3-2 x)",1,"-1/2*Log[3 - 2*x]","A",1
557,1,21,10,0.0032471,"\int \frac{81+36 x^2+16 x^4}{729-64 x^6} \, dx","Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6),x]","\frac{1}{12} \log (2 x+3)-\frac{1}{12} \log (3-2 x)","\frac{1}{6} \tanh ^{-1}\left(\frac{2 x}{3}\right)",1,"-1/12*Log[3 - 2*x] + Log[3 + 2*x]/12","B",1
558,1,24,24,0.0076506,"\int \frac{81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx","Integrate[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6),x]","\frac{\tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{3 \sqrt{3}}",1,"ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])","A",1
559,1,50,50,0.019835,"\int \frac{3-2 x}{729-64 x^6} \, dx","Integrate[(3 - 2*x)/(729 - 64*x^6),x]","-\frac{1}{972} \log \left(4 x^2-6 x+9\right)+\frac{1}{486} \log (2 x+3)+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{162 \sqrt{3}}","-\frac{1}{972} \log \left(4 x^2-6 x+9\right)+\frac{1}{486} \log (2 x+3)+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{162 \sqrt{3}}",1,"ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(162*Sqrt[3]) + Log[3 + 2*x]/486 - Log[9 - 6*x + 4*x^2]/972","A",1
560,1,46,50,0.0162439,"\int \frac{3+2 x}{729-64 x^6} \, dx","Integrate[(3 + 2*x)/(729 - 64*x^6),x]","\frac{1}{972} \left(\log \left(4 x^2+6 x+9\right)-2 \log (3-2 x)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)\right)","\frac{1}{972} \log \left(4 x^2+6 x+9\right)-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{162 \sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Log[3 - 2*x] + Log[9 + 6*x + 4*x^2])/972","A",1
561,1,56,60,0.0135743,"\int \frac{9-6 x+4 x^2}{729-64 x^6} \, dx","Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6),x]","\frac{1}{324} \left(-\log \left(4 x^2+6 x+9\right)-\log (3-2 x)+3 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)\right)","-\frac{1}{324} \log \left(4 x^2+6 x+9\right)-\frac{1}{324} \log (3-2 x)+\frac{1}{108} \log (2 x+3)+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{54 \sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - Log[3 - 2*x] + 3*Log[3 + 2*x] - Log[9 + 6*x + 4*x^2])/324","A",1
562,1,52,60,0.0130581,"\int \frac{9+6 x+4 x^2}{729-64 x^6} \, dx","Integrate[(9 + 6*x + 4*x^2)/(729 - 64*x^6),x]","\frac{1}{324} \left(\log \left(4 x^2-6 x+9\right)-3 \log (3-2 x)+\log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)\right)","\frac{1}{324} \log \left(4 x^2-6 x+9\right)-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (2 x+3)-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{54 \sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 3*Log[3 - 2*x] + Log[3 + 2*x] + Log[9 - 6*x + 4*x^2])/324","A",1
563,1,50,50,0.0065374,"\int \frac{27-8 x^3}{729-64 x^6} \, dx","Integrate[(27 - 8*x^3)/(729 - 64*x^6),x]","-\frac{1}{108} \log \left(4 x^2-6 x+9\right)+\frac{1}{54} \log (2 x+3)+\frac{\tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)}{18 \sqrt{3}}","-\frac{1}{108} \log \left(4 x^2-6 x+9\right)+\frac{1}{54} \log (2 x+3)-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{18 \sqrt{3}}",1,"ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*x^2]/108","A",1
564,1,50,50,0.01163,"\int \frac{27+36 x+24 x^2+8 x^3}{729-64 x^6} \, dx","Integrate[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6),x]","\frac{1}{36} \log \left(4 x^2-6 x+9\right)-\frac{1}{18} \log (3-2 x)+\frac{\tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)}{18 \sqrt{3}}","\frac{1}{36} \log \left(4 x^2-6 x+9\right)-\frac{1}{18} \log (3-2 x)-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{18 \sqrt{3}}",1,"ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) - Log[3 - 2*x]/18 + Log[9 - 6*x + 4*x^2]/36","A",1
565,1,100,110,0.1080699,"\int \frac{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{\left(729-64 x^6\right)^2} \, dx","Integrate[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6)^2,x]","\frac{-3 \log \left(4 x^2-6 x+9\right)-3 \log \left(4 x^2+6 x+9\right)-\frac{18}{2 x+3}-3 \log (3-2 x)+15 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{52488}","-\frac{\log \left(4 x^2-6 x+9\right)}{17496}-\frac{\log \left(4 x^2+6 x+9\right)}{17496}-\frac{1}{2916 (2 x+3)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (2 x+3)}{17496}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{2916 \sqrt{3}}",1,"(-18/(3 + 2*x) + 2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 3*Log[3 - 2*x] + 15*Log[3 + 2*x] - 3*Log[9 - 6*x + 4*x^2] - 3*Log[9 + 6*x + 4*x^2])/52488","A",1
566,1,97,110,0.0934424,"\int \frac{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{\left(729-64 x^6\right)^2} \, dx","Integrate[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6)^2,x]","\frac{3 \left(\log \left(4 x^2-6 x+9\right)+\log \left(4 x^2+6 x+9\right)+\frac{6}{3-2 x}-5 \log (3-2 x)+\log (2 x+3)\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{52488}","\frac{\log \left(4 x^2-6 x+9\right)}{17496}+\frac{\log \left(4 x^2+6 x+9\right)}{17496}+\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (2 x+3)}{17496}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{2916 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{8748 \sqrt{3}}",1,"(6*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] + 3*(6/(3 - 2*x) - 5*Log[3 - 2*x] + Log[3 + 2*x] + Log[9 - 6*x + 4*x^2] + Log[9 + 6*x + 4*x^2]))/52488","A",1
567,1,122,81,0.5688978,"\int \frac{81+36 x^2+16 x^4}{\left(729-64 x^6\right)^2} \, dx","Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]","\frac{\frac{36 x}{9-4 x^2}-9 \log (3-2 x)+9 \log (2 x+3)+3 \sqrt{3} \tan ^{-1}\left(\frac{1}{3} \left(\sqrt{3}-i\right) x\right)+4 i \sqrt{3} \tanh ^{-1}\left(\frac{1}{3} \left(1-i \sqrt{3}\right) x\right)+\left(-3+\frac{2}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}\right) \tanh ^{-1}\left(\frac{1}{3} \left(x+i \sqrt{3} x\right)\right)}{157464}","\frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{2 x}{3}\right)}{8748}",1,"((36*x)/(9 - 4*x^2) + 3*Sqrt[3]*ArcTan[((-I + Sqrt[3])*x)/3] + (4*I)*Sqrt[3]*ArcTanh[((1 - I*Sqrt[3])*x)/3] + (-3 + 2/Sqrt[(1 + I*Sqrt[3])/6])*ArcTanh[(x + I*Sqrt[3]*x)/3] - 9*Log[3 - 2*x] + 9*Log[3 + 2*x])/157464","C",0
568,1,84,92,0.0357122,"\int \frac{81+54 x-24 x^3-16 x^4}{\left(729-64 x^6\right)^2} \, dx","Integrate[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6)^2,x]","\frac{\frac{36 x}{4 x^2-6 x+9}-\log \left(4 x^2-6 x+9\right)+3 \log \left(4 x^2+6 x+9\right)-6 \log (3-2 x)+2 \log (2 x+3)+12 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)}{157464}","\frac{x}{4374 \left(4 x^2-6 x+9\right)}-\frac{\log \left(4 x^2-6 x+9\right)}{157464}+\frac{\log \left(4 x^2+6 x+9\right)}{52488}-\frac{\log (3-2 x)}{26244}+\frac{\log (2 x+3)}{78732}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{4374 \sqrt{3}}",1,"((36*x)/(9 - 6*x + 4*x^2) + 12*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/157464","A",1
569,1,119,148,0.0849171,"\int \frac{3-2 x}{\left(729-64 x^6\right)^2} \, dx","Integrate[(3 - 2*x)/(729 - 64*x^6)^2,x]","\frac{-9 \log \left(4 x^2-6 x+9\right)+\log \left(4 x^2+6 x+9\right)+\frac{1944 x}{32 x^5+48 x^4+72 x^3+108 x^2+162 x+243}-2 \log (3-2 x)+18 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+18 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{8503056}","\frac{3-x}{708588 \left(4 x^2-6 x+9\right)}+\frac{x}{236196 \left(4 x^2+6 x+9\right)}-\frac{\log \left(4 x^2-6 x+9\right)}{944784}+\frac{\log \left(4 x^2+6 x+9\right)}{8503056}-\frac{1}{708588 (2 x+3)}-\frac{\log (3-2 x)}{4251528}+\frac{\log (2 x+3)}{472392}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{1417176 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{157464 \sqrt{3}}",1,"((1944*x)/(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5) + 2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 18*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 2*Log[3 - 2*x] + 18*Log[3 + 2*x] - 9*Log[9 - 6*x + 4*x^2] + Log[9 + 6*x + 4*x^2])/8503056","A",1
570,1,121,146,0.0895147,"\int \frac{3+2 x}{\left(729-64 x^6\right)^2} \, dx","Integrate[(3 + 2*x)/(729 - 64*x^6)^2,x]","\frac{-\log \left(4 x^2-6 x+9\right)+9 \log \left(4 x^2+6 x+9\right)+\frac{1944 x}{-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243}-18 \log (3-2 x)+2 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{8503056}","\frac{x}{236196 \left(4 x^2-6 x+9\right)}-\frac{x+3}{708588 \left(4 x^2+6 x+9\right)}-\frac{\log \left(4 x^2-6 x+9\right)}{8503056}+\frac{\log \left(4 x^2+6 x+9\right)}{944784}+\frac{1}{708588 (3-2 x)}-\frac{\log (3-2 x)}{472392}+\frac{\log (2 x+3)}{4251528}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{1417176 \sqrt{3}}",1,"((1944*x)/(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5) + 18*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 18*Log[3 - 2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 9*Log[9 + 6*x + 4*x^2])/8503056","A",1
571,1,111,142,0.0722055,"\int \frac{9-6 x+4 x^2}{\left(729-64 x^6\right)^2} \, dx","Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6)^2,x]","\frac{-3 \log \left(4 x^2-6 x+9\right)-5 \log \left(4 x^2+6 x+9\right)+\frac{648 x}{-16 x^4-24 x^3+54 x+81}-8 \log (3-2 x)+24 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+18 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{2834352}","\frac{4 x+3}{236196 \left(4 x^2+6 x+9\right)}-\frac{\log \left(4 x^2-6 x+9\right)}{944784}-\frac{5 \log \left(4 x^2+6 x+9\right)}{2834352}+\frac{1}{472392 (3-2 x)}-\frac{1}{157464 (2 x+3)}-\frac{\log (3-2 x)}{354294}+\frac{\log (2 x+3)}{118098}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{472392 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{52488 \sqrt{3}}",1,"((648*x)/(81 + 54*x - 24*x^3 - 16*x^4) + 2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 18*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 8*Log[3 - 2*x] + 24*Log[3 + 2*x] - 3*Log[9 - 6*x + 4*x^2] - 5*Log[9 + 6*x + 4*x^2])/2834352","A",1
572,1,111,142,0.0742479,"\int \frac{9+6 x+4 x^2}{\left(729-64 x^6\right)^2} \, dx","Integrate[(9 + 6*x + 4*x^2)/(729 - 64*x^6)^2,x]","\frac{5 \log \left(4 x^2-6 x+9\right)+3 \log \left(4 x^2+6 x+9\right)+\frac{648 x}{-16 x^4+24 x^3-54 x+81}-24 \log (3-2 x)+8 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{2834352}","-\frac{3-4 x}{236196 \left(4 x^2-6 x+9\right)}+\frac{5 \log \left(4 x^2-6 x+9\right)}{2834352}+\frac{\log \left(4 x^2+6 x+9\right)}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{472392 \sqrt{3}}",1,"((648*x)/(81 - 54*x + 24*x^3 - 16*x^4) + 18*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 24*Log[3 - 2*x] + 8*Log[3 + 2*x] + 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/2834352","A",1
573,1,103,113,0.068498,"\int \frac{27-8 x^3}{\left(729-64 x^6\right)^2} \, dx","Integrate[(27 - 8*x^3)/(729 - 64*x^6)^2,x]","\frac{\frac{216 x}{8 x^3+27}-7 \log \left(4 x^2-6 x+9\right)+3 \log \left(4 x^2+6 x+9\right)-6 \log (3-2 x)+14 \log (2 x+3)+14 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{944784}","\frac{x}{4374 \left(8 x^3+27\right)}-\frac{7 \log \left(4 x^2-6 x+9\right)}{944784}+\frac{\log \left(4 x^2+6 x+9\right)}{314928}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (2 x+3)}{472392}-\frac{7 \tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{52488 \sqrt{3}}",1,"((216*x)/(27 + 8*x^3) + 14*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] + 14*Log[3 + 2*x] - 7*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784","A",1
574,1,111,131,0.0802525,"\int \frac{27+36 x+24 x^2+8 x^3}{\left(729-64 x^6\right)^2} \, dx","Integrate[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6)^2,x]","\frac{17 \log \left(4 x^2-6 x+9\right)+3 \log \left(4 x^2+6 x+9\right)+\frac{216 x}{-8 x^3+24 x^2-36 x+27}-42 \log (3-2 x)+2 \log (2 x+3)+22 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{944784}","-\frac{3-2 x}{26244 \left(4 x^2-6 x+9\right)}+\frac{17 \log \left(4 x^2-6 x+9\right)}{944784}+\frac{\log \left(4 x^2+6 x+9\right)}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{157464 \sqrt{3}}",1,"((216*x)/(27 - 36*x + 24*x^2 - 8*x^3) + 22*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 42*Log[3 - 2*x] + 2*Log[3 + 2*x] + 17*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784","A",1
575,1,91,99,0.0372889,"\int \frac{x \left(27-2 x^3\right)}{729-64 x^6} \, dx","Integrate[(x*(27 - 2*x^3))/(729 - 64*x^6),x]","\frac{1}{576} \left(5 \log \left(4 x^2-6 x+9\right)+3 \log \left(4 x^2+6 x+9\right)-6 \log (3-2 x)-10 \log (2 x+3)+10 \sqrt{3} \tan ^{-1}\left(\frac{4 x-3}{3 \sqrt{3}}\right)-6 \sqrt{3} \tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)\right)","\frac{5}{576} \log \left(4 x^2-6 x+9\right)+\frac{1}{192} \log \left(4 x^2+6 x+9\right)-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left(\frac{3-4 x}{3 \sqrt{3}}\right)}{96 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{4 x+3}{3 \sqrt{3}}\right)}{32 \sqrt{3}}",1,"(10*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] - 10*Log[3 + 2*x] + 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/576","A",1
576,1,130,162,0.4324823,"\int \frac{(c x)^m \left(d+e x^n+f x^{2 n}+g x^{3 n}\right)}{a+b x^n} \, dx","Integrate[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n),x]","\frac{x (c x)^m \left(\frac{a^2 g-a b f+b^2 e}{m+1}+\frac{\left(a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{b x^n (b f-a g)}{m+n+1}+\frac{b^2 g x^{2 n}}{m+2 n+1}\right)}{b^3}","\frac{(c x)^{m+1} \left(a^2 g-a b f+b^2 e\right)}{b^3 c (m+1)}+\frac{(c x)^{m+1} \left(a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a b^3 c (m+1)}+\frac{x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac{g x^{2 n+1} (c x)^m}{b (m+2 n+1)}",1,"(x*(c*x)^m*((b^2*e - a*b*f + a^2*g)/(1 + m) + (b*(b*f - a*g)*x^n)/(1 + m + n) + (b^2*g*x^(2*n))/(1 + m + 2*n) + ((b^3*d - a*b^2*e + a^2*b*f - a^3*g)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m))))/b^3","A",1
577,1,108,84,0.2276706,"\int \left(c+d x^{-1+n}\right) \left(a+b x^n\right)^3 \, dx","Integrate[(c + d*x^(-1 + n))*(a + b*x^n)^3,x]","\frac{x \left(c+d x^{n-1}\right) \left(4 a^3 c x+\frac{12 a^2 b c x^{n+1}}{n+1}+\frac{12 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left(a+b x^n\right)^4}{b n}+\frac{4 b^3 c x^{3 n+1}}{3 n+1}\right)}{4 \left(c x+d x^n\right)}","a^3 c x+\frac{3 a^2 b c x^{n+1}}{n+1}+\frac{3 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left(a+b x^n\right)^4}{4 b n}+\frac{b^3 c x^{3 n+1}}{3 n+1}",1,"(x*(c + d*x^(-1 + n))*(4*a^3*c*x + (12*a^2*b*c*x^(1 + n))/(1 + n) + (12*a*b^2*c*x^(1 + 2*n))/(1 + 2*n) + (4*b^3*c*x^(1 + 3*n))/(1 + 3*n) + (d*(a + b*x^n)^4)/(b*n)))/(4*(c*x + d*x^n))","A",1
578,1,120,61,0.152014,"\int \left(c+d x^{-1+n}\right) \left(a+b x^n\right)^2 \, dx","Integrate[(c + d*x^(-1 + n))*(a + b*x^n)^2,x]","\frac{a^3 d \left(2 n^2+3 n+1\right)+3 a^2 b \left(2 n^2+3 n+1\right) \left(c n x+d x^n\right)+3 a b^2 (2 n+1) x^n \left(2 c n x+d (n+1) x^n\right)+b^3 (n+1) x^{2 n} \left(3 c n x+d (2 n+1) x^n\right)}{3 b n (n+1) (2 n+1)}","a^2 c x+\frac{2 a b c x^{n+1}}{n+1}+\frac{d \left(a+b x^n\right)^3}{3 b n}+\frac{b^2 c x^{2 n+1}}{2 n+1}",1,"(a^3*d*(1 + 3*n + 2*n^2) + 3*a^2*b*(1 + 3*n + 2*n^2)*(c*n*x + d*x^n) + 3*a*b^2*(1 + 2*n)*x^n*(2*c*n*x + d*(1 + n)*x^n) + b^3*(1 + n)*x^(2*n)*(3*c*n*x + d*(1 + 2*n)*x^n))/(3*b*n*(1 + n)*(1 + 2*n))","A",1
579,1,42,41,0.1329278,"\int \left(c+d x^{-1+n}\right) \left(a+b x^n\right) \, dx","Integrate[(c + d*x^(-1 + n))*(a + b*x^n),x]","\frac{2 a \left(c n x+d x^n\right)+b x^n \left(\frac{2 c n x}{n+1}+d x^n\right)}{2 n}","a c x+\frac{a d x^n}{n}+\frac{b c x^{n+1}}{n+1}+\frac{b d x^{2 n}}{2 n}",1,"(2*a*(c*n*x + d*x^n) + b*x^n*((2*c*n*x)/(1 + n) + d*x^n))/(2*n)","A",1
580,1,12,12,0.0015748,"\int \left(c+d x^{-1+n}\right) \, dx","Integrate[c + d*x^(-1 + n),x]","c x+\frac{d x^n}{n}","c x+\frac{d x^n}{n}",1,"c*x + (d*x^n)/n","A",1
581,1,42,42,0.0655431,"\int \frac{c+d x^{-1+n}}{a+b x^n} \, dx","Integrate[(c + d*x^(-1 + n))/(a + b*x^n),x]","\frac{c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a}+\frac{d \log \left(a+b x^n\right)}{b n}","\frac{c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a}+\frac{d \log \left(a+b x^n\right)}{b n}",1,"(c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a + (d*Log[a + b*x^n])/(b*n)","A",1
582,1,44,44,0.1174005,"\int \frac{c+d x^{-1+n}}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^(-1 + n))/(a + b*x^n)^2,x]","\frac{c x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}-\frac{d}{a b n+b^2 n x^n}","\frac{c x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}-\frac{d}{b n \left(a+b x^n\right)}",1,"-(d/(a*b*n + b^2*n*x^n)) + (c*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2","A",1
583,1,63,46,0.1148368,"\int \frac{c+d x^{-1+n}}{\left(a+b x^n\right)^3} \, dx","Integrate[(c + d*x^(-1 + n))/(a + b*x^n)^3,x]","\frac{2 b c n x \left(a+b x^n\right)^2 \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)-a^3 d}{2 a^3 b n \left(a+b x^n\right)^2}","\frac{c x \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^3}-\frac{d}{2 b n \left(a+b x^n\right)^2}",1,"(-(a^3*d) + 2*b*c*n*x*(a + b*x^n)^2*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(2*a^3*b*n*(a + b*x^n)^2)","A",1
584,1,206,305,0.5609796,"\int \frac{(c x)^m \left(d+e x^n+f x^{2 n}+g x^{3 n}\right)}{\sqrt{a+b x^n}} \, dx","Integrate[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/Sqrt[a + b*x^n],x]","\frac{x (c x)^m \sqrt{\frac{b x^n}{a}+1} \left(\frac{d \, _2F_1\left(\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{m+1}+x^n \left(\frac{e \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{m+n+1}+x^n \left(\frac{f \, _2F_1\left(\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right)}{m+2 n+1}+\frac{g x^n \, _2F_1\left(\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right)}{m+3 n+1}\right)\right)\right)}{\sqrt{a+b x^n}}","\frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right)}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right)}{(m+3 n+1) \sqrt{a+b x^n}}",1,"(x*(c*x)^m*Sqrt[1 + (b*x^n)/a]*((d*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x^n*((e*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + x^n*((f*Hypergeometric2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n) + (g*x^n*Hypergeometric2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/(1 + m + 3*n)))))/Sqrt[a + b*x^n]","A",1
585,1,45,45,0.3054939,"\int \frac{-a h x^{-1+\frac{n}{4}}+b f x^{-1+\frac{n}{2}}+b g x^{-1+n}+b h x^{-1+\frac{5 n}{4}}}{\left(a+b x^n\right)^{3/2}} \, dx","Integrate[(-(a*h*x^(-1 + n/4)) + b*f*x^(-1 + n/2) + b*g*x^(-1 + n) + b*h*x^(-1 + (5*n)/4))/(a + b*x^n)^(3/2),x]","\frac{2 b f x^{n/2}-2 a \left(g+2 h x^{n/4}\right)}{a n \sqrt{a+b x^n}}","-\frac{2 \left(a g+2 a h x^{n/4}-b f x^{n/2}\right)}{a n \sqrt{a+b x^n}}",1,"(2*b*f*x^(n/2) - 2*a*(g + 2*h*x^(n/4)))/(a*n*Sqrt[a + b*x^n])","A",1
586,1,178,273,0.3177756,"\int (c x)^m \left(d+e x+f x^2+g x^3\right) \left(a+b x^n\right)^p \, dx","Integrate[(c*x)^m*(d + e*x + f*x^2 + g*x^3)*(a + b*x^n)^p,x]","x (c x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(\frac{d \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{m+1}+x \left(\frac{e \, _2F_1\left(\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right)}{m+2}+x \left(\frac{f \, _2F_1\left(\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right)}{m+3}+\frac{g x \, _2F_1\left(\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right)}{m+4}\right)\right)\right)","\frac{g (c x)^{m+4} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right)}{c^4 (m+4)}+\frac{f (c x)^{m+3} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right)}{c^3 (m+3)}+\frac{e (c x)^{m+2} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right)}{c^2 (m+2)}+\frac{d (c x)^{m+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{c (m+1)}",1,"(x*(c*x)^m*(a + b*x^n)^p*((d*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x*((e*Hypergeometric2F1[(2 + m)/n, -p, (2 + m + n)/n, -((b*x^n)/a)])/(2 + m) + x*((f*Hypergeometric2F1[(3 + m)/n, -p, (3 + m + n)/n, -((b*x^n)/a)])/(3 + m) + (g*x*Hypergeometric2F1[(4 + m)/n, -p, (4 + m + n)/n, -((b*x^n)/a)])/(4 + m)))))/(1 + (b*x^n)/a)^p","A",1
587,1,204,297,0.3017288,"\int (c x)^m \left(a+b x^n\right)^p \left(d+e x^n+f x^{2 n}+g x^{3 n}\right) \, dx","Integrate[(c*x)^m*(a + b*x^n)^p*(d + e*x^n + f*x^(2*n) + g*x^(3*n)),x]","x (c x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(\frac{d \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{m+1}+x^n \left(\frac{e \, _2F_1\left(\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{m+n+1}+x^n \left(\frac{f \, _2F_1\left(\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right)}{m+2 n+1}+\frac{g x^n \, _2F_1\left(\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right)}{m+3 n+1}\right)\right)\right)","\frac{d (c x)^{m+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{c (m+1)}+\frac{e x^{n+1} (c x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{m+n+1}+\frac{f x^{2 n+1} (c x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right)}{m+2 n+1}+\frac{g x^{3 n+1} (c x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right)}{m+3 n+1}",1,"(x*(c*x)^m*(a + b*x^n)^p*((d*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x^n*((e*Hypergeometric2F1[(1 + m + n)/n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + x^n*((f*Hypergeometric2F1[(1 + m + 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n) + (g*x^n*Hypergeometric2F1[(1 + m + 3*n)/n, -p, (1 + m + 4*n)/n, -((b*x^n)/a)])/(1 + m + 3*n)))))/(1 + (b*x^n)/a)^p","A",1
588,1,147,162,0.3929463,"\int \frac{c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^(n/2) + e*x^n + f*x^((3*n)/2))/(a + b*x^n)^2,x]","\frac{x \left((b c-a e) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)+\frac{2 x^{n/2} (b d-a f) \, _2F_1\left(2,\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-\frac{b x^n}{a}\right)}{n+2}+a e \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)+\frac{2 a f x^{n/2} \, _2F_1\left(1,\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-\frac{b x^n}{a}\right)}{n+2}\right)}{a^2 b}","\frac{x (a e-b c (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 b n}-\frac{x^{\frac{n+2}{2}} (b d (2-n)-a f (n+2)) \, _2F_1\left(1,\frac{1}{2} \left(1+\frac{2}{n}\right);\frac{1}{2} \left(3+\frac{2}{n}\right);-\frac{b x^n}{a}\right)}{a^2 b n (n+2)}+\frac{x \left(x^{n/2} (b d-a f)-a e+b c\right)}{a b n \left(a+b x^n\right)}",1,"(x*((2*a*f*x^(n/2)*Hypergeometric2F1[1, 1/2 + n^(-1), 3/2 + n^(-1), -((b*x^n)/a)])/(2 + n) + a*e*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)] + (2*(b*d - a*f)*x^(n/2)*Hypergeometric2F1[2, 1/2 + n^(-1), 3/2 + n^(-1), -((b*x^n)/a)])/(2 + n) + (b*c - a*e)*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(a^2*b)","A",1
589,1,24,24,0.1761916,"\int \frac{a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx","Integrate[(a*c + 2*(b*c + a*d)*x^2 + 3*b*d*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]","x \sqrt{a+b x^2} \sqrt{c+d x^2}","x \sqrt{a+b x^2} \sqrt{c+d x^2}",1,"x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]","A",1
590,1,93,103,0.2032689,"\int \frac{1+x^3}{\left(1-x^4\right) \sqrt[4]{1+x^4}} \, dx","Integrate[(1 + x^3)/((1 - x^4)*(1 + x^4)^(1/4)),x]","\frac{1}{4} x^4 F_1\left(1;\frac{1}{4},1;2;-x^4,x^4\right)+\frac{-\log \left(1-\frac{\sqrt[4]{2} x}{\sqrt[4]{x^4+1}}\right)+\log \left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^4+1}}+1\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^4+1}}\right)}{4 \sqrt[4]{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^4+1}}\right)}{2 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{x^4+1}}{\sqrt[4]{2}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^4+1}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{x^4+1}}{\sqrt[4]{2}}\right)}{2 \sqrt[4]{2}}",1,"(x^4*AppellF1[1, 1/4, 1, 2, -x^4, x^4])/4 + (2*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)] - Log[1 - (2^(1/4)*x)/(1 + x^4)^(1/4)] + Log[1 + (2^(1/4)*x)/(1 + x^4)^(1/4)])/(4*2^(1/4))","C",0
591,1,28,28,0.3455806,"\int \left(a+b x^n\right)^{\frac{-1-n}{n}} \left(c+d x^n\right)^{\frac{-1-n}{n}} \left(a c-b d x^{2 n}\right) \, dx","Integrate[(a + b*x^n)^((-1 - n)/n)*(c + d*x^n)^((-1 - n)/n)*(a*c - b*d*x^(2*n)),x]","x \left(a+b x^n\right)^{-1/n} \left(c+d x^n\right)^{-1/n}","x \left(a+b x^n\right)^{-1/n} \left(c+d x^n\right)^{-1/n}",1,"x/((a + b*x^n)^n^(-1)*(c + d*x^n)^n^(-1))","A",1
592,1,46,45,0.4097966,"\int (h x)^{-1-n-n p} \left(a+b x^n\right)^p \left(c+d x^n\right)^p \left(a c-b d x^{2 n}\right) \, dx","Integrate[(h*x)^(-1 - n - n*p)*(a + b*x^n)^p*(c + d*x^n)^p*(a*c - b*d*x^(2*n)),x]","-\frac{(h x)^{n (-p)-n} \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{h n p+h n}","-\frac{(h x)^{-n (p+1)} \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{h n (p+1)}",1,"-(((h*x)^(-n - n*p)*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(h*n + h*n*p))","A",1
593,1,31,31,0.6010819,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^p \left(e+\frac{(b c+a d) e (1+n+n p) x^n}{a c}+\frac{b d e (1+2 n+2 n p) x^{2 n}}{a c}\right) \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*x^(2*n))/(a*c)),x]","\frac{e x \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{a c}","\frac{e x \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{a c}",1,"(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)","A",1
594,1,41,45,0.8883853,"\int (h x)^m \left(a+b x^n\right)^p \left(c+d x^n\right)^p \left(e+\frac{(b c+a d) e (1+m+n+n p) x^n}{a c (1+m)}+\frac{b d e (1+m+2 n+2 n p) x^{2 n}}{a c (1+m)}\right) \, dx","Integrate[(h*x)^m*(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + m + n + n*p)*x^n)/(a*c*(1 + m)) + (b*d*e*(1 + m + 2*n + 2*n*p)*x^(2*n))/(a*c*(1 + m))),x]","\frac{e x (h x)^m \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{a c (m+1)}","\frac{e (h x)^{m+1} \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{p+1}}{a c h (m+1)}",1,"(e*x*(h*x)^m*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c*(1 + m))","A",1