1,1,436,0,0.4075481,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 (c+d x)^{7/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{7 d^7}-\frac{2 (c+d x)^{5/2} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{9 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7}","\frac{2 (c+d x)^{7/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{7 d^7}-\frac{2 (c+d x)^{5/2} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{9 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7}",1,"(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(3/2))/(3*d^7) - (2*(b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*(c + d*x)^(5/2))/(5*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(9/2))/(9*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*d^7)","A",2,1,32,0.03125,1,"{1620}"
2,1,324,0,0.2720802,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Int[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 (c+d x)^{5/2} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{5 d^6}+\frac{2 (c+d x)^{7/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{7 d^6}+\frac{2 (c+d x)^{3/2} (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{3 d^6}+\frac{2 \sqrt{c+d x} (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6}+\frac{2 b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{9 d^6}+\frac{2 b^2 D (c+d x)^{11/2}}{11 d^6}","\frac{2 (c+d x)^{5/2} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{5 d^6}+\frac{2 (c+d x)^{7/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{7 d^6}+\frac{2 (c+d x)^{3/2} (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{3 d^6}+\frac{2 \sqrt{c+d x} (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6}+\frac{2 b (c+d x)^{9/2} (2 a d D-5 b c D+b C d)}{9 d^6}+\frac{2 b^2 D (c+d x)^{11/2}}{11 d^6}",1,"(2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^6 + (2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(3/2))/(3*d^6) + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(5/2))/(5*d^6) + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(7/2))/(7*d^6) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(9/2))/(9*d^6) + (2*b^2*D*(c + d*x)^(11/2))/(11*d^6)","A",2,1,32,0.03125,1,"{1620}"
3,1,212,0,0.1730423,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","-\frac{2 (c+d x)^{3/2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5}+\frac{2 (c+d x)^{5/2} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5}","-\frac{2 (c+d x)^{3/2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5}+\frac{2 (c+d x)^{5/2} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5}",1,"(-2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^5 - (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))*(c + d*x)^(3/2))/(3*d^5) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*(c + d*x)^(5/2))/(5*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(7/2))/(7*d^5) + (2*b*D*(c + d*x)^(9/2))/(9*d^5)","A",2,1,30,0.03333,1,"{1620}"
4,1,115,0,0.0723718,"\int \frac{A+B x+C x^2+D x^3}{\sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/Sqrt[c + d*x],x]","\frac{2 \sqrt{c+d x} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4}-\frac{2 (c+d x)^{3/2} \left(-B d^2-3 c^2 D+2 c C d\right)}{3 d^4}+\frac{2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac{2 D (c+d x)^{7/2}}{7 d^4}","\frac{2 \sqrt{c+d x} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4}-\frac{2 (c+d x)^{3/2} \left(-B d^2-3 c^2 D+2 c C d\right)}{3 d^4}+\frac{2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac{2 D (c+d x)^{7/2}}{7 d^4}",1,"(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^4 - (2*(2*c*C*d - B*d^2 - 3*c^2*D)*(c + d*x)^(3/2))/(3*d^4) + (2*(C*d - 3*c*D)*(c + d*x)^(5/2))/(5*d^4) + (2*D*(c + d*x)^(7/2))/(7*d^4)","A",2,1,25,0.04000,1,"{1850}"
5,1,188,0,0.2055106,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) \sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*Sqrt[c + d*x]),x]","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{7/2} \sqrt{b c-a d}}+\frac{2 \sqrt{c+d x} \left(a^2 d^2 D-a b d (C d-c D)+b^2 \left(-\left(-B d^2+c^2 (-D)+c C d\right)\right)\right)}{b^3 d^3}+\frac{2 (c+d x)^{3/2} (-a d D-2 b c D+b C d)}{3 b^2 d^3}+\frac{2 D (c+d x)^{5/2}}{5 b d^3}","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{7/2} \sqrt{b c-a d}}+\frac{2 \sqrt{c+d x} \left(a^2 d^2 D-a b d (C d-c D)+b^2 \left(-\left(-B d^2+c^2 (-D)+c C d\right)\right)\right)}{b^3 d^3}+\frac{2 (c+d x)^{3/2} (-a d D-2 b c D+b C d)}{3 b^2 d^3}+\frac{2 D (c+d x)^{5/2}}{5 b d^3}",1,"(2*(a^2*d^2*D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*Sqrt[c + d*x])/(b^3*d^3) + (2*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(3/2))/(3*b^2*d^3) + (2*D*(c + d*x)^(5/2))/(5*b*d^3) - (2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(7/2)*Sqrt[b*c - a*d])","A",4,3,32,0.09375,1,"{1620, 63, 208}"
6,1,201,0,0.5181933,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*Sqrt[c + d*x]),x]","-\frac{\sqrt{c+d x} \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{(a+b x) (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b (2 c D+C d)-5 a^3 d D-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right)}{b^{7/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x} (-2 a d D-b c D+b C d)}{b^3 d^2}+\frac{2 D (c+d x)^{3/2}}{3 b^2 d^2}","-\frac{\sqrt{c+d x} \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{(a+b x) (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b (2 c D+C d)-5 a^3 d D-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right)}{b^{7/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x} (-2 a d D-b c D+b C d)}{b^3 d^2}+\frac{2 D (c+d x)^{3/2}}{3 b^2 d^2}",1,"(2*(b*C*d - b*c*D - 2*a*d*D)*Sqrt[c + d*x])/(b^3*d^2) - ((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*Sqrt[c + d*x])/((b*c - a*d)*(a + b*x)) + (2*D*(c + d*x)^(3/2))/(3*b^2*d^2) - ((b^3*(2*B*c - A*d) - a*b^2*(4*c*C + B*d) - 5*a^3*d*D + 3*a^2*b*(C*d + 2*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(7/2)*(b*c - a*d)^(3/2))","A",5,4,32,0.1250,1,"{1621, 897, 1153, 208}"
7,1,279,0,0.7409556,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 \sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*Sqrt[c + d*x]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d (12 c D+C d)-15 a^3 d^2 D-a b^2 \left(-B d^2+24 c^2 D+8 c C d\right)+b^3 \left(3 A d^2-4 B c d+8 c^2 C\right)\right)}{4 b^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x} \left(a^2 b (12 c D+5 C d)-9 a^3 d D-a b^2 (B d+8 c C)+b^3 (4 B c-3 A d)\right)}{4 b^3 (a+b x) (b c-a d)^2}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b^3 d}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d (12 c D+C d)-15 a^3 d^2 D-a b^2 \left(-B d^2+24 c^2 D+8 c C d\right)+b^3 \left(3 A d^2-4 B c d+8 c^2 C\right)\right)}{4 b^{7/2} (b c-a d)^{5/2}}-\frac{\sqrt{c+d x} \left(a^2 b (12 c D+5 C d)-9 a^3 d D-a b^2 (B d+8 c C)+b^3 (4 B c-3 A d)\right)}{4 b^3 (a+b x) (b c-a d)^2}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b^3 d}",1,"(2*D*Sqrt[c + d*x])/(b^3*d) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(2*b^3*(b*c - a*d)*(a + b*x)^2) - ((b^3*(4*B*c - 3*A*d) - a*b^2*(8*c*C + B*d) - 9*a^3*d*D + a^2*b*(5*C*d + 12*c*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)^2*(a + b*x)) - ((b^3*(8*c^2*C - 4*B*c*d + 3*A*d^2) - 15*a^3*d^2*D + 3*a^2*b*d*(C*d + 12*c*D) - a*b^2*(8*c*C*d - B*d^2 + 24*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(7/2)*(b*c - a*d)^(5/2))","A",5,5,32,0.1562,1,"{1621, 897, 1157, 388, 208}"
8,1,375,0,0.8212677,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*Sqrt[c + d*x]),x]","-\frac{\sqrt{c+d x} \left(a^2 b d (30 c D+C d)-11 a^3 d^2 D-a b^2 \left(-B d^2+24 c^2 D+4 c C d\right)+b^3 \left(5 A d^2-6 B c d+8 c^2 C\right)\right)}{8 b^3 (a+b x) (b c-a d)^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b d^2 (C d-18 c D)+5 a^3 d^3 D-a b^2 d \left(-B d^2-24 c^2 D+4 c C d\right)+b^3 \left(5 A d^3-6 B c d^2+8 c^2 C d-16 c^3 D\right)\right)}{8 b^{7/2} (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{3 b^3 (a+b x)^3 (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (18 c D+7 C d)-13 a^3 d D-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right)}{12 b^3 (a+b x)^2 (b c-a d)^2}","-\frac{\sqrt{c+d x} \left(a^2 b d (30 c D+C d)-11 a^3 d^2 D-a b^2 \left(-B d^2+24 c^2 D+4 c C d\right)+b^3 \left(5 A d^2-6 B c d+8 c^2 C\right)\right)}{8 b^3 (a+b x) (b c-a d)^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b d^2 (C d-18 c D)+5 a^3 d^3 D-a b^2 d \left(-B d^2-24 c^2 D+4 c C d\right)+b^3 \left(5 A d^3-6 B c d^2+8 c^2 C d-16 c^3 D\right)\right)}{8 b^{7/2} (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{3 b^3 (a+b x)^3 (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (18 c D+7 C d)-13 a^3 d D-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right)}{12 b^3 (a+b x)^2 (b c-a d)^2}",1,"-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(3*b^3*(b*c - a*d)*(a + b*x)^3) - ((b^3*(6*B*c - 5*A*d) - a*b^2*(12*c*C + B*d) - 13*a^3*d*D + a^2*b*(7*C*d + 18*c*D))*Sqrt[c + d*x])/(12*b^3*(b*c - a*d)^2*(a + b*x)^2) - ((b^3*(8*c^2*C - 6*B*c*d + 5*A*d^2) - 11*a^3*d^2*D + a^2*b*d*(C*d + 30*c*D) - a*b^2*(4*c*C*d - B*d^2 + 24*c^2*D))*Sqrt[c + d*x])/(8*b^3*(b*c - a*d)^3*(a + b*x)) + ((5*a^3*d^3*D + a^2*b*d^2*(C*d - 18*c*D) - a*b^2*d*(4*c*C*d - B*d^2 - 24*c^2*D) + b^3*(8*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 16*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*b^(7/2)*(b*c - a*d)^(7/2))","A",5,5,32,0.1562,1,"{1621, 897, 1157, 385, 208}"
9,1,495,0,1.0091634,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt{c+d x}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^5*Sqrt[c + d*x]),x]","\frac{\sqrt{c+d x} \left(3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left(-5 B d^2-48 c^2 D+16 c C d\right)+b^3 \left(35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right)\right)}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left(3 a^2 b d (56 c D+C d)-59 a^3 d^2 D-a b^2 \left(-5 B d^2+144 c^2 D+16 c C d\right)+b^3 \left(35 A d^2-40 B c d+48 c^2 C\right)\right)}{96 b^3 (a+b x)^2 (b c-a d)^3}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left(-5 B d^2-48 c^2 D+16 c C d\right)+b^3 \left(35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right)\right)}{64 b^{7/2} (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left(3 a^2 b (8 c D+3 C d)-17 a^3 d D-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right)}{24 b^3 (a+b x)^3 (b c-a d)^2}","\frac{\sqrt{c+d x} \left(3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left(-5 B d^2-48 c^2 D+16 c C d\right)+b^3 \left(35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right)\right)}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left(3 a^2 b d (56 c D+C d)-59 a^3 d^2 D-a b^2 \left(-5 B d^2+144 c^2 D+16 c C d\right)+b^3 \left(35 A d^2-40 B c d+48 c^2 C\right)\right)}{96 b^3 (a+b x)^2 (b c-a d)^3}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d^2 (C d-8 c D)+5 a^3 d^3 D-a b^2 d \left(-5 B d^2-48 c^2 D+16 c C d\right)+b^3 \left(35 A d^3-40 B c d^2+48 c^2 C d-64 c^3 D\right)\right)}{64 b^{7/2} (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{4 b^3 (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x} \left(3 a^2 b (8 c D+3 C d)-17 a^3 d D-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right)}{24 b^3 (a+b x)^3 (b c-a d)^2}",1,"-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(4*b^3*(b*c - a*d)*(a + b*x)^4) - ((b^3*(8*B*c - 7*A*d) - a*b^2*(16*c*C + B*d) - 17*a^3*d*D + 3*a^2*b*(3*C*d + 8*c*D))*Sqrt[c + d*x])/(24*b^3*(b*c - a*d)^2*(a + b*x)^3) - ((b^3*(48*c^2*C - 40*B*c*d + 35*A*d^2) - 59*a^3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C*d - 5*B*d^2 + 144*c^2*D))*Sqrt[c + d*x])/(96*b^3*(b*c - a*d)^3*(a + b*x)^2) + ((5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*Sqrt[c + d*x])/(64*b^3*(b*c - a*d)^4*(a + b*x)) - (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(64*b^(7/2)*(b*c - a*d)^(9/2))","A",6,6,32,0.1875,1,"{1621, 897, 1157, 385, 199, 208}"
10,1,434,0,0.3589114,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 (c+d x)^{5/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{5 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{3 d^7}+\frac{2 b (c+d x)^{7/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{7 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7}+\frac{2 (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac{2 b^3 D (c+d x)^{11/2}}{11 d^7}","\frac{2 (c+d x)^{5/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{5 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{3 d^7}+\frac{2 b (c+d x)^{7/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{7 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7}+\frac{2 (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac{2 b^3 D (c+d x)^{11/2}}{11 d^7}",1,"(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*Sqrt[c + d*x]) - (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*Sqrt[c + d*x])/d^7 - (2*(b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*(c + d*x)^(3/2))/(3*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(5/2))/(5*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(9/2))/(9*d^7) + (2*b^3*D*(c + d*x)^(11/2))/(11*d^7)","A",2,1,32,0.03125,1,"{1620}"
11,1,322,0,0.2585154,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Int[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 (c+d x)^{3/2} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{3 d^6}+\frac{2 (c+d x)^{5/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{5 d^6}+\frac{2 \sqrt{c+d x} (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6}-\frac{2 (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6 \sqrt{c+d x}}+\frac{2 b (c+d x)^{7/2} (2 a d D-5 b c D+b C d)}{7 d^6}+\frac{2 b^2 D (c+d x)^{9/2}}{9 d^6}","\frac{2 (c+d x)^{3/2} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{3 d^6}+\frac{2 (c+d x)^{5/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{5 d^6}+\frac{2 \sqrt{c+d x} (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6}-\frac{2 (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6 \sqrt{c+d x}}+\frac{2 b (c+d x)^{7/2} (2 a d D-5 b c D+b C d)}{7 d^6}+\frac{2 b^2 D (c+d x)^{9/2}}{9 d^6}",1,"(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^6*Sqrt[c + d*x]) + (2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*Sqrt[c + d*x])/d^6 + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(3/2))/(3*d^6) + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(5/2))/(5*d^6) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(7/2))/(7*d^6) + (2*b^2*D*(c + d*x)^(9/2))/(9*d^6)","A",2,1,32,0.03125,1,"{1620}"
12,1,210,0,0.1662339,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","-\frac{2 \sqrt{c+d x} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5}","-\frac{2 \sqrt{c+d x} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5}",1,"(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^5*Sqrt[c + d*x]) - (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))*Sqrt[c + d*x])/d^5 + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*(c + d*x)^(3/2))/(3*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(5/2))/(5*d^5) + (2*b*D*(c + d*x)^(7/2))/(7*d^5)","A",2,1,30,0.03333,1,"{1620}"
13,1,113,0,0.0810817,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{3/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(3/2),x]","-\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4}","-\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4}",1,"(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^4*Sqrt[c + d*x]) - (2*(2*c*C*d - B*d^2 - 3*c^2*D)*Sqrt[c + d*x])/d^4 + (2*(C*d - 3*c*D)*(c + d*x)^(3/2))/(3*d^4) + (2*D*(c + d*x)^(5/2))/(5*d^4)","A",2,1,25,0.04000,1,"{1850}"
14,1,193,0,0.2401374,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{3/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(3/2)),x]","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}+\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^3 \sqrt{c+d x} (b c-a d)}+\frac{2 \sqrt{c+d x} (-a d D-b c D+b C d)}{b^2 d^3}+\frac{2 D (c+d x)^{3/2}}{3 b d^3}-\frac{2 c D \sqrt{c+d x}}{b d^3}","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}+\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^3 \sqrt{c+d x} (b c-a d)}+\frac{2 \sqrt{c+d x} (-a d D-b c D+b C d)}{b^2 d^3}+\frac{2 D (c+d x)^{3/2}}{3 b d^3}-\frac{2 c D \sqrt{c+d x}}{b d^3}",1,"(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^3*(b*c - a*d)*Sqrt[c + d*x]) - (2*c*D*Sqrt[c + d*x])/(b*d^3) + (2*(b*C*d - b*c*D - a*d*D)*Sqrt[c + d*x])/(b^2*d^3) + (2*D*(c + d*x)^(3/2))/(3*b*d^3) - (2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3/2))","A",6,4,32,0.1250,1,"{1619, 43, 63, 208}"
15,1,253,0,0.6013816,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(3/2)),x]","\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(3 A d^3-2 B c d^2+2 c^2 C d-2 c^3 D\right)\right)}{b^3 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}}{(a+b x) \sqrt{c+d x} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b (6 c D+C d)-3 a^3 d D-a b^2 (4 c C-B d)+b^3 (2 B c-3 A d)\right)}{b^{5/2} (b c-a d)^{5/2}}+\frac{2 D \sqrt{c+d x}}{b^2 d^2}","\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(3 A d^3-2 B c d^2+2 c^2 C d-2 c^3 D\right)\right)}{b^3 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}}{(a+b x) \sqrt{c+d x} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b (6 c D+C d)-3 a^3 d D-a b^2 (4 c C-B d)+b^3 (2 B c-3 A d)\right)}{b^{5/2} (b c-a d)^{5/2}}+\frac{2 D \sqrt{c+d x}}{b^2 d^2}",1,"(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(2*c^2*C*d - 2*B*c*d^2 + 3*A*d^3 - 2*c^3*D))/(b^3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (A - (a*(b^2*B - a*b*C + a^2*D))/b^3)/((b*c - a*d)*(a + b*x)*Sqrt[c + d*x]) + (2*D*Sqrt[c + d*x])/(b^2*d^2) - ((b^3*(2*B*c - 3*A*d) - a*b^2*(4*c*C - B*d) - 3*a^3*d*D + a^2*b*(C*d + 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(5/2))","A",5,4,32,0.1250,1,"{1621, 897, 1261, 208}"
16,1,350,0,0.8323855,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{3/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(3/2)),x]","-\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(5 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right)\right)}{2 b^3 d \sqrt{c+d x} (b c-a d)^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-a^2 b d (C d-12 c D)-3 a^3 d^2 D+a b^2 \left(-3 B d^2-24 c^2 D+8 c C d\right)+b^3 \left(15 A d^2-12 B c d+8 c^2 C\right)\right)}{4 b^{5/2} (b c-a d)^{7/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{2 b^3 (a+b x)^2 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left(3 a^2 b (4 c D+C d)-7 a^3 d D-a b^2 (8 c C-B d)+b^3 (4 B c-5 A d)\right)}{4 b^2 (a+b x) (b c-a d)^3}","-\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(5 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right)\right)}{2 b^3 d \sqrt{c+d x} (b c-a d)^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-a^2 b d (C d-12 c D)-3 a^3 d^2 D+a b^2 \left(-3 B d^2-24 c^2 D+8 c C d\right)+b^3 \left(15 A d^2-12 B c d+8 c^2 C\right)\right)}{4 b^{5/2} (b c-a d)^{7/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{2 b^3 (a+b x)^2 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left(3 a^2 b (4 c D+C d)-7 a^3 d D-a b^2 (8 c C-B d)+b^3 (4 B c-5 A d)\right)}{4 b^2 (a+b x) (b c-a d)^3}",1,"-(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 5*A*d^3 - 4*c^3*D))/(2*b^3*d*(b*c - a*d)^3*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*Sqrt[c + d*x]) - ((b^3*(4*B*c - 5*A*d) - a*b^2*(8*c*C - B*d) - 7*a^3*d*D + 3*a^2*b*(C*d + 4*c*D))*Sqrt[c + d*x])/(4*b^2*(b*c - a*d)^3*(a + b*x)) - ((b^3*(8*c^2*C - 12*B*c*d + 15*A*d^2) - 3*a^3*d^2*D - a^2*b*d*(C*d - 12*c*D) + a*b^2*(8*c*C*d - 3*B*d^2 - 24*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(5/2)*(b*c - a*d)^(7/2))","A",5,5,32,0.1562,1,"{1621, 897, 1259, 453, 208}"
17,1,463,0,1.1499908,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*(c + d*x)^(3/2)),x]","-\frac{\sqrt{c+d x} \left(-a^2 b d (11 C d-18 c D)+5 a^3 d^2 D+a b^2 \left(-7 B d^2-72 c^2 D+36 c C d\right)+b^3 \left(49 A d^2-42 B c d+24 c^2 C\right)\right)}{24 b^2 (a+b x) (b c-a d)^4}+\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(7 A d^3-6 B c d^2+6 c^2 C d-6 c^3 D\right)\right)}{3 b^3 \sqrt{c+d x} (b c-a d)^4}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b d^2 (C d-6 c D)+a^3 d^3 D-a b^2 d \left(-5 B d^2-24 c^2 D+12 c C d\right)+b^3 \left(-\left(35 A d^3-30 B c d^2+24 c^2 C d-16 c^3 D\right)\right)\right)}{8 b^{5/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{3 b^3 (a+b x)^3 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (18 c D+5 C d)-11 a^3 d D-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right)}{12 b^2 (a+b x)^2 (b c-a d)^3}","-\frac{\sqrt{c+d x} \left(-a^2 b d (11 C d-18 c D)+5 a^3 d^2 D+a b^2 \left(-7 B d^2-72 c^2 D+36 c C d\right)+b^3 \left(49 A d^2-42 B c d+24 c^2 C\right)\right)}{24 b^2 (a+b x) (b c-a d)^4}+\frac{-a^2 b C d^3+a^3 d^3 D+a b^2 B d^3+b^3 \left(-\left(7 A d^3-6 B c d^2+6 c^2 C d-6 c^3 D\right)\right)}{3 b^3 \sqrt{c+d x} (b c-a d)^4}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^2 b d^2 (C d-6 c D)+a^3 d^3 D-a b^2 d \left(-5 B d^2-24 c^2 D+12 c C d\right)+b^3 \left(-\left(35 A d^3-30 B c d^2+24 c^2 C d-16 c^3 D\right)\right)\right)}{8 b^{5/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{3 b^3 (a+b x)^3 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (18 c D+5 C d)-11 a^3 d D-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right)}{12 b^2 (a+b x)^2 (b c-a d)^3}",1,"(a*b^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(6*c^2*C*d - 6*B*c*d^2 + 7*A*d^3 - 6*c^3*D))/(3*b^3*(b*c - a*d)^4*Sqrt[c + d*x]) - (A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(3*b^3*(b*c - a*d)*(a + b*x)^3*Sqrt[c + d*x]) - ((b^3*(6*B*c - 7*A*d) - a*b^2*(12*c*C - B*d) - 11*a^3*d*D + a^2*b*(5*C*d + 18*c*D))*Sqrt[c + d*x])/(12*b^2*(b*c - a*d)^3*(a + b*x)^2) - ((b^3*(24*c^2*C - 42*B*c*d + 49*A*d^2) + 5*a^3*d^2*D - a^2*b*d*(11*C*d - 18*c*D) + a*b^2*(36*c*C*d - 7*B*d^2 - 72*c^2*D))*Sqrt[c + d*x])/(24*b^2*(b*c - a*d)^4*(a + b*x)) - ((a^3*d^3*D + a^2*b*d^2*(C*d - 6*c*D) - a*b^2*d*(12*c*C*d - 5*B*d^2 - 24*c^2*D) - b^3*(24*c^2*C*d - 30*B*c*d^2 + 35*A*d^3 - 16*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(8*b^(5/2)*(b*c - a*d)^(9/2))","A",6,6,32,0.1875,1,"{1621, 897, 1259, 456, 453, 208}"
18,1,434,0,0.3525484,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 (c+d x)^{3/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{d^7}+\frac{2 b (c+d x)^{5/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{5 d^7}+\frac{2 (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7 \sqrt{c+d x}}+\frac{2 (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^7 (c+d x)^{3/2}}+\frac{2 b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7}","\frac{2 (c+d x)^{3/2} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d) \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{d^7}+\frac{2 b (c+d x)^{5/2} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{5 d^7}+\frac{2 (b c-a d)^2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7 \sqrt{c+d x}}+\frac{2 (b c-a d)^3 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^7 (c+d x)^{3/2}}+\frac{2 b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7}",1,"(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x)^(3/2)) + (2*(b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D)))/(d^7*Sqrt[c + d*x]) - (2*(b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*Sqrt[c + d*x])/d^7 + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(3/2))/(3*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(5/2))/(5*d^7) + (2*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(7/2))/(7*d^7) + (2*b^3*D*(c + d*x)^(9/2))/(9*d^7)","A",2,1,32,0.03125,1,"{1620}"
19,1,322,0,0.2425888,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Int[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \sqrt{c+d x} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{d^6}+\frac{2 (c+d x)^{3/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{3 d^6}-\frac{2 (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6 \sqrt{c+d x}}-\frac{2 (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^6 (c+d x)^{3/2}}+\frac{2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6}","\frac{2 \sqrt{c+d x} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{d^6}+\frac{2 (c+d x)^{3/2} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{3 d^6}-\frac{2 (b c-a d) \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6 \sqrt{c+d x}}-\frac{2 (b c-a d)^2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^6 (c+d x)^{3/2}}+\frac{2 b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{5 d^6}+\frac{2 b^2 D (c+d x)^{7/2}}{7 d^6}",1,"(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^6*(c + d*x)^(3/2)) - (2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D)))/(d^6*Sqrt[c + d*x]) + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*Sqrt[c + d*x])/d^6 + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5/2))/(5*d^6) + (2*b^2*D*(c + d*x)^(7/2))/(7*d^6)","A",2,1,32,0.03125,1,"{1620}"
20,1,210,0,0.172371,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5}","\frac{2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5}",1,"(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^5*(c + d*x)^(3/2)) + (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D)))/(d^5*Sqrt[c + d*x]) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*Sqrt[c + d*x])/d^5 + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(3/2))/(3*d^5) + (2*b*D*(c + d*x)^(5/2))/(5*d^5)","A",2,1,30,0.03333,1,"{1620}"
21,1,113,0,0.0721717,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{5/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(5/2),x]","-\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^4 (c+d x)^{3/2}}+\frac{2 \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4 \sqrt{c+d x}}+\frac{2 \sqrt{c+d x} (C d-3 c D)}{d^4}+\frac{2 D (c+d x)^{3/2}}{3 d^4}","-\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^4 (c+d x)^{3/2}}+\frac{2 \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4 \sqrt{c+d x}}+\frac{2 \sqrt{c+d x} (C d-3 c D)}{d^4}+\frac{2 D (c+d x)^{3/2}}{3 d^4}",1,"(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^4*(c + d*x)^(3/2)) + (2*(2*c*C*d - B*d^2 - 3*c^2*D))/(d^4*Sqrt[c + d*x]) + (2*(C*d - 3*c*D)*Sqrt[c + d*x])/d^4 + (2*D*(c + d*x)^(3/2))/(3*d^4)","A",2,1,25,0.04000,1,"{1850}"
22,1,210,0,0.3130433,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{5/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(5/2)),x]","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{3/2} (b c-a d)^{5/2}}+\frac{2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(-A d^3+c^2 C d-2 c^3 D\right)\right)}{d^3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^3 (c+d x)^{3/2} (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b d^3}","-\frac{2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{b^{3/2} (b c-a d)^{5/2}}+\frac{2 \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(-A d^3+c^2 C d-2 c^3 D\right)\right)}{d^3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{3 d^3 (c+d x)^{3/2} (b c-a d)}+\frac{2 D \sqrt{c+d x}}{b d^3}",1,"(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^3*(b*c - a*d)*(c + d*x)^(3/2)) + (2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(c^2*C*d - A*d^3 - 2*c^3*D)))/(d^3*(b*c - a*d)^2*Sqrt[c + d*x]) + (2*D*Sqrt[c + d*x])/(b*d^3) - (2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(5/2))","A",4,3,32,0.09375,1,"{1619, 63, 208}"
23,1,336,0,0.7911749,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(5/2)),x]","-\frac{a^2 b C d^3-a^3 d^3 D+a b^2 d \left(-3 B d^2-6 c^2 D+4 c C d\right)+b^3 \left(-\left(-5 A d^3+2 B c d^2-2 c^3 D\right)\right)}{b^2 d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left(-\left(5 A d^3-2 B c d^2+2 c^2 C d-2 c^3 D\right)\right)}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-a^2 b (C d-6 c D)+a^3 (-d) D-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right)}{b^{3/2} (b c-a d)^{7/2}}","-\frac{a^2 b C d^3-a^3 d^3 D+a b^2 d \left(-3 B d^2-6 c^2 D+4 c C d\right)+b^3 \left(-\left(-5 A d^3+2 B c d^2-2 c^3 D\right)\right)}{b^2 d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left(-\left(5 A d^3-2 B c d^2+2 c^2 C d-2 c^3 D\right)\right)}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-a^2 b (C d-6 c D)+a^3 (-d) D-a b^2 (4 c C-3 B d)+b^3 (2 B c-5 A d)\right)}{b^{3/2} (b c-a d)^{7/2}}",1,"(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d^3*D - b^3*(2*c^2*C*d - 2*B*c*d^2 + 5*A*d^3 - 2*c^3*D))/(3*b^3*d^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (A - (a*(b^2*B - a*b*C + a^2*D))/b^3)/((b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - (a^2*b*C*d^3 - a^3*d^3*D + a*b^2*d*(4*c*C*d - 3*B*d^2 - 6*c^2*D) - b^3*(2*B*c*d^2 - 5*A*d^3 - 2*c^3*D))/(b^2*d^2*(b*c - a*d)^3*Sqrt[c + d*x]) - ((b^3*(2*B*c - 5*A*d) - a*b^2*(4*c*C - 3*B*d) - a^3*d*D - a^2*b*(C*d - 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(7/2))","A",5,4,32,0.1250,1,"{1621, 897, 1261, 208}"
24,1,438,0,1.213876,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx","Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(5/2)),x]","\frac{a^2 b C d^2+a^3 \left(-d^2\right) D+a b^2 \left(-3 B d^2-6 c^2 D+4 c C d\right)+b^3 \left(7 A d^2-4 B c d+2 c^2 C\right)}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left(-\left(7 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right)\right)}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d (C d-4 c D)+a^3 d^2 D+3 a b^2 \left(-5 B d^2-8 c^2 D+8 c C d\right)+b^3 \left(35 A d^2-20 B c d+8 c^2 C\right)\right)}{4 b^{3/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (12 c D+C d)-5 a^3 d D-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right)}{4 b (a+b x) (b c-a d)^4}","\frac{a^2 b C d^2+a^3 \left(-d^2\right) D+a b^2 \left(-3 B d^2-6 c^2 D+4 c C d\right)+b^3 \left(7 A d^2-4 B c d+2 c^2 C\right)}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left(-\left(7 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right)\right)}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(3 a^2 b d (C d-4 c D)+a^3 d^2 D+3 a b^2 \left(-5 B d^2-8 c^2 D+8 c C d\right)+b^3 \left(35 A d^2-20 B c d+8 c^2 C\right)\right)}{4 b^{3/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left(a^2 D-a b C+b^2 B\right)}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{c+d x} \left(a^2 b (12 c D+C d)-5 a^3 d D-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right)}{4 b (a+b x) (b c-a d)^4}",1,"-(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 7*A*d^3 - 4*c^3*D))/(6*b^3*d*(b*c - a*d)^3*(c + d*x)^(3/2)) - (A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(2*b^3*(b*c - a*d)*(a + b*x)^2*(c + d*x)^(3/2)) + (a^2*b*C*d^2 + b^3*(2*c^2*C - 4*B*c*d + 7*A*d^2) - a^3*d^2*D + a*b^2*(4*c*C*d - 3*B*d^2 - 6*c^2*D))/(b^2*(b*c - a*d)^4*Sqrt[c + d*x]) - ((b^3*(4*B*c - 7*A*d) - a*b^2*(8*c*C - 3*B*d) - 5*a^3*d*D + a^2*b*(C*d + 12*c*D))*Sqrt[c + d*x])/(4*b*(b*c - a*d)^4*(a + b*x)) - ((b^3*(8*c^2*C - 20*B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4*c*D) + 3*a*b^2*(8*c*C*d - 5*B*d^2 - 8*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(9/2))","A",6,5,32,0.1562,1,"{1621, 897, 1259, 1261, 208}"
25,1,455,0,0.3379235,"\int (a+b x)^3 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Int[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","-\frac{(b c-a d) (c+d x)^{n+3} \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{d^7 (n+3)}+\frac{(c+d x)^{n+4} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{d^7 (n+4)}+\frac{b (c+d x)^{n+5} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{d^7 (n+5)}-\frac{(b c-a d)^3 (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7 (n+1)}-\frac{(b c-a d)^2 (c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7 (n+2)}+\frac{b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac{b^3 D (c+d x)^{n+7}}{d^7 (n+7)}","-\frac{(b c-a d) (c+d x)^{n+3} \left(a^2 d^2 (C d-3 c D)-a b d \left(-3 B d^2-15 c^2 D+8 c C d\right)+b^2 \left(3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right)\right)}{d^7 (n+3)}+\frac{(c+d x)^{n+4} \left(3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left(-B d^2-10 c^2 D+4 c C d\right)+b^3 \left(A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right)\right)}{d^7 (n+4)}+\frac{b (c+d x)^{n+5} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(-\left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{d^7 (n+5)}-\frac{(b c-a d)^3 (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^7 (n+1)}-\frac{(b c-a d)^2 (c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right)\right)}{d^7 (n+2)}+\frac{b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac{b^3 D (c+d x)^{n+7}}{d^7 (n+7)}",1,"-(((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^7*(1 + n))) - ((b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(2 + n))/(d^7*(2 + n)) - ((b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*(c + d*x)^(3 + n))/(d^7*(3 + n)) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(4 + n))/(d^7*(4 + n)) + (b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(5 + n))/(d^7*(5 + n)) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*D*(c + d*x)^(7 + n))/(d^7*(7 + n))","A",2,1,30,0.03333,1,"{1620}"
26,1,338,0,0.2465979,"\int (a+b x)^2 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Int[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+3} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)}","\frac{(c+d x)^{n+3} \left(a^2 d^2 (C d-3 c D)-2 a b d \left(-B d^2-6 c^2 D+3 c C d\right)+b^2 \left(A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right)\right)}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(-\left(-B d^2-10 c^2 D+4 c C d\right)\right)\right)}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right)\right)}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)}",1,"((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6*(1 + n)) + ((b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^2*D*(c + d*x)^(6 + n))/(d^6*(6 + n))","A",2,1,30,0.03333,1,"{1620}"
27,1,226,0,0.1611385,"\int (a+b x) (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Int[(a + b*x)*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","-\frac{(b c-a d) (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5 (n+1)}-\frac{(c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5 (n+2)}+\frac{(c+d x)^{n+3} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{d^5 (n+3)}+\frac{(c+d x)^{n+4} (a d D-4 b c D+b C d)}{d^5 (n+4)}+\frac{b D (c+d x)^{n+5}}{d^5 (n+5)}","-\frac{(b c-a d) (c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^5 (n+1)}-\frac{(c+d x)^{n+2} \left(a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(A d^3-2 B c d^2+3 c^2 C d-4 c^3 D\right)\right)}{d^5 (n+2)}+\frac{(c+d x)^{n+3} \left(a d (C d-3 c D)-b \left(-B d^2-6 c^2 D+3 c C d\right)\right)}{d^5 (n+3)}+\frac{(c+d x)^{n+4} (a d D-4 b c D+b C d)}{d^5 (n+4)}+\frac{b D (c+d x)^{n+5}}{d^5 (n+5)}",1,"-(((b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^5*(1 + n))) - ((a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))*(c + d*x)^(2 + n))/(d^5*(2 + n)) + ((a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*(c + d*x)^(3 + n))/(d^5*(3 + n)) + ((b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(4 + n))/(d^5*(4 + n)) + (b*D*(c + d*x)^(5 + n))/(d^5*(5 + n))","A",2,1,28,0.03571,1,"{1620}"
28,1,126,0,0.0718049,"\int (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Int[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)}","\frac{(c+d x)^{n+1} \left(A d^3-B c d^2+c^2 C d+c^3 (-D)\right)}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left(-B d^2-3 c^2 D+2 c C d\right)}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)}",1,"((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^4*(1 + n)) - ((2*c*C*d - B*d^2 - 3*c^2*D)*(c + d*x)^(2 + n))/(d^4*(2 + n)) + ((C*d - 3*c*D)*(c + d*x)^(3 + n))/(d^4*(3 + n)) + (D*(c + d*x)^(4 + n))/(d^4*(4 + n))","A",2,1,23,0.04348,1,"{1850}"
29,1,203,0,0.1795607,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{a+b x} \, dx","Int[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]","-\frac{(c+d x)^{n+1} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{b^3 (n+1) (b c-a d)}+\frac{(c+d x)^{n+1} \left(a^2 d^2 D-a b d (C d-c D)+b^2 \left(-\left(-B d^2+c^2 (-D)+c C d\right)\right)\right)}{b^3 d^3 (n+1)}+\frac{(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac{D (c+d x)^{n+3}}{b d^3 (n+3)}","-\frac{(c+d x)^{n+1} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{b^3 (n+1) (b c-a d)}+\frac{(c+d x)^{n+1} \left(a^2 d^2 D-a b d (C d-c D)+b^2 \left(-\left(-B d^2+c^2 (-D)+c C d\right)\right)\right)}{b^3 d^3 (n+1)}+\frac{(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac{D (c+d x)^{n+3}}{b d^3 (n+3)}",1,"((a^2*d^2*D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*(c + d*x)^(1 + n))/(b^3*d^3*(1 + n)) + ((b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(2 + n))/(b^2*d^3*(2 + n)) + (D*(c + d*x)^(3 + n))/(b*d^3*(3 + n)) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b^3*(b*c - a*d)*(1 + n))","A",3,2,30,0.06667,1,"{1620, 68}"
30,1,220,0,0.5237266,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^2} \, dx","Int[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]","\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right) \left(-a^2 b (3 c D+C d (n+2))+a^3 d D (n+3)+a b^2 (B d (n+1)+2 c C)-b^3 (A d n+B c)\right)}{b^3 (n+1) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{(a+b x) (b c-a d)}+\frac{(c+d x)^{n+1} (-2 a d D-b c D+b C d)}{b^3 d^2 (n+1)}+\frac{D (c+d x)^{n+2}}{b^2 d^2 (n+2)}","\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right) \left(-a^2 b (3 c D+C d (n+2))+a^3 d D (n+3)+a b^2 (B d (n+1)+2 c C)-b^3 (A d n+B c)\right)}{b^3 (n+1) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left(A-\frac{a \left(a^2 D-a b C+b^2 B\right)}{b^3}\right)}{(a+b x) (b c-a d)}+\frac{(c+d x)^{n+1} (-2 a d D-b c D+b C d)}{b^3 d^2 (n+1)}+\frac{D (c+d x)^{n+2}}{b^2 d^2 (n+2)}",1,"((b*C*d - b*c*D - 2*a*d*D)*(c + d*x)^(1 + n))/(b^3*d^2*(1 + n)) - ((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*(c + d*x)^(1 + n))/((b*c - a*d)*(a + b*x)) + (D*(c + d*x)^(2 + n))/(b^2*d^2*(2 + n)) + ((a^3*d*D*(3 + n) - b^3*(B*c + A*d*n) + a*b^2*(2*c*C + B*d*(1 + n)) - a^2*b*(3*c*D + C*d*(2 + n)))*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b^3*(b*c - a*d)^2*(1 + n))","A",4,4,30,0.1333,1,"{1621, 951, 80, 68}"
31,1,329,0,0.6189601,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^3} \, dx","Int[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^3,x]","-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right) \left(a^2 b d (n+2) (6 c D+C d (n+1))+a^3 \left(-d^2\right) D \left(n^2+5 n+6\right)-a b^2 \left(B d^2 n (n+1)+6 c^2 D+4 c C d (n+1)\right)+b^3 \left(-A d^2 (1-n) n+2 B c d n+2 c^2 C\right)\right)}{2 b^3 (n+1) (b c-a d)^3}-\frac{(c+d x)^{n+1} \left(a^2 b (6 c D+C d (n+3))+a^3 (-d) D (n+5)-a b^2 (B d (n+1)+4 c C)+b^3 (2 B c-A d (1-n))\right)}{2 b^3 (a+b x) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{D (c+d x)^{n+1}}{b^3 d (n+1)}","-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right) \left(a^2 b d (n+2) (6 c D+C d (n+1))+a^3 \left(-d^2\right) D \left(n^2+5 n+6\right)-a b^2 \left(B d^2 n (n+1)+6 c^2 D+4 c C d (n+1)\right)+b^3 \left(-A d^2 (1-n) n+2 B c d n+2 c^2 C\right)\right)}{2 b^3 (n+1) (b c-a d)^3}-\frac{(c+d x)^{n+1} \left(a^2 b (6 c D+C d (n+3))+a^3 (-d) D (n+5)-a b^2 (B d (n+1)+4 c C)+b^3 (2 B c-A d (1-n))\right)}{2 b^3 (a+b x) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{D (c+d x)^{n+1}}{b^3 d (n+1)}",1,"(D*(c + d*x)^(1 + n))/(b^3*d*(1 + n)) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(1 + n))/(2*b^3*(b*c - a*d)*(a + b*x)^2) - ((b^3*(2*B*c - A*d*(1 - n)) - a^3*d*D*(5 + n) - a*b^2*(4*c*C + B*d*(1 + n)) + a^2*b*(6*c*D + C*d*(3 + n)))*(c + d*x)^(1 + n))/(2*b^3*(b*c - a*d)^2*(a + b*x)) - ((b^3*(2*c^2*C + 2*B*c*d*n - A*d^2*(1 - n)*n) - a^3*d^2*D*(6 + 5*n + n^2) + a^2*b*d*(2 + n)*(6*c*D + C*d*(1 + n)) - a*b^2*(6*c^2*D + 4*c*C*d*(1 + n) + B*d^2*n*(1 + n)))*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(2*b^3*(b*c - a*d)^3*(1 + n))","A",4,4,30,0.1333,1,"{1621, 949, 80, 68}"
32,1,141,0,0.0890701,"\int (a+b x)^m (A+B x) (c+d x)^n \, dx","Int[(a + b*x)^m*(A + B*x)*(c + d*x)^n,x]","\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right)}{b^2 d (m+1) (m+n+2)}+\frac{B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}","\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right)}{b^2 d (m+1) (m+n+2)}+\frac{B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}",1,"(B*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((A*b*d*(2 + m + n) - B*(b*c*(1 + m) + a*d*(1 + n)))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^2*d*(1 + m)*(2 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)","A",3,3,20,0.1500,1,"{80, 70, 69}"
33,1,266,0,0.3094898,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2\right) \, dx","Int[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2),x]","-\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right) \left(d (m+n+2) \left(a^2 C d (n+1)+a b c C (m+2)-A b^2 d (m+n+3)\right)-(a d (n+1)+b c (m+1)) (a C d (m+2 n+4)-b B d (m+n+3)+b c C (m+2))\right)}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a C d (m+2 n+4)-b B d (m+n+3)+b c C (m+2))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{C (a+b x)^{m+2} (c+d x)^{n+1}}{b^2 d (m+n+3)}","-\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right) \left(d (m+n+2) \left(a^2 C d (n+1)+a b c C (m+2)-A b^2 d (m+n+3)\right)-(a d (n+1)+b c (m+1)) (a C d (m+2 n+4)+b (c C (m+2)-B d (m+n+3)))\right)}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (a C d (m+2 n+4)+b (c C (m+2)-B d (m+n+3)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac{C (a+b x)^{m+2} (c+d x)^{n+1}}{b^2 d (m+n+3)}",1,"-(((b*c*C*(2 + m) - b*B*d*(3 + m + n) + a*C*d*(4 + m + 2*n))*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b^2*d^2*(2 + m + n)*(3 + m + n))) + (C*(a + b*x)^(2 + m)*(c + d*x)^(1 + n))/(b^2*d*(3 + m + n)) - ((d*(2 + m + n)*(a*b*c*C*(2 + m) + a^2*C*d*(1 + n) - A*b^2*d*(3 + m + n)) - (b*c*(1 + m) + a*d*(1 + n))*(b*c*C*(2 + m) - b*B*d*(3 + m + n) + a*C*d*(4 + m + 2*n)))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^3*d^2*(1 + m)*(2 + m + n)*(3 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)","A",4,4,25,0.1600,1,"{951, 80, 70, 69}"
34,1,605,0,1.0691182,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Int[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right) \left(-\frac{(a d (n+1)+b c (m+1)) \left(a^2 d^2 D \left(m^2+m (3 n+8)+3 \left(n^2+5 n+6\right)\right)+a b d \left(c D (m+2) (m+3 n+6)-C d \left(m^2+m (3 n+8)+2 \left(n^2+6 n+8\right)\right)\right)+b^2 \left(B d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)+c^2 D \left(m^2+5 m+6\right)-c C d (m+2) (m+n+4)\right)\right)}{d (m+n+2)}-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 n+6))+a^3 d^2 D (n+1) (m+2 n+6)+a b^2 c (m+2) (c D (m+3)-C d (m+n+4))+A b^3 d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)\right)}{b^4 d^2 (m+1) (m+n+3) (m+n+4)}+\frac{(a+b x)^{m+1} (c+d x)^{n+1} \left(a^2 d^2 D \left(m^2+m (3 n+8)+3 \left(n^2+5 n+6\right)\right)+a b d \left(c D (m+2) (m+3 n+6)-C d \left(m^2+m (3 n+8)+2 \left(n^2+6 n+8\right)\right)\right)+b^2 \left(B d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)+c^2 D \left(m^2+5 m+6\right)-c C d (m+2) (m+n+4)\right)\right)}{b^3 d^3 (m+n+2) (m+n+3) (m+n+4)}-\frac{(a+b x)^{m+2} (c+d x)^{n+1} (a d D (2 m+3 n+9)+b c D (m+3)-b C d (m+n+4))}{b^3 d^2 (m+n+3) (m+n+4)}+\frac{D (a+b x)^{m+3} (c+d x)^{n+1}}{b^3 d (m+n+4)}","\frac{(a+b x)^{m+1} (c+d x)^n \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right) \left(d (m+n+2) \left(-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 n+6))+a^3 d^2 D (n+1) (m+2 n+6)+a b^2 c (m+2) (c D (m+3)-C d (m+n+4))+A b^3 d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)\right)-(a d (n+1)+b c (m+1)) \left(a^2 d^2 D \left(m^2+m (3 n+8)+3 \left(n^2+5 n+6\right)\right)+a b d \left(c D (m+2) (m+3 n+6)-C d \left(m^2+m (3 n+8)+2 \left(n^2+6 n+8\right)\right)\right)+b^2 \left(B d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)+c^2 D \left(m^2+5 m+6\right)-c C d (m+2) (m+n+4)\right)\right)\right)}{b^4 d^3 (m+1) (m+n+2) (m+n+3) (m+n+4)}+\frac{(a+b x)^{m+1} (c+d x)^{n+1} \left(a^2 d^2 D \left(m^2+m (3 n+8)+3 \left(n^2+5 n+6\right)\right)+a b d \left(c D (m+2) (m+3 n+6)-C d \left(m^2+m (3 n+8)+2 \left(n^2+6 n+8\right)\right)\right)+b^2 \left(B d^2 \left(m^2+m (2 n+7)+n^2+7 n+12\right)+c^2 D \left(m^2+5 m+6\right)-c C d (m+2) (m+n+4)\right)\right)}{b^3 d^3 (m+n+2) (m+n+3) (m+n+4)}-\frac{(a+b x)^{m+2} (c+d x)^{n+1} (a d D (2 m+3 n+9)+b (c D (m+3)-C d (m+n+4)))}{b^3 d^2 (m+n+3) (m+n+4)}+\frac{D (a+b x)^{m+3} (c+d x)^{n+1}}{b^3 d (m+n+4)}",1,"((a^2*d^2*D*(m^2 + m*(8 + 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) + B*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(8 + 6*n + n^2))))*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b^3*d^3*(2 + m + n)*(3 + m + n)*(4 + m + n)) - ((b*c*D*(3 + m) - b*C*d*(4 + m + n) + a*d*D*(9 + 2*m + 3*n))*(a + b*x)^(2 + m)*(c + d*x)^(1 + n))/(b^3*d^2*(3 + m + n)*(4 + m + n)) + (D*(a + b*x)^(3 + m)*(c + d*x)^(1 + n))/(b^3*d*(4 + m + n)) + ((a^3*d^2*D*(1 + n)*(6 + m + 2*n) + a*b^2*c*(2 + m)*(c*D*(3 + m) - C*d*(4 + m + n)) + A*b^3*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n)) - a^2*b*d*(C*d*(1 + n)*(4 + m + n) - c*D*(2 + m)*(6 + m + 3*n)) - ((b*c*(1 + m) + a*d*(1 + n))*(a^2*d^2*D*(m^2 + m*(8 + 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) + B*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(8 + 6*n + n^2)))))/(d*(2 + m + n)))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^4*d^2*(1 + m)*(3 + m + n)*(4 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)","A",5,5,30,0.1667,1,"{1623, 951, 80, 70, 69}"