1,1,391,436,1.3093018,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 \sqrt{c+d x} \left(-9009 (c+d x)^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^2-15 c^3 D+10 c^2 C d\right)\right)+5005 b (c+d x)^4 \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(B d^2+15 c^2 D-5 c C d\right)\right)+6435 (c+d x)^3 \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)-15015 (c+d x) (b c-a d)^2 \left(b \left(-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+45045 (b c-a d)^3 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)+4095 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+3465 b^3 D (c+d x)^6\right)}{45045 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^2-15 c^3 D+10 c^2 C 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)-\left(b^2 \left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{9 d^7}+\frac{2 (c+d x)^{7/2} \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)}{7 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^2-6 c^3 D+5 c^2 C 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^3 (-D)+c^2 C 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*Sqrt[c + d*x]*(45045*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) - 15015*(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) - 9009*(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)^2 + 6435*(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 + 5005*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)^4 + 4095*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 3465*b^3*D*(c + d*x)^6))/(45045*d^7)","A",1
2,1,287,324,0.6899323,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Integrate[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 \sqrt{c+d x} \left(693 (c+d x)^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^2-10 c^3 D+6 c^2 C d\right)\right)+495 (c+d x)^3 \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(B d^2+10 c^2 D-4 c C d\right)\right)+1155 (c+d x) (b c-a d) \left(b \left(-2 A d^3+3 B c d^2+5 c^3 D-4 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+3465 (b c-a d)^2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)+385 b (c+d x)^4 (2 a d D-5 b c D+b C d)+315 b^2 D (c+d x)^5\right)}{3465 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^2-10 c^3 D+6 c^2 C 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)-\left(b^2 \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^2-5 c^3 D+4 c^2 C 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^3 (-D)+c^2 C 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*Sqrt[c + d*x]*(3465*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) + 1155*(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) + 693*(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)^2 + 495*(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 + 385*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4 + 315*b^2*D*(c + d*x)^5))/(3465*d^6)","A",1
3,1,184,212,0.3137941,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 \sqrt{c+d x} \left(3 a d \left(d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right)+b \left(-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+24 c^2 d^2 (7 B+x (3 C+2 D x))+128 c^4 D-16 c^3 d (9 C+4 D x)\right)\right)}{315 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^2-4 c^3 D+3 c^2 C 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^3 (-D)+c^2 C 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*Sqrt[c + d*x]*(3*a*d*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + b*(128*c^4*D - 16*c^3*d*(9*C + 4*D*x) + 24*c^2*d^2*(7*B + x*(3*C + 2*D*x)) + d^4*x*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^3*(105*A + x*(42*B + x*(27*C + 20*D*x))))))/(315*d^5)","A",1
4,1,82,115,0.0967186,"\int \frac{A+B x+C x^2+D x^3}{\sqrt{c+d x}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/Sqrt[c + d*x],x]","\frac{2 \sqrt{c+d x} \left(d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right)}{105 d^4}","\frac{2 \sqrt{c+d x} \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*Sqrt[c + d*x]*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))))/(105*d^4)","A",1
5,1,185,188,0.2702771,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) \sqrt{c+d x}} \, dx","Integrate[(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(B d^2+c^2 D-c C d\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)-\left(b^2 \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",1
6,1,244,201,0.441787,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt{c+d x}} \, dx","Integrate[(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 \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{b^3 (a+b x) (b c-a d)}+\frac{d \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} (b c-a d)^{3/2}}-\frac{2 \left(3 a^2 D-2 a b C+b^2 B\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} (-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(-5 a^3 d D+3 a^2 b (2 c D+C 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*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(b^3*(b*c - a*d)*(a + b*x)) + (2*D*(c + d*x)^(3/2))/(3*b^2*d^2) - (2*(b^2*B - 2*a*b*C + 3*a^2*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(7/2)*Sqrt[b*c - a*d]) + (d*(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)*(b*c - a*d)^(3/2))","A",1
7,1,359,279,0.8660551,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 \sqrt{c+d x}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*Sqrt[c + d*x]),x]","\frac{\frac{2 \sqrt{b} \sqrt{c+d x} \left(a \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{(a+b x)^2 (b c-a d)}-\frac{3 d \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right)}{(a+b x) (b c-a d)^{5/2}}-\frac{4 \sqrt{b} \sqrt{c+d x} \left(3 a^2 D-2 a b C+b^2 B\right)}{(a+b x) (b c-a d)}+\frac{4 d \left(3 a^2 D-2 a b C+b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{8 (b C-3 a D) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{8 \sqrt{b} D \sqrt{c+d x}}{d}}{4 b^{7/2}}","-\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{\sqrt{c+d x} \left(-9 a^3 d D+a^2 b (12 c D+5 C 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{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-15 a^3 d^2 D+3 a^2 b d (12 c D+C 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{2 D \sqrt{c+d x}}{b^3 d}",1,"((8*Sqrt[b]*D*Sqrt[c + d*x])/d + (2*Sqrt[b]*(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/((b*c - a*d)*(a + b*x)^2) - (4*Sqrt[b]*(b^2*B - 2*a*b*C + 3*a^2*D)*Sqrt[c + d*x])/((b*c - a*d)*(a + b*x)) - (8*(b*C - 3*a*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d] + (4*d*(b^2*B - 2*a*b*C + 3*a^2*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2) - (3*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(-(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]) + d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/((b*c - a*d)^(5/2)*(a + b*x)))/(4*b^(7/2))","A",1
8,1,493,375,1.3944297,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt{c+d x}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*Sqrt[c + d*x]),x]","\frac{5 d (a+b x) \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(3 d^2 (a+b x)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right)-8 \sqrt{b} \sqrt{c+d x} (b c-a d)^{5/2} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)-12 \sqrt{b} (a+b x) \sqrt{c+d x} (b c-a d)^{5/2} \left(3 a^2 D-2 a b C+b^2 B\right)+18 d (a+b x)^2 (b c-a d) \left(3 a^2 D-2 a b C+b^2 B\right) \left(\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}-d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)\right)-24 \sqrt{b} (a+b x)^2 \sqrt{c+d x} (b c-a d)^{5/2} (b C-3 a D)+24 d (a+b x)^3 (b c-a d)^2 (b C-3 a D) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)-48 D (a+b x)^3 (b c-a d)^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{24 b^{7/2} (a+b x)^3 (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(-11 a^3 d^2 D+a^2 b d (30 c D+C 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{\sqrt{c+d x} \left(-13 a^3 d D+a^2 b (18 c D+7 C 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{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(5 a^3 d^3 D+a^2 b d^2 (C d-18 c 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^2-16 c^3 D+8 c^2 C d\right)\right)}{8 b^{7/2} (b c-a d)^{7/2}}",1,"(-8*Sqrt[b]*(b*c - a*d)^(5/2)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x] - 12*Sqrt[b]*(b*c - a*d)^(5/2)*(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)*Sqrt[c + d*x] - 24*Sqrt[b]*(b*c - a*d)^(5/2)*(b*C - 3*a*D)*(a + b*x)^2*Sqrt[c + d*x] - 48*(b*c - a*d)^3*D*(a + b*x)^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]] + 24*d*(b*c - a*d)^2*(b*C - 3*a*D)*(a + b*x)^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]] + 18*d*(b*c - a*d)*(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)^2*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x] - d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]) + 5*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(a + b*x)*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]*(2*b*c - 5*a*d - 3*b*d*x) + 3*d^2*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(24*b^(7/2)*(b*c - a*d)^(7/2)*(a + b*x)^3)","A",1
9,1,621,495,2.4194809,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt{c+d x}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^5*Sqrt[c + d*x]),x]","\frac{7 d (a+b x) \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(8 \sqrt{b} \sqrt{c+d x} (b c-a d)^{5/2}-5 d (a+b x) \left(3 d^2 (a+b x)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right)\right)-48 \sqrt{b} \sqrt{c+d x} (b c-a d)^{7/2} \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right)+40 d (a+b x)^2 (b c-a d) \left(3 a^2 D-2 a b C+b^2 B\right) \left(3 d^2 (a+b x)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right)-64 \sqrt{b} (a+b x) \sqrt{c+d x} (b c-a d)^{7/2} \left(3 a^2 D-2 a b C+b^2 B\right)-96 \sqrt{b} (a+b x)^2 \sqrt{c+d x} (b c-a d)^{7/2} (b C-3 a D)+144 d (a+b x)^3 (b c-a d)^2 (b C-3 a D) \left(\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}-d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)\right)-192 \sqrt{b} D (a+b x)^3 \sqrt{c+d x} (b c-a d)^{7/2}+192 d D (a+b x)^4 (b c-a d)^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)}{192 b^{7/2} (a+b x)^4 (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(-59 a^3 d^2 D+3 a^2 b d (56 c D+C 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{\sqrt{c+d x} \left(5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c 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^2-64 c^3 D+48 c^2 C d\right)\right)}{64 b^3 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left(-17 a^3 d D+3 a^2 b (8 c D+3 C 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{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c 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^2-64 c^3 D+48 c^2 C d\right)\right)}{64 b^{7/2} (b c-a d)^{9/2}}",1,"(-48*Sqrt[b]*(b*c - a*d)^(7/2)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x] - 64*Sqrt[b]*(b*c - a*d)^(7/2)*(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)*Sqrt[c + d*x] - 96*Sqrt[b]*(b*c - a*d)^(7/2)*(b*C - 3*a*D)*(a + b*x)^2*Sqrt[c + d*x] - 192*Sqrt[b]*(b*c - a*d)^(7/2)*D*(a + b*x)^3*Sqrt[c + d*x] + 192*d*(b*c - a*d)^3*D*(a + b*x)^4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]] + 144*d*(b*c - a*d)^2*(b*C - 3*a*D)*(a + b*x)^3*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x] - d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]) + 40*d*(b*c - a*d)*(b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)^2*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]*(2*b*c - 5*a*d - 3*b*d*x) + 3*d^2*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]) + 7*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(a + b*x)*(8*Sqrt[b]*(b*c - a*d)^(5/2)*Sqrt[c + d*x] - 5*d*(a + b*x)*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]*(2*b*c - 5*a*d - 3*b*d*x) + 3*d^2*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])))/(192*b^(7/2)*(b*c - a*d)^(9/2)*(a + b*x)^4)","A",1
10,1,391,434,1.1607463,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 \left(-1155 (c+d x)^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^2-15 c^3 D+10 c^2 C d\right)\right)+495 b (c+d x)^4 \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(B d^2+15 c^2 D-5 c C d\right)\right)+693 (c+d x)^3 \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)-3465 (c+d x) (b c-a d)^2 \left(b \left(-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+3465 (b c-a d)^3 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)+385 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+315 b^3 D (c+d x)^6\right)}{3465 d^7 \sqrt{c+d x}}","-\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^2-15 c^3 D+10 c^2 C 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)-\left(b^2 \left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{7 d^7}+\frac{2 (c+d x)^{5/2} \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)}{5 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^2-6 c^3 D+5 c^2 C d\right)\right)}{d^7}+\frac{2 (b c-a d)^3 \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*(3465*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) - 3465*(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) - 1155*(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)^2 + 693*(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 + 495*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)^4 + 385*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 315*b^3*D*(c + d*x)^6))/(3465*d^7*Sqrt[c + d*x])","A",1
11,1,287,322,0.6671196,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Integrate[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 \left(105 (c+d x)^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^2-10 c^3 D+6 c^2 C d\right)\right)+63 (c+d x)^3 \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(B d^2+10 c^2 D-4 c C d\right)\right)+315 (c+d x) (b c-a d) \left(b \left(-2 A d^3+3 B c d^2+5 c^3 D-4 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+315 (b c-a d)^2 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)+45 b (c+d x)^4 (2 a d D-5 b c D+b C d)+35 b^2 D (c+d x)^5\right)}{315 d^6 \sqrt{c+d 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^2-10 c^3 D+6 c^2 C 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)-\left(b^2 \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^2-5 c^3 D+4 c^2 C d\right)\right)}{d^6}-\frac{2 (b c-a d)^2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*(315*(b*c - a*d)^2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) + 315*(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) + 105*(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)^2 + 63*(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 + 45*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4 + 35*b^2*D*(c + d*x)^5))/(315*d^6*Sqrt[c + d*x])","A",1
12,1,188,210,0.3278954,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{14 a d \left(d^3 \left(x \left(15 B+5 C x+3 D x^2\right)-15 A\right)+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right)+b \left(4 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+2 d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))+16 c^2 d^2 (3 x (7 C+2 D x)-35 B)-768 c^4 D+96 c^3 d (7 C-4 D x)\right)}{105 d^5 \sqrt{c+d 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^2-4 c^3 D+3 c^2 C d\right)\right)}{d^5}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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,"(14*a*d*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))) + b*(-768*c^4*D + 96*c^3*d*(7*C - 4*D*x) + 16*c^2*d^2*(-35*B + 3*x*(7*C + 2*D*x)) + 4*c*d^3*(105*A - x*(70*B + 3*x*(7*C + 4*D*x))) + 2*d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))))/(105*d^5*Sqrt[c + d*x])","A",1
13,1,82,113,0.0929802,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{3/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(3/2),x]","\frac{2 \left(d^3 \left(x \left(15 B+5 C x+3 D x^2\right)-15 A\right)+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right)}{15 d^4 \sqrt{c+d x}}","-\frac{2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))))/(15*d^4*Sqrt[c + d*x])","A",1
14,1,174,193,0.5273085,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{3/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(3/2)),x]","2 \left(-\frac{\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{A d^3-B c d^2+c^3 (-D)+c^2 C d}{d^3 \sqrt{c+d x} (b c-a d)}+\frac{\sqrt{c+d x} (-a d D-2 b c D+b C d)}{b^2 d^3}+\frac{D (c+d x)^{3/2}}{3 b d^3}\right)","-\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^3 (-D)+c^2 C 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]) + ((b*C*d - 2*b*c*D - a*d*D)*Sqrt[c + d*x])/(b^2*d^3) + (D*(c + d*x)^(3/2))/(3*b*d^3) - ((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",1
15,1,283,253,0.7202756,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(3/2)),x]","\frac{\sqrt{c+d x} \left(a \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{b^2 (a+b x) (b c-a d)^2}+\frac{d \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)^{5/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-2 a^3 d D+a^2 b (3 c D+C d)-2 a b^2 c C+b^3 (B c-A d)\right)}{b^{5/2} (b c-a d)^{5/2}}+\frac{2 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)}{d^2 \sqrt{c+d x} (b c-a d)^2}+\frac{2 D \sqrt{c+d x}}{b^2 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{a^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left(b^3 \left(3 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right)\right)}{b^3 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-3 a^3 d D+a^2 b (6 c D+C 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,"(2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(d^2*(b*c - a*d)^2*Sqrt[c + d*x]) + (2*D*Sqrt[c + d*x])/(b^2*d^2) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(b^2*(b*c - a*d)^2*(a + b*x)) + (d*(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)^(5/2)) - (2*(-2*a*b^2*c*C + b^3*(B*c - A*d) - 2*a^3*d*D + a^2*b*(C*d + 3*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(5/2))","A",1
16,1,482,350,1.4178424,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{3/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(3/2)),x]","\frac{\sqrt{c+d x} \left(a \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{2 b^2 (a+b x)^2 (b c-a d)^2}-\frac{3 d \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right)}{4 b^{5/2} (a+b x) (b c-a d)^{7/2}}-\frac{\sqrt{c+d x} \left(-2 a^3 d D+a^2 b (3 c D+C d)-2 a b^2 c C+b^3 (B c-A d)\right)}{b^2 (a+b x) (b c-a d)^3}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 \left(-d^2\right) D+3 a^2 b c d D-3 a b^2 c^2 D+b^3 \left(A d^2-B c d+c^2 C\right)\right)}{b^{5/2} (b c-a d)^{7/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-2 a^3 d D+a^2 b (3 c D+C d)-2 a b^2 c C+b^3 (B c-A d)\right)}{b^{5/2} (b c-a d)^{7/2}}+\frac{2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)}{d \sqrt{c+d x} (b c-a d)^3}","-\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{a^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left(b^3 \left(5 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right)\right)}{2 b^3 d \sqrt{c+d x} (b c-a d)^3}-\frac{\sqrt{c+d x} \left(-7 a^3 d D+3 a^2 b (4 c D+C 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{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(-3 a^3 d^2 D-a^2 b d (C d-12 c 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}}",1,"(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d*(b*c - a*d)^3*Sqrt[c + d*x]) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(2*b^2*(b*c - a*d)^2*(a + b*x)^2) - ((-2*a*b^2*c*C + b^3*(B*c - A*d) - 2*a^3*d*D + a^2*b*(C*d + 3*c*D))*Sqrt[c + d*x])/(b^2*(b*c - a*d)^3*(a + b*x)) - (2*(b^3*(c^2*C - B*c*d + A*d^2) - 3*a*b^2*c^2*D + 3*a^2*b*c*d*D - a^3*d^2*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(7/2)) + (d*(-2*a*b^2*c*C + b^3*(B*c - A*d) - 2*a^3*d*D + a^2*b*(C*d + 3*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(7/2)) - (3*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(-(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]) + d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(4*b^(5/2)*(b*c - a*d)^(7/2)*(a + b*x))","A",1
17,1,697,463,2.3444652,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx","Integrate[(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 \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{3 b^2 (a+b x)^3 (b c-a d)^2}+\frac{5 d \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(3 d^2 (a+b x)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)+\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d} (-5 a d+2 b c-3 b d x)\right)}{24 b^{5/2} (a+b x)^2 (b c-a d)^{9/2}}+\frac{\sqrt{c+d x} \left(a^3 d^2 D-3 a^2 b c d D+3 a b^2 c^2 D-b^3 \left(A d^2-B c d+c^2 C\right)\right)}{b^2 (a+b x) (b c-a d)^4}-\frac{\sqrt{c+d x} \left(-2 a^3 d D+a^2 b (3 c D+C d)-2 a b^2 c C+b^3 (B c-A d)\right)}{2 b^2 (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(a^3 \left(-d^2\right) D+3 a^2 b c d D-3 a b^2 c^2 D+b^3 \left(A d^2-B c d+c^2 C\right)\right)}{b^{5/2} (b c-a d)^{9/2}}+\frac{3 d \left(d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right) \left(2 a^3 d D-a^2 b (3 c D+C d)+2 a b^2 c C+b^3 (A d-B c)\right)}{4 b^{5/2} (a+b x) (b c-a d)^{9/2}}+\frac{2 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)}{\sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)}{(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(5 a^3 d^2 D-a^2 b d (11 C d-18 c 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^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left(b^3 \left(7 A d^3-6 B c d^2-6 c^3 D+6 c^2 C d\right)\right)}{3 b^3 \sqrt{c+d x} (b c-a d)^4}-\frac{\sqrt{c+d x} \left(-11 a^3 d D+a^2 b (18 c D+5 C 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{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left(-5 B d^2-24 c^2 D+12 c C d\right)-\left(b^3 \left(35 A d^3-30 B c d^2-16 c^3 D+24 c^2 C d\right)\right)\right)}{8 b^{5/2} (b c-a d)^{9/2}}",1,"(2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/((b*c - a*d)^4*Sqrt[c + d*x]) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(3*b^2*(b*c - a*d)^2*(a + b*x)^3) - ((-2*a*b^2*c*C + b^3*(B*c - A*d) - 2*a^3*d*D + a^2*b*(C*d + 3*c*D))*Sqrt[c + d*x])/(2*b^2*(b*c - a*d)^3*(a + b*x)^2) + ((-(b^3*(c^2*C - B*c*d + A*d^2)) + 3*a*b^2*c^2*D - 3*a^2*b*c*d*D + a^3*d^2*D)*Sqrt[c + d*x])/(b^2*(b*c - a*d)^4*(a + b*x)) + (2*Sqrt[b]*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(9/2) + (d*(b^3*(c^2*C - B*c*d + A*d^2) - 3*a*b^2*c^2*D + 3*a^2*b*c*d*D - a^3*d^2*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(9/2)) + (3*d*(2*a*b^2*c*C + b^3*(-(B*c) + A*d) + 2*a^3*d*D - a^2*b*(C*d + 3*c*D))*(-(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]) + d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(4*b^(5/2)*(b*c - a*d)^(9/2)*(a + b*x)) + (5*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]*(2*b*c - 5*a*d - 3*b*d*x) + 3*d^2*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(24*b^(5/2)*(b*c - a*d)^(9/2)*(a + b*x)^2)","A",1
18,1,391,434,0.7270438,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(-315 (c+d x)^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^2-15 c^3 D+10 c^2 C d\right)\right)+63 b (c+d x)^4 \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(B d^2+15 c^2 D-5 c C d\right)\right)+105 (c+d x)^3 \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)+315 (c+d x) (b c-a d)^2 \left(b \left(-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+105 (b c-a d)^3 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)+45 b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)+35 b^3 D (c+d x)^6\right)}{315 d^7 (c+d x)^{3/2}}","-\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^2-15 c^3 D+10 c^2 C 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)-\left(b^2 \left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{5 d^7}+\frac{2 (c+d x)^{3/2} \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)}{3 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^2-6 c^3 D+5 c^2 C 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^3 (-D)+c^2 C 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*(105*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) + 315*(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) - 315*(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)^2 + 105*(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 + 63*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)^4 + 45*b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5 + 35*b^3*D*(c + d*x)^6))/(315*d^7*(c + d*x)^(3/2))","A",1
19,1,287,322,0.7026396,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Integrate[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(105 (c+d x)^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^2-10 c^3 D+6 c^2 C d\right)\right)+35 (c+d x)^3 \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(B d^2+10 c^2 D-4 c C d\right)\right)-105 (c+d x) (b c-a d) \left(b \left(-2 A d^3+3 B c d^2+5 c^3 D-4 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)+35 (b c-a d)^2 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)+21 b (c+d x)^4 (2 a d D-5 b c D+b C d)+15 b^2 D (c+d x)^5\right)}{105 d^6 (c+d x)^{3/2}}","\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^2-10 c^3 D+6 c^2 C 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)-\left(b^2 \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^2-5 c^3 D+4 c^2 C 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^3 (-D)+c^2 C 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*(35*(b*c - a*d)^2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D) - 105*(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) + 105*(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)^2 + 35*(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 + 21*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4 + 15*b^2*D*(c + d*x)^5))/(105*d^6*(c + d*x)^(3/2))","A",1
20,1,177,210,0.3090734,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(b \left(-2 c d^3 \left(5 A+x \left(-30 B+15 C x+4 D x^2\right)\right)+d^4 x \left(x \left(15 B+5 C x+3 D x^2\right)-15 A\right)+8 c^2 d^2 (5 B+3 x (2 D x-5 C))+128 c^4 D+c^3 (192 d D x-80 C d)\right)-5 a d \left(d^3 \left(A+3 B x-\left(x^2 (3 C+D x)\right)\right)+2 c d^2 (B+3 x (D x-2 C))+16 c^3 D-8 c^2 d (C-3 D x)\right)\right)}{15 d^5 (c+d x)^{3/2}}","\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^2-4 c^3 D+3 c^2 C d\right)\right)}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*(-5*a*d*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) + d^3*(A + 3*B*x - x^2*(3*C + D*x))) + b*(128*c^4*D + c^3*(-80*C*d + 192*d*D*x) + 8*c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(-15*A + x*(15*B + 5*C*x + 3*D*x^2)) - 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4*D*x^2)))))/(15*d^5*(c + d*x)^(3/2))","A",1
21,1,75,113,0.0942035,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{5/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(5/2),x]","-\frac{2 \left(d^3 \left(A+3 B x-\left(x^2 (3 C+D x)\right)\right)+2 c d^2 (B+3 x (D x-2 C))+16 c^3 D-8 c^2 d (C-3 D x)\right)}{3 d^4 (c+d x)^{3/2}}","-\frac{2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) + d^3*(A + 3*B*x - x^2*(3*C + D*x))))/(3*d^4*(c + d*x)^(3/2))","A",1
22,1,210,210,0.6612741,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{5/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(5/2)),x]","2 \left(-\frac{\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{a d \left(-B d^2-3 c^2 D+2 c C d\right)-b \left(-A d^3-2 c^3 D+c^2 C d\right)}{d^3 \sqrt{c+d x} (b c-a d)^2}+\frac{A d^3-B c d^2+c^3 (-D)+c^2 C d}{3 d^3 (c+d x)^{3/2} (b c-a d)}+\frac{D \sqrt{c+d x}}{b d^3}\right)","-\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^3-2 c^3 D+c^2 C 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^3 (-D)+c^2 C 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)) + (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]) + (D*Sqrt[c + d*x])/(b*d^3) - ((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",1
23,1,334,336,1.0450953,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(5/2)),x]","\frac{\sqrt{c+d x} \left(a \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{b (a+b x) (b c-a d)^3}+\frac{d \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)^{7/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 (-d) D+3 a^2 b c D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right)}{b^{3/2} (b c-a d)^{7/2}}+\frac{2 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)}{3 d^2 (c+d x)^{3/2} (b c-a d)^2}+\frac{b \left(4 A d^3-2 B c d^2+2 c^3 D\right)-2 a d \left(B d^2+3 c^2 D-2 c C d\right)}{d^2 \sqrt{c+d x} (a d-b c)^3}","-\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{-a^3 d^3 D+a^2 b C d^3+a b^2 d \left(-3 B d^2-6 c^2 D+4 c C d\right)-\left(b^3 \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^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left(b^3 \left(5 A d^3-2 B c d^2-2 c^3 D+2 c^2 C d\right)\right)}{3 b^3 d^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 (-d) D-a^2 b (C d-6 c 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,"(2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(3*d^2*(b*c - a*d)^2*(c + d*x)^(3/2)) + (-2*a*d*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*(-2*B*c*d^2 + 4*A*d^3 + 2*c^3*D))/(d^2*(-(b*c) + a*d)^3*Sqrt[c + d*x]) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(b*(b*c - a*d)^3*(a + b*x)) - (2*(b^3*(B*c - 2*A*d) + a*b^2*(-2*c*C + B*d) + 3*a^2*b*c*D - a^3*d*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(7/2)) + (d*(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)^(7/2))","A",1
24,1,536,438,2.09703,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx","Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(5/2)),x]","\frac{\sqrt{c+d x} \left(a \left(a^2 D-a b C+b^2 B\right)-A b^3\right)}{2 b (a+b x)^2 (b c-a d)^3}-\frac{3 d \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \left(d (a+b x) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right)-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right)}{4 b^{3/2} (a+b x) (b c-a d)^{9/2}}-\frac{\sqrt{c+d x} \left(a^3 (-d) D+3 a^2 b c D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right)}{b (a+b x) (b c-a d)^4}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 (-d) D+3 a^2 b c D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right)}{b^{3/2} (b c-a d)^{9/2}}+\frac{2 \left(b \left(3 A d^2-2 B c d+c^2 C\right)-a \left(B d^2+3 c^2 D-2 c C d\right)\right)}{\sqrt{c+d x} (b c-a d)^4}-\frac{2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(b \left(3 A d^2-2 B c d+c^2 C\right)-a \left(B d^2+3 c^2 D-2 c C d\right)\right)}{(b c-a d)^{9/2}}+\frac{2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)}{3 d (c+d x)^{3/2} (b c-a d)^3}","-\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{a^3 \left(-d^2\right) D+a^2 b C d^2+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^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left(b^3 \left(7 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right)\right)}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\sqrt{c+d x} \left(-5 a^3 d D+a^2 b (12 c D+C 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{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right) \left(a^3 d^2 D+3 a^2 b d (C d-4 c 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}}",1,"(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d*(b*c - a*d)^3*(c + d*x)^(3/2)) + (2*(b*(c^2*C - 2*B*c*d + 3*A*d^2) - a*(-2*c*C*d + B*d^2 + 3*c^2*D)))/((b*c - a*d)^4*Sqrt[c + d*x]) + ((-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(2*b*(b*c - a*d)^3*(a + b*x)^2) - ((b^3*(B*c - 2*A*d) + a*b^2*(-2*c*C + B*d) + 3*a^2*b*c*D - a^3*d*D)*Sqrt[c + d*x])/(b*(b*c - a*d)^4*(a + b*x)) + (d*(b^3*(B*c - 2*A*d) + a*b^2*(-2*c*C + B*d) + 3*a^2*b*c*D - a^3*d*D)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(9/2)) - (2*Sqrt[b]*(b*(c^2*C - 2*B*c*d + 3*A*d^2) - a*(-2*c*C*d + B*d^2 + 3*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(9/2) - (3*d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(-(Sqrt[b]*Sqrt[b*c - a*d]*Sqrt[c + d*x]) + d*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(4*b^(3/2)*(b*c - a*d)^(9/2)*(a + b*x))","A",1
25,1,418,455,0.6799478,"\int (a+b x)^3 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+1} \left(\frac{(c+d x)^2 (b c-a d) \left(a^2 d^2 (3 c D-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+15 c^3 D-10 c^2 C d\right)\right)}{n+3}+\frac{b (c+d x)^4 \left(3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left(B d^2+15 c^2 D-5 c C d\right)\right)}{n+5}+\frac{(c+d x)^3 \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)}{n+4}-\frac{(c+d x) (b c-a d)^2 \left(b \left(-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)}{n+2}+\frac{(b c-a d)^3 \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)}{n+1}+\frac{b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)}{n+6}+\frac{b^3 D (c+d x)^6}{n+7}\right)}{d^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^2-15 c^3 D+10 c^2 C d\right)\right)}{d^7 (n+3)}+\frac{b (c+d x)^{n+5} \left(3 a^2 d^2 D+3 a b d (C d-5 c D)-\left(b^2 \left(-B d^2-15 c^2 D+5 c C d\right)\right)\right)}{d^7 (n+5)}+\frac{(c+d x)^{n+4} \left(a^3 d^3 D+3 a^2 b d^2 (C d-4 c 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^2-20 c^3 D+10 c^2 C d\right)\right)}{d^7 (n+4)}-\frac{(b c-a d)^3 (c+d x)^{n+1} \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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^2-6 c^3 D+5 c^2 C 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,"((c + d*x)^(1 + n)*(((b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(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) + ((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)^2)/(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)^3)/(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)^4)/(5 + n) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5)/(6 + n) + (b^3*D*(c + d*x)^6)/(7 + n)))/d^7","A",1
26,1,308,338,0.4862203,"\int (a+b x)^2 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+1} \left(\frac{(c+d x)^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^2-10 c^3 D+6 c^2 C d\right)\right)}{n+3}+\frac{(c+d x)^3 \left(a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left(B d^2+10 c^2 D-4 c C d\right)\right)}{n+4}+\frac{(c+d x) (b c-a d) \left(b \left(-2 A d^3+3 B c d^2+5 c^3 D-4 c^2 C d\right)-a d \left(B d^2+3 c^2 D-2 c C d\right)\right)}{n+2}+\frac{(b c-a d)^2 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right)}{n+1}+\frac{b (c+d x)^4 (2 a d D-5 b c D+b C d)}{n+5}+\frac{b^2 D (c+d x)^5}{n+6}\right)}{d^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^2-10 c^3 D+6 c^2 C 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)-\left(b^2 \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^3 (-D)+c^2 C 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^2-5 c^3 D+4 c^2 C 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,"((c + d*x)^(1 + n)*(((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(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) + ((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)^2)/(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)^3)/(4 + n) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4)/(5 + n) + (b^2*D*(c + d*x)^5)/(6 + n)))/d^6","A",1
27,1,199,226,0.2703198,"\int (a+b x) (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(a + b*x)*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+1} \left(\frac{(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^2-4 c^3 D+3 c^2 C d\right)\right)}{n+2}+\frac{(b c-a d) \left(-A d^3+B c d^2+c^3 D-c^2 C d\right)}{n+1}+\frac{(c+d x)^2 \left(a d (C d-3 c D)+b \left(B d^2+6 c^2 D-3 c C d\right)\right)}{n+3}+\frac{(c+d x)^3 (a d D-4 b c D+b C d)}{n+4}+\frac{b D (c+d x)^4}{n+5}\right)}{d^5}","-\frac{(b c-a d) (c+d x)^{n+1} \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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^2-4 c^3 D+3 c^2 C 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,"((c + d*x)^(1 + n)*(((b*c - a*d)*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(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) + ((a*d*(C*d - 3*c*D) + b*(-3*c*C*d + B*d^2 + 6*c^2*D))*(c + d*x)^2)/(3 + n) + ((b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^3)/(4 + n) + (b*D*(c + d*x)^4)/(5 + n)))/d^5","A",1
28,1,108,126,0.1127391,"\int (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\frac{(c+d x)^{n+1} \left(\frac{A d^3-B c d^2+c^3 (-D)+c^2 C d}{n+1}+\frac{(c+d x) \left(B d^2+3 c^2 D-2 c C d\right)}{n+2}+\frac{(c+d x)^2 (C d-3 c D)}{n+3}+\frac{D (c+d x)^3}{n+4}\right)}{d^4}","\frac{(c+d x)^{n+1} \left(A d^3-B c d^2+c^3 (-D)+c^2 C 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 + d*x)^(1 + n)*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/(1 + n) + ((-2*c*C*d + B*d^2 + 3*c^2*D)*(c + d*x))/(2 + n) + ((C*d - 3*c*D)*(c + d*x)^2)/(3 + n) + (D*(c + d*x)^3)/(4 + n)))/d^4","A",1
29,1,181,203,0.2590119,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{a+b x} \, dx","Integrate[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]","\frac{(c+d x)^{n+1} \left(-\frac{\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)}{(n+1) (b c-a d)}+\frac{a^2 d^2 D+a b d (c D-C d)+b^2 \left(B d^2+c^2 D-c C d\right)}{d^3 (n+1)}+\frac{b (c+d x) (-a d D-2 b c D+b C d)}{d^3 (n+2)}+\frac{b^2 D (c+d x)^2}{d^3 (n+3)}\right)}{b^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)-\left(b^2 \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,"((c + d*x)^(1 + n)*((a^2*d^2*D + a*b*d*(-(C*d) + c*D) + b^2*(-(c*C*d) + B*d^2 + c^2*D))/(d^3*(1 + n)) + (b*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x))/(d^3*(2 + n)) + (b^2*D*(c + d*x)^2)/(d^3*(3 + n)) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)*(1 + n))))/b^3","A",1
30,1,180,220,0.2016336,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^2} \, dx","Integrate[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]","\frac{(c+d x)^{n+1} \left(\frac{d \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \, _2F_1\left(2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{(n+1) (b c-a d)^2}-\frac{\left(3 a^2 D-2 a b C+b^2 B\right) \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{(n+1) (b c-a d)}+\frac{-2 a d D-b c D+b C d}{d^2 (n+1)}+\frac{b D (c+d x)}{d^2 (n+2)}\right)}{b^3}","-\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} \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right) \left(a^3 d D (n+3)-a^2 b (3 c D+C d (n+2))+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} (-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,"((c + d*x)^(1 + n)*((b*C*d - b*c*D - 2*a*d*D)/(d^2*(1 + n)) + (b*D*(c + d*x))/(d^2*(2 + n)) - ((b^2*B - 2*a*b*C + 3*a^2*D)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)*(1 + n)) + (d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*(1 + n))))/b^3","A",1
31,1,188,329,0.1960307,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^3} \, dx","Integrate[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^3,x]","\frac{(c+d x)^{n+1} \left(-\frac{d^2 \left(A b^3-a \left(a^2 D-a b C+b^2 B\right)\right) \, _2F_1\left(3,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{(b c-a d)^3}+\frac{d \left(3 a^2 D-2 a b C+b^2 B\right) \, _2F_1\left(2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{(b c-a d)^2}-\frac{(b C-3 a D) \, _2F_1\left(1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right)}{b c-a d}+\frac{D}{d}\right)}{b^3 (n+1)}","-\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{(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^3 \left(-d^2\right) D \left(n^2+5 n+6\right)+a^2 b d (n+2) (6 c D+C d (n+1))-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^3 (-d) D (n+5)+a^2 b (6 c D+C d (n+3))-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{D (c+d x)^{n+1}}{b^3 d (n+1)}",1,"((c + d*x)^(1 + n)*(D/d - ((b*C - 3*a*D)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b*c - a*d) + (d*(b^2*B - 2*a*b*C + 3*a^2*D)*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b*c - a*d)^2 - (d^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[3, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b*c - a*d)^3))/(b^3*(1 + n))","A",1
32,1,117,141,0.1103911,"\int (a+b x)^m (A+B x) (c+d x)^n \, dx","Integrate[(a + b*x)^m*(A + B*x)*(c + d*x)^n,x]","\frac{(a+b x)^{m+1} (c+d x)^n \left(b B (c+d x)-\frac{\left(\frac{b (c+d x)}{b c-a d}\right)^{-n} (a B d (n+1)-A b d (m+n+2)+b B c (m+1)) \, _2F_1\left(m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right)}{m+1}\right)}{b^2 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,"((a + b*x)^(1 + m)*(c + d*x)^n*(b*B*(c + d*x) - ((b*B*c*(1 + m) + a*B*d*(1 + n) - A*b*d*(2 + m + n))*Hypergeometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])/((1 + m)*((b*(c + d*x))/(b*c - a*d))^n)))/(b^2*d*(2 + m + n))","A",1
33,1,187,268,0.2086117,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2\right) \, dx","Integrate[(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} \left(b \left(b \left(A d^2-B c d+c^2 C\right) \, _2F_1\left(m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right)-(b c-a d) (2 c C-B d) \, _2F_1\left(m+1,-n-1;m+2;\frac{d (a+b x)}{a d-b c}\right)\right)+C (b c-a d)^2 \, _2F_1\left(m+1,-n-2;m+2;\frac{d (a+b x)}{a d-b c}\right)\right)}{b^3 d^2 (m+1)}","-\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,"((a + b*x)^(1 + m)*(c + d*x)^n*(C*(b*c - a*d)^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)] + b*(-((b*c - a*d)*(2*c*C - B*d)*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)]) + b*(c^2*C - B*c*d + A*d^2)*Hypergeometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])))/(b^3*d^2*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n)","A",1
34,1,254,610,0.2647356,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","Integrate[(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} \left(b^3 \left(A d^3-B c d^2+c^3 (-D)+c^2 C d\right) \, _2F_1\left(m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right)+b^2 (b c-a d) \left(B d^2+3 c^2 D-2 c C d\right) \, _2F_1\left(m+1,-n-1;m+2;\frac{d (a+b x)}{a d-b c}\right)+b (b c-a d)^2 (C d-3 c D) \, _2F_1\left(m+1,-n-2;m+2;\frac{d (a+b x)}{a d-b c}\right)+D (b c-a d)^3 \, _2F_1\left(m+1,-n-3;m+2;\frac{d (a+b x)}{a d-b c}\right)\right)}{b^4 d^3 (m+1)}","\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+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^3 d^2 D (n+1) (m+2 n+6)-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 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+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 + b*x)^(1 + m)*(c + d*x)^n*((b*c - a*d)^3*D*Hypergeometric2F1[1 + m, -3 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)] + b*(b*c - a*d)^2*(C*d - 3*c*D)*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)] + b^2*(b*c - a*d)*(-2*c*C*d + B*d^2 + 3*c^2*D)*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)] + b^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Hypergeometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)]))/(b^4*d^3*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n)","A",1