1,1,1261,436,0.2856979,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","IntegrateAlgebraic[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 \left(3465 b^3 D (c+d x)^{13/2}+4095 b^3 C d (c+d x)^{11/2}-24570 b^3 c D (c+d x)^{11/2}+12285 a b^2 d D (c+d x)^{11/2}+5005 b^3 B d^2 (c+d x)^{9/2}+15015 a b^2 C d^2 (c+d x)^{9/2}-25025 b^3 c C d (c+d x)^{9/2}+75075 b^3 c^2 D (c+d x)^{9/2}+15015 a^2 b d^2 D (c+d x)^{9/2}-75075 a b^2 c d D (c+d x)^{9/2}+6435 A b^3 d^3 (c+d x)^{7/2}+19305 a b^2 B d^3 (c+d x)^{7/2}+19305 a^2 b C d^3 (c+d x)^{7/2}-25740 b^3 B c d^2 (c+d x)^{7/2}-77220 a b^2 c C d^2 (c+d x)^{7/2}+64350 b^3 c^2 C d (c+d x)^{7/2}-128700 b^3 c^3 D (c+d x)^{7/2}+6435 a^3 d^3 D (c+d x)^{7/2}-77220 a^2 b c d^2 D (c+d x)^{7/2}+193050 a b^2 c^2 d D (c+d x)^{7/2}+27027 a A b^2 d^4 (c+d x)^{5/2}+27027 a^2 b B d^4 (c+d x)^{5/2}+9009 a^3 C d^4 (c+d x)^{5/2}-27027 A b^3 c d^3 (c+d x)^{5/2}-81081 a b^2 B c d^3 (c+d x)^{5/2}-81081 a^2 b c C d^3 (c+d x)^{5/2}+54054 b^3 B c^2 d^2 (c+d x)^{5/2}+162162 a b^2 c^2 C d^2 (c+d x)^{5/2}-90090 b^3 c^3 C d (c+d x)^{5/2}+135135 b^3 c^4 D (c+d x)^{5/2}-27027 a^3 c d^3 D (c+d x)^{5/2}+162162 a^2 b c^2 d^2 D (c+d x)^{5/2}-270270 a b^2 c^3 d D (c+d x)^{5/2}+45045 a^2 A b d^5 (c+d x)^{3/2}+15015 a^3 B d^5 (c+d x)^{3/2}-90090 a A b^2 c d^4 (c+d x)^{3/2}-90090 a^2 b B c d^4 (c+d x)^{3/2}-30030 a^3 c C d^4 (c+d x)^{3/2}+45045 A b^3 c^2 d^3 (c+d x)^{3/2}+135135 a b^2 B c^2 d^3 (c+d x)^{3/2}+135135 a^2 b c^2 C d^3 (c+d x)^{3/2}-60060 b^3 B c^3 d^2 (c+d x)^{3/2}-180180 a b^2 c^3 C d^2 (c+d x)^{3/2}+75075 b^3 c^4 C d (c+d x)^{3/2}-90090 b^3 c^5 D (c+d x)^{3/2}+45045 a^3 c^2 d^3 D (c+d x)^{3/2}-180180 a^2 b c^3 d^2 D (c+d x)^{3/2}+225225 a b^2 c^4 d D (c+d x)^{3/2}+45045 a^3 A d^6 \sqrt{c+d x}-135135 a^2 A b c d^5 \sqrt{c+d x}-45045 a^3 B c d^5 \sqrt{c+d x}+135135 a A b^2 c^2 d^4 \sqrt{c+d x}+135135 a^2 b B c^2 d^4 \sqrt{c+d x}+45045 a^3 c^2 C d^4 \sqrt{c+d x}-45045 A b^3 c^3 d^3 \sqrt{c+d x}-135135 a b^2 B c^3 d^3 \sqrt{c+d x}-135135 a^2 b c^3 C d^3 \sqrt{c+d x}+45045 b^3 B c^4 d^2 \sqrt{c+d x}+135135 a b^2 c^4 C d^2 \sqrt{c+d x}-45045 b^3 c^5 C d \sqrt{c+d x}+45045 b^3 c^6 D \sqrt{c+d x}-45045 a^3 c^3 d^3 D \sqrt{c+d x}+135135 a^2 b c^4 d^2 D \sqrt{c+d x}-135135 a b^2 c^5 d D \sqrt{c+d x}\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*(-45045*b^3*c^5*C*d*Sqrt[c + d*x] + 45045*b^3*B*c^4*d^2*Sqrt[c + d*x] + 135135*a*b^2*c^4*C*d^2*Sqrt[c + d*x] - 45045*A*b^3*c^3*d^3*Sqrt[c + d*x] - 135135*a*b^2*B*c^3*d^3*Sqrt[c + d*x] - 135135*a^2*b*c^3*C*d^3*Sqrt[c + d*x] + 135135*a*A*b^2*c^2*d^4*Sqrt[c + d*x] + 135135*a^2*b*B*c^2*d^4*Sqrt[c + d*x] + 45045*a^3*c^2*C*d^4*Sqrt[c + d*x] - 135135*a^2*A*b*c*d^5*Sqrt[c + d*x] - 45045*a^3*B*c*d^5*Sqrt[c + d*x] + 45045*a^3*A*d^6*Sqrt[c + d*x] + 45045*b^3*c^6*D*Sqrt[c + d*x] - 135135*a*b^2*c^5*d*D*Sqrt[c + d*x] + 135135*a^2*b*c^4*d^2*D*Sqrt[c + d*x] - 45045*a^3*c^3*d^3*D*Sqrt[c + d*x] + 75075*b^3*c^4*C*d*(c + d*x)^(3/2) - 60060*b^3*B*c^3*d^2*(c + d*x)^(3/2) - 180180*a*b^2*c^3*C*d^2*(c + d*x)^(3/2) + 45045*A*b^3*c^2*d^3*(c + d*x)^(3/2) + 135135*a*b^2*B*c^2*d^3*(c + d*x)^(3/2) + 135135*a^2*b*c^2*C*d^3*(c + d*x)^(3/2) - 90090*a*A*b^2*c*d^4*(c + d*x)^(3/2) - 90090*a^2*b*B*c*d^4*(c + d*x)^(3/2) - 30030*a^3*c*C*d^4*(c + d*x)^(3/2) + 45045*a^2*A*b*d^5*(c + d*x)^(3/2) + 15015*a^3*B*d^5*(c + d*x)^(3/2) - 90090*b^3*c^5*D*(c + d*x)^(3/2) + 225225*a*b^2*c^4*d*D*(c + d*x)^(3/2) - 180180*a^2*b*c^3*d^2*D*(c + d*x)^(3/2) + 45045*a^3*c^2*d^3*D*(c + d*x)^(3/2) - 90090*b^3*c^3*C*d*(c + d*x)^(5/2) + 54054*b^3*B*c^2*d^2*(c + d*x)^(5/2) + 162162*a*b^2*c^2*C*d^2*(c + d*x)^(5/2) - 27027*A*b^3*c*d^3*(c + d*x)^(5/2) - 81081*a*b^2*B*c*d^3*(c + d*x)^(5/2) - 81081*a^2*b*c*C*d^3*(c + d*x)^(5/2) + 27027*a*A*b^2*d^4*(c + d*x)^(5/2) + 27027*a^2*b*B*d^4*(c + d*x)^(5/2) + 9009*a^3*C*d^4*(c + d*x)^(5/2) + 135135*b^3*c^4*D*(c + d*x)^(5/2) - 270270*a*b^2*c^3*d*D*(c + d*x)^(5/2) + 162162*a^2*b*c^2*d^2*D*(c + d*x)^(5/2) - 27027*a^3*c*d^3*D*(c + d*x)^(5/2) + 64350*b^3*c^2*C*d*(c + d*x)^(7/2) - 25740*b^3*B*c*d^2*(c + d*x)^(7/2) - 77220*a*b^2*c*C*d^2*(c + d*x)^(7/2) + 6435*A*b^3*d^3*(c + d*x)^(7/2) + 19305*a*b^2*B*d^3*(c + d*x)^(7/2) + 19305*a^2*b*C*d^3*(c + d*x)^(7/2) - 128700*b^3*c^3*D*(c + d*x)^(7/2) + 193050*a*b^2*c^2*d*D*(c + d*x)^(7/2) - 77220*a^2*b*c*d^2*D*(c + d*x)^(7/2) + 6435*a^3*d^3*D*(c + d*x)^(7/2) - 25025*b^3*c*C*d*(c + d*x)^(9/2) + 5005*b^3*B*d^2*(c + d*x)^(9/2) + 15015*a*b^2*C*d^2*(c + d*x)^(9/2) + 75075*b^3*c^2*D*(c + d*x)^(9/2) - 75075*a*b^2*c*d*D*(c + d*x)^(9/2) + 15015*a^2*b*d^2*D*(c + d*x)^(9/2) + 4095*b^3*C*d*(c + d*x)^(11/2) - 24570*b^3*c*D*(c + d*x)^(11/2) + 12285*a*b^2*d*D*(c + d*x)^(11/2) + 3465*b^3*D*(c + d*x)^(13/2)))/(45045*d^7)","B",1
2,1,780,324,0.1800484,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","IntegrateAlgebraic[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","-\frac{2 \left(-3465 a^2 A d^5 \sqrt{c+d x}-1155 a^2 B d^4 (c+d x)^{3/2}+3465 a^2 B c d^4 \sqrt{c+d x}+3465 a^2 c^3 d^2 D \sqrt{c+d x}-3465 a^2 c^2 C d^3 \sqrt{c+d x}-3465 a^2 c^2 d^2 D (c+d x)^{3/2}-693 a^2 C d^3 (c+d x)^{5/2}+2310 a^2 c C d^3 (c+d x)^{3/2}-495 a^2 d^2 D (c+d x)^{7/2}+2079 a^2 c d^2 D (c+d x)^{5/2}-2310 a A b d^4 (c+d x)^{3/2}+6930 a A b c d^4 \sqrt{c+d x}-6930 a b B c^2 d^3 \sqrt{c+d x}-1386 a b B d^3 (c+d x)^{5/2}+4620 a b B c d^3 (c+d x)^{3/2}-6930 a b c^4 d D \sqrt{c+d x}+6930 a b c^3 C d^2 \sqrt{c+d x}+9240 a b c^3 d D (c+d x)^{3/2}-6930 a b c^2 C d^2 (c+d x)^{3/2}-8316 a b c^2 d D (c+d x)^{5/2}-990 a b C d^2 (c+d x)^{7/2}+4158 a b c C d^2 (c+d x)^{5/2}-770 a b d D (c+d x)^{9/2}+3960 a b c d D (c+d x)^{7/2}-3465 A b^2 c^2 d^3 \sqrt{c+d x}-693 A b^2 d^3 (c+d x)^{5/2}+2310 A b^2 c d^3 (c+d x)^{3/2}+3465 b^2 B c^3 d^2 \sqrt{c+d x}-3465 b^2 B c^2 d^2 (c+d x)^{3/2}-495 b^2 B d^2 (c+d x)^{7/2}+2079 b^2 B c d^2 (c+d x)^{5/2}+3465 b^2 c^5 D \sqrt{c+d x}-3465 b^2 c^4 C d \sqrt{c+d x}-5775 b^2 c^4 D (c+d x)^{3/2}+4620 b^2 c^3 C d (c+d x)^{3/2}+6930 b^2 c^3 D (c+d x)^{5/2}-4158 b^2 c^2 C d (c+d x)^{5/2}-4950 b^2 c^2 D (c+d x)^{7/2}-385 b^2 C d (c+d x)^{9/2}+1980 b^2 c C d (c+d x)^{7/2}-315 b^2 D (c+d x)^{11/2}+1925 b^2 c D (c+d x)^{9/2}\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*(-3465*b^2*c^4*C*d*Sqrt[c + d*x] + 3465*b^2*B*c^3*d^2*Sqrt[c + d*x] + 6930*a*b*c^3*C*d^2*Sqrt[c + d*x] - 3465*A*b^2*c^2*d^3*Sqrt[c + d*x] - 6930*a*b*B*c^2*d^3*Sqrt[c + d*x] - 3465*a^2*c^2*C*d^3*Sqrt[c + d*x] + 6930*a*A*b*c*d^4*Sqrt[c + d*x] + 3465*a^2*B*c*d^4*Sqrt[c + d*x] - 3465*a^2*A*d^5*Sqrt[c + d*x] + 3465*b^2*c^5*D*Sqrt[c + d*x] - 6930*a*b*c^4*d*D*Sqrt[c + d*x] + 3465*a^2*c^3*d^2*D*Sqrt[c + d*x] + 4620*b^2*c^3*C*d*(c + d*x)^(3/2) - 3465*b^2*B*c^2*d^2*(c + d*x)^(3/2) - 6930*a*b*c^2*C*d^2*(c + d*x)^(3/2) + 2310*A*b^2*c*d^3*(c + d*x)^(3/2) + 4620*a*b*B*c*d^3*(c + d*x)^(3/2) + 2310*a^2*c*C*d^3*(c + d*x)^(3/2) - 2310*a*A*b*d^4*(c + d*x)^(3/2) - 1155*a^2*B*d^4*(c + d*x)^(3/2) - 5775*b^2*c^4*D*(c + d*x)^(3/2) + 9240*a*b*c^3*d*D*(c + d*x)^(3/2) - 3465*a^2*c^2*d^2*D*(c + d*x)^(3/2) - 4158*b^2*c^2*C*d*(c + d*x)^(5/2) + 2079*b^2*B*c*d^2*(c + d*x)^(5/2) + 4158*a*b*c*C*d^2*(c + d*x)^(5/2) - 693*A*b^2*d^3*(c + d*x)^(5/2) - 1386*a*b*B*d^3*(c + d*x)^(5/2) - 693*a^2*C*d^3*(c + d*x)^(5/2) + 6930*b^2*c^3*D*(c + d*x)^(5/2) - 8316*a*b*c^2*d*D*(c + d*x)^(5/2) + 2079*a^2*c*d^2*D*(c + d*x)^(5/2) + 1980*b^2*c*C*d*(c + d*x)^(7/2) - 495*b^2*B*d^2*(c + d*x)^(7/2) - 990*a*b*C*d^2*(c + d*x)^(7/2) - 4950*b^2*c^2*D*(c + d*x)^(7/2) + 3960*a*b*c*d*D*(c + d*x)^(7/2) - 495*a^2*d^2*D*(c + d*x)^(7/2) - 385*b^2*C*d*(c + d*x)^(9/2) + 1925*b^2*c*D*(c + d*x)^(9/2) - 770*a*b*d*D*(c + d*x)^(9/2) - 315*b^2*D*(c + d*x)^(11/2)))/(3465*d^6)","B",1
3,1,395,212,0.1492128,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{\sqrt{c+d x}} \, dx","IntegrateAlgebraic[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]","\frac{2 \left(315 a A d^4 \sqrt{c+d x}+105 a B d^3 (c+d x)^{3/2}-315 a B c d^3 \sqrt{c+d x}-315 a c^3 d D \sqrt{c+d x}+315 a c^2 C d^2 \sqrt{c+d x}+315 a c^2 d D (c+d x)^{3/2}+63 a C d^2 (c+d x)^{5/2}-210 a c C d^2 (c+d x)^{3/2}+45 a d D (c+d x)^{7/2}-189 a c d D (c+d x)^{5/2}+105 A b d^3 (c+d x)^{3/2}-315 A b c d^3 \sqrt{c+d x}+315 b B c^2 d^2 \sqrt{c+d x}+63 b B d^2 (c+d x)^{5/2}-210 b B c d^2 (c+d x)^{3/2}+315 b c^4 D \sqrt{c+d x}-315 b c^3 C d \sqrt{c+d x}-420 b c^3 D (c+d x)^{3/2}+315 b c^2 C d (c+d x)^{3/2}+378 b c^2 D (c+d x)^{5/2}+45 b C d (c+d x)^{7/2}-189 b c C d (c+d x)^{5/2}+35 b D (c+d x)^{9/2}-180 b c D (c+d x)^{7/2}\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*(-315*b*c^3*C*d*Sqrt[c + d*x] + 315*b*B*c^2*d^2*Sqrt[c + d*x] + 315*a*c^2*C*d^2*Sqrt[c + d*x] - 315*A*b*c*d^3*Sqrt[c + d*x] - 315*a*B*c*d^3*Sqrt[c + d*x] + 315*a*A*d^4*Sqrt[c + d*x] + 315*b*c^4*D*Sqrt[c + d*x] - 315*a*c^3*d*D*Sqrt[c + d*x] + 315*b*c^2*C*d*(c + d*x)^(3/2) - 210*b*B*c*d^2*(c + d*x)^(3/2) - 210*a*c*C*d^2*(c + d*x)^(3/2) + 105*A*b*d^3*(c + d*x)^(3/2) + 105*a*B*d^3*(c + d*x)^(3/2) - 420*b*c^3*D*(c + d*x)^(3/2) + 315*a*c^2*d*D*(c + d*x)^(3/2) - 189*b*c*C*d*(c + d*x)^(5/2) + 63*b*B*d^2*(c + d*x)^(5/2) + 63*a*C*d^2*(c + d*x)^(5/2) + 378*b*c^2*D*(c + d*x)^(5/2) - 189*a*c*d*D*(c + d*x)^(5/2) + 45*b*C*d*(c + d*x)^(7/2) - 180*b*c*D*(c + d*x)^(7/2) + 45*a*d*D*(c + d*x)^(7/2) + 35*b*D*(c + d*x)^(9/2)))/(315*d^5)","A",1
4,1,152,115,0.056311,"\int \frac{A+B x+C x^2+D x^3}{\sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/Sqrt[c + d*x],x]","-\frac{2 \left(-105 A d^3 \sqrt{c+d x}-35 B d^2 (c+d x)^{3/2}+105 B c d^2 \sqrt{c+d x}+105 c^3 D \sqrt{c+d x}-105 c^2 C d \sqrt{c+d x}-105 c^2 D (c+d x)^{3/2}-21 C d (c+d x)^{5/2}+70 c C d (c+d x)^{3/2}-15 D (c+d x)^{7/2}+63 c D (c+d x)^{5/2}\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*(-105*c^2*C*d*Sqrt[c + d*x] + 105*B*c*d^2*Sqrt[c + d*x] - 105*A*d^3*Sqrt[c + d*x] + 105*c^3*D*Sqrt[c + d*x] + 70*c*C*d*(c + d*x)^(3/2) - 35*B*d^2*(c + d*x)^(3/2) - 105*c^2*D*(c + d*x)^(3/2) - 21*C*d*(c + d*x)^(5/2) + 63*c*D*(c + d*x)^(5/2) - 15*D*(c + d*x)^(7/2)))/(105*d^4)","A",1
5,1,260,188,0.2001062,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) \sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*Sqrt[c + d*x]),x]","\frac{2 \left(15 a^2 d^2 D \sqrt{c+d x}-15 a b C d^2 \sqrt{c+d x}-5 a b d D (c+d x)^{3/2}+15 a b c d D \sqrt{c+d x}+15 b^2 B d^2 \sqrt{c+d x}+15 b^2 c^2 D \sqrt{c+d x}+5 b^2 C d (c+d x)^{3/2}-15 b^2 c C d \sqrt{c+d x}+3 b^2 D (c+d x)^{5/2}-10 b^2 c D (c+d x)^{3/2}\right)}{15 b^3 d^3}-\frac{2 \left(a^3 (-D)+a^2 b C-a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right)}{b^{7/2} \sqrt{a d-b c}}","-\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*(-15*b^2*c*C*d*Sqrt[c + d*x] + 15*b^2*B*d^2*Sqrt[c + d*x] - 15*a*b*C*d^2*Sqrt[c + d*x] + 15*b^2*c^2*D*Sqrt[c + d*x] + 15*a*b*c*d*D*Sqrt[c + d*x] + 15*a^2*d^2*D*Sqrt[c + d*x] + 5*b^2*C*d*(c + d*x)^(3/2) - 10*b^2*c*D*(c + d*x)^(3/2) - 5*a*b*d*D*(c + d*x)^(3/2) + 3*b^2*D*(c + d*x)^(5/2)))/(15*b^3*d^3) - (2*(A*b^3 - a*b^2*B + a^2*b*C - a^3*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(7/2)*Sqrt[-(b*c) + a*d])","A",1
6,1,425,201,0.3068451,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*Sqrt[c + d*x]),x]","\frac{-15 a^3 d^3 D \sqrt{c+d x}+9 a^2 b C d^3 \sqrt{c+d x}-10 a^2 b d^2 D (c+d x)^{3/2}+18 a^2 b c d^2 D \sqrt{c+d x}-3 a b^2 B d^3 \sqrt{c+d x}+6 a b^2 C d^2 (c+d x)^{3/2}-12 a b^2 c C d^2 \sqrt{c+d x}+2 a b^2 d D (c+d x)^{5/2}+2 a b^2 c d D (c+d x)^{3/2}+3 A b^3 d^3 \sqrt{c+d x}-6 b^3 c^3 D \sqrt{c+d x}+6 b^3 c^2 C d \sqrt{c+d x}+8 b^3 c^2 D (c+d x)^{3/2}-6 b^3 c C d (c+d x)^{3/2}-2 b^3 c D (c+d x)^{5/2}}{3 b^3 d^2 (b c-a d) (-a d-b (c+d x)+b c)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(-5 a^3 d D+6 a^2 b c D+3 a^2 b C d-a b^2 B d-4 a b^2 c C-A b^3 d+2 b^3 B c\right)}{b^{7/2} (a d-b c)^{3/2}}","-\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,"(6*b^3*c^2*C*d*Sqrt[c + d*x] - 12*a*b^2*c*C*d^2*Sqrt[c + d*x] + 3*A*b^3*d^3*Sqrt[c + d*x] - 3*a*b^2*B*d^3*Sqrt[c + d*x] + 9*a^2*b*C*d^3*Sqrt[c + d*x] - 6*b^3*c^3*D*Sqrt[c + d*x] + 18*a^2*b*c*d^2*D*Sqrt[c + d*x] - 15*a^3*d^3*D*Sqrt[c + d*x] - 6*b^3*c*C*d*(c + d*x)^(3/2) + 6*a*b^2*C*d^2*(c + d*x)^(3/2) + 8*b^3*c^2*D*(c + d*x)^(3/2) + 2*a*b^2*c*d*D*(c + d*x)^(3/2) - 10*a^2*b*d^2*D*(c + d*x)^(3/2) - 2*b^3*c*D*(c + d*x)^(5/2) + 2*a*b^2*d*D*(c + d*x)^(5/2))/(3*b^3*d^2*(b*c - a*d)*(b*c - a*d - b*(c + d*x))) + ((2*b^3*B*c - 4*a*b^2*c*C - A*b^3*d - a*b^2*B*d + 3*a^2*b*C*d + 6*a^2*b*c*D - 5*a^3*d*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(7/2)*(-(b*c) + a*d)^(3/2))","B",1
7,1,528,279,0.5085166,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 \sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*Sqrt[c + d*x]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(15 a^3 d^2 D-36 a^2 b c d D-3 a^2 b C d^2-a b^2 B d^2+24 a b^2 c^2 D+8 a b^2 c C d-3 A b^3 d^2+4 b^3 B c d-8 b^3 c^2 C\right)}{4 b^{7/2} (a d-b c)^{5/2}}+\frac{\sqrt{c+d x} \left(15 a^4 d^4 D+25 a^3 b d^3 D (c+d x)-51 a^3 b c d^3 D-3 a^3 b C d^4-a^2 b^2 B d^4+60 a^2 b^2 c^2 d^2 D-5 a^2 b^2 C d^3 (c+d x)+11 a^2 b^2 c C d^3+8 a^2 b^2 d^2 D (c+d x)^2-60 a^2 b^2 c d^2 D (c+d x)+5 a A b^3 d^4+a b^3 B d^3 (c+d x)-3 a b^3 B c d^3-32 a b^3 c^3 d D-8 a b^3 c^2 C d^2+48 a b^3 c^2 d D (c+d x)+8 a b^3 c C d^2 (c+d x)-16 a b^3 c d D (c+d x)^2+3 A b^4 d^3 (c+d x)-5 A b^4 c d^3+4 b^4 B c^2 d^2-4 b^4 B c d^2 (c+d x)+8 b^4 c^4 D-16 b^4 c^3 D (c+d x)+8 b^4 c^2 D (c+d x)^2\right)}{4 b^3 d (b c-a d)^2 (-a d-b (c+d x)+b c)^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,"(Sqrt[c + d*x]*(4*b^4*B*c^2*d^2 - 8*a*b^3*c^2*C*d^2 - 5*A*b^4*c*d^3 - 3*a*b^3*B*c*d^3 + 11*a^2*b^2*c*C*d^3 + 5*a*A*b^3*d^4 - a^2*b^2*B*d^4 - 3*a^3*b*C*d^4 + 8*b^4*c^4*D - 32*a*b^3*c^3*d*D + 60*a^2*b^2*c^2*d^2*D - 51*a^3*b*c*d^3*D + 15*a^4*d^4*D - 4*b^4*B*c*d^2*(c + d*x) + 8*a*b^3*c*C*d^2*(c + d*x) + 3*A*b^4*d^3*(c + d*x) + a*b^3*B*d^3*(c + d*x) - 5*a^2*b^2*C*d^3*(c + d*x) - 16*b^4*c^3*D*(c + d*x) + 48*a*b^3*c^2*d*D*(c + d*x) - 60*a^2*b^2*c*d^2*D*(c + d*x) + 25*a^3*b*d^3*D*(c + d*x) + 8*b^4*c^2*D*(c + d*x)^2 - 16*a*b^3*c*d*D*(c + d*x)^2 + 8*a^2*b^2*d^2*D*(c + d*x)^2))/(4*b^3*d*(b*c - a*d)^2*(b*c - a*d - b*(c + d*x))^2) + ((-8*b^3*c^2*C + 4*b^3*B*c*d + 8*a*b^2*c*C*d - 3*A*b^3*d^2 - a*b^2*B*d^2 - 3*a^2*b*C*d^2 + 24*a*b^2*c^2*D - 36*a^2*b*c*d*D + 15*a^3*d^2*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(4*b^(7/2)*(-(b*c) + a*d)^(5/2))","A",1
8,1,791,375,1.2385532,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*Sqrt[c + d*x]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(-5 a^3 d^3 D+18 a^2 b c d^2 D-a^2 b C d^3-a b^2 B d^3-24 a b^2 c^2 d D+4 a b^2 c C d^2-5 A b^3 d^3+6 b^3 B c d^2+16 b^3 c^3 D-8 b^3 c^2 C d\right)}{8 b^{7/2} (a d-b c)^{7/2}}+\frac{\sqrt{c+d x} \left(-15 a^5 d^5 D-40 a^4 b d^4 D (c+d x)+84 a^4 b c d^4 D-3 a^4 b C d^5-3 a^3 b^2 B d^5-195 a^3 b^2 c^2 d^3 D-8 a^3 b^2 C d^4 (c+d x)+18 a^3 b^2 c C d^4-33 a^3 b^2 d^3 D (c+d x)^2+184 a^3 b^2 c d^3 D (c+d x)+33 a^2 A b^3 d^5+8 a^2 b^3 B d^4 (c+d x)-24 a^2 b^3 B c d^4+198 a^2 b^3 c^3 d^2 D-3 a^2 b^3 c^2 C d^3-288 a^2 b^3 c^2 d^2 D (c+d x)+3 a^2 b^3 C d^3 (c+d x)^2+8 a^2 b^3 c C d^3 (c+d x)+90 a^2 b^3 c d^2 D (c+d x)^2+40 a A b^4 d^4 (c+d x)-66 a A b^4 c d^4+57 a b^4 B c^2 d^3+3 a b^4 B d^3 (c+d x)^2-56 a b^4 B c d^3 (c+d x)-72 a b^4 c^4 d D-36 a b^4 c^3 C d^2+144 a b^4 c^3 d D (c+d x)+48 a b^4 c^2 C d^2 (c+d x)-72 a b^4 c^2 d D (c+d x)^2-12 a b^4 c C d^2 (c+d x)^2+33 A b^5 c^2 d^3+15 A b^5 d^3 (c+d x)^2-40 A b^5 c d^3 (c+d x)-30 b^5 B c^3 d^2+48 b^5 B c^2 d^2 (c+d x)-18 b^5 B c d^2 (c+d x)^2+24 b^5 c^4 C d-48 b^5 c^3 C d (c+d x)+24 b^5 c^2 C d (c+d x)^2\right)}{24 b^3 (b c-a d)^3 (-a d-b (c+d x)+b c)^3}","-\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,"(Sqrt[c + d*x]*(24*b^5*c^4*C*d - 30*b^5*B*c^3*d^2 - 36*a*b^4*c^3*C*d^2 + 33*A*b^5*c^2*d^3 + 57*a*b^4*B*c^2*d^3 - 3*a^2*b^3*c^2*C*d^3 - 66*a*A*b^4*c*d^4 - 24*a^2*b^3*B*c*d^4 + 18*a^3*b^2*c*C*d^4 + 33*a^2*A*b^3*d^5 - 3*a^3*b^2*B*d^5 - 3*a^4*b*C*d^5 - 72*a*b^4*c^4*d*D + 198*a^2*b^3*c^3*d^2*D - 195*a^3*b^2*c^2*d^3*D + 84*a^4*b*c*d^4*D - 15*a^5*d^5*D - 48*b^5*c^3*C*d*(c + d*x) + 48*b^5*B*c^2*d^2*(c + d*x) + 48*a*b^4*c^2*C*d^2*(c + d*x) - 40*A*b^5*c*d^3*(c + d*x) - 56*a*b^4*B*c*d^3*(c + d*x) + 8*a^2*b^3*c*C*d^3*(c + d*x) + 40*a*A*b^4*d^4*(c + d*x) + 8*a^2*b^3*B*d^4*(c + d*x) - 8*a^3*b^2*C*d^4*(c + d*x) + 144*a*b^4*c^3*d*D*(c + d*x) - 288*a^2*b^3*c^2*d^2*D*(c + d*x) + 184*a^3*b^2*c*d^3*D*(c + d*x) - 40*a^4*b*d^4*D*(c + d*x) + 24*b^5*c^2*C*d*(c + d*x)^2 - 18*b^5*B*c*d^2*(c + d*x)^2 - 12*a*b^4*c*C*d^2*(c + d*x)^2 + 15*A*b^5*d^3*(c + d*x)^2 + 3*a*b^4*B*d^3*(c + d*x)^2 + 3*a^2*b^3*C*d^3*(c + d*x)^2 - 72*a*b^4*c^2*d*D*(c + d*x)^2 + 90*a^2*b^3*c*d^2*D*(c + d*x)^2 - 33*a^3*b^2*d^3*D*(c + d*x)^2))/(24*b^3*(b*c - a*d)^3*(b*c - a*d - b*(c + d*x))^3) + ((-8*b^3*c^2*C*d + 6*b^3*B*c*d^2 + 4*a*b^2*c*C*d^2 - 5*A*b^3*d^3 - a*b^2*B*d^3 - a^2*b*C*d^3 + 16*b^3*c^3*D - 24*a*b^2*c^2*d*D + 18*a^2*b*c*d^2*D - 5*a^3*d^3*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(8*b^(7/2)*(-(b*c) + a*d)^(7/2))","B",1
9,1,1256,495,1.9531429,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt{c+d x}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^5*Sqrt[c + d*x]),x]","\frac{\left(-35 A b^3 d^4-5 a b^2 B d^4-3 a^2 b C d^4-5 a^3 D d^4+40 b^3 B c d^3+16 a b^2 c C d^3+24 a^2 b c D d^3-48 b^3 c^2 C d^2-48 a b^2 c^2 D d^2+64 b^3 c^3 D d\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{a d-b c} \sqrt{c+d x}}{b c-a d}\right)}{64 b^{7/2} (a d-b c)^{9/2}}-\frac{\sqrt{c+d x} \left(-279 a^3 A b^3 d^7+15 a^4 b^2 B d^7+9 a^5 b C d^7+15 a^6 D d^7+837 a^2 A b^4 c d^6+219 a^3 b^3 B c d^6-75 a^4 b^2 c C d^6-117 a^5 b c D d^6-511 a^2 A b^4 (c+d x) d^6-73 a^3 b^3 B (c+d x) d^6+33 a^4 b^2 C (c+d x) d^6+55 a^5 b D (c+d x) d^6-837 a A b^5 c^2 d^5-747 a^2 b^4 B c^2 d^5-385 a A b^5 (c+d x)^2 d^5-55 a^2 b^4 B (c+d x)^2 d^5-33 a^3 b^3 C (c+d x)^2 d^5+73 a^4 b^2 D (c+d x)^2 d^5-69 a^3 b^3 c^2 C d^5+405 a^4 b^2 c^2 D d^5+1022 a A b^5 c (c+d x) d^5+730 a^2 b^4 B c (c+d x) d^5+14 a^3 b^3 c C (c+d x) d^5-374 a^4 b^2 c D (c+d x) d^5+279 A b^6 c^3 d^4+777 a b^5 B c^3 d^4-105 A b^6 (c+d x)^3 d^4-15 a b^5 B (c+d x)^3 d^4-9 a^2 b^4 C (c+d x)^3 d^4-15 a^3 b^3 D (c+d x)^3 d^4+385 A b^6 c (c+d x)^2 d^4+495 a b^5 B c (c+d x)^2 d^4+209 a^2 b^4 c C (c+d x)^2 d^4-193 a^3 b^3 c D (c+d x)^2 d^4+567 a^2 b^4 c^3 C d^4-471 a^3 b^3 c^3 D d^4-511 A b^6 c^2 (c+d x) d^4-1241 a b^5 B c^2 (c+d x) d^4-751 a^2 b^4 c^2 C (c+d x) d^4+727 a^3 b^3 c^2 D (c+d x) d^4-264 b^6 B c^4 d^3+120 b^6 B c (c+d x)^3 d^3+48 a b^5 c C (c+d x)^3 d^3+72 a^2 b^4 c D (c+d x)^3 d^3-440 b^6 B c^2 (c+d x)^2 d^3-704 a b^5 c^2 C (c+d x)^2 d^3-24 a^2 b^4 c^2 D (c+d x)^2 d^3-672 a b^5 c^4 C d^3-72 a^2 b^4 c^4 D d^3+584 b^6 B c^3 (c+d x) d^3+1328 a b^5 c^3 C (c+d x) d^3+24 a^2 b^4 c^3 D (c+d x) d^3-144 b^6 c^2 C (c+d x)^3 d^2-144 a b^5 c^2 D (c+d x)^3 d^2+528 b^6 c^3 C (c+d x)^2 d^2+720 a b^5 c^3 D (c+d x)^2 d^2+240 b^6 c^5 C d^2+432 a b^5 c^5 D d^2-624 b^6 c^4 C (c+d x) d^2-1008 a b^5 c^4 D (c+d x) d^2+192 b^6 c^3 D (c+d x)^3 d-576 b^6 c^4 D (c+d x)^2 d-192 b^6 c^6 D d+576 b^6 c^5 D (c+d x) d\right)}{192 b^3 (b c-a d)^4 (b c-a d-b (c+d x))^4}","-\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,"-1/192*(Sqrt[c + d*x]*(240*b^6*c^5*C*d^2 - 264*b^6*B*c^4*d^3 - 672*a*b^5*c^4*C*d^3 + 279*A*b^6*c^3*d^4 + 777*a*b^5*B*c^3*d^4 + 567*a^2*b^4*c^3*C*d^4 - 837*a*A*b^5*c^2*d^5 - 747*a^2*b^4*B*c^2*d^5 - 69*a^3*b^3*c^2*C*d^5 + 837*a^2*A*b^4*c*d^6 + 219*a^3*b^3*B*c*d^6 - 75*a^4*b^2*c*C*d^6 - 279*a^3*A*b^3*d^7 + 15*a^4*b^2*B*d^7 + 9*a^5*b*C*d^7 - 192*b^6*c^6*d*D + 432*a*b^5*c^5*d^2*D - 72*a^2*b^4*c^4*d^3*D - 471*a^3*b^3*c^3*d^4*D + 405*a^4*b^2*c^2*d^5*D - 117*a^5*b*c*d^6*D + 15*a^6*d^7*D - 624*b^6*c^4*C*d^2*(c + d*x) + 584*b^6*B*c^3*d^3*(c + d*x) + 1328*a*b^5*c^3*C*d^3*(c + d*x) - 511*A*b^6*c^2*d^4*(c + d*x) - 1241*a*b^5*B*c^2*d^4*(c + d*x) - 751*a^2*b^4*c^2*C*d^4*(c + d*x) + 1022*a*A*b^5*c*d^5*(c + d*x) + 730*a^2*b^4*B*c*d^5*(c + d*x) + 14*a^3*b^3*c*C*d^5*(c + d*x) - 511*a^2*A*b^4*d^6*(c + d*x) - 73*a^3*b^3*B*d^6*(c + d*x) + 33*a^4*b^2*C*d^6*(c + d*x) + 576*b^6*c^5*d*D*(c + d*x) - 1008*a*b^5*c^4*d^2*D*(c + d*x) + 24*a^2*b^4*c^3*d^3*D*(c + d*x) + 727*a^3*b^3*c^2*d^4*D*(c + d*x) - 374*a^4*b^2*c*d^5*D*(c + d*x) + 55*a^5*b*d^6*D*(c + d*x) + 528*b^6*c^3*C*d^2*(c + d*x)^2 - 440*b^6*B*c^2*d^3*(c + d*x)^2 - 704*a*b^5*c^2*C*d^3*(c + d*x)^2 + 385*A*b^6*c*d^4*(c + d*x)^2 + 495*a*b^5*B*c*d^4*(c + d*x)^2 + 209*a^2*b^4*c*C*d^4*(c + d*x)^2 - 385*a*A*b^5*d^5*(c + d*x)^2 - 55*a^2*b^4*B*d^5*(c + d*x)^2 - 33*a^3*b^3*C*d^5*(c + d*x)^2 - 576*b^6*c^4*d*D*(c + d*x)^2 + 720*a*b^5*c^3*d^2*D*(c + d*x)^2 - 24*a^2*b^4*c^2*d^3*D*(c + d*x)^2 - 193*a^3*b^3*c*d^4*D*(c + d*x)^2 + 73*a^4*b^2*d^5*D*(c + d*x)^2 - 144*b^6*c^2*C*d^2*(c + d*x)^3 + 120*b^6*B*c*d^3*(c + d*x)^3 + 48*a*b^5*c*C*d^3*(c + d*x)^3 - 105*A*b^6*d^4*(c + d*x)^3 - 15*a*b^5*B*d^4*(c + d*x)^3 - 9*a^2*b^4*C*d^4*(c + d*x)^3 + 192*b^6*c^3*d*D*(c + d*x)^3 - 144*a*b^5*c^2*d^2*D*(c + d*x)^3 + 72*a^2*b^4*c*d^3*D*(c + d*x)^3 - 15*a^3*b^3*d^4*D*(c + d*x)^3))/(b^3*(b*c - a*d)^4*(b*c - a*d - b*(c + d*x))^4) + ((-48*b^3*c^2*C*d^2 + 40*b^3*B*c*d^3 + 16*a*b^2*c*C*d^3 - 35*A*b^3*d^4 - 5*a*b^2*B*d^4 - 3*a^2*b*C*d^4 + 64*b^3*c^3*d*D - 48*a*b^2*c^2*d^2*D + 24*a^2*b*c*d^3*D - 5*a^3*d^4*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(64*b^(7/2)*(-(b*c) + a*d)^(9/2))","B",1
10,1,1000,434,0.4915044,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 \left(-3465 b^3 D c^6+3465 b^3 C d c^5+10395 a b^2 d D c^5-20790 b^3 D (c+d x) c^5-3465 b^3 B d^2 c^4-10395 a b^2 C d^2 c^4+17325 b^3 D (c+d x)^2 c^4-10395 a^2 b d^2 D c^4+17325 b^3 C d (c+d x) c^4+51975 a b^2 d D (c+d x) c^4+3465 A b^3 d^3 c^3+10395 a b^2 B d^3 c^3+10395 a^2 b C d^3 c^3-13860 b^3 D (c+d x)^3 c^3-11550 b^3 C d (c+d x)^2 c^3-34650 a b^2 d D (c+d x)^2 c^3+3465 a^3 d^3 D c^3-13860 b^3 B d^2 (c+d x) c^3-41580 a b^2 C d^2 (c+d x) c^3-41580 a^2 b d^2 D (c+d x) c^3-10395 a A b^2 d^4 c^2-10395 a^2 b B d^4 c^2-3465 a^3 C d^4 c^2+7425 b^3 D (c+d x)^4 c^2+6930 b^3 C d (c+d x)^3 c^2+20790 a b^2 d D (c+d x)^3 c^2+6930 b^3 B d^2 (c+d x)^2 c^2+20790 a b^2 C d^2 (c+d x)^2 c^2+20790 a^2 b d^2 D (c+d x)^2 c^2+10395 A b^3 d^3 (c+d x) c^2+31185 a b^2 B d^3 (c+d x) c^2+31185 a^2 b C d^3 (c+d x) c^2+10395 a^3 d^3 D (c+d x) c^2+10395 a^2 A b d^5 c+3465 a^3 B d^5 c-2310 b^3 D (c+d x)^5 c-2475 b^3 C d (c+d x)^4 c-7425 a b^2 d D (c+d x)^4 c-2772 b^3 B d^2 (c+d x)^3 c-8316 a b^2 C d^2 (c+d x)^3 c-8316 a^2 b d^2 D (c+d x)^3 c-3465 A b^3 d^3 (c+d x)^2 c-10395 a b^2 B d^3 (c+d x)^2 c-10395 a^2 b C d^3 (c+d x)^2 c-3465 a^3 d^3 D (c+d x)^2 c-20790 a A b^2 d^4 (c+d x) c-20790 a^2 b B d^4 (c+d x) c-6930 a^3 C d^4 (c+d x) c-3465 a^3 A d^6+315 b^3 D (c+d x)^6+385 b^3 C d (c+d x)^5+1155 a b^2 d D (c+d x)^5+495 b^3 B d^2 (c+d x)^4+1485 a b^2 C d^2 (c+d x)^4+1485 a^2 b d^2 D (c+d x)^4+693 A b^3 d^3 (c+d x)^3+2079 a b^2 B d^3 (c+d x)^3+2079 a^2 b C d^3 (c+d x)^3+693 a^3 d^3 D (c+d x)^3+3465 a A b^2 d^4 (c+d x)^2+3465 a^2 b B d^4 (c+d x)^2+1155 a^3 C d^4 (c+d x)^2+10395 a^2 A b d^5 (c+d x)+3465 a^3 B d^5 (c+d x)\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^3*c^5*C*d - 3465*b^3*B*c^4*d^2 - 10395*a*b^2*c^4*C*d^2 + 3465*A*b^3*c^3*d^3 + 10395*a*b^2*B*c^3*d^3 + 10395*a^2*b*c^3*C*d^3 - 10395*a*A*b^2*c^2*d^4 - 10395*a^2*b*B*c^2*d^4 - 3465*a^3*c^2*C*d^4 + 10395*a^2*A*b*c*d^5 + 3465*a^3*B*c*d^5 - 3465*a^3*A*d^6 - 3465*b^3*c^6*D + 10395*a*b^2*c^5*d*D - 10395*a^2*b*c^4*d^2*D + 3465*a^3*c^3*d^3*D + 17325*b^3*c^4*C*d*(c + d*x) - 13860*b^3*B*c^3*d^2*(c + d*x) - 41580*a*b^2*c^3*C*d^2*(c + d*x) + 10395*A*b^3*c^2*d^3*(c + d*x) + 31185*a*b^2*B*c^2*d^3*(c + d*x) + 31185*a^2*b*c^2*C*d^3*(c + d*x) - 20790*a*A*b^2*c*d^4*(c + d*x) - 20790*a^2*b*B*c*d^4*(c + d*x) - 6930*a^3*c*C*d^4*(c + d*x) + 10395*a^2*A*b*d^5*(c + d*x) + 3465*a^3*B*d^5*(c + d*x) - 20790*b^3*c^5*D*(c + d*x) + 51975*a*b^2*c^4*d*D*(c + d*x) - 41580*a^2*b*c^3*d^2*D*(c + d*x) + 10395*a^3*c^2*d^3*D*(c + d*x) - 11550*b^3*c^3*C*d*(c + d*x)^2 + 6930*b^3*B*c^2*d^2*(c + d*x)^2 + 20790*a*b^2*c^2*C*d^2*(c + d*x)^2 - 3465*A*b^3*c*d^3*(c + d*x)^2 - 10395*a*b^2*B*c*d^3*(c + d*x)^2 - 10395*a^2*b*c*C*d^3*(c + d*x)^2 + 3465*a*A*b^2*d^4*(c + d*x)^2 + 3465*a^2*b*B*d^4*(c + d*x)^2 + 1155*a^3*C*d^4*(c + d*x)^2 + 17325*b^3*c^4*D*(c + d*x)^2 - 34650*a*b^2*c^3*d*D*(c + d*x)^2 + 20790*a^2*b*c^2*d^2*D*(c + d*x)^2 - 3465*a^3*c*d^3*D*(c + d*x)^2 + 6930*b^3*c^2*C*d*(c + d*x)^3 - 2772*b^3*B*c*d^2*(c + d*x)^3 - 8316*a*b^2*c*C*d^2*(c + d*x)^3 + 693*A*b^3*d^3*(c + d*x)^3 + 2079*a*b^2*B*d^3*(c + d*x)^3 + 2079*a^2*b*C*d^3*(c + d*x)^3 - 13860*b^3*c^3*D*(c + d*x)^3 + 20790*a*b^2*c^2*d*D*(c + d*x)^3 - 8316*a^2*b*c*d^2*D*(c + d*x)^3 + 693*a^3*d^3*D*(c + d*x)^3 - 2475*b^3*c*C*d*(c + d*x)^4 + 495*b^3*B*d^2*(c + d*x)^4 + 1485*a*b^2*C*d^2*(c + d*x)^4 + 7425*b^3*c^2*D*(c + d*x)^4 - 7425*a*b^2*c*d*D*(c + d*x)^4 + 1485*a^2*b*d^2*D*(c + d*x)^4 + 385*b^3*C*d*(c + d*x)^5 - 2310*b^3*c*D*(c + d*x)^5 + 1155*a*b^2*d*D*(c + d*x)^5 + 315*b^3*D*(c + d*x)^6))/(3465*d^7*Sqrt[c + d*x])","B",1
11,1,599,322,0.3020312,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 \left(-315 a^2 A d^5+315 a^2 B d^4 (c+d x)+315 a^2 B c d^4+315 a^2 c^3 d^2 D-315 a^2 c^2 C d^3+945 a^2 c^2 d^2 D (c+d x)-630 a^2 c C d^3 (c+d x)+105 a^2 C d^3 (c+d x)^2-315 a^2 c d^2 D (c+d x)^2+63 a^2 d^2 D (c+d x)^3+630 a A b d^4 (c+d x)+630 a A b c d^4-630 a b B c^2 d^3-1260 a b B c d^3 (c+d x)+210 a b B d^3 (c+d x)^2-630 a b c^4 d D+630 a b c^3 C d^2-2520 a b c^3 d D (c+d x)+1890 a b c^2 C d^2 (c+d x)+1260 a b c^2 d D (c+d x)^2-630 a b c C d^2 (c+d x)^2+126 a b C d^2 (c+d x)^3-504 a b c d D (c+d x)^3+90 a b d D (c+d x)^4-315 A b^2 c^2 d^3-630 A b^2 c d^3 (c+d x)+105 A b^2 d^3 (c+d x)^2+315 b^2 B c^3 d^2+945 b^2 B c^2 d^2 (c+d x)-315 b^2 B c d^2 (c+d x)^2+63 b^2 B d^2 (c+d x)^3+315 b^2 c^5 D-315 b^2 c^4 C d+1575 b^2 c^4 D (c+d x)-1260 b^2 c^3 C d (c+d x)-1050 b^2 c^3 D (c+d x)^2+630 b^2 c^2 C d (c+d x)^2+630 b^2 c^2 D (c+d x)^3-252 b^2 c C d (c+d x)^3+45 b^2 C d (c+d x)^4-225 b^2 c D (c+d x)^4+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^2*c^4*C*d + 315*b^2*B*c^3*d^2 + 630*a*b*c^3*C*d^2 - 315*A*b^2*c^2*d^3 - 630*a*b*B*c^2*d^3 - 315*a^2*c^2*C*d^3 + 630*a*A*b*c*d^4 + 315*a^2*B*c*d^4 - 315*a^2*A*d^5 + 315*b^2*c^5*D - 630*a*b*c^4*d*D + 315*a^2*c^3*d^2*D - 1260*b^2*c^3*C*d*(c + d*x) + 945*b^2*B*c^2*d^2*(c + d*x) + 1890*a*b*c^2*C*d^2*(c + d*x) - 630*A*b^2*c*d^3*(c + d*x) - 1260*a*b*B*c*d^3*(c + d*x) - 630*a^2*c*C*d^3*(c + d*x) + 630*a*A*b*d^4*(c + d*x) + 315*a^2*B*d^4*(c + d*x) + 1575*b^2*c^4*D*(c + d*x) - 2520*a*b*c^3*d*D*(c + d*x) + 945*a^2*c^2*d^2*D*(c + d*x) + 630*b^2*c^2*C*d*(c + d*x)^2 - 315*b^2*B*c*d^2*(c + d*x)^2 - 630*a*b*c*C*d^2*(c + d*x)^2 + 105*A*b^2*d^3*(c + d*x)^2 + 210*a*b*B*d^3*(c + d*x)^2 + 105*a^2*C*d^3*(c + d*x)^2 - 1050*b^2*c^3*D*(c + d*x)^2 + 1260*a*b*c^2*d*D*(c + d*x)^2 - 315*a^2*c*d^2*D*(c + d*x)^2 - 252*b^2*c*C*d*(c + d*x)^3 + 63*b^2*B*d^2*(c + d*x)^3 + 126*a*b*C*d^2*(c + d*x)^3 + 630*b^2*c^2*D*(c + d*x)^3 - 504*a*b*c*d*D*(c + d*x)^3 + 63*a^2*d^2*D*(c + d*x)^3 + 45*b^2*C*d*(c + d*x)^4 - 225*b^2*c*D*(c + d*x)^4 + 90*a*b*d*D*(c + d*x)^4 + 35*b^2*D*(c + d*x)^5))/(315*d^6*Sqrt[c + d*x])","A",1
12,1,286,210,0.214624,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{3/2}} \, dx","IntegrateAlgebraic[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]","\frac{2 \left(-105 a A d^4+105 a B d^3 (c+d x)+105 a B c d^3+105 a c^3 d D-105 a c^2 C d^2+315 a c^2 d D (c+d x)-210 a c C d^2 (c+d x)+35 a C d^2 (c+d x)^2-105 a c d D (c+d x)^2+21 a d D (c+d x)^3+105 A b d^3 (c+d x)+105 A b c d^3-105 b B c^2 d^2-210 b B c d^2 (c+d x)+35 b B d^2 (c+d x)^2-105 b c^4 D+105 b c^3 C d-420 b c^3 D (c+d x)+315 b c^2 C d (c+d x)+210 b c^2 D (c+d x)^2-105 b c C d (c+d x)^2+21 b C d (c+d x)^3-84 b c D (c+d x)^3+15 b D (c+d x)^4\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,"(2*(105*b*c^3*C*d - 105*b*B*c^2*d^2 - 105*a*c^2*C*d^2 + 105*A*b*c*d^3 + 105*a*B*c*d^3 - 105*a*A*d^4 - 105*b*c^4*D + 105*a*c^3*d*D + 315*b*c^2*C*d*(c + d*x) - 210*b*B*c*d^2*(c + d*x) - 210*a*c*C*d^2*(c + d*x) + 105*A*b*d^3*(c + d*x) + 105*a*B*d^3*(c + d*x) - 420*b*c^3*D*(c + d*x) + 315*a*c^2*d*D*(c + d*x) - 105*b*c*C*d*(c + d*x)^2 + 35*b*B*d^2*(c + d*x)^2 + 35*a*C*d^2*(c + d*x)^2 + 210*b*c^2*D*(c + d*x)^2 - 105*a*c*d*D*(c + d*x)^2 + 21*b*C*d*(c + d*x)^3 - 84*b*c*D*(c + d*x)^3 + 21*a*d*D*(c + d*x)^3 + 15*b*D*(c + d*x)^4))/(105*d^5*Sqrt[c + d*x])","A",1
13,1,107,113,0.075935,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(3/2),x]","\frac{2 \left(-15 A d^3+15 B d^2 (c+d x)+15 B c d^2+15 c^3 D-15 c^2 C d+45 c^2 D (c+d x)-30 c C d (c+d x)+5 C d (c+d x)^2-15 c D (c+d x)^2+3 D (c+d x)^3\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*(-15*c^2*C*d + 15*B*c*d^2 - 15*A*d^3 + 15*c^3*D - 30*c*C*d*(c + d*x) + 15*B*d^2*(c + d*x) + 45*c^2*D*(c + d*x) + 5*C*d*(c + d*x)^2 - 15*c*D*(c + d*x)^2 + 3*D*(c + d*x)^3))/(15*d^4*Sqrt[c + d*x])","A",1
14,1,243,193,0.219295,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(3/2)),x]","\frac{2 \left(3 a^2 d^2 D (c+d x)-3 a b C d^2 (c+d x)+3 a b c d D (c+d x)-a b d D (c+d x)^2+3 A b^2 d^3-3 b^2 B c d^2-3 b^2 c^3 D+3 b^2 c^2 C d-6 b^2 c^2 D (c+d x)+3 b^2 c C d (c+d x)+b^2 c D (c+d x)^2\right)}{3 b^2 d^3 \sqrt{c+d x} (b c-a d)}+\frac{2 \left(a^3 (-D)+a^2 b C-a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right)}{b^{5/2} (a d-b c)^{3/2}}","-\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*(3*b^2*c^2*C*d - 3*b^2*B*c*d^2 + 3*A*b^2*d^3 - 3*b^2*c^3*D + 3*b^2*c*C*d*(c + d*x) - 3*a*b*C*d^2*(c + d*x) - 6*b^2*c^2*D*(c + d*x) + 3*a*b*c*d*D*(c + d*x) + 3*a^2*d^2*D*(c + d*x) + b^2*c*D*(c + d*x)^2 - a*b*d*D*(c + d*x)^2))/(3*b^2*d^3*(b*c - a*d)*Sqrt[c + d*x]) + (2*(A*b^3 - a*b^2*B + a^2*b*C - a^3*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(5/2)*(-(b*c) + a*d)^(3/2))","A",1
15,1,409,253,0.3820259,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(3/2)),x]","\frac{3 a^3 d^3 D (c+d x)-a^2 b C d^3 (c+d x)-6 a^2 b c d^2 D (c+d x)+2 a^2 b d^2 D (c+d x)^2-2 a A b^2 d^4+a b^2 B d^3 (c+d x)+2 a b^2 B c d^3+2 a b^2 c^3 d D-2 a b^2 c^2 C d^2+6 a b^2 c^2 d D (c+d x)-4 a b^2 c d D (c+d x)^2-3 A b^3 d^3 (c+d x)+2 A b^3 c d^3-2 b^3 B c^2 d^2+2 b^3 B c d^2 (c+d x)-2 b^3 c^4 D+2 b^3 c^3 C d-2 b^3 c^2 C d (c+d x)+2 b^3 c^2 D (c+d x)^2}{b^2 d^2 \sqrt{c+d x} (b c-a d)^2 (a d+b (c+d x)-b c)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(3 a^3 d D-6 a^2 b c D-a^2 b C d-a b^2 B d+4 a b^2 c C+3 A b^3 d-2 b^3 B c\right)}{b^{5/2} (a d-b c)^{5/2}}","-\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*b^3*c^3*C*d - 2*b^3*B*c^2*d^2 - 2*a*b^2*c^2*C*d^2 + 2*A*b^3*c*d^3 + 2*a*b^2*B*c*d^3 - 2*a*A*b^2*d^4 - 2*b^3*c^4*D + 2*a*b^2*c^3*d*D - 2*b^3*c^2*C*d*(c + d*x) + 2*b^3*B*c*d^2*(c + d*x) - 3*A*b^3*d^3*(c + d*x) + a*b^2*B*d^3*(c + d*x) - a^2*b*C*d^3*(c + d*x) + 6*a*b^2*c^2*d*D*(c + d*x) - 6*a^2*b*c*d^2*D*(c + d*x) + 3*a^3*d^3*D*(c + d*x) + 2*b^3*c^2*D*(c + d*x)^2 - 4*a*b^2*c*d*D*(c + d*x)^2 + 2*a^2*b*d^2*D*(c + d*x)^2)/(b^2*d^2*(b*c - a*d)^2*Sqrt[c + d*x]*(-(b*c) + a*d + b*(c + d*x))) + ((-2*b^3*B*c + 4*a*b^2*c*C + 3*A*b^3*d - a*b^2*B*d - a^2*b*C*d - 6*a^2*b*c*D + 3*a^3*d*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(5/2)*(-(b*c) + a*d)^(5/2))","A",1
16,1,719,350,0.9707793,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(3/2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(-3 a^3 d^2 D+12 a^2 b c d D-a^2 b C d^2-3 a b^2 B d^2-24 a b^2 c^2 D+8 a b^2 c C d+15 A b^3 d^2-12 b^3 B c d+8 b^3 c^2 C\right)}{4 b^{5/2} (a d-b c)^{7/2}}+\frac{3 a^4 d^4 D (c+d x)+a^3 b C d^4 (c+d x)-15 a^3 b c d^3 D (c+d x)+5 a^3 b d^3 D (c+d x)^2+8 a^2 A b^2 d^5-5 a^2 b^2 B d^4 (c+d x)-8 a^2 b^2 B c d^4-8 a^2 b^2 c^3 d^2 D+8 a^2 b^2 c^2 C d^3+12 a^2 b^2 c^2 d^2 D (c+d x)+7 a^2 b^2 c C d^3 (c+d x)-a^2 b^2 C d^3 (c+d x)^2-12 a^2 b^2 c d^2 D (c+d x)^2+25 a A b^3 d^4 (c+d x)-16 a A b^3 c d^4+16 a b^3 B c^2 d^3-15 a b^3 B c d^3 (c+d x)-3 a b^3 B d^3 (c+d x)^2+16 a b^3 c^4 d D-16 a b^3 c^3 C d^2-16 a b^3 c^3 d D (c+d x)+8 a b^3 c^2 C d^2 (c+d x)+8 a b^3 c C d^2 (c+d x)^2+8 A b^4 c^2 d^3-25 A b^4 c d^3 (c+d x)+15 A b^4 d^3 (c+d x)^2-8 b^4 B c^3 d^2+20 b^4 B c^2 d^2 (c+d x)-12 b^4 B c d^2 (c+d x)^2-8 b^4 c^5 D+8 b^4 c^4 C d+16 b^4 c^4 D (c+d x)-16 b^4 c^3 C d (c+d x)-8 b^4 c^3 D (c+d x)^2+8 b^4 c^2 C d (c+d x)^2}{4 b^2 d \sqrt{c+d x} (b c-a d)^3 (a d+b (c+d x)-b c)^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{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,"(8*b^4*c^4*C*d - 8*b^4*B*c^3*d^2 - 16*a*b^3*c^3*C*d^2 + 8*A*b^4*c^2*d^3 + 16*a*b^3*B*c^2*d^3 + 8*a^2*b^2*c^2*C*d^3 - 16*a*A*b^3*c*d^4 - 8*a^2*b^2*B*c*d^4 + 8*a^2*A*b^2*d^5 - 8*b^4*c^5*D + 16*a*b^3*c^4*d*D - 8*a^2*b^2*c^3*d^2*D - 16*b^4*c^3*C*d*(c + d*x) + 20*b^4*B*c^2*d^2*(c + d*x) + 8*a*b^3*c^2*C*d^2*(c + d*x) - 25*A*b^4*c*d^3*(c + d*x) - 15*a*b^3*B*c*d^3*(c + d*x) + 7*a^2*b^2*c*C*d^3*(c + d*x) + 25*a*A*b^3*d^4*(c + d*x) - 5*a^2*b^2*B*d^4*(c + d*x) + a^3*b*C*d^4*(c + d*x) + 16*b^4*c^4*D*(c + d*x) - 16*a*b^3*c^3*d*D*(c + d*x) + 12*a^2*b^2*c^2*d^2*D*(c + d*x) - 15*a^3*b*c*d^3*D*(c + d*x) + 3*a^4*d^4*D*(c + d*x) + 8*b^4*c^2*C*d*(c + d*x)^2 - 12*b^4*B*c*d^2*(c + d*x)^2 + 8*a*b^3*c*C*d^2*(c + d*x)^2 + 15*A*b^4*d^3*(c + d*x)^2 - 3*a*b^3*B*d^3*(c + d*x)^2 - a^2*b^2*C*d^3*(c + d*x)^2 - 8*b^4*c^3*D*(c + d*x)^2 - 12*a^2*b^2*c*d^2*D*(c + d*x)^2 + 5*a^3*b*d^3*D*(c + d*x)^2)/(4*b^2*d*(b*c - a*d)^3*Sqrt[c + d*x]*(-(b*c) + a*d + b*(c + d*x))^2) + ((8*b^3*c^2*C - 12*b^3*B*c*d + 8*a*b^2*c*C*d + 15*A*b^3*d^2 - 3*a*b^2*B*d^2 - a^2*b*C*d^2 - 24*a*b^2*c^2*D + 12*a^2*b*c*d*D - 3*a^3*d^2*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(4*b^(5/2)*(-(b*c) + a*d)^(7/2))","B",1
17,1,1135,463,1.6706667,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*(c + d*x)^(3/2)),x]","\frac{-48 b^5 D c^6+48 b^5 C d c^5+144 a b^4 d D c^5+144 b^5 D (c+d x) c^5-48 b^5 B d^2 c^4-144 a b^4 C d^2 c^4-144 b^5 D (c+d x)^2 c^4-144 a^2 b^3 d^2 D c^4-168 b^5 C d (c+d x) c^4-216 a b^4 d D (c+d x) c^4+48 A b^5 d^3 c^3+144 a b^4 B d^3 c^3+144 a^2 b^3 C d^3 c^3+48 b^5 D (c+d x)^3 c^3+192 b^5 C d (c+d x)^2 c^3+48 a^3 b^2 d^3 D c^3+198 b^5 B d^2 (c+d x) c^3+276 a b^4 C d^2 (c+d x) c^3+18 a^2 b^3 d^2 D (c+d x) c^3-144 a A b^4 d^4 c^2-144 a^2 b^3 B d^4 c^2-48 a^3 b^2 C d^4 c^2-72 b^5 C d (c+d x)^3 c^2+72 a b^4 d D (c+d x)^3 c^2-240 b^5 B d^2 (c+d x)^2 c^2-96 a b^4 C d^2 (c+d x)^2 c^2+144 a^2 b^3 d^2 D (c+d x)^2 c^2-231 A b^5 d^3 (c+d x) c^2-363 a b^4 B d^3 (c+d x) c^2-51 a^2 b^3 C d^3 (c+d x) c^2+33 a^3 b^2 d^3 D (c+d x) c^2+144 a^2 A b^3 d^5 c+48 a^3 b^2 B d^5 c+90 b^5 B d^2 (c+d x)^3 c-36 a b^4 C d^2 (c+d x)^3 c-18 a^2 b^3 d^2 D (c+d x)^3 c+280 A b^5 d^3 (c+d x)^2 c+200 a b^4 B d^3 (c+d x)^2 c-104 a^2 b^3 C d^3 (c+d x)^2 c+8 a^3 b^2 d^3 D (c+d x)^2 c+462 a A b^4 d^4 (c+d x) c+132 a^2 b^3 B d^4 (c+d x) c-54 a^3 b^2 C d^4 (c+d x) c+24 a^4 b d^4 D (c+d x) c-48 a^3 A b^2 d^6-105 A b^5 d^3 (c+d x)^3+15 a b^4 B d^3 (c+d x)^3+3 a^2 b^3 C d^3 (c+d x)^3+3 a^3 b^2 d^3 D (c+d x)^3-280 a A b^4 d^4 (c+d x)^2+40 a^2 b^3 B d^4 (c+d x)^2+8 a^3 b^2 C d^4 (c+d x)^2-8 a^4 b d^4 D (c+d x)^2-231 a^2 A b^3 d^5 (c+d x)+33 a^3 b^2 B d^5 (c+d x)-3 a^4 b C d^5 (c+d x)-3 a^5 d^5 D (c+d x)}{24 b^2 (b c-a d)^4 \sqrt{c+d x} (-b c+a d+b (c+d x))^3}+\frac{\left(35 A d^3 b^3-30 B c d^2 b^3+24 c^2 C d b^3-16 c^3 D b^3-5 a B d^3 b^2+12 a c C d^2 b^2-24 a c^2 d D b^2-a^2 C d^3 b+6 a^2 c d^2 D b-a^3 d^3 D\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{a d-b c} \sqrt{c+d x}}{b c-a d}\right)}{8 b^{5/2} (a d-b c)^{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,"(48*b^5*c^5*C*d - 48*b^5*B*c^4*d^2 - 144*a*b^4*c^4*C*d^2 + 48*A*b^5*c^3*d^3 + 144*a*b^4*B*c^3*d^3 + 144*a^2*b^3*c^3*C*d^3 - 144*a*A*b^4*c^2*d^4 - 144*a^2*b^3*B*c^2*d^4 - 48*a^3*b^2*c^2*C*d^4 + 144*a^2*A*b^3*c*d^5 + 48*a^3*b^2*B*c*d^5 - 48*a^3*A*b^2*d^6 - 48*b^5*c^6*D + 144*a*b^4*c^5*d*D - 144*a^2*b^3*c^4*d^2*D + 48*a^3*b^2*c^3*d^3*D - 168*b^5*c^4*C*d*(c + d*x) + 198*b^5*B*c^3*d^2*(c + d*x) + 276*a*b^4*c^3*C*d^2*(c + d*x) - 231*A*b^5*c^2*d^3*(c + d*x) - 363*a*b^4*B*c^2*d^3*(c + d*x) - 51*a^2*b^3*c^2*C*d^3*(c + d*x) + 462*a*A*b^4*c*d^4*(c + d*x) + 132*a^2*b^3*B*c*d^4*(c + d*x) - 54*a^3*b^2*c*C*d^4*(c + d*x) - 231*a^2*A*b^3*d^5*(c + d*x) + 33*a^3*b^2*B*d^5*(c + d*x) - 3*a^4*b*C*d^5*(c + d*x) + 144*b^5*c^5*D*(c + d*x) - 216*a*b^4*c^4*d*D*(c + d*x) + 18*a^2*b^3*c^3*d^2*D*(c + d*x) + 33*a^3*b^2*c^2*d^3*D*(c + d*x) + 24*a^4*b*c*d^4*D*(c + d*x) - 3*a^5*d^5*D*(c + d*x) + 192*b^5*c^3*C*d*(c + d*x)^2 - 240*b^5*B*c^2*d^2*(c + d*x)^2 - 96*a*b^4*c^2*C*d^2*(c + d*x)^2 + 280*A*b^5*c*d^3*(c + d*x)^2 + 200*a*b^4*B*c*d^3*(c + d*x)^2 - 104*a^2*b^3*c*C*d^3*(c + d*x)^2 - 280*a*A*b^4*d^4*(c + d*x)^2 + 40*a^2*b^3*B*d^4*(c + d*x)^2 + 8*a^3*b^2*C*d^4*(c + d*x)^2 - 144*b^5*c^4*D*(c + d*x)^2 + 144*a^2*b^3*c^2*d^2*D*(c + d*x)^2 + 8*a^3*b^2*c*d^3*D*(c + d*x)^2 - 8*a^4*b*d^4*D*(c + d*x)^2 - 72*b^5*c^2*C*d*(c + d*x)^3 + 90*b^5*B*c*d^2*(c + d*x)^3 - 36*a*b^4*c*C*d^2*(c + d*x)^3 - 105*A*b^5*d^3*(c + d*x)^3 + 15*a*b^4*B*d^3*(c + d*x)^3 + 3*a^2*b^3*C*d^3*(c + d*x)^3 + 48*b^5*c^3*D*(c + d*x)^3 + 72*a*b^4*c^2*d*D*(c + d*x)^3 - 18*a^2*b^3*c*d^2*D*(c + d*x)^3 + 3*a^3*b^2*d^3*D*(c + d*x)^3)/(24*b^2*(b*c - a*d)^4*Sqrt[c + d*x]*(-(b*c) + a*d + b*(c + d*x))^3) + ((24*b^3*c^2*C*d - 30*b^3*B*c*d^2 + 12*a*b^2*c*C*d^2 + 35*A*b^3*d^3 - 5*a*b^2*B*d^3 - a^2*b*C*d^3 - 16*b^3*c^3*D - 24*a*b^2*c^2*d*D + 6*a^2*b*c*d^2*D - a^3*d^3*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(8*b^(5/2)*(-(b*c) + a*d)^(9/2))","B",1
18,1,1000,434,0.4075604,"\int \frac{(a+b x)^3 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(-105 b^3 D c^6+105 b^3 C d c^5+315 a b^2 d D c^5+1890 b^3 D (c+d x) c^5-105 b^3 B d^2 c^4-315 a b^2 C d^2 c^4+4725 b^3 D (c+d x)^2 c^4-315 a^2 b d^2 D c^4-1575 b^3 C d (c+d x) c^4-4725 a b^2 d D (c+d x) c^4+105 A b^3 d^3 c^3+315 a b^2 B d^3 c^3+315 a^2 b C d^3 c^3-2100 b^3 D (c+d x)^3 c^3-3150 b^3 C d (c+d x)^2 c^3-9450 a b^2 d D (c+d x)^2 c^3+105 a^3 d^3 D c^3+1260 b^3 B d^2 (c+d x) c^3+3780 a b^2 C d^2 (c+d x) c^3+3780 a^2 b d^2 D (c+d x) c^3-315 a A b^2 d^4 c^2-315 a^2 b B d^4 c^2-105 a^3 C d^4 c^2+945 b^3 D (c+d x)^4 c^2+1050 b^3 C d (c+d x)^3 c^2+3150 a b^2 d D (c+d x)^3 c^2+1890 b^3 B d^2 (c+d x)^2 c^2+5670 a b^2 C d^2 (c+d x)^2 c^2+5670 a^2 b d^2 D (c+d x)^2 c^2-945 A b^3 d^3 (c+d x) c^2-2835 a b^2 B d^3 (c+d x) c^2-2835 a^2 b C d^3 (c+d x) c^2-945 a^3 d^3 D (c+d x) c^2+315 a^2 A b d^5 c+105 a^3 B d^5 c-270 b^3 D (c+d x)^5 c-315 b^3 C d (c+d x)^4 c-945 a b^2 d D (c+d x)^4 c-420 b^3 B d^2 (c+d x)^3 c-1260 a b^2 C d^2 (c+d x)^3 c-1260 a^2 b d^2 D (c+d x)^3 c-945 A b^3 d^3 (c+d x)^2 c-2835 a b^2 B d^3 (c+d x)^2 c-2835 a^2 b C d^3 (c+d x)^2 c-945 a^3 d^3 D (c+d x)^2 c+1890 a A b^2 d^4 (c+d x) c+1890 a^2 b B d^4 (c+d x) c+630 a^3 C d^4 (c+d x) c-105 a^3 A d^6+35 b^3 D (c+d x)^6+45 b^3 C d (c+d x)^5+135 a b^2 d D (c+d x)^5+63 b^3 B d^2 (c+d x)^4+189 a b^2 C d^2 (c+d x)^4+189 a^2 b d^2 D (c+d x)^4+105 A b^3 d^3 (c+d x)^3+315 a b^2 B d^3 (c+d x)^3+315 a^2 b C d^3 (c+d x)^3+105 a^3 d^3 D (c+d x)^3+945 a A b^2 d^4 (c+d x)^2+945 a^2 b B d^4 (c+d x)^2+315 a^3 C d^4 (c+d x)^2-945 a^2 A b d^5 (c+d x)-315 a^3 B d^5 (c+d x)\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^3*c^5*C*d - 105*b^3*B*c^4*d^2 - 315*a*b^2*c^4*C*d^2 + 105*A*b^3*c^3*d^3 + 315*a*b^2*B*c^3*d^3 + 315*a^2*b*c^3*C*d^3 - 315*a*A*b^2*c^2*d^4 - 315*a^2*b*B*c^2*d^4 - 105*a^3*c^2*C*d^4 + 315*a^2*A*b*c*d^5 + 105*a^3*B*c*d^5 - 105*a^3*A*d^6 - 105*b^3*c^6*D + 315*a*b^2*c^5*d*D - 315*a^2*b*c^4*d^2*D + 105*a^3*c^3*d^3*D - 1575*b^3*c^4*C*d*(c + d*x) + 1260*b^3*B*c^3*d^2*(c + d*x) + 3780*a*b^2*c^3*C*d^2*(c + d*x) - 945*A*b^3*c^2*d^3*(c + d*x) - 2835*a*b^2*B*c^2*d^3*(c + d*x) - 2835*a^2*b*c^2*C*d^3*(c + d*x) + 1890*a*A*b^2*c*d^4*(c + d*x) + 1890*a^2*b*B*c*d^4*(c + d*x) + 630*a^3*c*C*d^4*(c + d*x) - 945*a^2*A*b*d^5*(c + d*x) - 315*a^3*B*d^5*(c + d*x) + 1890*b^3*c^5*D*(c + d*x) - 4725*a*b^2*c^4*d*D*(c + d*x) + 3780*a^2*b*c^3*d^2*D*(c + d*x) - 945*a^3*c^2*d^3*D*(c + d*x) - 3150*b^3*c^3*C*d*(c + d*x)^2 + 1890*b^3*B*c^2*d^2*(c + d*x)^2 + 5670*a*b^2*c^2*C*d^2*(c + d*x)^2 - 945*A*b^3*c*d^3*(c + d*x)^2 - 2835*a*b^2*B*c*d^3*(c + d*x)^2 - 2835*a^2*b*c*C*d^3*(c + d*x)^2 + 945*a*A*b^2*d^4*(c + d*x)^2 + 945*a^2*b*B*d^4*(c + d*x)^2 + 315*a^3*C*d^4*(c + d*x)^2 + 4725*b^3*c^4*D*(c + d*x)^2 - 9450*a*b^2*c^3*d*D*(c + d*x)^2 + 5670*a^2*b*c^2*d^2*D*(c + d*x)^2 - 945*a^3*c*d^3*D*(c + d*x)^2 + 1050*b^3*c^2*C*d*(c + d*x)^3 - 420*b^3*B*c*d^2*(c + d*x)^3 - 1260*a*b^2*c*C*d^2*(c + d*x)^3 + 105*A*b^3*d^3*(c + d*x)^3 + 315*a*b^2*B*d^3*(c + d*x)^3 + 315*a^2*b*C*d^3*(c + d*x)^3 - 2100*b^3*c^3*D*(c + d*x)^3 + 3150*a*b^2*c^2*d*D*(c + d*x)^3 - 1260*a^2*b*c*d^2*D*(c + d*x)^3 + 105*a^3*d^3*D*(c + d*x)^3 - 315*b^3*c*C*d*(c + d*x)^4 + 63*b^3*B*d^2*(c + d*x)^4 + 189*a*b^2*C*d^2*(c + d*x)^4 + 945*b^3*c^2*D*(c + d*x)^4 - 945*a*b^2*c*d*D*(c + d*x)^4 + 189*a^2*b*d^2*D*(c + d*x)^4 + 45*b^3*C*d*(c + d*x)^5 - 270*b^3*c*D*(c + d*x)^5 + 135*a*b^2*d*D*(c + d*x)^5 + 35*b^3*D*(c + d*x)^6))/(315*d^7*(c + d*x)^(3/2))","B",1
19,1,599,322,0.2280916,"\int \frac{(a+b x)^2 \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(-35 a^2 A d^5-105 a^2 B d^4 (c+d x)+35 a^2 B c d^4+35 a^2 c^3 d^2 D-35 a^2 c^2 C d^3-315 a^2 c^2 d^2 D (c+d x)+210 a^2 c C d^3 (c+d x)+105 a^2 C d^3 (c+d x)^2-315 a^2 c d^2 D (c+d x)^2+35 a^2 d^2 D (c+d x)^3-210 a A b d^4 (c+d x)+70 a A b c d^4-70 a b B c^2 d^3+420 a b B c d^3 (c+d x)+210 a b B d^3 (c+d x)^2-70 a b c^4 d D+70 a b c^3 C d^2+840 a b c^3 d D (c+d x)-630 a b c^2 C d^2 (c+d x)+1260 a b c^2 d D (c+d x)^2-630 a b c C d^2 (c+d x)^2+70 a b C d^2 (c+d x)^3-280 a b c d D (c+d x)^3+42 a b d D (c+d x)^4-35 A b^2 c^2 d^3+210 A b^2 c d^3 (c+d x)+105 A b^2 d^3 (c+d x)^2+35 b^2 B c^3 d^2-315 b^2 B c^2 d^2 (c+d x)-315 b^2 B c d^2 (c+d x)^2+35 b^2 B d^2 (c+d x)^3+35 b^2 c^5 D-35 b^2 c^4 C d-525 b^2 c^4 D (c+d x)+420 b^2 c^3 C d (c+d x)-1050 b^2 c^3 D (c+d x)^2+630 b^2 c^2 C d (c+d x)^2+350 b^2 c^2 D (c+d x)^3-140 b^2 c C d (c+d x)^3+21 b^2 C d (c+d x)^4-105 b^2 c D (c+d x)^4+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^2*c^4*C*d + 35*b^2*B*c^3*d^2 + 70*a*b*c^3*C*d^2 - 35*A*b^2*c^2*d^3 - 70*a*b*B*c^2*d^3 - 35*a^2*c^2*C*d^3 + 70*a*A*b*c*d^4 + 35*a^2*B*c*d^4 - 35*a^2*A*d^5 + 35*b^2*c^5*D - 70*a*b*c^4*d*D + 35*a^2*c^3*d^2*D + 420*b^2*c^3*C*d*(c + d*x) - 315*b^2*B*c^2*d^2*(c + d*x) - 630*a*b*c^2*C*d^2*(c + d*x) + 210*A*b^2*c*d^3*(c + d*x) + 420*a*b*B*c*d^3*(c + d*x) + 210*a^2*c*C*d^3*(c + d*x) - 210*a*A*b*d^4*(c + d*x) - 105*a^2*B*d^4*(c + d*x) - 525*b^2*c^4*D*(c + d*x) + 840*a*b*c^3*d*D*(c + d*x) - 315*a^2*c^2*d^2*D*(c + d*x) + 630*b^2*c^2*C*d*(c + d*x)^2 - 315*b^2*B*c*d^2*(c + d*x)^2 - 630*a*b*c*C*d^2*(c + d*x)^2 + 105*A*b^2*d^3*(c + d*x)^2 + 210*a*b*B*d^3*(c + d*x)^2 + 105*a^2*C*d^3*(c + d*x)^2 - 1050*b^2*c^3*D*(c + d*x)^2 + 1260*a*b*c^2*d*D*(c + d*x)^2 - 315*a^2*c*d^2*D*(c + d*x)^2 - 140*b^2*c*C*d*(c + d*x)^3 + 35*b^2*B*d^2*(c + d*x)^3 + 70*a*b*C*d^2*(c + d*x)^3 + 350*b^2*c^2*D*(c + d*x)^3 - 280*a*b*c*d*D*(c + d*x)^3 + 35*a^2*d^2*D*(c + d*x)^3 + 21*b^2*C*d*(c + d*x)^4 - 105*b^2*c*D*(c + d*x)^4 + 42*a*b*d*D*(c + d*x)^4 + 15*b^2*D*(c + d*x)^5))/(105*d^6*(c + d*x)^(3/2))","A",1
20,1,286,210,0.1841345,"\int \frac{(a+b x) \left(A+B x+C x^2+D x^3\right)}{(c+d x)^{5/2}} \, dx","IntegrateAlgebraic[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]","\frac{2 \left(-5 a A d^4-15 a B d^3 (c+d x)+5 a B c d^3+5 a c^3 d D-5 a c^2 C d^2-45 a c^2 d D (c+d x)+30 a c C d^2 (c+d x)+15 a C d^2 (c+d x)^2-45 a c d D (c+d x)^2+5 a d D (c+d x)^3-15 A b d^3 (c+d x)+5 A b c d^3-5 b B c^2 d^2+30 b B c d^2 (c+d x)+15 b B d^2 (c+d x)^2-5 b c^4 D+5 b c^3 C d+60 b c^3 D (c+d x)-45 b c^2 C d (c+d x)+90 b c^2 D (c+d x)^2-45 b c C d (c+d x)^2+5 b C d (c+d x)^3-20 b c D (c+d x)^3+3 b D (c+d x)^4\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*b*c^3*C*d - 5*b*B*c^2*d^2 - 5*a*c^2*C*d^2 + 5*A*b*c*d^3 + 5*a*B*c*d^3 - 5*a*A*d^4 - 5*b*c^4*D + 5*a*c^3*d*D - 45*b*c^2*C*d*(c + d*x) + 30*b*B*c*d^2*(c + d*x) + 30*a*c*C*d^2*(c + d*x) - 15*A*b*d^3*(c + d*x) - 15*a*B*d^3*(c + d*x) + 60*b*c^3*D*(c + d*x) - 45*a*c^2*d*D*(c + d*x) - 45*b*c*C*d*(c + d*x)^2 + 15*b*B*d^2*(c + d*x)^2 + 15*a*C*d^2*(c + d*x)^2 + 90*b*c^2*D*(c + d*x)^2 - 45*a*c*d*D*(c + d*x)^2 + 5*b*C*d*(c + d*x)^3 - 20*b*c*D*(c + d*x)^3 + 5*a*d*D*(c + d*x)^3 + 3*b*D*(c + d*x)^4))/(15*d^5*(c + d*x)^(3/2))","A",1
21,1,104,113,0.0666986,"\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(5/2),x]","\frac{2 \left(-A d^3-3 B d^2 (c+d x)+B c d^2+c^3 D-c^2 C d-9 c^2 D (c+d x)+6 c C d (c+d x)+3 C d (c+d x)^2-9 c D (c+d x)^2+D (c+d x)^3\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*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D + 6*c*C*d*(c + d*x) - 3*B*d^2*(c + d*x) - 9*c^2*D*(c + d*x) + 3*C*d*(c + d*x)^2 - 9*c*D*(c + d*x)^2 + D*(c + d*x)^3))/(3*d^4*(c + d*x)^(3/2))","A",1
22,1,317,210,0.2760125,"\int \frac{A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(5/2)),x]","\frac{2 \left(3 a^2 d^2 D (c+d x)^2-a A b d^4-3 a b B d^3 (c+d x)+a b B c d^3+a b c^3 d D-a b c^2 C d^2-9 a b c^2 d D (c+d x)+6 a b c C d^2 (c+d x)-6 a b c d D (c+d x)^2+3 A b^2 d^3 (c+d x)+A b^2 c d^3-b^2 B c^2 d^2-b^2 c^4 D+b^2 c^3 C d+6 b^2 c^3 D (c+d x)-3 b^2 c^2 C d (c+d x)+3 b^2 c^2 D (c+d x)^2\right)}{3 b d^3 (c+d x)^{3/2} (b c-a d)^2}-\frac{2 \left(a^3 (-D)+a^2 b C-a b^2 B+A b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right)}{b^{3/2} (a d-b c)^{5/2}}","-\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*(b^2*c^3*C*d - b^2*B*c^2*d^2 - a*b*c^2*C*d^2 + A*b^2*c*d^3 + a*b*B*c*d^3 - a*A*b*d^4 - b^2*c^4*D + a*b*c^3*d*D - 3*b^2*c^2*C*d*(c + d*x) + 6*a*b*c*C*d^2*(c + d*x) + 3*A*b^2*d^3*(c + d*x) - 3*a*b*B*d^3*(c + d*x) + 6*b^2*c^3*D*(c + d*x) - 9*a*b*c^2*d*D*(c + d*x) + 3*b^2*c^2*D*(c + d*x)^2 - 6*a*b*c*d*D*(c + d*x)^2 + 3*a^2*d^2*D*(c + d*x)^2))/(3*b*d^3*(b*c - a*d)^2*(c + d*x)^(3/2)) - (2*(A*b^3 - a*b^2*B + a^2*b*C - a^3*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(3/2)*(-(b*c) + a*d)^(5/2))","A",1
23,1,606,336,0.7049077,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^2*(c + d*x)^(5/2)),x]","\frac{3 a^3 d^3 D (c+d x)^2+2 a^2 A b d^5+6 a^2 b B d^4 (c+d x)-2 a^2 b B c d^4-2 a^2 b c^3 d^2 D+2 a^2 b c^2 C d^3+18 a^2 b c^2 d^2 D (c+d x)-12 a^2 b c C d^3 (c+d x)-3 a^2 b C d^3 (c+d x)^2-10 a A b^2 d^4 (c+d x)-4 a A b^2 c d^4+4 a b^2 B c^2 d^3-2 a b^2 B c d^3 (c+d x)+9 a b^2 B d^3 (c+d x)^2+4 a b^2 c^4 d D-4 a b^2 c^3 C d^2-26 a b^2 c^3 d D (c+d x)+14 a b^2 c^2 C d^2 (c+d x)+18 a b^2 c^2 d D (c+d x)^2-12 a b^2 c C d^2 (c+d x)^2+2 A b^3 c^2 d^3+10 A b^3 c d^3 (c+d x)-15 A b^3 d^3 (c+d x)^2-2 b^3 B c^3 d^2-4 b^3 B c^2 d^2 (c+d x)+6 b^3 B c d^2 (c+d x)^2-2 b^3 c^5 D+2 b^3 c^4 C d+8 b^3 c^4 D (c+d x)-2 b^3 c^3 C d (c+d x)-6 b^3 c^3 D (c+d x)^2}{3 b d^2 (c+d x)^{3/2} (b c-a d)^3 (a d+b (c+d x)-b c)}+\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x} \sqrt{a d-b c}}{b c-a d}\right) \left(a^3 (-d) D+6 a^2 b c D-a^2 b C d+3 a b^2 B d-4 a b^2 c C-5 A b^3 d+2 b^3 B c\right)}{b^{3/2} (a d-b c)^{7/2}}","-\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{-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*b^3*c^4*C*d - 2*b^3*B*c^3*d^2 - 4*a*b^2*c^3*C*d^2 + 2*A*b^3*c^2*d^3 + 4*a*b^2*B*c^2*d^3 + 2*a^2*b*c^2*C*d^3 - 4*a*A*b^2*c*d^4 - 2*a^2*b*B*c*d^4 + 2*a^2*A*b*d^5 - 2*b^3*c^5*D + 4*a*b^2*c^4*d*D - 2*a^2*b*c^3*d^2*D - 2*b^3*c^3*C*d*(c + d*x) - 4*b^3*B*c^2*d^2*(c + d*x) + 14*a*b^2*c^2*C*d^2*(c + d*x) + 10*A*b^3*c*d^3*(c + d*x) - 2*a*b^2*B*c*d^3*(c + d*x) - 12*a^2*b*c*C*d^3*(c + d*x) - 10*a*A*b^2*d^4*(c + d*x) + 6*a^2*b*B*d^4*(c + d*x) + 8*b^3*c^4*D*(c + d*x) - 26*a*b^2*c^3*d*D*(c + d*x) + 18*a^2*b*c^2*d^2*D*(c + d*x) + 6*b^3*B*c*d^2*(c + d*x)^2 - 12*a*b^2*c*C*d^2*(c + d*x)^2 - 15*A*b^3*d^3*(c + d*x)^2 + 9*a*b^2*B*d^3*(c + d*x)^2 - 3*a^2*b*C*d^3*(c + d*x)^2 - 6*b^3*c^3*D*(c + d*x)^2 + 18*a*b^2*c^2*d*D*(c + d*x)^2 + 3*a^3*d^3*D*(c + d*x)^2)/(3*b*d^2*(b*c - a*d)^3*(c + d*x)^(3/2)*(-(b*c) + a*d + b*(c + d*x))) + ((2*b^3*B*c - 4*a*b^2*c*C - 5*A*b^3*d + 3*a*b^2*B*d - a^2*b*C*d + 6*a^2*b*c*D - a^3*d*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(b^(3/2)*(-(b*c) + a*d)^(7/2))","A",1
24,1,1057,438,1.2210515,"\int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx","IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^3*(c + d*x)^(5/2)),x]","\frac{-8 b^4 D c^6+8 b^4 C d c^5+24 a b^3 d D c^5+16 b^4 D (c+d x) c^5-8 b^4 B d^2 c^4-24 a b^3 C d^2 c^4-8 b^4 D (c+d x)^2 c^4-24 a^2 b^2 d^2 D c^4+8 b^4 C d (c+d x) c^4-104 a b^3 d D (c+d x) c^4+8 A b^4 d^3 c^3+24 a b^3 B d^3 c^3+24 a^2 b^2 C d^3 c^3-40 b^4 C d (c+d x)^2 c^3+152 a b^3 d D (c+d x)^2 c^3+8 a^3 b d^3 D c^3-32 b^4 B d^2 (c+d x) c^3+32 a b^3 C d^2 (c+d x) c^3+160 a^2 b^2 d^2 D (c+d x) c^3-24 a A b^3 d^4 c^2-24 a^2 b^2 B d^4 c^2-8 a^3 b C d^4 c^2+24 b^4 C d (c+d x)^3 c^2-72 a b^3 d D (c+d x)^3 c^2+100 b^4 B d^2 (c+d x)^2 c^2-80 a b^3 C d^2 (c+d x)^2 c^2-108 a^2 b^2 d^2 D (c+d x)^2 c^2+56 A b^4 d^3 (c+d x) c^2+40 a b^3 B d^3 (c+d x) c^2-88 a^2 b^2 C d^3 (c+d x) c^2-72 a^3 b d^3 D (c+d x) c^2+24 a^2 A b^2 d^5 c+8 a^3 b B d^5 c-60 b^4 B d^2 (c+d x)^3 c+72 a b^3 C d^2 (c+d x)^3 c-36 a^2 b^2 d^2 D (c+d x)^3 c-175 A b^4 d^3 (c+d x)^2 c-25 a b^3 B d^3 (c+d x)^2 c+105 a^2 b^2 C d^3 (c+d x)^2 c-33 a^3 b d^3 D (c+d x)^2 c-112 a A b^3 d^4 (c+d x) c+16 a^2 b^2 B d^4 (c+d x) c+48 a^3 b C d^4 (c+d x) c-8 a^3 A b d^6+105 A b^4 d^3 (c+d x)^3-45 a b^3 B d^3 (c+d x)^3+9 a^2 b^2 C d^3 (c+d x)^3+3 a^3 b d^3 D (c+d x)^3+175 a A b^3 d^4 (c+d x)^2-75 a^2 b^2 B d^4 (c+d x)^2+15 a^3 b C d^4 (c+d x)^2-3 a^4 d^4 D (c+d x)^2+56 a^2 A b^2 d^5 (c+d x)-24 a^3 b B d^5 (c+d x)}{12 b d (b c-a d)^4 (c+d x)^{3/2} (-b c+a d+b (c+d x))^2}+\frac{\left(-d^2 D a^3-3 b C d^2 a^2+12 b c d D a^2+15 b^2 B d^2 a-24 b^2 c C d a+24 b^2 c^2 D a-35 A b^3 d^2-8 b^3 c^2 C+20 b^3 B c d\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{a d-b c} \sqrt{c+d x}}{b c-a d}\right)}{4 b^{3/2} (a d-b c)^{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{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,"(8*b^4*c^5*C*d - 8*b^4*B*c^4*d^2 - 24*a*b^3*c^4*C*d^2 + 8*A*b^4*c^3*d^3 + 24*a*b^3*B*c^3*d^3 + 24*a^2*b^2*c^3*C*d^3 - 24*a*A*b^3*c^2*d^4 - 24*a^2*b^2*B*c^2*d^4 - 8*a^3*b*c^2*C*d^4 + 24*a^2*A*b^2*c*d^5 + 8*a^3*b*B*c*d^5 - 8*a^3*A*b*d^6 - 8*b^4*c^6*D + 24*a*b^3*c^5*d*D - 24*a^2*b^2*c^4*d^2*D + 8*a^3*b*c^3*d^3*D + 8*b^4*c^4*C*d*(c + d*x) - 32*b^4*B*c^3*d^2*(c + d*x) + 32*a*b^3*c^3*C*d^2*(c + d*x) + 56*A*b^4*c^2*d^3*(c + d*x) + 40*a*b^3*B*c^2*d^3*(c + d*x) - 88*a^2*b^2*c^2*C*d^3*(c + d*x) - 112*a*A*b^3*c*d^4*(c + d*x) + 16*a^2*b^2*B*c*d^4*(c + d*x) + 48*a^3*b*c*C*d^4*(c + d*x) + 56*a^2*A*b^2*d^5*(c + d*x) - 24*a^3*b*B*d^5*(c + d*x) + 16*b^4*c^5*D*(c + d*x) - 104*a*b^3*c^4*d*D*(c + d*x) + 160*a^2*b^2*c^3*d^2*D*(c + d*x) - 72*a^3*b*c^2*d^3*D*(c + d*x) - 40*b^4*c^3*C*d*(c + d*x)^2 + 100*b^4*B*c^2*d^2*(c + d*x)^2 - 80*a*b^3*c^2*C*d^2*(c + d*x)^2 - 175*A*b^4*c*d^3*(c + d*x)^2 - 25*a*b^3*B*c*d^3*(c + d*x)^2 + 105*a^2*b^2*c*C*d^3*(c + d*x)^2 + 175*a*A*b^3*d^4*(c + d*x)^2 - 75*a^2*b^2*B*d^4*(c + d*x)^2 + 15*a^3*b*C*d^4*(c + d*x)^2 - 8*b^4*c^4*D*(c + d*x)^2 + 152*a*b^3*c^3*d*D*(c + d*x)^2 - 108*a^2*b^2*c^2*d^2*D*(c + d*x)^2 - 33*a^3*b*c*d^3*D*(c + d*x)^2 - 3*a^4*d^4*D*(c + d*x)^2 + 24*b^4*c^2*C*d*(c + d*x)^3 - 60*b^4*B*c*d^2*(c + d*x)^3 + 72*a*b^3*c*C*d^2*(c + d*x)^3 + 105*A*b^4*d^3*(c + d*x)^3 - 45*a*b^3*B*d^3*(c + d*x)^3 + 9*a^2*b^2*C*d^3*(c + d*x)^3 - 72*a*b^3*c^2*d*D*(c + d*x)^3 - 36*a^2*b^2*c*d^2*D*(c + d*x)^3 + 3*a^3*b*d^3*D*(c + d*x)^3)/(12*b*d*(b*c - a*d)^4*(c + d*x)^(3/2)*(-(b*c) + a*d + b*(c + d*x))^2) + ((-8*b^3*c^2*C + 20*b^3*B*c*d - 24*a*b^2*c*C*d - 35*A*b^3*d^2 + 15*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + 24*a*b^2*c^2*D + 12*a^2*b*c*d*D - a^3*d^2*D)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b*c - a*d)])/(4*b^(3/2)*(-(b*c) + a*d)^(9/2))","B",1
25,0,0,455,1.1405156,"\int (a+b x)^3 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\int (a+b x)^3 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","-\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,"Could not integrate","F",-1
26,0,0,338,0.487935,"\int (a+b x)^2 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\int (a+b x)^2 (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","\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,"Could not integrate","F",-1
27,0,0,226,0.2224443,"\int (a+b x) (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x)*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\int (a+b x) (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","-\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,"Could not integrate","F",-1
28,0,0,126,0.0533287,"\int (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\int (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","\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,"Could not integrate","F",-1
29,0,0,203,0.246114,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{a+b x} \, dx","IntegrateAlgebraic[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]","\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{a+b x} \, dx","-\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,"Could not integrate","F",-1
30,0,0,220,0.5505037,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^2} \, dx","IntegrateAlgebraic[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]","\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^2} \, dx","-\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,"Could not integrate","F",-1
31,0,0,329,0.7875342,"\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^3} \, dx","IntegrateAlgebraic[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^3,x]","\int \frac{(c+d x)^n \left(A+B x+C x^2+D x^3\right)}{(a+b x)^3} \, dx","-\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,"Could not integrate","F",-1
32,0,0,141,0.0427279,"\int (a+b x)^m (A+B x) (c+d x)^n \, dx","IntegrateAlgebraic[(a + b*x)^m*(A + B*x)*(c + d*x)^n,x]","\int (a+b x)^m (A+B x) (c+d x)^n \, dx","\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,"Could not integrate","F",-1
33,0,0,268,0.0477923,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2\right) \, dx","IntegrateAlgebraic[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2),x]","\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2\right) \, dx","-\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,"Could not integrate","F",-1
34,0,0,610,0.0649825,"\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","IntegrateAlgebraic[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]","\int (a+b x)^m (c+d x)^n \left(A+B x+C x^2+D x^3\right) \, dx","\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,"Could not integrate","F",-1