Integrand size = 26, antiderivative size = 267 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{3 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{5/2}}{5 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{11/2}}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \]
-2/3*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(3/2)/e^6+2/5*d*(-b*e+c*d)*(B*d*( -3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(5/2)/e^6+2/7*(A*e*(b^2*e^2-6*b* c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(7/2)/e^6- 2/9*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(9/2)/ e^6-2/11*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(11/2)/e^6+2/13*B*c^2*(e*x+d)^ (13/2)/e^6
Time = 0.17 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (13 A e \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (143 b^2 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+26 b c e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )-5 c^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]
(2*(d + e*x)^(3/2)*(13*A*e*(33*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 2 2*b*c*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) + B*(143*b ^2*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 26*b*c*e*(128* d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) - 5*c^2 *(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^ 4 - 693*e^5*x^5))))/(45045*e^6)
Time = 0.41 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{7/2} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^5}-\frac {d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^5}+\frac {c (d+e x)^{9/2} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5}+\frac {B c^2 (d+e x)^{11/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac {2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac {2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac {2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac {2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6}\) |
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(3/2))/(3*e^6) + (2*d*(c*d - b *e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(5/2))/(5*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(7/2))/(7*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10 *c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d + e*x)^(9/2))/(9*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(11/2))/(11*e^6) + (2*B*c^2*(d + e*x)^(13/2)) /(13*e^6)
3.13.24.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {5 \left (-\frac {105}{44} d \,e^{4} x^{4}+\frac {70}{33} d^{2} e^{3} x^{3}-\frac {32}{33} d^{5}+\frac {21}{8} e^{5} x^{5}-\frac {20}{11} d^{3} e^{2} x^{2}+\frac {16}{11} d^{4} e x \right ) B}{13}+\frac {16 A \left (\frac {315}{128} e^{4} x^{4}-\frac {35}{16} d \,e^{3} x^{3}+\frac {15}{8} d^{2} e^{2} x^{2}-\frac {3}{2} d^{3} e x +d^{4}\right ) e}{33}\right ) c^{2}-\frac {4 \left (\frac {\left (-\frac {315}{16} e^{4} x^{4}+\frac {35}{2} d \,e^{3} x^{3}-15 d^{2} e^{2} x^{2}+12 d^{3} e x -8 d^{4}\right ) B}{11}+A e \left (-\frac {35}{16} e^{3} x^{3}+\frac {15}{8} d \,e^{2} x^{2}-\frac {3}{2} d^{2} e x +d^{3}\right )\right ) e b c}{3}+\left (\left (-\frac {5}{4} d \,e^{2} x^{2}+\frac {35}{24} e^{3} x^{3}-\frac {2}{3} d^{3}+d^{2} e x \right ) B +A e \left (\frac {15}{8} e^{2} x^{2}-\frac {3}{2} d e x +d^{2}\right )\right ) e^{2} b^{2}\right )}{105 e^{6}}\) | \(257\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) \left (b e -c d \right ) c +B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right ) c +\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} \left (b e -c d \right ) c +\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(278\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) \left (b e -c d \right ) c +B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right ) c +\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} \left (b e -c d \right ) c +\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(278\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,x^{5} c^{2} e^{5}+4095 A \,x^{4} c^{2} e^{5}+8190 B \,x^{4} b c \,e^{5}-3150 B \,x^{4} c^{2} d \,e^{4}+10010 A \,x^{3} b c \,e^{5}-3640 A \,x^{3} c^{2} d \,e^{4}+5005 B \,x^{3} b^{2} e^{5}-7280 B \,x^{3} b c d \,e^{4}+2800 B \,x^{3} c^{2} d^{2} e^{3}+6435 A \,x^{2} b^{2} e^{5}-8580 A \,x^{2} b c d \,e^{4}+3120 A \,x^{2} c^{2} d^{2} e^{3}-4290 B \,x^{2} b^{2} d \,e^{4}+6240 B \,x^{2} b c \,d^{2} e^{3}-2400 B \,x^{2} c^{2} d^{3} e^{2}-5148 A x \,b^{2} d \,e^{4}+6864 A x b c \,d^{2} e^{3}-2496 A x \,c^{2} d^{3} e^{2}+3432 B x \,b^{2} d^{2} e^{3}-4992 B x b c \,d^{3} e^{2}+1920 B x \,c^{2} d^{4} e +3432 A \,b^{2} d^{2} e^{3}-4576 A b c \,d^{3} e^{2}+1664 A \,c^{2} d^{4} e -2288 B \,b^{2} d^{3} e^{2}+3328 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{45045 e^{6}}\) | \(341\) |
| trager | \(\frac {2 \left (3465 B \,c^{2} e^{6} x^{6}+4095 A \,c^{2} e^{6} x^{5}+8190 B b c \,e^{6} x^{5}+315 B \,c^{2} d \,e^{5} x^{5}+10010 A b c \,e^{6} x^{4}+455 A \,c^{2} d \,e^{5} x^{4}+5005 B \,b^{2} e^{6} x^{4}+910 B b c d \,e^{5} x^{4}-350 B \,c^{2} d^{2} e^{4} x^{4}+6435 A \,b^{2} e^{6} x^{3}+1430 A b c d \,e^{5} x^{3}-520 A \,c^{2} d^{2} e^{4} x^{3}+715 B \,b^{2} d \,e^{5} x^{3}-1040 B b c \,d^{2} e^{4} x^{3}+400 B \,c^{2} d^{3} e^{3} x^{3}+1287 A \,b^{2} d \,e^{5} x^{2}-1716 A b c \,d^{2} e^{4} x^{2}+624 A \,c^{2} d^{3} e^{3} x^{2}-858 B \,b^{2} d^{2} e^{4} x^{2}+1248 B b c \,d^{3} e^{3} x^{2}-480 B \,c^{2} d^{4} e^{2} x^{2}-1716 A \,b^{2} d^{2} e^{4} x +2288 A b c \,d^{3} e^{3} x -832 A \,c^{2} d^{4} e^{2} x +1144 B \,b^{2} d^{3} e^{3} x -1664 B b c \,d^{4} e^{2} x +640 B \,c^{2} d^{5} e x +3432 A \,b^{2} d^{3} e^{3}-4576 A b c \,d^{4} e^{2}+1664 A \,c^{2} d^{5} e -2288 B \,b^{2} d^{4} e^{2}+3328 B b c \,d^{5} e -1280 B \,c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(429\) |
| risch | \(\frac {2 \left (3465 B \,c^{2} e^{6} x^{6}+4095 A \,c^{2} e^{6} x^{5}+8190 B b c \,e^{6} x^{5}+315 B \,c^{2} d \,e^{5} x^{5}+10010 A b c \,e^{6} x^{4}+455 A \,c^{2} d \,e^{5} x^{4}+5005 B \,b^{2} e^{6} x^{4}+910 B b c d \,e^{5} x^{4}-350 B \,c^{2} d^{2} e^{4} x^{4}+6435 A \,b^{2} e^{6} x^{3}+1430 A b c d \,e^{5} x^{3}-520 A \,c^{2} d^{2} e^{4} x^{3}+715 B \,b^{2} d \,e^{5} x^{3}-1040 B b c \,d^{2} e^{4} x^{3}+400 B \,c^{2} d^{3} e^{3} x^{3}+1287 A \,b^{2} d \,e^{5} x^{2}-1716 A b c \,d^{2} e^{4} x^{2}+624 A \,c^{2} d^{3} e^{3} x^{2}-858 B \,b^{2} d^{2} e^{4} x^{2}+1248 B b c \,d^{3} e^{3} x^{2}-480 B \,c^{2} d^{4} e^{2} x^{2}-1716 A \,b^{2} d^{2} e^{4} x +2288 A b c \,d^{3} e^{3} x -832 A \,c^{2} d^{4} e^{2} x +1144 B \,b^{2} d^{3} e^{3} x -1664 B b c \,d^{4} e^{2} x +640 B \,c^{2} d^{5} e x +3432 A \,b^{2} d^{3} e^{3}-4576 A b c \,d^{4} e^{2}+1664 A \,c^{2} d^{5} e -2288 B \,b^{2} d^{4} e^{2}+3328 B b c \,d^{5} e -1280 B \,c^{2} d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(429\) |
16/105*(e*x+d)^(3/2)*((5/13*(-105/44*d*e^4*x^4+70/33*d^2*e^3*x^3-32/33*d^5 +21/8*e^5*x^5-20/11*d^3*e^2*x^2+16/11*d^4*e*x)*B+16/33*A*(315/128*e^4*x^4- 35/16*d*e^3*x^3+15/8*d^2*e^2*x^2-3/2*d^3*e*x+d^4)*e)*c^2-4/3*(1/11*(-315/1 6*e^4*x^4+35/2*d*e^3*x^3-15*d^2*e^2*x^2+12*d^3*e*x-8*d^4)*B+A*e*(-35/16*e^ 3*x^3+15/8*d*e^2*x^2-3/2*d^2*e*x+d^3))*e*b*c+((-5/4*d*e^2*x^2+35/24*e^3*x^ 3-2/3*d^3+d^2*e*x)*B+A*e*(15/8*e^2*x^2-3/2*d*e*x+d^2))*e^2*b^2)/e^6
Time = 0.36 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.34 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 3432 \, A b^{2} d^{3} e^{3} + 1664 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e - 2288 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{2} + 315 \, {\left (B c^{2} d e^{5} + 13 \, {\left (2 \, B b c + A c^{2}\right )} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{2} e^{4} - 13 \, {\left (2 \, B b c + A c^{2}\right )} d e^{5} - 143 \, {\left (B b^{2} + 2 \, A b c\right )} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{3} e^{3} + 1287 \, A b^{2} e^{6} - 104 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{4} + 143 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{4} e^{2} - 429 \, A b^{2} d e^{5} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{3} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{5} e - 429 \, A b^{2} d^{2} e^{4} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{2} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
2/45045*(3465*B*c^2*e^6*x^6 - 1280*B*c^2*d^6 + 3432*A*b^2*d^3*e^3 + 1664*( 2*B*b*c + A*c^2)*d^5*e - 2288*(B*b^2 + 2*A*b*c)*d^4*e^2 + 315*(B*c^2*d*e^5 + 13*(2*B*b*c + A*c^2)*e^6)*x^5 - 35*(10*B*c^2*d^2*e^4 - 13*(2*B*b*c + A* c^2)*d*e^5 - 143*(B*b^2 + 2*A*b*c)*e^6)*x^4 + 5*(80*B*c^2*d^3*e^3 + 1287*A *b^2*e^6 - 104*(2*B*b*c + A*c^2)*d^2*e^4 + 143*(B*b^2 + 2*A*b*c)*d*e^5)*x^ 3 - 3*(160*B*c^2*d^4*e^2 - 429*A*b^2*d*e^5 - 208*(2*B*b*c + A*c^2)*d^3*e^3 + 286*(B*b^2 + 2*A*b*c)*d^2*e^4)*x^2 + 4*(160*B*c^2*d^5*e - 429*A*b^2*d^2 *e^4 - 208*(2*B*b*c + A*c^2)*d^4*e^2 + 286*(B*b^2 + 2*A*b*c)*d^3*e^3)*x)*s qrt(e*x + d)/e^6
Time = 1.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.64 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*c**2*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*c* *2*e + 2*B*b*c*e - 5*B*c**2*d)/(11*e**5) + (d + e*x)**(9/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(9*e**5) + (d + e*x)**(7/2)*(A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2* d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(7*e**5) + (d + e*x)**(5/2)*(-2 *A*b**2*d*e**3 + 6*A*b*c*d**2*e**2 - 4*A*c**2*d**3*e + 3*B*b**2*d**2*e**2 - 8*B*b*c*d**3*e + 5*B*c**2*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*b**2*d**2 *e**3 - 2*A*b*c*d**3*e**2 + A*c**2*d**4*e - B*b**2*d**3*e**2 + 2*B*b*c*d** 4*e - B*c**2*d**5)/(3*e**5))/e, Ne(e, 0)), (sqrt(d)*(A*b**2*x**3/3 + B*c** 2*x**6/6 + x**5*(A*c**2 + 2*B*b*c)/5 + x**4*(2*A*b*c + B*b**2)/4), True))
Time = 0.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{2} - 4095 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 6435 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \]
2/45045*(3465*(e*x + d)^(13/2)*B*c^2 - 4095*(5*B*c^2*d - (2*B*b*c + A*c^2) *e)*(e*x + d)^(11/2) + 5005*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b ^2 + 2*A*b*c)*e^2)*(e*x + d)^(9/2) - 6435*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2 *B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(7/2) + 9009* (5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A* b*c)*d^2*e^2)*(e*x + d)^(5/2) - 15015*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b* c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(3/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (243) = 486\).
Time = 0.27 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.95 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
2/45045*(3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*A*b^2*d/e^2 + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e *x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*b^2*d/e^3 + 2574*(5*(e*x + d)^ (7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d ^3)*A*b*c*d/e^3 + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*b^2/e^2 + 286*(35*(e*x + d)^(9/2 ) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)* d^3 + 315*sqrt(e*x + d)*d^4)*B*b*c*d/e^4 + 143*(35*(e*x + d)^(9/2) - 180*( e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315 *sqrt(e*x + d)*d^4)*A*c^2*d/e^4 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^ (7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B*b^2/e^3 + 286*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 3 78*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)* A*b*c/e^3 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d )^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sq rt(e*x + d)*d^5)*B*c^2*d/e^5 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9 /2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d )^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*b*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^ 3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*A*c^2/e^4 + 15*(2...
Time = 10.43 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.95 \[ \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{9\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{5\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \]
((d + e*x)^(11/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(11*e^6) + ((d + e *x)^(7/2)*(2*A*b^2*e^3 - 20*B*c^2*d^3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 1 2*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(7*e^6) + ((d + e*x)^(9/2)*(2*B*b^2*e^2 + 20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(9*e^6) + (2*B* c^2*(d + e*x)^(13/2))/(13*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(5/2)*(2*A*b*e ^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(5*e^6) + (2*d^2*(A*e - B*d)*(b*e - c*d)^2*(d + e*x)^(3/2))/(3*e^6)