Integrand size = 26, antiderivative size = 267 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{7/2}}{7 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{9/2}}{9 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{15/2}}{15 e^6}+\frac {2 B c^2 (d+e x)^{17/2}}{17 e^6} \]
-2/7*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(7/2)/e^6+2/9*d*(-b*e+c*d)*(B*d*( -3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(9/2)/e^6+2/11*(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)^(11/2)/e^ 6-2/13*(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)^(13 /2)/e^6-2/15*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(15/2)/e^6+2/17*B*c^2*(e*x +d)^(17/2)/e^6
Time = 0.19 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (17 A e \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+B \left (255 b^2 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+34 b c e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )-5 c^2 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \]
(2*(d + e*x)^(7/2)*(17*A*e*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3 0*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + c^2*(128*d^ 4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) + B*( 255*b^2*e^2*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 34*b*c* e*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^ 4) - 5*c^2*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5))))/(765765*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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{11/2} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^5}-\frac {d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{e^5}+\frac {c (d+e x)^{13/2} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5}+\frac {B c^2 (d+e x)^{15/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac {2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac {2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac {2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac {2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac {2 B c^2 (d+e x)^{17/2}}{17 e^6}\) |
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*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)^(13/2))/(13*e^6) - (2*c*(5*B* c*d - 2*b*B*e - A*c*e)*(d + e*x)^(15/2))/(15*e^6) + (2*B*c^2*(d + e*x)^(17 /2))/(17*e^6)
3.13.22.3.1 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 = 0.60 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\left (-\frac {231}{68} d \,e^{4} x^{4}+\frac {462}{221} d^{2} e^{3} x^{3}-\frac {32}{221} d^{5}+\frac {693}{136} e^{5} x^{5}-\frac {252}{221} d^{3} e^{2} x^{2}+\frac {112}{221} d^{4} e x \right ) B +\frac {16 \left (\frac {3003}{128} e^{4} x^{4}-\frac {231}{16} d \,e^{3} x^{3}+\frac {63}{8} d^{2} e^{2} x^{2}-\frac {7}{2} d^{3} e x +d^{4}\right ) A e}{65}\right ) c^{2}-\frac {12 \left (\left (-\frac {1001}{80} e^{4} x^{4}+\frac {77}{10} d \,e^{3} x^{3}-\frac {21}{5} d^{2} e^{2} x^{2}+\frac {28}{15} d^{3} e x -\frac {8}{15} d^{4}\right ) B +A e \left (-\frac {231}{16} e^{3} x^{3}+\frac {63}{8} d \,e^{2} x^{2}-\frac {7}{2} d^{2} e x +d^{3}\right )\right ) e b c}{13}+\left (\left (-\frac {189}{52} d \,e^{2} x^{2}+\frac {693}{104} e^{3} x^{3}-\frac {6}{13} d^{3}+\frac {21}{13} d^{2} e x \right ) B +A e \left (\frac {63}{8} e^{2} x^{2}-\frac {7}{2} d e x +d^{2}\right )\right ) e^{2} b^{2}\right )}{693 e^{6}}\) | \(256\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) \left (b e -c d \right ) c +B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right ) c +\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} \left (b e -c d \right ) c +\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(278\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) \left (b e -c d \right ) c +B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right ) c +\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} \left (b e -c d \right ) c +\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(278\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (45045 B \,x^{5} c^{2} e^{5}+51051 A \,x^{4} c^{2} e^{5}+102102 B \,x^{4} b c \,e^{5}-30030 B \,x^{4} c^{2} d \,e^{4}+117810 A \,x^{3} b c \,e^{5}-31416 A \,x^{3} c^{2} d \,e^{4}+58905 B \,x^{3} b^{2} e^{5}-62832 B \,x^{3} b c d \,e^{4}+18480 B \,x^{3} c^{2} d^{2} e^{3}+69615 A \,x^{2} b^{2} e^{5}-64260 A \,x^{2} b c d \,e^{4}+17136 A \,x^{2} c^{2} d^{2} e^{3}-32130 B \,x^{2} b^{2} d \,e^{4}+34272 B \,x^{2} b c \,d^{2} e^{3}-10080 B \,x^{2} c^{2} d^{3} e^{2}-30940 A x \,b^{2} d \,e^{4}+28560 A x b c \,d^{2} e^{3}-7616 A x \,c^{2} d^{3} e^{2}+14280 B x \,b^{2} d^{2} e^{3}-15232 B x b c \,d^{3} e^{2}+4480 B x \,c^{2} d^{4} e +8840 A \,b^{2} d^{2} e^{3}-8160 A b c \,d^{3} e^{2}+2176 A \,c^{2} d^{4} e -4080 B \,b^{2} d^{3} e^{2}+4352 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{765765 e^{6}}\) | \(341\) |
| trager | \(\frac {2 \left (45045 B \,e^{8} c^{2} x^{8}+51051 A \,c^{2} e^{8} x^{7}+102102 B b c \,e^{8} x^{7}+105105 B \,c^{2} d \,e^{7} x^{7}+117810 A b c \,e^{8} x^{6}+121737 A \,c^{2} d \,e^{7} x^{6}+58905 B \,b^{2} e^{8} x^{6}+243474 B b c d \,e^{7} x^{6}+63525 B \,c^{2} d^{2} e^{6} x^{6}+69615 A \,b^{2} e^{8} x^{5}+289170 A b c d \,e^{7} x^{5}+76041 A \,c^{2} d^{2} e^{6} x^{5}+144585 B \,b^{2} d \,e^{7} x^{5}+152082 B b c \,d^{2} e^{6} x^{5}+315 B \,c^{2} d^{3} e^{5} x^{5}+177905 A \,b^{2} d \,e^{7} x^{4}+189210 A b c \,d^{2} e^{6} x^{4}+595 A \,c^{2} d^{3} e^{5} x^{4}+94605 B \,b^{2} d^{2} e^{6} x^{4}+1190 B b c \,d^{3} e^{5} x^{4}-350 B \,c^{2} d^{4} e^{4} x^{4}+124865 A \,b^{2} d^{2} e^{6} x^{3}+2550 A b c \,d^{3} e^{5} x^{3}-680 A \,c^{2} d^{4} e^{4} x^{3}+1275 B \,b^{2} d^{3} e^{5} x^{3}-1360 B b c \,d^{4} e^{4} x^{3}+400 B \,c^{2} d^{5} e^{3} x^{3}+3315 A \,b^{2} d^{3} e^{5} x^{2}-3060 A b c \,d^{4} e^{4} x^{2}+816 A \,c^{2} d^{5} e^{3} x^{2}-1530 B \,b^{2} d^{4} e^{4} x^{2}+1632 B b c \,d^{5} e^{3} x^{2}-480 B \,c^{2} d^{6} e^{2} x^{2}-4420 A \,b^{2} d^{4} e^{4} x +4080 A b c \,d^{5} e^{3} x -1088 A \,c^{2} d^{6} e^{2} x +2040 B \,b^{2} d^{5} e^{3} x -2176 B b c \,d^{6} e^{2} x +640 B \,c^{2} d^{7} e x +8840 A \,b^{2} d^{5} e^{3}-8160 A b c \,d^{6} e^{2}+2176 A \,c^{2} d^{7} e -4080 B \,b^{2} d^{6} e^{2}+4352 B b c \,d^{7} e -1280 B \,c^{2} d^{8}\right ) \sqrt {e x +d}}{765765 e^{6}}\) | \(605\) |
| risch | \(\frac {2 \left (45045 B \,e^{8} c^{2} x^{8}+51051 A \,c^{2} e^{8} x^{7}+102102 B b c \,e^{8} x^{7}+105105 B \,c^{2} d \,e^{7} x^{7}+117810 A b c \,e^{8} x^{6}+121737 A \,c^{2} d \,e^{7} x^{6}+58905 B \,b^{2} e^{8} x^{6}+243474 B b c d \,e^{7} x^{6}+63525 B \,c^{2} d^{2} e^{6} x^{6}+69615 A \,b^{2} e^{8} x^{5}+289170 A b c d \,e^{7} x^{5}+76041 A \,c^{2} d^{2} e^{6} x^{5}+144585 B \,b^{2} d \,e^{7} x^{5}+152082 B b c \,d^{2} e^{6} x^{5}+315 B \,c^{2} d^{3} e^{5} x^{5}+177905 A \,b^{2} d \,e^{7} x^{4}+189210 A b c \,d^{2} e^{6} x^{4}+595 A \,c^{2} d^{3} e^{5} x^{4}+94605 B \,b^{2} d^{2} e^{6} x^{4}+1190 B b c \,d^{3} e^{5} x^{4}-350 B \,c^{2} d^{4} e^{4} x^{4}+124865 A \,b^{2} d^{2} e^{6} x^{3}+2550 A b c \,d^{3} e^{5} x^{3}-680 A \,c^{2} d^{4} e^{4} x^{3}+1275 B \,b^{2} d^{3} e^{5} x^{3}-1360 B b c \,d^{4} e^{4} x^{3}+400 B \,c^{2} d^{5} e^{3} x^{3}+3315 A \,b^{2} d^{3} e^{5} x^{2}-3060 A b c \,d^{4} e^{4} x^{2}+816 A \,c^{2} d^{5} e^{3} x^{2}-1530 B \,b^{2} d^{4} e^{4} x^{2}+1632 B b c \,d^{5} e^{3} x^{2}-480 B \,c^{2} d^{6} e^{2} x^{2}-4420 A \,b^{2} d^{4} e^{4} x +4080 A b c \,d^{5} e^{3} x -1088 A \,c^{2} d^{6} e^{2} x +2040 B \,b^{2} d^{5} e^{3} x -2176 B b c \,d^{6} e^{2} x +640 B \,c^{2} d^{7} e x +8840 A \,b^{2} d^{5} e^{3}-8160 A b c \,d^{6} e^{2}+2176 A \,c^{2} d^{7} e -4080 B \,b^{2} d^{6} e^{2}+4352 B b c \,d^{7} e -1280 B \,c^{2} d^{8}\right ) \sqrt {e x +d}}{765765 e^{6}}\) | \(605\) |
16/693*(e*x+d)^(7/2)*(((-231/68*d*e^4*x^4+462/221*d^2*e^3*x^3-32/221*d^5+6 93/136*e^5*x^5-252/221*d^3*e^2*x^2+112/221*d^4*e*x)*B+16/65*(3003/128*e^4* x^4-231/16*d*e^3*x^3+63/8*d^2*e^2*x^2-7/2*d^3*e*x+d^4)*A*e)*c^2-12/13*((-1 001/80*e^4*x^4+77/10*d*e^3*x^3-21/5*d^2*e^2*x^2+28/15*d^3*e*x-8/15*d^4)*B+ A*e*(-231/16*e^3*x^3+63/8*d*e^2*x^2-7/2*d^2*e*x+d^3))*e*b*c+((-189/52*d*e^ 2*x^2+693/104*e^3*x^3-6/13*d^3+21/13*d^2*e*x)*B+A*e*(63/8*e^2*x^2-7/2*d*e* x+d^2))*e^2*b^2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (243) = 486\).
Time = 0.32 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.85 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, B c^{2} e^{8} x^{8} - 1280 \, B c^{2} d^{8} + 8840 \, A b^{2} d^{5} e^{3} + 2176 \, {\left (2 \, B b c + A c^{2}\right )} d^{7} e - 4080 \, {\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{2} + 3003 \, {\left (35 \, B c^{2} d e^{7} + 17 \, {\left (2 \, B b c + A c^{2}\right )} e^{8}\right )} x^{7} + 231 \, {\left (275 \, B c^{2} d^{2} e^{6} + 527 \, {\left (2 \, B b c + A c^{2}\right )} d e^{7} + 255 \, {\left (B b^{2} + 2 \, A b c\right )} e^{8}\right )} x^{6} + 63 \, {\left (5 \, B c^{2} d^{3} e^{5} + 1105 \, A b^{2} e^{8} + 1207 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{6} + 2295 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{7}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{4} e^{4} - 5083 \, A b^{2} d e^{7} - 17 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{5} - 2703 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{5} e^{3} + 24973 \, A b^{2} d^{2} e^{6} - 136 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{4} + 255 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{6} e^{2} - 1105 \, A b^{2} d^{3} e^{5} - 272 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e^{3} + 510 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{7} e - 1105 \, A b^{2} d^{4} e^{4} - 272 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} e^{2} + 510 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{3}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{6}} \]
2/765765*(45045*B*c^2*e^8*x^8 - 1280*B*c^2*d^8 + 8840*A*b^2*d^5*e^3 + 2176 *(2*B*b*c + A*c^2)*d^7*e - 4080*(B*b^2 + 2*A*b*c)*d^6*e^2 + 3003*(35*B*c^2 *d*e^7 + 17*(2*B*b*c + A*c^2)*e^8)*x^7 + 231*(275*B*c^2*d^2*e^6 + 527*(2*B *b*c + A*c^2)*d*e^7 + 255*(B*b^2 + 2*A*b*c)*e^8)*x^6 + 63*(5*B*c^2*d^3*e^5 + 1105*A*b^2*e^8 + 1207*(2*B*b*c + A*c^2)*d^2*e^6 + 2295*(B*b^2 + 2*A*b*c )*d*e^7)*x^5 - 35*(10*B*c^2*d^4*e^4 - 5083*A*b^2*d*e^7 - 17*(2*B*b*c + A*c ^2)*d^3*e^5 - 2703*(B*b^2 + 2*A*b*c)*d^2*e^6)*x^4 + 5*(80*B*c^2*d^5*e^3 + 24973*A*b^2*d^2*e^6 - 136*(2*B*b*c + A*c^2)*d^4*e^4 + 255*(B*b^2 + 2*A*b*c )*d^3*e^5)*x^3 - 3*(160*B*c^2*d^6*e^2 - 1105*A*b^2*d^3*e^5 - 272*(2*B*b*c + A*c^2)*d^5*e^3 + 510*(B*b^2 + 2*A*b*c)*d^4*e^4)*x^2 + 4*(160*B*c^2*d^7*e - 1105*A*b^2*d^4*e^4 - 272*(2*B*b*c + A*c^2)*d^6*e^2 + 510*(B*b^2 + 2*A*b *c)*d^5*e^3)*x)*sqrt(e*x + d)/e^6
Time = 1.44 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.64 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{5}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{7 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*c**2*(d + e*x)**(17/2)/(17*e**5) + (d + e*x)**(15/2)*(A*c* *2*e + 2*B*b*c*e - 5*B*c**2*d)/(15*e**5) + (d + e*x)**(13/2)*(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)/(13*e**5) + (d + e*x)**(11/2)*(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)/(11*e**5) + (d + e*x)**(9/2) *(-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)/(9*e**5) + (d + e*x)**(7/2)*(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)/(7*e**5))/e, Ne(e, 0)), (d**(5/2)*(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), Tr ue))
Time = 0.21 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (45045 \, {\left (e x + d\right )}^{\frac {17}{2}} B c^{2} - 51051 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 69615 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{765765 \, e^{6}} \]
2/765765*(45045*(e*x + d)^(17/2)*B*c^2 - 51051*(5*B*c^2*d - (2*B*b*c + A*c ^2)*e)*(e*x + d)^(15/2) + 58905*(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)^(13/2) - 69615*(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)^(11/2) + 85085*(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)^(9/2) - 109395*(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)^(7/2))/e^ 6
Leaf count of result is larger than twice the leaf count of optimal. 1888 vs. \(2 (243) = 486\).
Time = 0.30 (sec) , antiderivative size = 1888, normalized size of antiderivative = 7.07 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
2/765765*(51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b^2*d^3/e^2 + 21879*(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^3/e^3 + 43758*(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^3/e^3 + 65637*(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*d^2/e^2 + 4862*(3 5*(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^3/e^4 + 2431*(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^3/e^4 + 7293*(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*d^2/e^3 + 14586*(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*b*c*d^2/e^3 + 7293*(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 + 3 15*sqrt(e*x + d)*d^4)*A*b^2*d/e^2 + 1105*(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*c^2*d^3/e^5 + 6630*(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...
Time = 10.42 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{15\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{13\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{9\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6} \]
((d + e*x)^(15/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(15*e^6) + ((d + e *x)^(11/2)*(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 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(11*e^6) + ((d + e*x)^(13/2)*(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))/(13*e^6) + ( 2*B*c^2*(d + e*x)^(17/2))/(17*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(9/2)*(2*A *b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(9*e^6) + (2*d^2*(A*e - B*d)* (b*e - c*d)^2*(d + e*x)^(7/2))/(7*e^6)