Integrand size = 26, antiderivative size = 267 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{9/2}}{9 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{15/2}}{15 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{17/2}}{17 e^6}+\frac {2 B c^2 (d+e x)^{19/2}}{19 e^6} \] Output:
-2/9*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(9/2)/e^6+2/11*d*(-b*e+c*d)*(B*d* (-3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(11/2)/e^6+2/13*(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)^(13/2)/ e^6-2/15*(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)^( 15/2)/e^6-2/17*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(17/2)/e^6+2/19*B*c^2*(e *x+d)^(19/2)/e^6
Time = 0.38 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{9/2} \left (19 A e \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )+B \left (323 b^2 e^2 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+38 b c e \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )-5 c^2 \left (256 d^5-1152 d^4 e x+3168 d^3 e^2 x^2-6864 d^2 e^3 x^3+12870 d e^4 x^4-21879 e^5 x^5\right )\right )\right )}{2078505 e^6} \] Input:
Integrate[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]
Output:
(2*(d + e*x)^(9/2)*(19*A*e*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 3 4*b*c*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + c^2*(128*d^ 4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)) + B*( 323*b^2*e^2*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 38*b*c* e*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^ 4) - 5*c^2*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x^2 - 6864*d^2*e^3*x^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5))))/(2078505*e^6)
Time = 0.81 (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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{13/2} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^5}-\frac {d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{e^5}+\frac {c (d+e x)^{15/2} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5}+\frac {B c^2 (d+e x)^{17/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{15/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{15 e^6}+\frac {2 (d+e x)^{13/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{13 e^6}-\frac {2 d^2 (d+e x)^{9/2} (B d-A e) (c d-b e)^2}{9 e^6}-\frac {2 c (d+e x)^{17/2} (-A c e-2 b B e+5 B c d)}{17 e^6}+\frac {2 d (d+e x)^{11/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{11 e^6}+\frac {2 B c^2 (d+e x)^{19/2}}{19 e^6}\) |
Input:
Int[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]
Output:
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*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)^(11/2))/(11*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)^(13/2))/(13*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)^(15/2))/(15*e^6) - (2*c*(5* B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(17/2))/(17*e^6) + (2*B*c^2*(d + e*x)^( 19/2))/(19*e^6)
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 = 2.34 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (e x +d \right )^{\frac {9}{2}} \left (\left (\frac {\left (\frac {1287}{8} e^{5} x^{5}-\frac {6435}{68} d \,e^{4} x^{4}-\frac {396}{17} d^{3} e^{2} x^{2}+\frac {858}{17} d^{2} e^{3} x^{3}-\frac {32}{17} d^{5}+\frac {144}{17} d^{4} e x \right ) B}{19}+\frac {16 e \left (\frac {6435}{128} e^{4} x^{4}-\frac {429}{16} d \,e^{3} x^{3}+\frac {99}{8} d^{2} e^{2} x^{2}-\frac {9}{2} d^{3} e x +d^{4}\right ) A}{85}\right ) c^{2}-\frac {4 e \left (\frac {\left (-\frac {6435}{16} e^{4} x^{4}+\frac {429}{2} d \,e^{3} x^{3}-99 d^{2} e^{2} x^{2}+36 d^{3} e x -8 d^{4}\right ) B}{17}+A e \left (-\frac {429}{16} e^{3} x^{3}+\frac {99}{8} d \,e^{2} x^{2}-\frac {9}{2} d^{2} e x +d^{3}\right )\right ) b c}{5}+e^{2} b^{2} \left (\frac {\left (\frac {429}{8} e^{3} x^{3}-2 d^{3}-\frac {99}{4} d \,e^{2} x^{2}+9 d^{2} e x \right ) B}{5}+A e \left (\frac {99}{8} e^{2} x^{2}+d^{2}-\frac {9}{2} d e x \right )\right )\right )}{1287 e^{6}}\) | \(259\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) c \left (b e -c d \right )+B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c \left (b e -c d \right )+\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} c \left (b e -c d \right )+\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{6}}\) | \(278\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) c \left (b e -c d \right )+B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c \left (b e -c d \right )+\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} c \left (b e -c d \right )+\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{6}}\) | \(278\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (109395 B \,x^{5} c^{2} e^{5}+122265 A \,x^{4} c^{2} e^{5}+244530 B \,x^{4} b c \,e^{5}-64350 B \,x^{4} c^{2} d \,e^{4}+277134 A \,x^{3} b c \,e^{5}-65208 A \,x^{3} c^{2} d \,e^{4}+138567 B \,x^{3} b^{2} e^{5}-130416 B \,x^{3} b c d \,e^{4}+34320 B \,x^{3} c^{2} d^{2} e^{3}+159885 A \,x^{2} b^{2} e^{5}-127908 A \,x^{2} b c d \,e^{4}+30096 A \,x^{2} c^{2} d^{2} e^{3}-63954 B \,x^{2} b^{2} d \,e^{4}+60192 B \,x^{2} b c \,d^{2} e^{3}-15840 B \,x^{2} c^{2} d^{3} e^{2}-58140 A x \,b^{2} d \,e^{4}+46512 A x b c \,d^{2} e^{3}-10944 A x \,c^{2} d^{3} e^{2}+23256 B x \,b^{2} d^{2} e^{3}-21888 B x b c \,d^{3} e^{2}+5760 B x \,c^{2} d^{4} e +12920 A \,b^{2} d^{2} e^{3}-10336 A b c \,d^{3} e^{2}+2432 A \,c^{2} d^{4} e -5168 B \,b^{2} d^{3} e^{2}+4864 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{2078505 e^{6}}\) | \(341\) |
| orering | \(\frac {2 \left (109395 B \,x^{5} c^{2} e^{5}+122265 A \,x^{4} c^{2} e^{5}+244530 B \,x^{4} b c \,e^{5}-64350 B \,x^{4} c^{2} d \,e^{4}+277134 A \,x^{3} b c \,e^{5}-65208 A \,x^{3} c^{2} d \,e^{4}+138567 B \,x^{3} b^{2} e^{5}-130416 B \,x^{3} b c d \,e^{4}+34320 B \,x^{3} c^{2} d^{2} e^{3}+159885 A \,x^{2} b^{2} e^{5}-127908 A \,x^{2} b c d \,e^{4}+30096 A \,x^{2} c^{2} d^{2} e^{3}-63954 B \,x^{2} b^{2} d \,e^{4}+60192 B \,x^{2} b c \,d^{2} e^{3}-15840 B \,x^{2} c^{2} d^{3} e^{2}-58140 A x \,b^{2} d \,e^{4}+46512 A x b c \,d^{2} e^{3}-10944 A x \,c^{2} d^{3} e^{2}+23256 B x \,b^{2} d^{2} e^{3}-21888 B x b c \,d^{3} e^{2}+5760 B x \,c^{2} d^{4} e +12920 A \,b^{2} d^{2} e^{3}-10336 A b c \,d^{3} e^{2}+2432 A \,c^{2} d^{4} e -5168 B \,b^{2} d^{3} e^{2}+4864 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \left (e x +d \right )^{\frac {9}{2}} \left (c \,x^{2}+b x \right )^{2}}{2078505 e^{6} \left (c x +b \right )^{2} x^{2}}\) | \(362\) |
| trager | \(\frac {2 \left (109395 B \,c^{2} e^{9} x^{9}+122265 A \,c^{2} e^{9} x^{8}+244530 B b c \,e^{9} x^{8}+373230 B \,c^{2} d \,e^{8} x^{8}+277134 A b c \,e^{9} x^{7}+423852 A \,c^{2} d \,e^{8} x^{7}+138567 B \,b^{2} e^{9} x^{7}+847704 B b c d \,e^{8} x^{7}+433290 B \,c^{2} d^{2} e^{7} x^{7}+159885 A \,b^{2} e^{9} x^{6}+980628 A b c d \,e^{8} x^{6}+502854 A \,c^{2} d^{2} e^{7} x^{6}+490314 B \,b^{2} d \,e^{8} x^{6}+1005708 B b c \,d^{2} e^{7} x^{6}+172920 B \,c^{2} d^{3} e^{6} x^{6}+581400 A \,b^{2} d \,e^{8} x^{5}+1197684 A b c \,d^{2} e^{7} x^{5}+207252 A \,c^{2} d^{3} e^{6} x^{5}+598842 B \,b^{2} d^{2} e^{7} x^{5}+414504 B b c \,d^{3} e^{6} x^{5}+315 B \,c^{2} d^{4} e^{5} x^{5}+739670 A \,b^{2} d^{2} e^{7} x^{4}+516800 A b c \,d^{3} e^{6} x^{4}+665 A \,c^{2} d^{4} e^{5} x^{4}+258400 B \,b^{2} d^{3} e^{6} x^{4}+1330 B b c \,d^{4} e^{5} x^{4}-350 B \,c^{2} d^{5} e^{4} x^{4}+342380 A \,b^{2} d^{3} e^{6} x^{3}+3230 A b c \,d^{4} e^{5} x^{3}-760 A \,c^{2} d^{5} e^{4} x^{3}+1615 B \,b^{2} d^{4} e^{5} x^{3}-1520 B b c \,d^{5} e^{4} x^{3}+400 B \,c^{2} d^{6} e^{3} x^{3}+4845 A \,b^{2} d^{4} e^{5} x^{2}-3876 A b c \,d^{5} e^{4} x^{2}+912 A \,c^{2} d^{6} e^{3} x^{2}-1938 B \,b^{2} d^{5} e^{4} x^{2}+1824 B b c \,d^{6} e^{3} x^{2}-480 B \,c^{2} d^{7} e^{2} x^{2}-6460 A \,b^{2} d^{5} e^{4} x +5168 A b c \,d^{6} e^{3} x -1216 A \,c^{2} d^{7} e^{2} x +2584 B \,b^{2} d^{6} e^{3} x -2432 B b c \,d^{7} e^{2} x +640 B \,c^{2} d^{8} e x +12920 A \,b^{2} d^{6} e^{3}-10336 A b c \,d^{7} e^{2}+2432 A \,c^{2} d^{8} e -5168 B \,b^{2} d^{7} e^{2}+4864 B b c \,d^{8} e -1280 B \,c^{2} d^{9}\right ) \sqrt {e x +d}}{2078505 e^{6}}\) | \(693\) |
| risch | \(\frac {2 \left (109395 B \,c^{2} e^{9} x^{9}+122265 A \,c^{2} e^{9} x^{8}+244530 B b c \,e^{9} x^{8}+373230 B \,c^{2} d \,e^{8} x^{8}+277134 A b c \,e^{9} x^{7}+423852 A \,c^{2} d \,e^{8} x^{7}+138567 B \,b^{2} e^{9} x^{7}+847704 B b c d \,e^{8} x^{7}+433290 B \,c^{2} d^{2} e^{7} x^{7}+159885 A \,b^{2} e^{9} x^{6}+980628 A b c d \,e^{8} x^{6}+502854 A \,c^{2} d^{2} e^{7} x^{6}+490314 B \,b^{2} d \,e^{8} x^{6}+1005708 B b c \,d^{2} e^{7} x^{6}+172920 B \,c^{2} d^{3} e^{6} x^{6}+581400 A \,b^{2} d \,e^{8} x^{5}+1197684 A b c \,d^{2} e^{7} x^{5}+207252 A \,c^{2} d^{3} e^{6} x^{5}+598842 B \,b^{2} d^{2} e^{7} x^{5}+414504 B b c \,d^{3} e^{6} x^{5}+315 B \,c^{2} d^{4} e^{5} x^{5}+739670 A \,b^{2} d^{2} e^{7} x^{4}+516800 A b c \,d^{3} e^{6} x^{4}+665 A \,c^{2} d^{4} e^{5} x^{4}+258400 B \,b^{2} d^{3} e^{6} x^{4}+1330 B b c \,d^{4} e^{5} x^{4}-350 B \,c^{2} d^{5} e^{4} x^{4}+342380 A \,b^{2} d^{3} e^{6} x^{3}+3230 A b c \,d^{4} e^{5} x^{3}-760 A \,c^{2} d^{5} e^{4} x^{3}+1615 B \,b^{2} d^{4} e^{5} x^{3}-1520 B b c \,d^{5} e^{4} x^{3}+400 B \,c^{2} d^{6} e^{3} x^{3}+4845 A \,b^{2} d^{4} e^{5} x^{2}-3876 A b c \,d^{5} e^{4} x^{2}+912 A \,c^{2} d^{6} e^{3} x^{2}-1938 B \,b^{2} d^{5} e^{4} x^{2}+1824 B b c \,d^{6} e^{3} x^{2}-480 B \,c^{2} d^{7} e^{2} x^{2}-6460 A \,b^{2} d^{5} e^{4} x +5168 A b c \,d^{6} e^{3} x -1216 A \,c^{2} d^{7} e^{2} x +2584 B \,b^{2} d^{6} e^{3} x -2432 B b c \,d^{7} e^{2} x +640 B \,c^{2} d^{8} e x +12920 A \,b^{2} d^{6} e^{3}-10336 A b c \,d^{7} e^{2}+2432 A \,c^{2} d^{8} e -5168 B \,b^{2} d^{7} e^{2}+4864 B b c \,d^{8} e -1280 B \,c^{2} d^{9}\right ) \sqrt {e x +d}}{2078505 e^{6}}\) | \(693\) |
Input:
int((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
16/1287*(e*x+d)^(9/2)*((1/19*(1287/8*e^5*x^5-6435/68*d*e^4*x^4-396/17*d^3* e^2*x^2+858/17*d^2*e^3*x^3-32/17*d^5+144/17*d^4*e*x)*B+16/85*e*(6435/128*e ^4*x^4-429/16*d*e^3*x^3+99/8*d^2*e^2*x^2-9/2*d^3*e*x+d^4)*A)*c^2-4/5*e*(1/ 17*(-6435/16*e^4*x^4+429/2*d*e^3*x^3-99*d^2*e^2*x^2+36*d^3*e*x-8*d^4)*B+A* e*(-429/16*e^3*x^3+99/8*d*e^2*x^2-9/2*d^2*e*x+d^3))*b*c+e^2*b^2*(1/5*(429/ 8*e^3*x^3-2*d^3-99/4*d*e^2*x^2+9*d^2*e*x)*B+A*e*(99/8*e^2*x^2+d^2-9/2*d*e* x)))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (243) = 486\).
Time = 0.08 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.10 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (109395 \, B c^{2} e^{9} x^{9} - 1280 \, B c^{2} d^{9} + 12920 \, A b^{2} d^{6} e^{3} + 2432 \, {\left (2 \, B b c + A c^{2}\right )} d^{8} e - 5168 \, {\left (B b^{2} + 2 \, A b c\right )} d^{7} e^{2} + 6435 \, {\left (58 \, B c^{2} d e^{8} + 19 \, {\left (2 \, B b c + A c^{2}\right )} e^{9}\right )} x^{8} + 429 \, {\left (1010 \, B c^{2} d^{2} e^{7} + 988 \, {\left (2 \, B b c + A c^{2}\right )} d e^{8} + 323 \, {\left (B b^{2} + 2 \, A b c\right )} e^{9}\right )} x^{7} + 33 \, {\left (5240 \, B c^{2} d^{3} e^{6} + 4845 \, A b^{2} e^{9} + 15238 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{7} + 14858 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{8}\right )} x^{6} + 9 \, {\left (35 \, B c^{2} d^{4} e^{5} + 64600 \, A b^{2} d e^{8} + 23028 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{6} + 66538 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{7}\right )} x^{5} - 5 \, {\left (70 \, B c^{2} d^{5} e^{4} - 147934 \, A b^{2} d^{2} e^{7} - 133 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{5} - 51680 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{6} e^{3} + 68476 \, A b^{2} d^{3} e^{6} - 152 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e^{4} + 323 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{7} e^{2} - 1615 \, A b^{2} d^{4} e^{5} - 304 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} e^{3} + 646 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{8} e - 1615 \, A b^{2} d^{5} e^{4} - 304 \, {\left (2 \, B b c + A c^{2}\right )} d^{7} e^{2} + 646 \, {\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{3}\right )} x\right )} \sqrt {e x + d}}{2078505 \, e^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
2/2078505*(109395*B*c^2*e^9*x^9 - 1280*B*c^2*d^9 + 12920*A*b^2*d^6*e^3 + 2 432*(2*B*b*c + A*c^2)*d^8*e - 5168*(B*b^2 + 2*A*b*c)*d^7*e^2 + 6435*(58*B* c^2*d*e^8 + 19*(2*B*b*c + A*c^2)*e^9)*x^8 + 429*(1010*B*c^2*d^2*e^7 + 988* (2*B*b*c + A*c^2)*d*e^8 + 323*(B*b^2 + 2*A*b*c)*e^9)*x^7 + 33*(5240*B*c^2* d^3*e^6 + 4845*A*b^2*e^9 + 15238*(2*B*b*c + A*c^2)*d^2*e^7 + 14858*(B*b^2 + 2*A*b*c)*d*e^8)*x^6 + 9*(35*B*c^2*d^4*e^5 + 64600*A*b^2*d*e^8 + 23028*(2 *B*b*c + A*c^2)*d^3*e^6 + 66538*(B*b^2 + 2*A*b*c)*d^2*e^7)*x^5 - 5*(70*B*c ^2*d^5*e^4 - 147934*A*b^2*d^2*e^7 - 133*(2*B*b*c + A*c^2)*d^4*e^5 - 51680* (B*b^2 + 2*A*b*c)*d^3*e^6)*x^4 + 5*(80*B*c^2*d^6*e^3 + 68476*A*b^2*d^3*e^6 - 152*(2*B*b*c + A*c^2)*d^5*e^4 + 323*(B*b^2 + 2*A*b*c)*d^4*e^5)*x^3 - 3* (160*B*c^2*d^7*e^2 - 1615*A*b^2*d^4*e^5 - 304*(2*B*b*c + A*c^2)*d^6*e^3 + 646*(B*b^2 + 2*A*b*c)*d^5*e^4)*x^2 + 4*(160*B*c^2*d^8*e - 1615*A*b^2*d^5*e ^4 - 304*(2*B*b*c + A*c^2)*d^7*e^2 + 646*(B*b^2 + 2*A*b*c)*d^6*e^3)*x)*sqr t(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (272) = 544\).
Time = 1.04 (sec) , antiderivative size = 1352, normalized size of antiderivative = 5.06 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(e*x+d)**(7/2)*(c*x**2+b*x)**2,x)
Output:
Piecewise((16*A*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt (d + e*x)/(1287*e**2) + 2*A*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*A*b **2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/12 87 + 80*A*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*A*b*c*d**7*sqrt(d + e*x)/(6435*e**4) + 32*A*b*c*d**6*x*sqrt( d + e*x)/(6435*e**3) - 8*A*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4*A*b *c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 640*A*b*c*d**3*x**4*sqrt(d + e*x)/12 87 + 824*A*b*c*d**2*e*x**5*sqrt(d + e*x)/715 + 184*A*b*c*d*e**2*x**6*sqrt( d + e*x)/195 + 4*A*b*c*e**3*x**7*sqrt(d + e*x)/15 + 256*A*c**2*d**8*sqrt(d + e*x)/(109395*e**5) - 128*A*c**2*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32 *A*c**2*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*A*c**2*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*A*c**2*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*A *c**2*d**3*x**5*sqrt(d + e*x)/12155 + 1604*A*c**2*d**2*e*x**6*sqrt(d + e*x )/3315 + 104*A*c**2*d*e**2*x**7*sqrt(d + e*x)/255 + 2*A*c**2*e**3*x**8*sqr t(d + e*x)/17 - 32*B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*b**2*d**6* x*sqrt(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*b**2*d*e**2 *x**6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15 + 512*B*b*c* d**8*sqrt(d + e*x)/(109395*e**5) - 256*B*b*c*d**7*x*sqrt(d + e*x)/(1093...
Time = 0.04 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (109395 \, {\left (e x + d\right )}^{\frac {19}{2}} B c^{2} - 122265 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 138567 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 159885 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 188955 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 230945 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{2078505 \, e^{6}} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
2/2078505*(109395*(e*x + d)^(19/2)*B*c^2 - 122265*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(17/2) + 138567*(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)^(15/2) - 159885*(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)^(1 3/2) + 188955*(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)^(11/2) - 230945*(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)^(9 /2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 2546 vs. \(2 (243) = 486\).
Time = 0.31 (sec) , antiderivative size = 2546, normalized size of antiderivative = 9.54 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
2/14549535*(969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b^2*d^4/e^2 + 415701*(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^4/e^3 + 831402* (5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sq rt(e*x + d)*d^3)*A*b*c*d^4/e^3 + 1662804*(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^3/e^2 + 92378*(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^4/e^4 + 46189 *(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 4 20*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*c^2*d^4/e^4 + 184756*(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^3/e^3 + 369512*(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^3/e^3 + 277134*(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^2*d^2/e^2 + 20995*(63*(e*x + d)^(1 1/2) - 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^4/e^5 + 167960*(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(...
Time = 10.96 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{17\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{15\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{19/2}}{19\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{11\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6} \] Input:
int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^(7/2),x)
Output:
((d + e*x)^(17/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(17*e^6) + ((d + e *x)^(13/2)*(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))/(13*e^6) + ((d + e*x)^(15/2)*(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))/(15*e^6) + ( 2*B*c^2*(d + e*x)^(19/2))/(19*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(11/2)*(2* A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(11*e^6) + (2*d^2*(A*e - B*d )*(b*e - c*d)^2*(d + e*x)^(9/2))/(9*e^6)
Time = 0.24 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.59 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (109395 b \,c^{2} e^{9} x^{9}+122265 a \,c^{2} e^{9} x^{8}+244530 b^{2} c \,e^{9} x^{8}+373230 b \,c^{2} d \,e^{8} x^{8}+277134 a b c \,e^{9} x^{7}+423852 a \,c^{2} d \,e^{8} x^{7}+138567 b^{3} e^{9} x^{7}+847704 b^{2} c d \,e^{8} x^{7}+433290 b \,c^{2} d^{2} e^{7} x^{7}+159885 a \,b^{2} e^{9} x^{6}+980628 a b c d \,e^{8} x^{6}+502854 a \,c^{2} d^{2} e^{7} x^{6}+490314 b^{3} d \,e^{8} x^{6}+1005708 b^{2} c \,d^{2} e^{7} x^{6}+172920 b \,c^{2} d^{3} e^{6} x^{6}+581400 a \,b^{2} d \,e^{8} x^{5}+1197684 a b c \,d^{2} e^{7} x^{5}+207252 a \,c^{2} d^{3} e^{6} x^{5}+598842 b^{3} d^{2} e^{7} x^{5}+414504 b^{2} c \,d^{3} e^{6} x^{5}+315 b \,c^{2} d^{4} e^{5} x^{5}+739670 a \,b^{2} d^{2} e^{7} x^{4}+516800 a b c \,d^{3} e^{6} x^{4}+665 a \,c^{2} d^{4} e^{5} x^{4}+258400 b^{3} d^{3} e^{6} x^{4}+1330 b^{2} c \,d^{4} e^{5} x^{4}-350 b \,c^{2} d^{5} e^{4} x^{4}+342380 a \,b^{2} d^{3} e^{6} x^{3}+3230 a b c \,d^{4} e^{5} x^{3}-760 a \,c^{2} d^{5} e^{4} x^{3}+1615 b^{3} d^{4} e^{5} x^{3}-1520 b^{2} c \,d^{5} e^{4} x^{3}+400 b \,c^{2} d^{6} e^{3} x^{3}+4845 a \,b^{2} d^{4} e^{5} x^{2}-3876 a b c \,d^{5} e^{4} x^{2}+912 a \,c^{2} d^{6} e^{3} x^{2}-1938 b^{3} d^{5} e^{4} x^{2}+1824 b^{2} c \,d^{6} e^{3} x^{2}-480 b \,c^{2} d^{7} e^{2} x^{2}-6460 a \,b^{2} d^{5} e^{4} x +5168 a b c \,d^{6} e^{3} x -1216 a \,c^{2} d^{7} e^{2} x +2584 b^{3} d^{6} e^{3} x -2432 b^{2} c \,d^{7} e^{2} x +640 b \,c^{2} d^{8} e x +12920 a \,b^{2} d^{6} e^{3}-10336 a b c \,d^{7} e^{2}+2432 a \,c^{2} d^{8} e -5168 b^{3} d^{7} e^{2}+4864 b^{2} c \,d^{8} e -1280 b \,c^{2} d^{9}\right )}{2078505 e^{6}} \] Input:
int((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x)^2,x)
Output:
(2*sqrt(d + e*x)*(12920*a*b**2*d**6*e**3 - 6460*a*b**2*d**5*e**4*x + 4845* a*b**2*d**4*e**5*x**2 + 342380*a*b**2*d**3*e**6*x**3 + 739670*a*b**2*d**2* e**7*x**4 + 581400*a*b**2*d*e**8*x**5 + 159885*a*b**2*e**9*x**6 - 10336*a* b*c*d**7*e**2 + 5168*a*b*c*d**6*e**3*x - 3876*a*b*c*d**5*e**4*x**2 + 3230* a*b*c*d**4*e**5*x**3 + 516800*a*b*c*d**3*e**6*x**4 + 1197684*a*b*c*d**2*e* *7*x**5 + 980628*a*b*c*d*e**8*x**6 + 277134*a*b*c*e**9*x**7 + 2432*a*c**2* d**8*e - 1216*a*c**2*d**7*e**2*x + 912*a*c**2*d**6*e**3*x**2 - 760*a*c**2* d**5*e**4*x**3 + 665*a*c**2*d**4*e**5*x**4 + 207252*a*c**2*d**3*e**6*x**5 + 502854*a*c**2*d**2*e**7*x**6 + 423852*a*c**2*d*e**8*x**7 + 122265*a*c**2 *e**9*x**8 - 5168*b**3*d**7*e**2 + 2584*b**3*d**6*e**3*x - 1938*b**3*d**5* e**4*x**2 + 1615*b**3*d**4*e**5*x**3 + 258400*b**3*d**3*e**6*x**4 + 598842 *b**3*d**2*e**7*x**5 + 490314*b**3*d*e**8*x**6 + 138567*b**3*e**9*x**7 + 4 864*b**2*c*d**8*e - 2432*b**2*c*d**7*e**2*x + 1824*b**2*c*d**6*e**3*x**2 - 1520*b**2*c*d**5*e**4*x**3 + 1330*b**2*c*d**4*e**5*x**4 + 414504*b**2*c*d **3*e**6*x**5 + 1005708*b**2*c*d**2*e**7*x**6 + 847704*b**2*c*d*e**8*x**7 + 244530*b**2*c*e**9*x**8 - 1280*b*c**2*d**9 + 640*b*c**2*d**8*e*x - 480*b *c**2*d**7*e**2*x**2 + 400*b*c**2*d**6*e**3*x**3 - 350*b*c**2*d**5*e**4*x* *4 + 315*b*c**2*d**4*e**5*x**5 + 172920*b*c**2*d**3*e**6*x**6 + 433290*b*c **2*d**2*e**7*x**7 + 373230*b*c**2*d*e**8*x**8 + 109395*b*c**2*e**9*x**9)) /(2078505*e**6)