Integrand size = 28, antiderivative size = 427 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{3 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{15/2}}{5 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{17/2}}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \]
-2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(5/2)/e^8+2/7*(a*e^2-b*d*e +c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(7/2)/e^8-2/3* (-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e *x+d)^(9/2)/e^8+2/11*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3 *d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^( 11/2)/e^8-10/13*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x +d)^(13/2)/e^8+2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^( 15/2)/e^8-14/17*c^3*(-b*e+2*c*d)*(e*x+d)^(17/2)/e^8+4/19*c^4*(e*x+d)^(19/2 )/e^8
Time = 0.42 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.41 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (-14 c^4 \left (2048 d^7-5120 d^6 e x+8960 d^5 e^2 x^2-13440 d^4 e^3 x^3+18480 d^3 e^4 x^4-24024 d^2 e^5 x^5+30030 d e^6 x^6-36465 e^7 x^7\right )+4199 b e^4 \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+323 c e^3 \left (858 a^3 e^3 (-2 d+5 e x)+429 a^2 b e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+156 a b^2 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b^3 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-969 c^2 e^2 \left (26 a^2 e^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-5 a b e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^2 \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )+19 c^3 e \left (34 a e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+7 b \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )\right )}{4849845 e^8} \]
(2*(d + e*x)^(5/2)*(-14*c^4*(2048*d^7 - 5120*d^6*e*x + 8960*d^5*e^2*x^2 - 13440*d^4*e^3*x^3 + 18480*d^3*e^4*x^4 - 24024*d^2*e^5*x^5 + 30030*d*e^6*x^ 6 - 36465*e^7*x^7) + 4199*b*e^4*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)) + 323*c*e^3*(858*a^3*e^3*(-2*d + 5*e*x) + 4 29*a^2*b*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 156*a*b^2*e*(-16*d^3 + 40*d ^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b^3*(128*d^4 - 320*d^3*e*x + 560* d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)) - 969*c^2*e^2*(26*a^2*e^2*(16 *d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3) - 5*a*b*e*(128*d^4 - 320*d ^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b^2*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2 - 1680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003* e^5*x^5)) + 19*c^3*e*(34*a*e*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5) + 7*b*(1024*d^6 - 2560*d ^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d* e^5*x^5 + 15015*e^6*x^6))))/(4849845*e^8)
Time = 0.63 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^{11/2} (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {(d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {(d+e x)^{3/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {7 c^3 (d+e x)^{15/2} (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^{17/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac {2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8}\) |
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^8) + (2* (c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*( d + e*x)^(7/2))/(7*e^8) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2* d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(9/2))/(3*e^8) + (2*(70*c^4 *d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e ) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(11/2))/(11*e ^8) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a *e))*(d + e*x)^(15/2))/(5*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(17/2))/( 17*e^8) + (4*c^4*(d + e*x)^(19/2))/(19*e^8)
3.17.8.3.1 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 = 0.78 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (\left (\frac {10 c^{4} x^{7}}{19}+\left (\frac {35}{17} b \,x^{6}+2 a \,x^{5}\right ) c^{3}+3 \left (b^{2} x^{5}+\frac {10}{11} a^{2} x^{3}+\frac {25}{13} a b \,x^{4}\right ) c^{2}+5 \left (\frac {5}{13} b^{3} x^{4}+\frac {2}{7} a^{3} x +\frac {12}{11} a \,b^{2} x^{3}+a^{2} b \,x^{2}\right ) c +b \left (\frac {5}{3} a \,b^{2} x^{2}+\frac {15}{7} b \,a^{2} x +\frac {5}{11} x^{3} b^{3}+a^{3}\right )\right ) e^{7}-\frac {4 d \left (\frac {245 c^{4} x^{6}}{323}+\frac {35 \left (\frac {91 b x}{85}+a \right ) x^{4} c^{3}}{13}+\frac {35 \left (\frac {33}{26} b^{2} x^{2}+\frac {30}{13} a b x +a^{2}\right ) x^{2} c^{2}}{11}+\left (\frac {350}{143} x^{3} b^{3}+\frac {70}{11} a \,b^{2} x^{2}+5 b \,a^{2} x +a^{3}\right ) c +\frac {35 b^{4} x^{2}}{66}+\frac {5 a \,b^{3} x}{3}+\frac {3 b^{2} a^{2}}{2}\right ) e^{6}}{7}+\frac {8 d^{2} \left (\frac {98 c^{4} x^{5}}{323}+\frac {140 \left (\frac {77 b x}{68}+a \right ) x^{3} c^{3}}{143}+\frac {10 \left (\frac {21}{13} b^{2} x^{2}+\frac {35}{13} a b x +a^{2}\right ) x \,c^{2}}{11}+b \left (\frac {350}{429} b^{2} x^{2}+\frac {20}{11} a b x +a^{2}\right ) c +\frac {b^{3} \left (\frac {5 b x}{11}+a \right )}{3}\right ) e^{5}}{7}-\frac {32 d^{3} \left (\frac {2695 c^{4} x^{4}}{4199}+\frac {70 \left (\frac {21 b x}{17}+a \right ) x^{2} c^{3}}{39}+\left (\frac {35}{13} b^{2} x^{2}+\frac {50}{13} a b x +a^{2}\right ) c^{2}+2 \left (\frac {25}{39} b^{3} x +b^{2} a \right ) c +\frac {b^{4}}{6}\right ) e^{4}}{77}+\frac {640 d^{4} \left (\frac {98 c^{3} x^{3}}{323}+\frac {2 \left (\frac {49 b x}{34}+a \right ) x \,c^{2}}{3}+b \left (b x +a \right ) c +\frac {b^{3}}{3}\right ) c \,e^{3}}{1001}-\frac {512 d^{5} \left (\frac {245 c^{2} x^{2}}{323}+\left (\frac {35 b x}{17}+a \right ) c +\frac {3 b^{2}}{2}\right ) c^{2} e^{2}}{3003}+\frac {1024 d^{6} \left (\frac {10 c x}{19}+b \right ) c^{3} e}{7293}-\frac {4096 c^{4} d^{7}}{138567}\right )}{5 e^{8}}\) | \(500\) |
| derivativedivides | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(713\) |
| default | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} c^{2}+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right )+2 c \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right )+2 c \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right )+6 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(713\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (510510 x^{7} c^{4} e^{7}+1996995 x^{6} b \,c^{3} e^{7}-420420 x^{6} c^{4} d \,e^{6}+1939938 x^{5} a \,c^{3} e^{7}+2909907 x^{5} b^{2} c^{2} e^{7}-1597596 x^{5} b \,c^{3} d \,e^{6}+336336 x^{5} c^{4} d^{2} e^{5}+5595975 x^{4} a b \,c^{2} e^{7}-1492260 x^{4} a \,c^{3} d \,e^{6}+1865325 x^{4} b^{3} c \,e^{7}-2238390 x^{4} b^{2} c^{2} d \,e^{6}+1228920 x^{4} b \,c^{3} d^{2} e^{5}-258720 x^{4} c^{4} d^{3} e^{4}+2645370 x^{3} a^{2} c^{2} e^{7}+5290740 x^{3} a \,b^{2} c \,e^{7}-4069800 x^{3} a b \,c^{2} d \,e^{6}+1085280 x^{3} a \,c^{3} d^{2} e^{5}+440895 x^{3} b^{4} e^{7}-1356600 x^{3} b^{3} c d \,e^{6}+1627920 x^{3} b^{2} c^{2} d^{2} e^{5}-893760 x^{3} b \,c^{3} d^{3} e^{4}+188160 x^{3} c^{4} d^{4} e^{3}+4849845 x^{2} a^{2} b c \,e^{7}-1763580 x^{2} a^{2} c^{2} d \,e^{6}+1616615 x^{2} a \,b^{3} e^{7}-3527160 x^{2} a \,b^{2} c d \,e^{6}+2713200 x^{2} a b \,c^{2} d^{2} e^{5}-723520 x^{2} a \,c^{3} d^{3} e^{4}-293930 x^{2} b^{4} d \,e^{6}+904400 x^{2} b^{3} c \,d^{2} e^{5}-1085280 x^{2} b^{2} c^{2} d^{3} e^{4}+595840 x^{2} b \,c^{3} d^{4} e^{3}-125440 x^{2} c^{4} d^{5} e^{2}+1385670 x \,a^{3} c \,e^{7}+2078505 x \,a^{2} b^{2} e^{7}-2771340 x \,a^{2} b c d \,e^{6}+1007760 x \,a^{2} c^{2} d^{2} e^{5}-923780 x a \,b^{3} d \,e^{6}+2015520 x a \,b^{2} c \,d^{2} e^{5}-1550400 x a b \,c^{2} d^{3} e^{4}+413440 x a \,c^{3} d^{4} e^{3}+167960 x \,b^{4} d^{2} e^{5}-516800 x \,b^{3} c \,d^{3} e^{4}+620160 x \,b^{2} c^{2} d^{4} e^{3}-340480 x b \,c^{3} d^{5} e^{2}+71680 x \,c^{4} d^{6} e +969969 a^{3} b \,e^{7}-554268 a^{3} c d \,e^{6}-831402 a^{2} b^{2} d \,e^{6}+1108536 a^{2} b c \,d^{2} e^{5}-403104 a^{2} c^{2} d^{3} e^{4}+369512 a \,b^{3} d^{2} e^{5}-806208 a \,b^{2} c \,d^{3} e^{4}+620160 a b \,c^{2} d^{4} e^{3}-165376 a \,c^{3} d^{5} e^{2}-67184 b^{4} d^{3} e^{4}+206720 b^{3} c \,d^{4} e^{3}-248064 b^{2} c^{2} d^{5} e^{2}+136192 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{4849845 e^{8}}\) | \(795\) |
| trager | \(\text {Expression too large to display}\) | \(1225\) |
| risch | \(\text {Expression too large to display}\) | \(1225\) |
2/5*(e*x+d)^(5/2)*((10/19*c^4*x^7+(35/17*b*x^6+2*a*x^5)*c^3+3*(b^2*x^5+10/ 11*a^2*x^3+25/13*a*b*x^4)*c^2+5*(5/13*b^3*x^4+2/7*a^3*x+12/11*a*b^2*x^3+a^ 2*b*x^2)*c+b*(5/3*a*b^2*x^2+15/7*b*a^2*x+5/11*x^3*b^3+a^3))*e^7-4/7*d*(245 /323*c^4*x^6+35/13*(91/85*b*x+a)*x^4*c^3+35/11*(33/26*b^2*x^2+30/13*a*b*x+ a^2)*x^2*c^2+(350/143*x^3*b^3+70/11*a*b^2*x^2+5*b*a^2*x+a^3)*c+35/66*b^4*x ^2+5/3*a*b^3*x+3/2*b^2*a^2)*e^6+8/7*d^2*(98/323*c^4*x^5+140/143*(77/68*b*x +a)*x^3*c^3+10/11*(21/13*b^2*x^2+35/13*a*b*x+a^2)*x*c^2+b*(350/429*b^2*x^2 +20/11*a*b*x+a^2)*c+1/3*b^3*(5/11*b*x+a))*e^5-32/77*d^3*(2695/4199*c^4*x^4 +70/39*(21/17*b*x+a)*x^2*c^3+(35/13*b^2*x^2+50/13*a*b*x+a^2)*c^2+2*(25/39* b^3*x+b^2*a)*c+1/6*b^4)*e^4+640/1001*d^4*(98/323*c^3*x^3+2/3*(49/34*b*x+a) *x*c^2+b*(b*x+a)*c+1/3*b^3)*c*e^3-512/3003*d^5*(245/323*c^2*x^2+(35/17*b*x +a)*c+3/2*b^2)*c^2*e^2+1024/7293*d^6*(10/19*c*x+b)*c^3*e-4096/138567*c^4*d ^7)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (395) = 790\).
Time = 0.26 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.24 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (510510 \, c^{4} e^{9} x^{9} - 28672 \, c^{4} d^{9} + 136192 \, b c^{3} d^{8} e + 969969 \, a^{3} b d^{2} e^{7} - 82688 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{7} e^{2} + 206720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{6} e^{3} - 67184 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{5} e^{4} + 369512 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} e^{5} - 277134 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e^{6} + 15015 \, {\left (40 \, c^{4} d e^{8} + 133 \, b c^{3} e^{9}\right )} x^{8} + 3003 \, {\left (2 \, c^{4} d^{2} e^{7} + 798 \, b c^{3} d e^{8} + 323 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{9}\right )} x^{7} - 231 \, {\left (28 \, c^{4} d^{3} e^{6} - 133 \, b c^{3} d^{2} e^{7} - 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{8} - 8075 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{9}\right )} x^{6} + 21 \, {\left (336 \, c^{4} d^{4} e^{5} - 1596 \, b c^{3} d^{3} e^{6} + 969 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{7} + 113050 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{8} + 20995 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{9}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{5} e^{4} - 1064 \, b c^{3} d^{4} e^{5} + 646 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{6} - 1615 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{7} - 16796 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{8} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{9}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{6} e^{3} - 8512 \, b c^{3} d^{5} e^{4} + 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{5} - 12920 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{6} + 4199 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{7} + 461890 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{8} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{9}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{7} e^{2} - 17024 \, b c^{3} d^{6} e^{3} - 323323 \, a^{3} b e^{9} + 10336 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{4} - 25840 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{5} + 8398 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{6} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{7} - 369512 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{8}\right )} x^{2} + {\left (14336 \, c^{4} d^{8} e - 68096 \, b c^{3} d^{7} e^{2} + 1939938 \, a^{3} b d e^{8} + 41344 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{3} - 103360 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{4} + 33592 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{5} - 184756 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{6} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{8}} \]
2/4849845*(510510*c^4*e^9*x^9 - 28672*c^4*d^9 + 136192*b*c^3*d^8*e + 96996 9*a^3*b*d^2*e^7 - 82688*(3*b^2*c^2 + 2*a*c^3)*d^7*e^2 + 206720*(b^3*c + 3* a*b*c^2)*d^6*e^3 - 67184*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 369512*( a*b^3 + 3*a^2*b*c)*d^4*e^5 - 277134*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6 + 15015* (40*c^4*d*e^8 + 133*b*c^3*e^9)*x^8 + 3003*(2*c^4*d^2*e^7 + 798*b*c^3*d*e^8 + 323*(3*b^2*c^2 + 2*a*c^3)*e^9)*x^7 - 231*(28*c^4*d^3*e^6 - 133*b*c^3*d^ 2*e^7 - 5168*(3*b^2*c^2 + 2*a*c^3)*d*e^8 - 8075*(b^3*c + 3*a*b*c^2)*e^9)*x ^6 + 21*(336*c^4*d^4*e^5 - 1596*b*c^3*d^3*e^6 + 969*(3*b^2*c^2 + 2*a*c^3)* d^2*e^7 + 113050*(b^3*c + 3*a*b*c^2)*d*e^8 + 20995*(b^4 + 12*a*b^2*c + 6*a ^2*c^2)*e^9)*x^5 - 35*(224*c^4*d^5*e^4 - 1064*b*c^3*d^4*e^5 + 646*(3*b^2*c ^2 + 2*a*c^3)*d^3*e^6 - 1615*(b^3*c + 3*a*b*c^2)*d^2*e^7 - 16796*(b^4 + 12 *a*b^2*c + 6*a^2*c^2)*d*e^8 - 46189*(a*b^3 + 3*a^2*b*c)*e^9)*x^4 + 5*(1792 *c^4*d^6*e^3 - 8512*b*c^3*d^5*e^4 + 5168*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 1 2920*(b^3*c + 3*a*b*c^2)*d^3*e^6 + 4199*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2 *e^7 + 461890*(a*b^3 + 3*a^2*b*c)*d*e^8 + 138567*(3*a^2*b^2 + 2*a^3*c)*e^9 )*x^3 - 3*(3584*c^4*d^7*e^2 - 17024*b*c^3*d^6*e^3 - 323323*a^3*b*e^9 + 103 36*(3*b^2*c^2 + 2*a*c^3)*d^5*e^4 - 25840*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 839 8*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 46189*(a*b^3 + 3*a^2*b*c)*d^2*e ^7 - 369512*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*x^2 + (14336*c^4*d^8*e - 68096*b* c^3*d^7*e^2 + 1939938*a^3*b*d*e^8 + 41344*(3*b^2*c^2 + 2*a*c^3)*d^6*e^3...
Time = 1.89 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.02 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 18 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 6 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 60 a b c^{2} d^{3} e^{3} + 30 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 20 b^{3} c d^{3} e^{3} + 45 b^{2} c^{2} d^{4} e^{2} - 42 b c^{3} d^{5} e + 14 c^{4} d^{6}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 9 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 3 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 15 a b c^{2} d^{4} e^{3} - 6 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 5 b^{3} c d^{4} e^{3} - 9 b^{2} c^{2} d^{5} e^{2} + 7 b c^{3} d^{6} e - 2 c^{4} d^{7}\right )}{5 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((2*(2*c**4*(d + e*x)**(19/2)/(19*e**7) + (d + e*x)**(17/2)*(7*b* c**3*e - 14*c**4*d)/(17*e**7) + (d + e*x)**(15/2)*(6*a*c**3*e**2 + 9*b**2* c**2*e**2 - 42*b*c**3*d*e + 42*c**4*d**2)/(15*e**7) + (d + e*x)**(13/2)*(1 5*a*b*c**2*e**3 - 30*a*c**3*d*e**2 + 5*b**3*c*e**3 - 45*b**2*c**2*d*e**2 + 105*b*c**3*d**2*e - 70*c**4*d**3)/(13*e**7) + (d + e*x)**(11/2)*(6*a**2*c **2*e**4 + 12*a*b**2*c*e**4 - 60*a*b*c**2*d*e**3 + 60*a*c**3*d**2*e**2 + b **4*e**4 - 20*b**3*c*d*e**3 + 90*b**2*c**2*d**2*e**2 - 140*b*c**3*d**3*e + 70*c**4*d**4)/(11*e**7) + (d + e*x)**(9/2)*(9*a**2*b*c*e**5 - 18*a**2*c** 2*d*e**4 + 3*a*b**3*e**5 - 36*a*b**2*c*d*e**4 + 90*a*b*c**2*d**2*e**3 - 60 *a*c**3*d**3*e**2 - 3*b**4*d*e**4 + 30*b**3*c*d**2*e**3 - 90*b**2*c**2*d** 3*e**2 + 105*b*c**3*d**4*e - 42*c**4*d**5)/(9*e**7) + (d + e*x)**(7/2)*(2* a**3*c*e**6 + 3*a**2*b**2*e**6 - 18*a**2*b*c*d*e**5 + 18*a**2*c**2*d**2*e* *4 - 6*a*b**3*d*e**5 + 36*a*b**2*c*d**2*e**4 - 60*a*b*c**2*d**3*e**3 + 30* a*c**3*d**4*e**2 + 3*b**4*d**2*e**4 - 20*b**3*c*d**3*e**3 + 45*b**2*c**2*d **4*e**2 - 42*b*c**3*d**5*e + 14*c**4*d**6)/(7*e**7) + (d + e*x)**(5/2)*(a **3*b*e**7 - 2*a**3*c*d*e**6 - 3*a**2*b**2*d*e**6 + 9*a**2*b*c*d**2*e**5 - 6*a**2*c**2*d**3*e**4 + 3*a*b**3*d**2*e**5 - 12*a*b**2*c*d**3*e**4 + 15*a *b*c**2*d**4*e**3 - 6*a*c**3*d**5*e**2 - b**4*d**3*e**4 + 5*b**3*c*d**4*e* *3 - 9*b**2*c**2*d**5*e**2 + 7*b*c**3*d**6*e - 2*c**4*d**7)/(5*e**7))/e, N e(e, 0)), (d**(3/2)*(a + b*x + c*x**2)**4/4, True))
Time = 0.21 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.51 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (510510 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{4} - 1996995 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1865325 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 440895 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 969969 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{4849845 \, e^{8}} \]
2/4849845*(510510*(e*x + d)^(19/2)*c^4 - 1996995*(2*c^4*d - b*c^3*e)*(e*x + d)^(17/2) + 969969*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^ 2)*(e*x + d)^(15/2) - 1865325*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(13/2) + 440895*(70* c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(11/2) - 1616615*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - ( a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(9/2) + 692835*(14*c^4*d^6 - 42*b*c^3*d^ 5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + ( 3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(7/2) - 969969*(2*c^4*d^7 - 7*b*c^3*d^ 6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)* d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d ^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(5/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 2941 vs. \(2 (395) = 790\).
Time = 0.31 (sec) , antiderivative size = 2941, normalized size of antiderivative = 6.89 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*a^3*b*d^2 + 9699690*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*b*d + 14549535*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d )*a^2*b^2*d^2/e + 9699690*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*c*d^2/ e + 969969*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^ 2)*a^3*b + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^3*d^2/e^2 + 8729721*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2) *d + 15*sqrt(e*x + d)*d^2)*a^2*b*c*d^2/e^2 + 5819814*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2*d/e + 3879876*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*c*d/e + 415 701*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 3 5*sqrt(e*x + d)*d^3)*b^4*d^2/e^3 + 4988412*(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*c*d^2/e^ 3 + 2494206*(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^2*c^2*d^2/e^3 + 2494206*(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^3*d/e^2 + 7482618*(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^2*b*c*d/e^2 + 1247103*(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^2*b^2/e + 831402*(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^3*c/e + 230945*(35*(e*x + d)^...
Time = 0.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{15\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{11\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{3\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{7\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{13\,e^8} \]
((d + e*x)^(15/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d *e))/(15*e^8) + (4*c^4*(d + e*x)^(19/2))/(19*e^8) - ((28*c^4*d - 14*b*c^3* e)*(d + e*x)^(17/2))/(17*e^8) + ((d + e*x)^(11/2)*(2*b^4*e^4 + 140*c^4*d^4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e ^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e^3 - 120*a*b*c^2*d*e^3))/(11*e^8) + (2* (b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(5*e^8) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^2*e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/ (3*e^8) + (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2 + 14*c^2 *d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(7*e^8) + (10*c*(b*e - 2*c*d)*(d + e*x)^(1 3/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(13*e^8)