Integrand size = 28, antiderivative size = 252 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{9 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{11 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6} \]
-2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(5/2)/e^6+4/7*(a*e^2-b*d*e +c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(7/2)/e^6-2/9*(-b*e+2 *c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(9/2)/e^6+8/11*c*( 5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(11/2)/e^6-10/13*c^2*(-b*e+2*c *d)*(e*x+d)^(13/2)/e^6+4/15*c^3*(e*x+d)^(15/2)/e^6
Time = 0.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{5/2} \left (c^3 \left (-512 d^5+1280 d^4 e x-2240 d^3 e^2 x^2+3360 d^2 e^3 x^3-4620 d e^4 x^4+6006 e^5 x^5\right )+143 b e^3 \left (63 a^2 e^2+18 a b e (-2 d+5 e x)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-78 c e^2 \left (33 a^2 e^2 (2 d-5 e x)-11 a b e \left (8 d^2-20 d e x+35 e^2 x^2\right )+2 b^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )+3 c^2 e \left (52 a e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )\right )}{45045 e^6} \]
(2*(d + e*x)^(5/2)*(c^3*(-512*d^5 + 1280*d^4*e*x - 2240*d^3*e^2*x^2 + 3360 *d^2*e^3*x^3 - 4620*d*e^4*x^4 + 6006*e^5*x^5) + 143*b*e^3*(63*a^2*e^2 + 18 *a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 78*c*e^2*(3 3*a^2*e^2*(2*d - 5*e*x) - 11*a*b*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 2*b^2 *(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3)) + 3*c^2*e*(52*a*e*(-1 6*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b*(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))))/(45045*e^6)
Time = 0.42 (sec) , antiderivative size = 252, 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.
| \(\displaystyle \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {4 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^{7/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^{3/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {5 c^2 (d+e x)^{11/2} (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^{13/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^6}+\frac {4 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6}\) |
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^6) + (4* (c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x )^(7/2))/(7*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(9/2))/(9*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(11/2))/(11*e^6) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(13/ 2))/(13*e^6) + (4*c^3*(d + e*x)^(15/2))/(15*e^6)
3.17.2.3.1 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 = 0.46 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {2 c^{3} x^{5}}{3}+\frac {20 x^{3} \left (\frac {55 b x}{52}+a \right ) c^{2}}{11}+\frac {10 \left (\frac {14}{11} b^{2} x^{2}+\frac {7}{3} a b x +a^{2}\right ) x c}{7}+b \left (\frac {10}{7} a b x +\frac {5}{9} b^{2} x^{2}+a^{2}\right )\right ) e^{5}-\frac {4 d \left (\frac {35 c^{3} x^{4}}{39}+\frac {70 \left (\frac {15 b x}{13}+a \right ) x^{2} c^{2}}{33}+\left (\frac {70}{33} b^{2} x^{2}+\frac {10}{3} a b x +a^{2}\right ) c +b^{2} \left (\frac {5 b x}{9}+a \right )\right ) e^{4}}{7}+\frac {16 d^{2} \left (\frac {70 c^{3} x^{3}}{143}+\frac {10 x \left (\frac {35 b x}{26}+a \right ) c^{2}}{11}+b \left (\frac {10 b x}{11}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{21}-\frac {64 d^{3} \left (\frac {35 c^{2} x^{2}}{39}+\left (\frac {25 b x}{13}+a \right ) c +b^{2}\right ) c \,e^{2}}{231}+\frac {640 d^{4} \left (\frac {2 c x}{3}+b \right ) c^{2} e}{3003}-\frac {512 c^{3} d^{5}}{9009}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{6}}\) | \(234\) |
| derivativedivides | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {10 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{6}}\) | \(265\) |
| default | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {10 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} c +2 c \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (b e -2 c d \right ) \left (\left (b e -2 c d \right )^{2}+2 c \left (e^{2} a -b d e +c \,d^{2}\right )\right )+4 c \left (e^{2} a -b d e +c \,d^{2}\right ) \left (b e -2 c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (b e -2 c d \right )^{2} \left (e^{2} a -b d e +c \,d^{2}\right )+2 c \left (e^{2} a -b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (b e -2 c d \right ) \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{6}}\) | \(265\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (6006 x^{5} c^{3} e^{5}+17325 x^{4} b \,c^{2} e^{5}-4620 x^{4} c^{3} d \,e^{4}+16380 x^{3} a \,c^{2} e^{5}+16380 x^{3} b^{2} c \,e^{5}-12600 x^{3} b \,c^{2} d \,e^{4}+3360 x^{3} c^{3} d^{2} e^{3}+30030 x^{2} a b c \,e^{5}-10920 x^{2} a \,c^{2} d \,e^{4}+5005 x^{2} b^{3} e^{5}-10920 x^{2} b^{2} c d \,e^{4}+8400 x^{2} b \,c^{2} d^{2} e^{3}-2240 x^{2} c^{3} d^{3} e^{2}+12870 x \,a^{2} c \,e^{5}+12870 x a \,b^{2} e^{5}-17160 x a b c d \,e^{4}+6240 x a \,c^{2} d^{2} e^{3}-2860 x \,b^{3} d \,e^{4}+6240 x \,b^{2} c \,d^{2} e^{3}-4800 x b \,c^{2} d^{3} e^{2}+1280 x \,c^{3} d^{4} e +9009 a^{2} b \,e^{5}-5148 a^{2} c d \,e^{4}-5148 a \,b^{2} d \,e^{4}+6864 a b c \,d^{2} e^{3}-2496 a \,c^{2} d^{3} e^{2}+1144 b^{3} d^{2} e^{3}-2496 b^{2} c \,d^{3} e^{2}+1920 b \,c^{2} d^{4} e -512 c^{3} d^{5}\right )}{45045 e^{6}}\) | \(359\) |
| trager | \(\frac {2 \left (6006 c^{3} e^{7} x^{7}+17325 b \,c^{2} e^{7} x^{6}+7392 c^{3} d \,e^{6} x^{6}+16380 a \,c^{2} e^{7} x^{5}+16380 b^{2} c \,e^{7} x^{5}+22050 b \,c^{2} d \,e^{6} x^{5}+126 c^{3} d^{2} e^{5} x^{5}+30030 a b c \,e^{7} x^{4}+21840 a \,c^{2} d \,e^{6} x^{4}+5005 b^{3} e^{7} x^{4}+21840 b^{2} c d \,e^{6} x^{4}+525 b \,c^{2} d^{2} e^{5} x^{4}-140 c^{3} d^{3} e^{4} x^{4}+12870 a^{2} c \,e^{7} x^{3}+12870 a \,b^{2} e^{7} x^{3}+42900 a b c d \,e^{6} x^{3}+780 a \,c^{2} d^{2} e^{5} x^{3}+7150 b^{3} d \,e^{6} x^{3}+780 b^{2} c \,d^{2} e^{5} x^{3}-600 b \,c^{2} d^{3} e^{4} x^{3}+160 c^{3} d^{4} e^{3} x^{3}+9009 a^{2} b \,e^{7} x^{2}+20592 a^{2} c d \,e^{6} x^{2}+20592 a \,b^{2} d \,e^{6} x^{2}+2574 a b c \,d^{2} e^{5} x^{2}-936 a \,c^{2} d^{3} e^{4} x^{2}+429 b^{3} d^{2} e^{5} x^{2}-936 b^{2} c \,d^{3} e^{4} x^{2}+720 b \,c^{2} d^{4} e^{3} x^{2}-192 c^{3} d^{5} e^{2} x^{2}+18018 a^{2} b d \,e^{6} x +2574 a^{2} c \,d^{2} e^{5} x +2574 a \,b^{2} d^{2} e^{5} x -3432 a b c \,d^{3} e^{4} x +1248 a \,c^{2} d^{4} e^{3} x -572 b^{3} d^{3} e^{4} x +1248 b^{2} c \,d^{4} e^{3} x -960 b \,c^{2} d^{5} e^{2} x +256 c^{3} d^{6} e x +9009 a^{2} b \,d^{2} e^{5}-5148 a^{2} c \,d^{3} e^{4}-5148 a \,b^{2} d^{3} e^{4}+6864 a b c \,d^{4} e^{3}-2496 a \,c^{2} d^{5} e^{2}+1144 b^{3} d^{4} e^{3}-2496 b^{2} c \,d^{5} e^{2}+1920 b \,c^{2} d^{6} e -512 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(619\) |
| risch | \(\frac {2 \left (6006 c^{3} e^{7} x^{7}+17325 b \,c^{2} e^{7} x^{6}+7392 c^{3} d \,e^{6} x^{6}+16380 a \,c^{2} e^{7} x^{5}+16380 b^{2} c \,e^{7} x^{5}+22050 b \,c^{2} d \,e^{6} x^{5}+126 c^{3} d^{2} e^{5} x^{5}+30030 a b c \,e^{7} x^{4}+21840 a \,c^{2} d \,e^{6} x^{4}+5005 b^{3} e^{7} x^{4}+21840 b^{2} c d \,e^{6} x^{4}+525 b \,c^{2} d^{2} e^{5} x^{4}-140 c^{3} d^{3} e^{4} x^{4}+12870 a^{2} c \,e^{7} x^{3}+12870 a \,b^{2} e^{7} x^{3}+42900 a b c d \,e^{6} x^{3}+780 a \,c^{2} d^{2} e^{5} x^{3}+7150 b^{3} d \,e^{6} x^{3}+780 b^{2} c \,d^{2} e^{5} x^{3}-600 b \,c^{2} d^{3} e^{4} x^{3}+160 c^{3} d^{4} e^{3} x^{3}+9009 a^{2} b \,e^{7} x^{2}+20592 a^{2} c d \,e^{6} x^{2}+20592 a \,b^{2} d \,e^{6} x^{2}+2574 a b c \,d^{2} e^{5} x^{2}-936 a \,c^{2} d^{3} e^{4} x^{2}+429 b^{3} d^{2} e^{5} x^{2}-936 b^{2} c \,d^{3} e^{4} x^{2}+720 b \,c^{2} d^{4} e^{3} x^{2}-192 c^{3} d^{5} e^{2} x^{2}+18018 a^{2} b d \,e^{6} x +2574 a^{2} c \,d^{2} e^{5} x +2574 a \,b^{2} d^{2} e^{5} x -3432 a b c \,d^{3} e^{4} x +1248 a \,c^{2} d^{4} e^{3} x -572 b^{3} d^{3} e^{4} x +1248 b^{2} c \,d^{4} e^{3} x -960 b \,c^{2} d^{5} e^{2} x +256 c^{3} d^{6} e x +9009 a^{2} b \,d^{2} e^{5}-5148 a^{2} c \,d^{3} e^{4}-5148 a \,b^{2} d^{3} e^{4}+6864 a b c \,d^{4} e^{3}-2496 a \,c^{2} d^{5} e^{2}+1144 b^{3} d^{4} e^{3}-2496 b^{2} c \,d^{5} e^{2}+1920 b \,c^{2} d^{6} e -512 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(619\) |
2/5*((2/3*c^3*x^5+20/11*x^3*(55/52*b*x+a)*c^2+10/7*(14/11*b^2*x^2+7/3*a*b* x+a^2)*x*c+b*(10/7*a*b*x+5/9*b^2*x^2+a^2))*e^5-4/7*d*(35/39*c^3*x^4+70/33* (15/13*b*x+a)*x^2*c^2+(70/33*b^2*x^2+10/3*a*b*x+a^2)*c+b^2*(5/9*b*x+a))*e^ 4+16/21*d^2*(70/143*c^3*x^3+10/11*x*(35/26*b*x+a)*c^2+b*(10/11*b*x+a)*c+1/ 6*b^3)*e^3-64/231*d^3*(35/39*c^2*x^2+(25/13*b*x+a)*c+b^2)*c*e^2+640/3003*d ^4*(2/3*c*x+b)*c^2*e-512/9009*c^3*d^5)*(e*x+d)^(5/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (228) = 456\).
Time = 0.27 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.96 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6006 \, c^{3} e^{7} x^{7} - 512 \, c^{3} d^{7} + 1920 \, b c^{2} d^{6} e + 9009 \, a^{2} b d^{2} e^{5} - 2496 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} + 1144 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} - 5148 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \, {\left (32 \, c^{3} d e^{6} + 75 \, b c^{2} e^{7}\right )} x^{6} + 126 \, {\left (c^{3} d^{2} e^{5} + 175 \, b c^{2} d e^{6} + 130 \, {\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} - 35 \, {\left (4 \, c^{3} d^{3} e^{4} - 15 \, b c^{2} d^{2} e^{5} - 624 \, {\left (b^{2} c + a c^{2}\right )} d e^{6} - 143 \, {\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} + 10 \, {\left (16 \, c^{3} d^{4} e^{3} - 60 \, b c^{2} d^{3} e^{4} + 78 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} + 715 \, {\left (b^{3} + 6 \, a b c\right )} d e^{6} + 1287 \, {\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} - 3003 \, a^{2} b e^{7} + 312 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 143 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} - 6864 \, {\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} + 2 \, {\left (128 \, c^{3} d^{6} e - 480 \, b c^{2} d^{5} e^{2} + 9009 \, a^{2} b d e^{6} + 624 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 286 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 1287 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
2/45045*(6006*c^3*e^7*x^7 - 512*c^3*d^7 + 1920*b*c^2*d^6*e + 9009*a^2*b*d^ 2*e^5 - 2496*(b^2*c + a*c^2)*d^5*e^2 + 1144*(b^3 + 6*a*b*c)*d^4*e^3 - 5148 *(a*b^2 + a^2*c)*d^3*e^4 + 231*(32*c^3*d*e^6 + 75*b*c^2*e^7)*x^6 + 126*(c^ 3*d^2*e^5 + 175*b*c^2*d*e^6 + 130*(b^2*c + a*c^2)*e^7)*x^5 - 35*(4*c^3*d^3 *e^4 - 15*b*c^2*d^2*e^5 - 624*(b^2*c + a*c^2)*d*e^6 - 143*(b^3 + 6*a*b*c)* e^7)*x^4 + 10*(16*c^3*d^4*e^3 - 60*b*c^2*d^3*e^4 + 78*(b^2*c + a*c^2)*d^2* e^5 + 715*(b^3 + 6*a*b*c)*d*e^6 + 1287*(a*b^2 + a^2*c)*e^7)*x^3 - 3*(64*c^ 3*d^5*e^2 - 240*b*c^2*d^4*e^3 - 3003*a^2*b*e^7 + 312*(b^2*c + a*c^2)*d^3*e ^4 - 143*(b^3 + 6*a*b*c)*d^2*e^5 - 6864*(a*b^2 + a^2*c)*d*e^6)*x^2 + 2*(12 8*c^3*d^6*e - 480*b*c^2*d^5*e^2 + 9009*a^2*b*d*e^6 + 624*(b^2*c + a*c^2)*d ^4*e^3 - 286*(b^3 + 6*a*b*c)*d^3*e^4 + 1287*(a*b^2 + a^2*c)*d^2*e^5)*x)*sq rt(e*x + d)/e^6
Time = 1.49 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.68 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{5 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((2*(2*c**3*(d + e*x)**(15/2)/(15*e**5) + (d + e*x)**(13/2)*(5*b* c**2*e - 10*c**3*d)/(13*e**5) + (d + e*x)**(11/2)*(4*a*c**2*e**2 + 4*b**2* c*e**2 - 20*b*c**2*d*e + 20*c**3*d**2)/(11*e**5) + (d + e*x)**(9/2)*(6*a*b *c*e**3 - 12*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2 *e - 20*c**3*d**3)/(9*e**5) + (d + e*x)**(7/2)*(2*a**2*c*e**4 + 2*a*b**2*e **4 - 12*a*b*c*d*e**3 + 12*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 12*b**2*c*d* *2*e**2 - 20*b*c**2*d**3*e + 10*c**3*d**4)/(7*e**5) + (d + e*x)**(5/2)*(a* *2*b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4 + 6*a*b*c*d**2*e**3 - 4*a*c* *2*d**3*e**2 + b**3*d**2*e**3 - 4*b**2*c*d**3*e**2 + 5*b*c**2*d**4*e - 2*c **3*d**5)/(5*e**5))/e, Ne(e, 0)), (d**(3/2)*(a + b*x + c*x**2)**3/3, True) )
Time = 0.20 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.22 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (6006 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 17325 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 16380 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 12870 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 9009 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{6}} \]
2/45045*(6006*(e*x + d)^(15/2)*c^3 - 17325*(2*c^3*d - b*c^2*e)*(e*x + d)^( 13/2) + 16380*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(1 1/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*(e*x + d)^(9/2) + 12870*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*( b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(7/2) - 9009*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c ^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d) ^(5/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1556 vs. \(2 (228) = 456\).
Time = 0.28 (sec) , antiderivative size = 1556, normalized size of antiderivative = 6.17 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^2*b*d^2 + 30030*((e*x + d)^(3/2) - 3*sqrt(e *x + d)*d)*a^2*b*d + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b^2*d^2 /e + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*c*d^2/e + 3003*(3*(e* x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b + 3003*( 3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^3*d^2/e ^2 + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^ 2)*a*b*c*d^2/e^2 + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sq rt(e*x + d)*d^2)*a*b^2*d/e + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2) *d + 15*sqrt(e*x + d)*d^2)*a^2*c*d/e + 5148*(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^2*c*d^2/e^3 + 5148*(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*c^2*d^2/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)*b^3*d/e^2 + 15444*(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^2 + 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^2/e + 2574 *(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*s qrt(e*x + d)*d^3)*a^2*c/e + 715*(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*c^2*d^2/e^4 + 1144*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d +...
Time = 0.07 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.06 \[ \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{7\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^6} \]
((d + e*x)^(7/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 2 4*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(7* e^6) + (4*c^3*(d + e*x)^(15/2))/(15*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e *x)^(13/2))/(13*e^6) + ((d + e*x)^(11/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2 *c*e^2 - 40*b*c^2*d*e))/(11*e^6) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e ^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(9*e^6) + (2*(b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^6)