Integrand size = 28, antiderivative size = 427 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{5 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{13 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{15/2}}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8} \]
-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(3/2)/e^8+2/5*(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)^(5/2)/e^8-6/7* (-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)^(7/2)/e^8+2/9*(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)^(9 /2)/e^8-10/11*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 )^(11/2)/e^8+6/13*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(1 3/2)/e^8-14/15*c^3*(-b*e+2*c*d)*(e*x+d)^(15/2)/e^8+4/17*c^4*(e*x+d)^(17/2) /e^8
Time = 0.39 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.41 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (-14 c^4 \left (2048 d^7-3072 d^6 e x+3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5+6006 d e^6 x^6-6435 e^7 x^7\right )+2431 b e^4 \left (105 a^3 e^3+63 a^2 b e^2 (-2 d+3 e x)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )+221 c e^3 \left (462 a^3 e^3 (-2 d+3 e x)+297 a^2 b e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+132 a b^2 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b^3 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-51 c^2 e^2 \left (286 a^2 e^2 \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-65 a b e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+15 b^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )+17 c^3 e \left (30 a e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+7 b \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )\right )}{765765 e^8} \]
(2*(d + e*x)^(3/2)*(-14*c^4*(2048*d^7 - 3072*d^6*e*x + 3840*d^5*e^2*x^2 - 4480*d^4*e^3*x^3 + 5040*d^3*e^4*x^4 - 5544*d^2*e^5*x^5 + 6006*d*e^6*x^6 - 6435*e^7*x^7) + 2431*b*e^4*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) + 9* a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d *e^2*x^2 + 35*e^3*x^3)) + 221*c*e^3*(462*a^3*e^3*(-2*d + 3*e*x) + 297*a^2* b*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 132*a*b^2*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b^3*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2* x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) - 51*c^2*e^2*(286*a^2*e^2*(16*d^3 - 24 *d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3) - 65*a*b*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 15*b^2*(256*d^5 - 384*d^4 *e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e^5*x^5)) + 17*c^3*e*(30*a*e*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3* x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + 7*b*(1024*d^6 - 1536*d^5*e*x + 1920*d ^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e ^6*x^6))))/(765765*e^8)
Time = 0.60 (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.
| \(\displaystyle \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{7/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^{11/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^{9/2} (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {(d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {\sqrt {d+e x} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {7 c^3 (d+e x)^{13/2} (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^{15/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^8}+\frac {6 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{13 e^8}-\frac {10 c (d+e x)^{11/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {14 c^3 (d+e x)^{15/2} (2 c d-b e)}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8}\) |
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(3/2))/(3*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)^(5/2))/(5*e^8) - (6*(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)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*e^8) + (6*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e ))*(d + e*x)^(13/2))/(13*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(15/2))/(1 5*e^8) + (4*c^4*(d + e*x)^(17/2))/(17*e^8)
3.17.9.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.49 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {6 c^{4} x^{7}}{17}+\left (\frac {7}{5} b \,x^{6}+\frac {18}{13} a \,x^{5}\right ) c^{3}+\left (\frac {27}{13} b^{2} x^{5}+2 a^{2} x^{3}+\frac {45}{11} a b \,x^{4}\right ) c^{2}+\left (\frac {15}{11} b^{3} x^{4}+\frac {6}{5} a^{3} x +4 a \,b^{2} x^{3}+\frac {27}{7} a^{2} b \,x^{2}\right ) c +b \left (\frac {9}{7} a \,b^{2} x^{2}+\frac {9}{5} b \,a^{2} x +\frac {1}{3} x^{3} b^{3}+a^{3}\right )\right ) e^{7}-\frac {4 d \left (\frac {7 c^{4} x^{6}}{17}+\frac {225 \left (\frac {77 b x}{75}+a \right ) x^{4} c^{3}}{143}+\left (\frac {675}{286} b^{2} x^{4}+\frac {50}{11} a b \,x^{3}+\frac {15}{7} a^{2} x^{2}\right ) c^{2}+\left (\frac {50}{33} x^{3} b^{3}+\frac {30}{7} a \,b^{2} x^{2}+\frac {27}{7} b \,a^{2} x +a^{3}\right ) c +\frac {5 b^{4} x^{2}}{14}+\frac {9 a \,b^{3} x}{7}+\frac {3 b^{2} a^{2}}{2}\right ) e^{6}}{5}+\frac {72 d^{2} \left (\frac {98 c^{4} x^{5}}{663}+\frac {700 \left (\frac {21 b x}{20}+a \right ) x^{3} c^{3}}{1287}+\left (\frac {350}{429} b^{2} x^{3}+\frac {50}{33} a b \,x^{2}+\frac {2}{3} a^{2} x \right ) c^{2}+b \left (\frac {50}{99} b^{2} x^{2}+\frac {4}{3} a b x +a^{2}\right ) c +\frac {b^{3} \left (\frac {b x}{3}+a \right )}{3}\right ) e^{5}}{35}-\frac {32 d^{3} \left (\frac {735 c^{4} x^{4}}{2431}+\frac {150 x^{2} \left (\frac {49 b x}{45}+a \right ) c^{3}}{143}+\left (\frac {225}{143} b^{2} x^{2}+\frac {30}{11} a b x +a^{2}\right ) c^{2}+2 b^{2} \left (\frac {5 b x}{11}+a \right ) c +\frac {b^{4}}{6}\right ) e^{4}}{35}+\frac {128 d^{4} \left (\frac {98 c^{3} x^{3}}{663}+\frac {6 \left (\frac {7 b x}{6}+a \right ) x \,c^{2}}{13}+b \left (\frac {9 b x}{13}+a \right ) c +\frac {b^{3}}{3}\right ) c \,e^{3}}{77}-\frac {512 d^{5} \left (\frac {7 c^{2} x^{2}}{17}+\left (\frac {7 b x}{5}+a \right ) c +\frac {3 b^{2}}{2}\right ) c^{2} e^{2}}{1001}+\frac {1024 d^{6} c^{3} \left (\frac {6 c x}{17}+b \right ) e}{2145}-\frac {4096 c^{4} d^{7}}{36465}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}\) | \(504\) |
| derivativedivides | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}}{e^{8}}\) | \(713\) |
| default | \(\frac {\frac {4 c^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {14 \left (b e -2 c d \right ) c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}}{e^{8}}\) | \(713\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (90090 x^{7} c^{4} e^{7}+357357 x^{6} b \,c^{3} e^{7}-84084 x^{6} c^{4} d \,e^{6}+353430 x^{5} a \,c^{3} e^{7}+530145 x^{5} b^{2} c^{2} e^{7}-329868 x^{5} b \,c^{3} d \,e^{6}+77616 x^{5} c^{4} d^{2} e^{5}+1044225 x^{4} a b \,c^{2} e^{7}-321300 x^{4} a \,c^{3} d \,e^{6}+348075 x^{4} b^{3} c \,e^{7}-481950 x^{4} b^{2} c^{2} d \,e^{6}+299880 x^{4} b \,c^{3} d^{2} e^{5}-70560 x^{4} c^{4} d^{3} e^{4}+510510 x^{3} a^{2} c^{2} e^{7}+1021020 x^{3} a \,b^{2} c \,e^{7}-928200 x^{3} a b \,c^{2} d \,e^{6}+285600 x^{3} a \,c^{3} d^{2} e^{5}+85085 x^{3} b^{4} e^{7}-309400 x^{3} b^{3} c d \,e^{6}+428400 x^{3} b^{2} c^{2} d^{2} e^{5}-266560 x^{3} b \,c^{3} d^{3} e^{4}+62720 x^{3} c^{4} d^{4} e^{3}+984555 x^{2} a^{2} b c \,e^{7}-437580 x^{2} a^{2} c^{2} d \,e^{6}+328185 x^{2} a \,b^{3} e^{7}-875160 x^{2} a \,b^{2} c d \,e^{6}+795600 x^{2} a b \,c^{2} d^{2} e^{5}-244800 x^{2} a \,c^{3} d^{3} e^{4}-72930 x^{2} b^{4} d \,e^{6}+265200 x^{2} b^{3} c \,d^{2} e^{5}-367200 x^{2} b^{2} c^{2} d^{3} e^{4}+228480 x^{2} b \,c^{3} d^{4} e^{3}-53760 x^{2} c^{4} d^{5} e^{2}+306306 x \,a^{3} c \,e^{7}+459459 x \,a^{2} b^{2} e^{7}-787644 x \,a^{2} b c d \,e^{6}+350064 x \,a^{2} c^{2} d^{2} e^{5}-262548 x a \,b^{3} d \,e^{6}+700128 x a \,b^{2} c \,d^{2} e^{5}-636480 x a b \,c^{2} d^{3} e^{4}+195840 x a \,c^{3} d^{4} e^{3}+58344 x \,b^{4} d^{2} e^{5}-212160 x \,b^{3} c \,d^{3} e^{4}+293760 x \,b^{2} c^{2} d^{4} e^{3}-182784 x b \,c^{3} d^{5} e^{2}+43008 x \,c^{4} d^{6} e +255255 a^{3} b \,e^{7}-204204 a^{3} c d \,e^{6}-306306 a^{2} b^{2} d \,e^{6}+525096 a^{2} b c \,d^{2} e^{5}-233376 a^{2} c^{2} d^{3} e^{4}+175032 a \,b^{3} d^{2} e^{5}-466752 a \,b^{2} c \,d^{3} e^{4}+424320 a b \,c^{2} d^{4} e^{3}-130560 a \,c^{3} d^{5} e^{2}-38896 b^{4} d^{3} e^{4}+141440 b^{3} c \,d^{4} e^{3}-195840 b^{2} c^{2} d^{5} e^{2}+121856 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{765765 e^{8}}\) | \(795\) |
| trager | \(\text {Expression too large to display}\) | \(1008\) |
| risch | \(\text {Expression too large to display}\) | \(1008\) |
2/3*((6/17*c^4*x^7+(7/5*b*x^6+18/13*a*x^5)*c^3+(27/13*b^2*x^5+2*a^2*x^3+45 /11*a*b*x^4)*c^2+(15/11*b^3*x^4+6/5*a^3*x+4*a*b^2*x^3+27/7*a^2*b*x^2)*c+b* (9/7*a*b^2*x^2+9/5*b*a^2*x+1/3*x^3*b^3+a^3))*e^7-4/5*d*(7/17*c^4*x^6+225/1 43*(77/75*b*x+a)*x^4*c^3+(675/286*b^2*x^4+50/11*a*b*x^3+15/7*a^2*x^2)*c^2+ (50/33*x^3*b^3+30/7*a*b^2*x^2+27/7*b*a^2*x+a^3)*c+5/14*b^4*x^2+9/7*a*b^3*x +3/2*b^2*a^2)*e^6+72/35*d^2*(98/663*c^4*x^5+700/1287*(21/20*b*x+a)*x^3*c^3 +(350/429*b^2*x^3+50/33*a*b*x^2+2/3*a^2*x)*c^2+b*(50/99*b^2*x^2+4/3*a*b*x+ a^2)*c+1/3*b^3*(1/3*b*x+a))*e^5-32/35*d^3*(735/2431*c^4*x^4+150/143*x^2*(4 9/45*b*x+a)*c^3+(225/143*b^2*x^2+30/11*a*b*x+a^2)*c^2+2*b^2*(5/11*b*x+a)*c +1/6*b^4)*e^4+128/77*d^4*(98/663*c^3*x^3+6/13*(7/6*b*x+a)*x*c^2+b*(9/13*b* x+a)*c+1/3*b^3)*c*e^3-512/1001*d^5*(7/17*c^2*x^2+(7/5*b*x+a)*c+3/2*b^2)*c^ 2*e^2+1024/2145*d^6*c^3*(6/17*c*x+b)*e-4096/36465*c^4*d^7)*(e*x+d)^(3/2)/e ^8
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (395) = 790\).
Time = 0.27 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.88 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (90090 \, c^{4} e^{8} x^{8} - 28672 \, c^{4} d^{8} + 121856 \, b c^{3} d^{7} e + 255255 \, a^{3} b d e^{7} - 65280 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 141440 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} - 38896 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 175032 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 102102 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 3003 \, {\left (2 \, c^{4} d e^{7} + 119 \, b c^{3} e^{8}\right )} x^{7} - 231 \, {\left (28 \, c^{4} d^{2} e^{6} - 119 \, b c^{3} d e^{7} - 765 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 63 \, {\left (112 \, c^{4} d^{3} e^{5} - 476 \, b c^{3} d^{2} e^{6} + 255 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 5525 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{4} e^{4} - 952 \, b c^{3} d^{3} e^{5} + 510 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 1105 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} - 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{5} e^{3} - 7616 \, b c^{3} d^{4} e^{4} + 4080 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 8840 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 65637 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{6} e^{2} - 15232 \, b c^{3} d^{5} e^{3} + 8160 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 17680 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 4862 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 21879 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} - 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + {\left (14336 \, c^{4} d^{7} e - 60928 \, b c^{3} d^{6} e^{2} + 255255 \, a^{3} b e^{8} + 32640 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 70720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 19448 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 87516 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{8}} \]
2/765765*(90090*c^4*e^8*x^8 - 28672*c^4*d^8 + 121856*b*c^3*d^7*e + 255255* a^3*b*d*e^7 - 65280*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 141440*(b^3*c + 3*a*b* c^2)*d^5*e^3 - 38896*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 175032*(a*b^ 3 + 3*a^2*b*c)*d^3*e^5 - 102102*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 3003*(2*c^ 4*d*e^7 + 119*b*c^3*e^8)*x^7 - 231*(28*c^4*d^2*e^6 - 119*b*c^3*d*e^7 - 765 *(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 63*(112*c^4*d^3*e^5 - 476*b*c^3*d^2*e^6 + 255*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 5525*(b^3*c + 3*a*b*c^2)*e^8)*x^5 - 35 *(224*c^4*d^4*e^4 - 952*b*c^3*d^3*e^5 + 510*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 1105*(b^3*c + 3*a*b*c^2)*d*e^7 - 2431*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8 )*x^4 + 5*(1792*c^4*d^5*e^3 - 7616*b*c^3*d^4*e^4 + 4080*(3*b^2*c^2 + 2*a*c ^3)*d^3*e^5 - 8840*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 2431*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 65637*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 3*(3584*c^4*d^6*e^ 2 - 15232*b*c^3*d^5*e^3 + 8160*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 17680*(b^3* c + 3*a*b*c^2)*d^3*e^5 + 4862*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 218 79*(a*b^3 + 3*a^2*b*c)*d*e^7 - 51051*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + (143 36*c^4*d^7*e - 60928*b*c^3*d^6*e^2 + 255255*a^3*b*e^8 + 32640*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 70720*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 19448*(b^4 + 12*a* b^2*c + 6*a^2*c^2)*d^3*e^5 - 87516*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 51051*(3* a^2*b^2 + 2*a^3*c)*d*e^7)*x)*sqrt(e*x + d)/e^8
Time = 1.76 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.02 \[ \int (b+2 c x) \sqrt {d+e x} \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 {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{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 )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{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 )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{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 )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} \left (a + b x + c x^{2}\right )^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((2*(2*c**4*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(7*b* c**3*e - 14*c**4*d)/(15*e**7) + (d + e*x)**(13/2)*(6*a*c**3*e**2 + 9*b**2* c**2*e**2 - 42*b*c**3*d*e + 42*c**4*d**2)/(13*e**7) + (d + e*x)**(11/2)*(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)/(11*e**7) + (d + e*x)**(9/2)*(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)/(9*e**7) + (d + e*x)**(7/2)*(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)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(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)/(3*e**7))/e, Ne( e, 0)), (sqrt(d)*(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) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {2 \, {\left (90090 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{4} - 357357 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\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 {13}{2}} - 348075 \, {\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 {11}{2}} + 85085 \, {\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 {9}{2}} - 328185 \, {\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 {7}{2}} + 153153 \, {\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 {5}{2}} - 255255 \, {\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 {3}{2}}\right )}}{765765 \, e^{8}} \]
2/765765*(90090*(e*x + d)^(17/2)*c^4 - 357357*(2*c^4*d - b*c^3*e)*(e*x + d )^(15/2) + 176715*(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)^(13/2) - 348075*(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)^(11/2) + 85085*(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)^(9/2) - 32818 5*(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)^(7/2) + 153153*(14*c^4*d^6 - 42*b*c^3*d^5*e + 1 5*(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)^(5/2) - 255255*(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)^(3/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1769 vs. \(2 (395) = 790\).
Time = 0.28 (sec) , antiderivative size = 1769, normalized size of antiderivative = 4.14 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*a^3*b*d + 255255*((e*x + d)^(3/2) - 3*sqrt( e*x + d)*d)*a^3*b + 765765*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b^2*d /e + 510510*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*c*d/e + 153153*(3*(e *x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^3*d/e^2 + 459459*(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/e^2 + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt (e*x + d)*d^2)*a^2*b^2/e + 102102*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)* d + 15*sqrt(e*x + d)*d^2)*a^3*c/e + 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^4*d/e^3 + 262 548*(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)*a*b^2*c*d/e^3 + 131274*(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/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^3/e^2 + 196911*(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/e^2 + 12155*(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^3*c*d/e^4 + 36465 *(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*b*c^2*d/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*...
Time = 0.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{13/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{13\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{9\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{7\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{5\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{11\,e^8} \]
((d + e*x)^(13/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d *e))/(13*e^8) + (4*c^4*(d + e*x)^(17/2))/(17*e^8) - ((28*c^4*d - 14*b*c^3* e)*(d + e*x)^(15/2))/(15*e^8) + ((d + e*x)^(9/2)*(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))/(9*e^8) + (2*(b *e - 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(3*e^8) + (6*(b*e - 2*c*d)*(d + e*x)^(7/2)*(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))/(7 *e^8) + (2*(d + e*x)^(5/2)*(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))/(5*e^8) + (10*c*(b*e - 2*c*d)*(d + e*x)^(11/ 2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(11*e^8)