Integrand size = 26, antiderivative size = 399 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) x}{e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^2}{2 e^8}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^3}{3 e^8}-\frac {5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^4}{4 e^8}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^8}-\frac {7 c^3 (2 c d-b e) (d+e x)^6}{6 e^8}+\frac {2 c^4 (d+e x)^7}{7 e^8}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8} \]
(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))*x/e^7-3/2* (-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)^2/e^8+1/3*(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)^3/e^8- 5/4*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)^4/e^8+3/ 5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^5/e^8-7/6*c^3*(-b* e+2*c*d)*(e*x+d)^6/e^8+2/7*c^4*(e*x+d)^7/e^8-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d ^2)^3*ln(e*x+d)/e^8
Time = 0.22 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.21 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {e x \left (2 c^4 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )+70 b^2 e^4 \left (18 a^2 e^2+9 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+35 c e^3 \left (24 a^3 e^3+54 a^2 b e^2 (-2 d+e x)+24 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )+21 c^2 e^2 \left (20 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+3 b^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+7 c^3 e \left (6 a e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )-7 b \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )\right )-420 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3 \log (d+e x)}{420 e^8} \]
(e*x*(2*c^4*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 8 4*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6) + 70*b^2*e^4*(18*a^2*e^2 + 9*a* b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 35*c*e^3*(24*a^3*e ^3 + 54*a^2*b*e^2*(-2*d + e*x) + 24*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) - 5*b^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3)) + 21*c^2*e^2*(20*a ^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b*e*(-12*d^3 + 6*d^2*e*x - 4*d *e^2*x^2 + 3*e^3*x^3) + 3*b^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d *e^3*x^3 + 12*e^4*x^4)) + 7*c^3*e*(6*a*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2 *x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) - 7*b*(60*d^5 - 30*d^4*e*x + 20*d^3*e^2* x^2 - 15*d^2*e^3*x^3 + 12*d*e^4*x^4 - 10*e^5*x^5))) - 420*(2*c*d - b*e)*(c *d^2 + e*(-(b*d) + a*e))^3*Log[d + e*x])/(420*e^8)
Time = 0.85 (sec) , antiderivative size = 399, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^3 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (d+e x) (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac {7 c^3 (d+e x)^5 (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^6}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac {3 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {5 c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{4 e^8}-\frac {3 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8}+\frac {x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {7 c^3 (d+e x)^6 (2 c d-b e)}{6 e^8}+\frac {2 c^4 (d+e x)^7}{7 e^8}\) |
((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))* x)/e^7 - (3*(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)^2)/(2*e^8) + ((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)^3)/(3*e^8) - (5*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)^4)/(4*e^8) + (3*c^2*(14* c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^5)/(5*e^8) - (7*c^3*( 2*c*d - b*e)*(d + e*x)^6)/(6*e^8) + (2*c^4*(d + e*x)^7)/(7*e^8) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^8
3.16.19.3.1 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 = 0.59 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.74
| method | result | size |
| norman | \(\frac {\left (2 a^{3} c \,e^{6}+3 a^{2} b^{2} e^{6}-9 a^{2} b c d \,e^{5}+6 a^{2} c^{2} d^{2} e^{4}-3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}-15 a b \,c^{2} d^{3} e^{3}+6 a \,c^{3} d^{4} e^{2}+b^{4} d^{2} e^{4}-5 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}-7 b \,c^{3} d^{5} e +2 c^{4} d^{6}\right ) x}{e^{7}}+\frac {2 c^{4} x^{7}}{7 e}+\frac {\left (9 a^{2} b c \,e^{5}-6 a^{2} c^{2} d \,e^{4}+3 a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}-6 a \,c^{3} d^{3} e^{2}-b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}-9 b^{2} c^{2} d^{3} e^{2}+7 b \,c^{3} d^{4} e -2 c^{4} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {\left (6 a^{2} c^{2} e^{4}+12 a \,b^{2} c \,e^{4}-15 a b \,c^{2} d \,e^{3}+6 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-5 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}-7 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {c \left (15 c \,e^{3} b a -6 a \,c^{2} d \,e^{2}+5 b^{3} e^{3}-9 b^{2} c d \,e^{2}+7 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-7 b c d e +2 c^{2} d^{2}\right ) x^{5}}{5 e^{3}}+\frac {c^{3} \left (7 b e -2 c d \right ) x^{6}}{6 e^{2}}+\frac {\left (a^{3} b \,e^{7}-2 a^{3} c d \,e^{6}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 a^{2} c^{2} d^{3} e^{4}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 a \,c^{3} d^{5} e^{2}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 c^{4} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(695\) |
| default | \(\frac {\frac {3}{2} a \,b^{3} e^{6} x^{2}-\frac {1}{2} b^{4} d \,e^{5} x^{2}-c^{4} d^{5} e \,x^{2}-\frac {1}{2} c^{4} d^{3} e^{3} x^{4}+2 a^{2} c^{2} e^{6} x^{3}+\frac {2}{3} c^{4} d^{4} e^{2} x^{3}+\frac {7}{6} b \,c^{3} e^{6} x^{6}-\frac {1}{3} c^{4} d \,e^{5} x^{6}+\frac {6}{5} a \,c^{3} e^{6} x^{5}+\frac {9}{5} b^{2} c^{2} e^{6} x^{5}+\frac {2}{5} c^{4} d^{2} e^{4} x^{5}+\frac {5}{4} b^{3} c \,e^{6} x^{4}+b^{4} d^{2} e^{4} x +2 a^{3} c \,e^{6} x +3 a^{2} b^{2} e^{6} x +\frac {15}{2} a b \,c^{2} d^{2} e^{4} x^{2}-6 a \,b^{2} c d \,e^{5} x^{2}-5 a b \,c^{2} d \,e^{5} x^{3}+12 a \,b^{2} c \,d^{2} e^{4} x -15 a b \,c^{2} d^{3} e^{3} x -9 a^{2} b c d \,e^{5} x +2 c^{4} d^{6} x +\frac {1}{3} b^{4} e^{6} x^{3}+6 a^{2} c^{2} d^{2} e^{4} x -3 a \,b^{3} d \,e^{5} x +6 a \,c^{3} d^{4} e^{2} x +\frac {2}{7} c^{4} x^{7} e^{6}-5 b^{3} c \,d^{3} e^{3} x +9 b^{2} c^{2} d^{4} e^{2} x -7 b \,c^{3} d^{5} e x +3 b^{2} c^{2} d^{2} e^{4} x^{3}+4 a \,b^{2} c \,e^{6} x^{3}+2 a \,c^{3} d^{2} e^{4} x^{3}-\frac {5}{3} b^{3} c d \,e^{5} x^{3}-3 a^{2} c^{2} d \,e^{5} x^{2}-3 a \,c^{3} d^{3} e^{3} x^{2}+\frac {5}{2} b^{3} c \,d^{2} e^{4} x^{2}-\frac {9}{2} b^{2} c^{2} d^{3} e^{3} x^{2}-\frac {7}{3} b \,c^{3} d^{3} e^{3} x^{3}+\frac {9}{2} a^{2} b c \,e^{6} x^{2}-\frac {7}{5} b \,c^{3} d \,e^{5} x^{5}+\frac {15}{4} a b \,c^{2} e^{6} x^{4}-\frac {3}{2} a \,c^{3} d \,e^{5} x^{4}-\frac {9}{4} b^{2} c^{2} d \,e^{5} x^{4}+\frac {7}{4} b \,c^{3} d^{2} e^{4} x^{4}+\frac {7}{2} b \,c^{3} d^{4} e^{2} x^{2}}{e^{7}}+\frac {\left (a^{3} b \,e^{7}-2 a^{3} c d \,e^{6}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 a^{2} c^{2} d^{3} e^{4}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 a \,c^{3} d^{5} e^{2}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 c^{4} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(792\) |
| risch | \(\frac {6 a \,c^{3} d^{4} x}{e^{5}}-\frac {5 b^{3} c \,d^{3} x}{e^{4}}+\frac {9 b^{2} c^{2} d^{4} x}{e^{5}}-\frac {7 b \,c^{3} d^{5} x}{e^{6}}+\frac {3 b^{2} c^{2} d^{2} x^{3}}{e^{3}}+\frac {4 a \,b^{2} c \,x^{3}}{e}+\frac {2 a \,c^{3} d^{2} x^{3}}{e^{3}}-\frac {5 b^{3} c d \,x^{3}}{3 e^{2}}-\frac {3 a^{2} c^{2} d \,x^{2}}{e^{2}}-\frac {3 a \,c^{3} d^{3} x^{2}}{e^{4}}+\frac {5 b^{3} c \,d^{2} x^{2}}{2 e^{3}}-\frac {9 b^{2} c^{2} d^{3} x^{2}}{2 e^{4}}-\frac {7 b \,c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {9 a^{2} b c \,x^{2}}{2 e}-\frac {7 b \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {15 a b \,c^{2} x^{4}}{4 e}-\frac {3 a \,c^{3} d \,x^{4}}{2 e^{2}}-\frac {9 b^{2} c^{2} d \,x^{4}}{4 e^{2}}+\frac {7 b \,c^{3} d^{2} x^{4}}{4 e^{3}}+\frac {7 b \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {b^{4} d \,x^{2}}{2 e^{2}}-\frac {c^{4} d^{5} x^{2}}{e^{6}}-\frac {c^{4} d^{3} x^{4}}{2 e^{4}}+\frac {2 a^{2} c^{2} x^{3}}{e}+\frac {2 c^{4} d^{4} x^{3}}{3 e^{5}}+\frac {7 b \,c^{3} x^{6}}{6 e}-\frac {c^{4} d \,x^{6}}{3 e^{2}}+\frac {6 a \,c^{3} x^{5}}{5 e}+\frac {9 b^{2} c^{2} x^{5}}{5 e}+\frac {2 c^{4} d^{2} x^{5}}{5 e^{3}}+\frac {5 b^{3} c \,x^{4}}{4 e}+\frac {b^{4} d^{2} x}{e^{3}}+\frac {2 a^{3} c x}{e}+\frac {3 a^{2} b^{2} x}{e}+\frac {2 c^{4} d^{6} x}{e^{7}}+\frac {\ln \left (e x +d \right ) a^{3} b}{e}-\frac {\ln \left (e x +d \right ) b^{4} d^{3}}{e^{4}}-\frac {2 \ln \left (e x +d \right ) c^{4} d^{7}}{e^{8}}+\frac {3 a \,b^{3} x^{2}}{2 e}+\frac {15 a b \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {6 a \,b^{2} c d \,x^{2}}{e^{2}}-\frac {5 a b \,c^{2} d \,x^{3}}{e^{2}}+\frac {12 a \,b^{2} c \,d^{2} x}{e^{3}}-\frac {15 a b \,c^{2} d^{3} x}{e^{4}}-\frac {9 a^{2} b c d x}{e^{2}}+\frac {9 \ln \left (e x +d \right ) a^{2} b c \,d^{2}}{e^{3}}-\frac {12 \ln \left (e x +d \right ) a \,b^{2} c \,d^{3}}{e^{4}}+\frac {15 \ln \left (e x +d \right ) a b \,c^{2} d^{4}}{e^{5}}-\frac {2 \ln \left (e x +d \right ) a^{3} c d}{e^{2}}-\frac {3 \ln \left (e x +d \right ) a^{2} b^{2} d}{e^{2}}-\frac {6 \ln \left (e x +d \right ) a^{2} c^{2} d^{3}}{e^{4}}+\frac {3 \ln \left (e x +d \right ) a \,b^{3} d^{2}}{e^{3}}-\frac {6 \ln \left (e x +d \right ) a \,c^{3} d^{5}}{e^{6}}+\frac {5 \ln \left (e x +d \right ) b^{3} c \,d^{4}}{e^{5}}-\frac {9 \ln \left (e x +d \right ) b^{2} c^{2} d^{5}}{e^{6}}+\frac {7 \ln \left (e x +d \right ) b \,c^{3} d^{6}}{e^{7}}+\frac {6 a^{2} c^{2} d^{2} x}{e^{3}}-\frac {3 a \,b^{3} d x}{e^{2}}+\frac {2 c^{4} x^{7}}{7 e}+\frac {b^{4} x^{3}}{3 e}\) | \(872\) |
| parallelrisch | \(\frac {-420 \ln \left (e x +d \right ) b^{4} d^{3} e^{4}+490 x^{6} b \,c^{3} e^{7}-140 x^{6} c^{4} d \,e^{6}+504 x^{5} a \,c^{3} e^{7}+756 x^{5} b^{2} c^{2} e^{7}+168 x^{5} c^{4} d^{2} e^{5}+525 x^{4} b^{3} c \,e^{7}-210 x^{4} c^{4} d^{3} e^{4}+840 x^{3} a^{2} c^{2} e^{7}+280 x^{3} c^{4} d^{4} e^{3}+630 x^{2} a \,b^{3} e^{7}-210 x^{2} b^{4} d \,e^{6}-420 x^{2} c^{4} d^{5} e^{2}+840 x \,a^{3} c \,e^{7}+1260 x \,a^{2} b^{2} e^{7}+420 x \,b^{4} d^{2} e^{5}+840 x \,c^{4} d^{6} e +420 \ln \left (e x +d \right ) a^{3} b \,e^{7}+5040 x a \,b^{2} c \,d^{2} e^{5}-6300 x a b \,c^{2} d^{3} e^{4}-5040 \ln \left (e x +d \right ) a \,b^{2} c \,d^{3} e^{4}+6300 \ln \left (e x +d \right ) a b \,c^{2} d^{4} e^{3}+3780 \ln \left (e x +d \right ) a^{2} b c \,d^{2} e^{5}-2100 x^{3} a b \,c^{2} d \,e^{6}-2520 x^{2} a \,b^{2} c d \,e^{6}+3150 x^{2} a b \,c^{2} d^{2} e^{5}-3780 x \,a^{2} b c d \,e^{6}-1260 \ln \left (e x +d \right ) a^{2} b^{2} d \,e^{6}-2520 \ln \left (e x +d \right ) a^{2} c^{2} d^{3} e^{4}+1260 \ln \left (e x +d \right ) a \,b^{3} d^{2} e^{5}-2520 \ln \left (e x +d \right ) a \,c^{3} d^{5} e^{2}+2100 \ln \left (e x +d \right ) b^{3} c \,d^{4} e^{3}-3780 \ln \left (e x +d \right ) b^{2} c^{2} d^{5} e^{2}+2940 \ln \left (e x +d \right ) b \,c^{3} d^{6} e +120 x^{7} c^{4} e^{7}-588 x^{5} b \,c^{3} d \,e^{6}+1575 x^{4} a b \,c^{2} e^{7}-630 x^{4} a \,c^{3} d \,e^{6}-945 x^{4} b^{2} c^{2} d \,e^{6}+735 x^{4} b \,c^{3} d^{2} e^{5}+1680 x^{3} a \,b^{2} c \,e^{7}+840 x^{3} a \,c^{3} d^{2} e^{5}-700 x^{3} b^{3} c d \,e^{6}+1260 x^{3} b^{2} c^{2} d^{2} e^{5}-980 x^{3} b \,c^{3} d^{3} e^{4}+1890 x^{2} a^{2} b c \,e^{7}-1260 x^{2} a^{2} c^{2} d \,e^{6}-1260 x^{2} a \,c^{3} d^{3} e^{4}+1050 x^{2} b^{3} c \,d^{2} e^{5}-1890 x^{2} b^{2} c^{2} d^{3} e^{4}+1470 x^{2} b \,c^{3} d^{4} e^{3}+2520 x \,a^{2} c^{2} d^{2} e^{5}-1260 x a \,b^{3} d \,e^{6}+2520 x a \,c^{3} d^{4} e^{3}-2100 x \,b^{3} c \,d^{3} e^{4}+3780 x \,b^{2} c^{2} d^{4} e^{3}-2940 x b \,c^{3} d^{5} e^{2}-840 \ln \left (e x +d \right ) a^{3} c d \,e^{6}+140 x^{3} b^{4} e^{7}-840 \ln \left (e x +d \right ) c^{4} d^{7}}{420 e^{8}}\) | \(872\) |
(2*a^3*c*e^6+3*a^2*b^2*e^6-9*a^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4-3*a*b^3*d*e^5 +12*a*b^2*c*d^2*e^4-15*a*b*c^2*d^3*e^3+6*a*c^3*d^4*e^2+b^4*d^2*e^4-5*b^3*c *d^3*e^3+9*b^2*c^2*d^4*e^2-7*b*c^3*d^5*e+2*c^4*d^6)/e^7*x+2/7*c^4/e*x^7+1/ 2/e^6*(9*a^2*b*c*e^5-6*a^2*c^2*d*e^4+3*a*b^3*e^5-12*a*b^2*c*d*e^4+15*a*b*c ^2*d^2*e^3-6*a*c^3*d^3*e^2-b^4*d*e^4+5*b^3*c*d^2*e^3-9*b^2*c^2*d^3*e^2+7*b *c^3*d^4*e-2*c^4*d^5)*x^2+1/3/e^5*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-15*a*b*c^2 *d*e^3+6*a*c^3*d^2*e^2+b^4*e^4-5*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-7*b*c^3*d^3 *e+2*c^4*d^4)*x^3+1/4*c/e^4*(15*a*b*c*e^3-6*a*c^2*d*e^2+5*b^3*e^3-9*b^2*c* d*e^2+7*b*c^2*d^2*e-2*c^3*d^3)*x^4+1/5*c^2/e^3*(6*a*c*e^2+9*b^2*e^2-7*b*c* d*e+2*c^2*d^2)*x^5+1/6*c^3/e^2*(7*b*e-2*c*d)*x^6+(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)/e^8*ln(e*x+d)
Time = 0.32 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.62 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {120 \, c^{4} e^{7} x^{7} - 70 \, {\left (2 \, c^{4} d e^{6} - 7 \, b c^{3} e^{7}\right )} x^{6} + 84 \, {\left (2 \, c^{4} d^{2} e^{5} - 7 \, b c^{3} d e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 105 \, {\left (2 \, c^{4} d^{3} e^{4} - 7 \, b c^{3} d^{2} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 140 \, {\left (2 \, c^{4} d^{4} e^{3} - 7 \, b c^{3} d^{3} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 210 \, {\left (2 \, c^{4} d^{5} e^{2} - 7 \, b c^{3} d^{4} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 420 \, {\left (2 \, c^{4} d^{6} e - 7 \, b c^{3} d^{5} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x - 420 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]
1/420*(120*c^4*e^7*x^7 - 70*(2*c^4*d*e^6 - 7*b*c^3*e^7)*x^6 + 84*(2*c^4*d^ 2*e^5 - 7*b*c^3*d*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 105*(2*c^4*d^3* e^4 - 7*b*c^3*d^2*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 5*(b^3*c + 3*a*b*c ^2)*e^7)*x^4 + 140*(2*c^4*d^4*e^3 - 7*b*c^3*d^3*e^4 + 3*(3*b^2*c^2 + 2*a*c ^3)*d^2*e^5 - 5*(b^3*c + 3*a*b*c^2)*d*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2) *e^7)*x^3 - 210*(2*c^4*d^5*e^2 - 7*b*c^3*d^4*e^3 + 3*(3*b^2*c^2 + 2*a*c^3) *d^3*e^4 - 5*(b^3*c + 3*a*b*c^2)*d^2*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)* d*e^6 - 3*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 + 420*(2*c^4*d^6*e - 7*b*c^3*d^5*e^ 2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 5*(b^3*c + 3*a*b*c^2)*d^3*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 3*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2* b^2 + 2*a^3*c)*e^7)*x - 420*(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)*log(e*x + d))/e^8
Time = 0.67 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.61 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {2 c^{4} x^{7}}{7 e} + x^{6} \cdot \left (\frac {7 b c^{3}}{6 e} - \frac {c^{4} d}{3 e^{2}}\right ) + x^{5} \cdot \left (\frac {6 a c^{3}}{5 e} + \frac {9 b^{2} c^{2}}{5 e} - \frac {7 b c^{3} d}{5 e^{2}} + \frac {2 c^{4} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {15 a b c^{2}}{4 e} - \frac {3 a c^{3} d}{2 e^{2}} + \frac {5 b^{3} c}{4 e} - \frac {9 b^{2} c^{2} d}{4 e^{2}} + \frac {7 b c^{3} d^{2}}{4 e^{3}} - \frac {c^{4} d^{3}}{2 e^{4}}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c^{2}}{e} + \frac {4 a b^{2} c}{e} - \frac {5 a b c^{2} d}{e^{2}} + \frac {2 a c^{3} d^{2}}{e^{3}} + \frac {b^{4}}{3 e} - \frac {5 b^{3} c d}{3 e^{2}} + \frac {3 b^{2} c^{2} d^{2}}{e^{3}} - \frac {7 b c^{3} d^{3}}{3 e^{4}} + \frac {2 c^{4} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {9 a^{2} b c}{2 e} - \frac {3 a^{2} c^{2} d}{e^{2}} + \frac {3 a b^{3}}{2 e} - \frac {6 a b^{2} c d}{e^{2}} + \frac {15 a b c^{2} d^{2}}{2 e^{3}} - \frac {3 a c^{3} d^{3}}{e^{4}} - \frac {b^{4} d}{2 e^{2}} + \frac {5 b^{3} c d^{2}}{2 e^{3}} - \frac {9 b^{2} c^{2} d^{3}}{2 e^{4}} + \frac {7 b c^{3} d^{4}}{2 e^{5}} - \frac {c^{4} d^{5}}{e^{6}}\right ) + x \left (\frac {2 a^{3} c}{e} + \frac {3 a^{2} b^{2}}{e} - \frac {9 a^{2} b c d}{e^{2}} + \frac {6 a^{2} c^{2} d^{2}}{e^{3}} - \frac {3 a b^{3} d}{e^{2}} + \frac {12 a b^{2} c d^{2}}{e^{3}} - \frac {15 a b c^{2} d^{3}}{e^{4}} + \frac {6 a c^{3} d^{4}}{e^{5}} + \frac {b^{4} d^{2}}{e^{3}} - \frac {5 b^{3} c d^{3}}{e^{4}} + \frac {9 b^{2} c^{2} d^{4}}{e^{5}} - \frac {7 b c^{3} d^{5}}{e^{6}} + \frac {2 c^{4} d^{6}}{e^{7}}\right ) + \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \]
2*c**4*x**7/(7*e) + x**6*(7*b*c**3/(6*e) - c**4*d/(3*e**2)) + x**5*(6*a*c* *3/(5*e) + 9*b**2*c**2/(5*e) - 7*b*c**3*d/(5*e**2) + 2*c**4*d**2/(5*e**3)) + x**4*(15*a*b*c**2/(4*e) - 3*a*c**3*d/(2*e**2) + 5*b**3*c/(4*e) - 9*b**2 *c**2*d/(4*e**2) + 7*b*c**3*d**2/(4*e**3) - c**4*d**3/(2*e**4)) + x**3*(2* a**2*c**2/e + 4*a*b**2*c/e - 5*a*b*c**2*d/e**2 + 2*a*c**3*d**2/e**3 + b**4 /(3*e) - 5*b**3*c*d/(3*e**2) + 3*b**2*c**2*d**2/e**3 - 7*b*c**3*d**3/(3*e* *4) + 2*c**4*d**4/(3*e**5)) + x**2*(9*a**2*b*c/(2*e) - 3*a**2*c**2*d/e**2 + 3*a*b**3/(2*e) - 6*a*b**2*c*d/e**2 + 15*a*b*c**2*d**2/(2*e**3) - 3*a*c** 3*d**3/e**4 - b**4*d/(2*e**2) + 5*b**3*c*d**2/(2*e**3) - 9*b**2*c**2*d**3/ (2*e**4) + 7*b*c**3*d**4/(2*e**5) - c**4*d**5/e**6) + x*(2*a**3*c/e + 3*a* *2*b**2/e - 9*a**2*b*c*d/e**2 + 6*a**2*c**2*d**2/e**3 - 3*a*b**3*d/e**2 + 12*a*b**2*c*d**2/e**3 - 15*a*b*c**2*d**3/e**4 + 6*a*c**3*d**4/e**5 + b**4* d**2/e**3 - 5*b**3*c*d**3/e**4 + 9*b**2*c**2*d**4/e**5 - 7*b*c**3*d**5/e** 6 + 2*c**4*d**6/e**7) + (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**8
Time = 0.20 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.61 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {120 \, c^{4} e^{6} x^{7} - 70 \, {\left (2 \, c^{4} d e^{5} - 7 \, b c^{3} e^{6}\right )} x^{6} + 84 \, {\left (2 \, c^{4} d^{2} e^{4} - 7 \, b c^{3} d e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{6}\right )} x^{5} - 105 \, {\left (2 \, c^{4} d^{3} e^{3} - 7 \, b c^{3} d^{2} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{5} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{6}\right )} x^{4} + 140 \, {\left (2 \, c^{4} d^{4} e^{2} - 7 \, b c^{3} d^{3} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{4} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{6}\right )} x^{3} - 210 \, {\left (2 \, c^{4} d^{5} e - 7 \, b c^{3} d^{4} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{3} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{5} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{6}\right )} x^{2} + 420 \, {\left (2 \, c^{4} d^{6} - 7 \, b c^{3} d^{5} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \]
1/420*(120*c^4*e^6*x^7 - 70*(2*c^4*d*e^5 - 7*b*c^3*e^6)*x^6 + 84*(2*c^4*d^ 2*e^4 - 7*b*c^3*d*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*e^6)*x^5 - 105*(2*c^4*d^3* e^3 - 7*b*c^3*d^2*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^5 - 5*(b^3*c + 3*a*b*c ^2)*e^6)*x^4 + 140*(2*c^4*d^4*e^2 - 7*b*c^3*d^3*e^3 + 3*(3*b^2*c^2 + 2*a*c ^3)*d^2*e^4 - 5*(b^3*c + 3*a*b*c^2)*d*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2) *e^6)*x^3 - 210*(2*c^4*d^5*e - 7*b*c^3*d^4*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d ^3*e^3 - 5*(b^3*c + 3*a*b*c^2)*d^2*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d* e^5 - 3*(a*b^3 + 3*a^2*b*c)*e^6)*x^2 + 420*(2*c^4*d^6 - 7*b*c^3*d^5*e + 3* (3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^3*e^3 + (b^4 + 12* a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*x)/e^7 - (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)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (387) = 774\).
Time = 0.26 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.99 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {120 \, c^{4} e^{6} x^{7} - 140 \, c^{4} d e^{5} x^{6} + 490 \, b c^{3} e^{6} x^{6} + 168 \, c^{4} d^{2} e^{4} x^{5} - 588 \, b c^{3} d e^{5} x^{5} + 756 \, b^{2} c^{2} e^{6} x^{5} + 504 \, a c^{3} e^{6} x^{5} - 210 \, c^{4} d^{3} e^{3} x^{4} + 735 \, b c^{3} d^{2} e^{4} x^{4} - 945 \, b^{2} c^{2} d e^{5} x^{4} - 630 \, a c^{3} d e^{5} x^{4} + 525 \, b^{3} c e^{6} x^{4} + 1575 \, a b c^{2} e^{6} x^{4} + 280 \, c^{4} d^{4} e^{2} x^{3} - 980 \, b c^{3} d^{3} e^{3} x^{3} + 1260 \, b^{2} c^{2} d^{2} e^{4} x^{3} + 840 \, a c^{3} d^{2} e^{4} x^{3} - 700 \, b^{3} c d e^{5} x^{3} - 2100 \, a b c^{2} d e^{5} x^{3} + 140 \, b^{4} e^{6} x^{3} + 1680 \, a b^{2} c e^{6} x^{3} + 840 \, a^{2} c^{2} e^{6} x^{3} - 420 \, c^{4} d^{5} e x^{2} + 1470 \, b c^{3} d^{4} e^{2} x^{2} - 1890 \, b^{2} c^{2} d^{3} e^{3} x^{2} - 1260 \, a c^{3} d^{3} e^{3} x^{2} + 1050 \, b^{3} c d^{2} e^{4} x^{2} + 3150 \, a b c^{2} d^{2} e^{4} x^{2} - 210 \, b^{4} d e^{5} x^{2} - 2520 \, a b^{2} c d e^{5} x^{2} - 1260 \, a^{2} c^{2} d e^{5} x^{2} + 630 \, a b^{3} e^{6} x^{2} + 1890 \, a^{2} b c e^{6} x^{2} + 840 \, c^{4} d^{6} x - 2940 \, b c^{3} d^{5} e x + 3780 \, b^{2} c^{2} d^{4} e^{2} x + 2520 \, a c^{3} d^{4} e^{2} x - 2100 \, b^{3} c d^{3} e^{3} x - 6300 \, a b c^{2} d^{3} e^{3} x + 420 \, b^{4} d^{2} e^{4} x + 5040 \, a b^{2} c d^{2} e^{4} x + 2520 \, a^{2} c^{2} d^{2} e^{4} x - 1260 \, a b^{3} d e^{5} x - 3780 \, a^{2} b c d e^{5} x + 1260 \, a^{2} b^{2} e^{6} x + 840 \, a^{3} c e^{6} x}{420 \, e^{7}} - \frac {{\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 9 \, b^{2} c^{2} d^{5} e^{2} + 6 \, a c^{3} d^{5} e^{2} - 5 \, b^{3} c d^{4} e^{3} - 15 \, a b c^{2} d^{4} e^{3} + b^{4} d^{3} e^{4} + 12 \, a b^{2} c d^{3} e^{4} + 6 \, a^{2} c^{2} d^{3} e^{4} - 3 \, a b^{3} d^{2} e^{5} - 9 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} - a^{3} b e^{7}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \]
1/420*(120*c^4*e^6*x^7 - 140*c^4*d*e^5*x^6 + 490*b*c^3*e^6*x^6 + 168*c^4*d ^2*e^4*x^5 - 588*b*c^3*d*e^5*x^5 + 756*b^2*c^2*e^6*x^5 + 504*a*c^3*e^6*x^5 - 210*c^4*d^3*e^3*x^4 + 735*b*c^3*d^2*e^4*x^4 - 945*b^2*c^2*d*e^5*x^4 - 6 30*a*c^3*d*e^5*x^4 + 525*b^3*c*e^6*x^4 + 1575*a*b*c^2*e^6*x^4 + 280*c^4*d^ 4*e^2*x^3 - 980*b*c^3*d^3*e^3*x^3 + 1260*b^2*c^2*d^2*e^4*x^3 + 840*a*c^3*d ^2*e^4*x^3 - 700*b^3*c*d*e^5*x^3 - 2100*a*b*c^2*d*e^5*x^3 + 140*b^4*e^6*x^ 3 + 1680*a*b^2*c*e^6*x^3 + 840*a^2*c^2*e^6*x^3 - 420*c^4*d^5*e*x^2 + 1470* b*c^3*d^4*e^2*x^2 - 1890*b^2*c^2*d^3*e^3*x^2 - 1260*a*c^3*d^3*e^3*x^2 + 10 50*b^3*c*d^2*e^4*x^2 + 3150*a*b*c^2*d^2*e^4*x^2 - 210*b^4*d*e^5*x^2 - 2520 *a*b^2*c*d*e^5*x^2 - 1260*a^2*c^2*d*e^5*x^2 + 630*a*b^3*e^6*x^2 + 1890*a^2 *b*c*e^6*x^2 + 840*c^4*d^6*x - 2940*b*c^3*d^5*e*x + 3780*b^2*c^2*d^4*e^2*x + 2520*a*c^3*d^4*e^2*x - 2100*b^3*c*d^3*e^3*x - 6300*a*b*c^2*d^3*e^3*x + 420*b^4*d^2*e^4*x + 5040*a*b^2*c*d^2*e^4*x + 2520*a^2*c^2*d^2*e^4*x - 1260 *a*b^3*d*e^5*x - 3780*a^2*b*c*d*e^5*x + 1260*a^2*b^2*e^6*x + 840*a^3*c*e^6 *x)/e^7 - (2*c^4*d^7 - 7*b*c^3*d^6*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e ^6 + 2*a^3*c*d*e^6 - a^3*b*e^7)*log(abs(e*x + d))/e^8
Time = 0.10 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.75 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx=x^3\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{3\,e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{3\,e}\right )-x^4\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{4\,e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{2\,e}-\frac {3\,a\,b\,\left (b^2+3\,a\,c\right )}{2\,e}\right )+x^6\,\left (\frac {7\,b\,c^3}{6\,e}-\frac {c^4\,d}{3\,e^2}\right )+x\,\left (\frac {2\,c\,a^3+3\,a^2\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e}+\frac {d\,\left (\frac {d\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{e}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )}{e}-\frac {3\,a\,b\,\left (b^2+3\,a\,c\right )}{e}\right )}{e}\right )+x^5\,\left (\frac {9\,b^2\,c^2+6\,a\,c^3}{5\,e}-\frac {d\,\left (\frac {7\,b\,c^3}{e}-\frac {2\,c^4\,d}{e^2}\right )}{5\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6-9\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4-3\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-15\,a\,b\,c^2\,d^4\,e^3+6\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-5\,b^3\,c\,d^4\,e^3+9\,b^2\,c^2\,d^5\,e^2-7\,b\,c^3\,d^6\,e+2\,c^4\,d^7\right )}{e^8}+\frac {2\,c^4\,x^7}{7\,e} \]
x^3*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(3*e) + (d*((d*((6*a*c^3 + 9*b^2*c^2)/ e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2))/e))/e - (5*b*c*(3*a*c + b^2))/e))/(3 *e)) - x^4*((d*((6*a*c^3 + 9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2) )/e))/(4*e) - (5*b*c*(3*a*c + b^2))/(4*e)) - x^2*((d*((b^4 + 6*a^2*c^2 + 1 2*a*b^2*c)/e + (d*((d*((6*a*c^3 + 9*b^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4* d)/e^2))/e))/e - (5*b*c*(3*a*c + b^2))/e))/e))/(2*e) - (3*a*b*(3*a*c + b^2 ))/(2*e)) + x^6*((7*b*c^3)/(6*e) - (c^4*d)/(3*e^2)) + x*((2*a^3*c + 3*a^2* b^2)/e + (d*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e + (d*((d*((6*a*c^3 + 9*b ^2*c^2)/e - (d*((7*b*c^3)/e - (2*c^4*d)/e^2))/e))/e - (5*b*c*(3*a*c + b^2) )/e))/e))/e - (3*a*b*(3*a*c + b^2))/e))/e) + x^5*((6*a*c^3 + 9*b^2*c^2)/(5 *e) - (d*((7*b*c^3)/e - (2*c^4*d)/e^2))/(5*e)) - (log(d + e*x)*(2*c^4*d^7 - a^3*b*e^7 + b^4*d^3*e^4 - 3*a*b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 + 6*a*c^3*d^ 5*e^2 - 5*b^3*c*d^4*e^3 + 6*a^2*c^2*d^3*e^4 + 9*b^2*c^2*d^5*e^2 + 2*a^3*c* d*e^6 - 7*b*c^3*d^6*e - 15*a*b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 - 9*a^2*b* c*d^2*e^5))/e^8 + (2*c^4*x^7)/(7*e)