Integrand size = 26, antiderivative size = 396 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {c \left (40 c^3 d^3-5 b^3 e^3-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)\right ) x}{e^7}+\frac {c^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2}{2 e^6}-\frac {c^3 (8 c d-7 b e) x^3}{3 e^5}+\frac {c^4 x^4}{2 e^4}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^8 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^8 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^8} \]
-c*(40*c^3*d^3-5*b^3*e^3-2*c^2*d*e*(-12*a*e+35*b*d)+3*b*c*e^2*(-5*a*e+12*b *d))*x/e^7+1/2*c^2*(6*a*c*e^2+9*b^2*e^2-28*b*c*d*e+20*c^2*d^2)*x^2/e^6-1/3 *c^3*(-7*b*e+8*c*d)*x^3/e^5+1/2*c^4*x^4/e^4+1/3*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)^3/e^8/(e*x+d)^3-1/2*(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^8/(e*x+d)^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^8/(e*x+d)+(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))*ln(e*x+d)/e^8
Time = 0.14 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.02 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {-6 c e \left (40 c^3 d^3-5 b^3 e^3+3 b c e^2 (12 b d-5 a e)+2 c^2 d e (-35 b d+12 a e)\right ) x+3 c^2 e^2 \left (20 c^2 d^2-28 b c d e+9 b^2 e^2+6 a c e^2\right ) x^2-2 c^3 e^3 (8 c d-7 b e) x^3+3 c^4 e^4 x^4+\frac {2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {3 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {18 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{d+e x}+6 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{6 e^8} \]
(-6*c*e*(40*c^3*d^3 - 5*b^3*e^3 + 3*b*c*e^2*(12*b*d - 5*a*e) + 2*c^2*d*e*( -35*b*d + 12*a*e))*x + 3*c^2*e^2*(20*c^2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6* a*c*e^2)*x^2 - 2*c^3*e^3*(8*c*d - 7*b*e)*x^3 + 3*c^4*e^4*x^4 + (2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (3*(14*c^2*d^2 + 3*b^2*e ^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (18 *(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x) + 6*(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))*Log[d + e*x])/(6*e^8)
Time = 0.83 (sec) , antiderivative size = 396, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^7 (d+e x)}+\frac {c \left (2 c^2 d e (35 b d-12 a e)-3 b c e^2 (12 b d-5 a e)+5 b^3 e^3-40 c^3 d^3\right )}{e^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7 (d+e x)^2}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 x \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{e^6}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^4}-\frac {c^3 x^2 (8 c d-7 b e)}{e^5}+\frac {2 c^4 x^3}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}-\frac {c x \left (-2 c^2 d e (35 b d-12 a e)+3 b c e^2 (12 b d-5 a e)-5 b^3 e^3+40 c^3 d^3\right )}{e^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac {c^2 x^2 \left (6 a c e^2+9 b^2 e^2-28 b c d e+20 c^2 d^2\right )}{2 e^6}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac {c^3 x^3 (8 c d-7 b e)}{3 e^5}+\frac {c^4 x^4}{2 e^4}\) |
-((c*(40*c^3*d^3 - 5*b^3*e^3 - 2*c^2*d*e*(35*b*d - 12*a*e) + 3*b*c*e^2*(12 *b*d - 5*a*e))*x)/e^7) + (c^2*(20*c^2*d^2 - 28*b*c*d*e + 9*b^2*e^2 + 6*a*c *e^2)*x^2)/(2*e^6) - (c^3*(8*c*d - 7*b*e)*x^3)/(3*e^5) + (c^4*x^4)/(2*e^4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^3) - ((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)))/(2*e^8 *(d + e*x)^2) + (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)))/(e^8*(d + e*x)) + ((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))*Log[d + e*x])/e^8
3.16.22.3.1 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 = 0.42 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.78
| method | result | size |
| norman | \(\frac {-\frac {2 a^{3} b \,e^{7}+2 a^{3} c d \,e^{6}+3 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}-66 a^{2} c^{2} d^{3} e^{4}+6 a \,b^{3} d^{2} e^{5}-132 a \,b^{2} c \,d^{3} e^{4}+660 a b \,c^{2} d^{4} e^{3}-660 a \,c^{3} d^{5} e^{2}-11 b^{4} d^{3} e^{4}+220 b^{3} c \,d^{4} e^{3}-990 b^{2} c^{2} d^{5} e^{2}+1540 b \,c^{3} d^{6} e -770 c^{4} d^{7}}{6 e^{8}}+\frac {c^{4} x^{7}}{2 e}-\frac {3 \left (3 a^{2} b c \,e^{5}-6 a^{2} c^{2} d \,e^{4}+a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-60 a \,c^{3} d^{3} e^{2}-b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-90 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -70 c^{4} d^{5}\right ) x^{2}}{e^{6}}-\frac {\left (2 a^{3} c \,e^{6}+3 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}-54 a^{2} c^{2} d^{2} e^{4}+6 a \,b^{3} d \,e^{5}-108 a \,b^{2} c \,d^{2} e^{4}+540 a b \,c^{2} d^{3} e^{3}-540 a \,c^{3} d^{4} e^{2}-9 b^{4} d^{2} e^{4}+180 b^{3} c \,d^{3} e^{3}-810 b^{2} c^{2} d^{4} e^{2}+1260 b \,c^{3} d^{5} e -630 c^{4} d^{6}\right ) x}{2 e^{7}}+\frac {5 c \left (6 c \,e^{3} b a -6 a \,c^{2} d \,e^{2}+2 b^{3} e^{3}-9 b^{2} c d \,e^{2}+14 b \,c^{2} d^{2} e -7 c^{3} d^{3}\right ) x^{4}}{2 e^{4}}+\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-14 b c d e +7 c^{2} d^{2}\right ) x^{5}}{2 e^{3}}+\frac {7 c^{3} \left (2 b e -c d \right ) x^{6}}{6 e^{2}}}{\left (e x +d \right )^{3}}+\frac {\left (6 a^{2} c^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(703\) |
| default | \(\frac {c \left (\frac {1}{2} c^{3} x^{4} e^{3}+\frac {7}{3} b \,c^{2} e^{3} x^{3}-\frac {8}{3} c^{3} d \,e^{2} x^{3}+3 a \,c^{2} e^{3} x^{2}+\frac {9}{2} b^{2} c \,e^{3} x^{2}-14 b \,c^{2} d \,e^{2} x^{2}+10 c^{3} d^{2} e \,x^{2}+15 c \,e^{3} b a x -24 a \,c^{2} d \,e^{2} x +5 b^{3} e^{3} x -36 b^{2} c d \,e^{2} x +70 b \,c^{2} d^{2} e x -40 c^{3} d^{3} x \right )}{e^{7}}-\frac {a^{3} b \,e^{7}-2 a^{3} c d \,e^{6}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 a^{2} c^{2} d^{3} e^{4}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 a \,c^{3} d^{5} e^{2}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 c^{4} d^{7}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {9 a^{2} b c \,e^{5}-18 a^{2} c^{2} d \,e^{4}+3 a \,b^{3} e^{5}-36 a \,b^{2} c d \,e^{4}+90 a b \,c^{2} d^{2} e^{3}-60 a \,c^{3} d^{3} e^{2}-3 b^{4} d \,e^{4}+30 b^{3} c \,d^{2} e^{3}-90 b^{2} c^{2} d^{3} e^{2}+105 b \,c^{3} d^{4} e -42 c^{4} d^{5}}{e^{8} \left (e x +d \right )}+\frac {\left (6 a^{2} c^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {2 a^{3} c \,e^{6}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 a^{2} c^{2} d^{2} e^{4}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 a \,c^{3} d^{4} e^{2}+3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 c^{4} d^{6}}{2 e^{8} \left (e x +d \right )^{2}}\) | \(725\) |
| risch | \(\frac {c^{4} x^{4}}{2 e^{4}}+\frac {7 c^{3} b \,x^{3}}{3 e^{4}}-\frac {8 c^{4} d \,x^{3}}{3 e^{5}}+\frac {3 c^{3} a \,x^{2}}{e^{4}}+\frac {9 c^{2} b^{2} x^{2}}{2 e^{4}}-\frac {14 c^{3} b d \,x^{2}}{e^{5}}+\frac {10 c^{4} d^{2} x^{2}}{e^{6}}+\frac {15 c^{2} b a x}{e^{4}}-\frac {24 c^{3} a d x}{e^{5}}+\frac {5 c \,b^{3} x}{e^{4}}-\frac {36 c^{2} b^{2} d x}{e^{5}}+\frac {70 c^{3} b \,d^{2} x}{e^{6}}-\frac {40 c^{4} d^{3} x}{e^{7}}+\frac {\left (-9 a^{2} b c \,e^{6}+18 a^{2} c^{2} d \,e^{5}-3 a \,b^{3} e^{6}+36 a \,b^{2} c d \,e^{5}-90 a b \,c^{2} d^{2} e^{4}+60 a \,c^{3} d^{3} e^{3}+3 b^{4} d \,e^{5}-30 b^{3} c \,d^{2} e^{4}+90 b^{2} c^{2} d^{3} e^{3}-105 b \,d^{4} c^{3} e^{2}+42 c^{4} d^{5} e \right ) x^{2}+\left (-a^{3} c \,e^{6}-\frac {3}{2} a^{2} b^{2} e^{6}-9 a^{2} b c d \,e^{5}+27 a^{2} c^{2} d^{2} e^{4}-3 a \,b^{3} d \,e^{5}+54 a \,b^{2} c \,d^{2} e^{4}-150 a b \,c^{2} d^{3} e^{3}+105 a \,c^{3} d^{4} e^{2}+\frac {9}{2} b^{4} d^{2} e^{4}-50 b^{3} c \,d^{3} e^{3}+\frac {315}{2} b^{2} c^{2} d^{4} e^{2}-189 b \,c^{3} d^{5} e +77 c^{4} d^{6}\right ) x -\frac {2 a^{3} b \,e^{7}+2 a^{3} c d \,e^{6}+3 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}-66 a^{2} c^{2} d^{3} e^{4}+6 a \,b^{3} d^{2} e^{5}-132 a \,b^{2} c \,d^{3} e^{4}+390 a b \,c^{2} d^{4} e^{3}-282 a \,c^{3} d^{5} e^{2}-11 b^{4} d^{3} e^{4}+130 b^{3} c \,d^{4} e^{3}-423 b^{2} c^{2} d^{5} e^{2}+518 b \,c^{3} d^{6} e -214 c^{4} d^{7}}{6 e}}{e^{7} \left (e x +d \right )^{3}}+\frac {6 \ln \left (e x +d \right ) a^{2} c^{2}}{e^{4}}+\frac {12 \ln \left (e x +d \right ) a \,b^{2} c}{e^{4}}-\frac {60 \ln \left (e x +d \right ) a b \,c^{2} d}{e^{5}}+\frac {60 \ln \left (e x +d \right ) a \,c^{3} d^{2}}{e^{6}}+\frac {\ln \left (e x +d \right ) b^{4}}{e^{4}}-\frac {20 \ln \left (e x +d \right ) b^{3} c d}{e^{5}}+\frac {90 \ln \left (e x +d \right ) b^{2} c^{2} d^{2}}{e^{6}}-\frac {140 \ln \left (e x +d \right ) b \,c^{3} d^{3}}{e^{7}}+\frac {70 \ln \left (e x +d \right ) c^{4} d^{4}}{e^{8}}\) | \(772\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1370\) |
(-1/6*(2*a^3*b*e^7+2*a^3*c*d*e^6+3*a^2*b^2*d*e^6+18*a^2*b*c*d^2*e^5-66*a^2 *c^2*d^3*e^4+6*a*b^3*d^2*e^5-132*a*b^2*c*d^3*e^4+660*a*b*c^2*d^4*e^3-660*a *c^3*d^5*e^2-11*b^4*d^3*e^4+220*b^3*c*d^4*e^3-990*b^2*c^2*d^5*e^2+1540*b*c ^3*d^6*e-770*c^4*d^7)/e^8+1/2*c^4/e*x^7-3*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a *b^3*e^5-12*a*b^2*c*d*e^4+60*a*b*c^2*d^2*e^3-60*a*c^3*d^3*e^2-b^4*d*e^4+20 *b^3*c*d^2*e^3-90*b^2*c^2*d^3*e^2+140*b*c^3*d^4*e-70*c^4*d^5)/e^6*x^2-1/2* (2*a^3*c*e^6+3*a^2*b^2*e^6+18*a^2*b*c*d*e^5-54*a^2*c^2*d^2*e^4+6*a*b^3*d*e ^5-108*a*b^2*c*d^2*e^4+540*a*b*c^2*d^3*e^3-540*a*c^3*d^4*e^2-9*b^4*d^2*e^4 +180*b^3*c*d^3*e^3-810*b^2*c^2*d^4*e^2+1260*b*c^3*d^5*e-630*c^4*d^6)/e^7*x +5/2*c*(6*a*b*c*e^3-6*a*c^2*d*e^2+2*b^3*e^3-9*b^2*c*d*e^2+14*b*c^2*d^2*e-7 *c^3*d^3)/e^4*x^4+1/2*c^2*(6*a*c*e^2+9*b^2*e^2-14*b*c*d*e+7*c^2*d^2)/e^3*x ^5+7/6*c^3*(2*b*e-c*d)/e^2*x^6)/(e*x+d)^3+1/e^8*(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)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (386) = 772\).
Time = 0.26 (sec) , antiderivative size = 1031, normalized size of antiderivative = 2.60 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {3 \, c^{4} e^{7} x^{7} + 214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e - 2 \, a^{3} b e^{7} + 141 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 130 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 11 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 6 \, {\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} - 7 \, {\left (c^{4} d e^{6} - 2 \, b c^{3} e^{7}\right )} x^{6} + 3 \, {\left (7 \, c^{4} d^{2} e^{5} - 14 \, b c^{3} d e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 15 \, {\left (7 \, c^{4} d^{3} e^{4} - 14 \, b c^{3} d^{2} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} - {\left (556 \, c^{4} d^{4} e^{3} - 1022 \, b c^{3} d^{3} e^{4} + 189 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 90 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6}\right )} x^{3} - 3 \, {\left (136 \, c^{4} d^{5} e^{2} - 182 \, b c^{3} d^{4} e^{3} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} - 6 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 3 \, {\left (74 \, c^{4} d^{6} e - 238 \, b c^{3} d^{5} e^{2} + 81 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 90 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 9 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 6 \, {\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 + 6 \, {\left (70 \, c^{4} d^{7} - 140 \, b c^{3} d^{6} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 20 \, {\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} + {\left (70 \, c^{4} d^{4} e^{3} - 140 \, b c^{3} d^{3} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 20 \, {\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} + 3 \, {\left (70 \, c^{4} d^{5} e^{2} - 140 \, b c^{3} d^{4} e^{3} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 20 \, {\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}\right )} x^{2} + 3 \, {\left (70 \, c^{4} d^{6} e - 140 \, b c^{3} d^{5} e^{2} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 20 \, {\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}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \]
1/6*(3*c^4*e^7*x^7 + 214*c^4*d^7 - 518*b*c^3*d^6*e - 2*a^3*b*e^7 + 141*(3* b^2*c^2 + 2*a*c^3)*d^5*e^2 - 130*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 11*(b^4 + 1 2*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - (3*a^2*b^ 2 + 2*a^3*c)*d*e^6 - 7*(c^4*d*e^6 - 2*b*c^3*e^7)*x^6 + 3*(7*c^4*d^2*e^5 - 14*b*c^3*d*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 15*(7*c^4*d^3*e^4 - 14 *b*c^3*d^2*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 2*(b^3*c + 3*a*b*c^2)*e^7 )*x^4 - (556*c^4*d^4*e^3 - 1022*b*c^3*d^3*e^4 + 189*(3*b^2*c^2 + 2*a*c^3)* d^2*e^5 - 90*(b^3*c + 3*a*b*c^2)*d*e^6)*x^3 - 3*(136*c^4*d^5*e^2 - 182*b*c ^3*d^4*e^3 + 9*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 + 30*(b^3*c + 3*a*b*c^2)*d^2* e^5 - 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 + 6*(a*b^3 + 3*a^2*b*c)*e^7)* x^2 + 3*(74*c^4*d^6*e - 238*b*c^3*d^5*e^2 + 81*(3*b^2*c^2 + 2*a*c^3)*d^4*e ^3 - 90*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2 *e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 - (3*a^2*b^2 + 2*a^3*c)*e^7)*x + 6*(70* c^4*d^7 - 140*b*c^3*d^6*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 20*(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 + (70*c^4*d^4 *e^3 - 140*b*c^3*d^3*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 20*(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 + 3*(70*c^4*d^5 *e^2 - 140*b*c^3*d^4*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 20*(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)*x^2 + 3*(70*c^4 *d^6*e - 140*b*c^3*d^5*e^2 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 20*(b^3...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.69 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e - 2 \, a^{3} b e^{7} + 141 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 130 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 11 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 6 \, {\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} + 18 \, {\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 10 \, {\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} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 3 \, {\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 105 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 100 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 9 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 6 \, {\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}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, c^{4} e^{3} x^{4} - 2 \, {\left (8 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (20 \, c^{4} d^{2} e - 28 \, b c^{3} d e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{2} - 6 \, {\left (40 \, c^{4} d^{3} - 70 \, b c^{3} d^{2} e + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x}{6 \, e^{7}} + \frac {{\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 )} \log \left (e x + d\right )}{e^{8}} \]
1/6*(214*c^4*d^7 - 518*b*c^3*d^6*e - 2*a^3*b*e^7 + 141*(3*b^2*c^2 + 2*a*c^ 3)*d^5*e^2 - 130*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 11*(b^4 + 12*a*b^2*c + 6*a^ 2*c^2)*d^3*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^5 - (3*a^2*b^2 + 2*a^3*c)*d*e ^6 + 18*(14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 10*(3*b^2*c^2 + 2*a*c^3)*d^3* e^4 - 10*(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 - (a*b^3 + 3*a^2*b*c)*e^7)*x^2 + 3*(154*c^4*d^6*e - 378*b*c^3*d^5*e^2 + 105*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 100*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 9*(b ^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 - (3*a^ 2*b^2 + 2*a^3*c)*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/6*(3*c^4*e^3*x^4 - 2*(8*c^4*d*e^2 - 7*b*c^3*e^3)*x^3 + 3*(20*c^4*d^2* e - 28*b*c^3*d*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*e^3)*x^2 - 6*(40*c^4*d^3 - 70 *b*c^3*d^2*e + 12*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - 5*(b^3*c + 3*a*b*c^2)*e^3) *x)/e^7 + (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)*log( e*x + d)/e^8
Time = 0.28 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.85 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {{\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {214 \, c^{4} d^{7} - 518 \, b c^{3} d^{6} e + 423 \, b^{2} c^{2} d^{5} e^{2} + 282 \, a c^{3} d^{5} e^{2} - 130 \, b^{3} c d^{4} e^{3} - 390 \, a b c^{2} d^{4} e^{3} + 11 \, b^{4} d^{3} e^{4} + 132 \, a b^{2} c d^{3} e^{4} + 66 \, a^{2} c^{2} d^{3} e^{4} - 6 \, a b^{3} d^{2} e^{5} - 18 \, a^{2} b c d^{2} e^{5} - 3 \, a^{2} b^{2} d e^{6} - 2 \, a^{3} c d e^{6} - 2 \, a^{3} b e^{7} + 18 \, {\left (14 \, c^{4} d^{5} e^{2} - 35 \, b c^{3} d^{4} e^{3} + 30 \, b^{2} c^{2} d^{3} e^{4} + 20 \, a c^{3} d^{3} e^{4} - 10 \, b^{3} c d^{2} e^{5} - 30 \, a b c^{2} d^{2} e^{5} + b^{4} d e^{6} + 12 \, a b^{2} c d e^{6} + 6 \, a^{2} c^{2} d e^{6} - a b^{3} e^{7} - 3 \, a^{2} b c e^{7}\right )} x^{2} + 3 \, {\left (154 \, c^{4} d^{6} e - 378 \, b c^{3} d^{5} e^{2} + 315 \, b^{2} c^{2} d^{4} e^{3} + 210 \, a c^{3} d^{4} e^{3} - 100 \, b^{3} c d^{3} e^{4} - 300 \, a b c^{2} d^{3} e^{4} + 9 \, b^{4} d^{2} e^{5} + 108 \, a b^{2} c d^{2} e^{5} + 54 \, a^{2} c^{2} d^{2} e^{5} - 6 \, a b^{3} d e^{6} - 18 \, a^{2} b c d e^{6} - 3 \, a^{2} b^{2} e^{7} - 2 \, a^{3} c e^{7}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{8}} + \frac {3 \, c^{4} e^{12} x^{4} - 16 \, c^{4} d e^{11} x^{3} + 14 \, b c^{3} e^{12} x^{3} + 60 \, c^{4} d^{2} e^{10} x^{2} - 84 \, b c^{3} d e^{11} x^{2} + 27 \, b^{2} c^{2} e^{12} x^{2} + 18 \, a c^{3} e^{12} x^{2} - 240 \, c^{4} d^{3} e^{9} x + 420 \, b c^{3} d^{2} e^{10} x - 216 \, b^{2} c^{2} d e^{11} x - 144 \, a c^{3} d e^{11} x + 30 \, b^{3} c e^{12} x + 90 \, a b c^{2} e^{12} x}{6 \, e^{16}} \]
(70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20 *b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 + b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4 )*log(abs(e*x + d))/e^8 + 1/6*(214*c^4*d^7 - 518*b*c^3*d^6*e + 423*b^2*c^2 *d^5*e^2 + 282*a*c^3*d^5*e^2 - 130*b^3*c*d^4*e^3 - 390*a*b*c^2*d^4*e^3 + 1 1*b^4*d^3*e^4 + 132*a*b^2*c*d^3*e^4 + 66*a^2*c^2*d^3*e^4 - 6*a*b^3*d^2*e^5 - 18*a^2*b*c*d^2*e^5 - 3*a^2*b^2*d*e^6 - 2*a^3*c*d*e^6 - 2*a^3*b*e^7 + 18 *(14*c^4*d^5*e^2 - 35*b*c^3*d^4*e^3 + 30*b^2*c^2*d^3*e^4 + 20*a*c^3*d^3*e^ 4 - 10*b^3*c*d^2*e^5 - 30*a*b*c^2*d^2*e^5 + b^4*d*e^6 + 12*a*b^2*c*d*e^6 + 6*a^2*c^2*d*e^6 - a*b^3*e^7 - 3*a^2*b*c*e^7)*x^2 + 3*(154*c^4*d^6*e - 378 *b*c^3*d^5*e^2 + 315*b^2*c^2*d^4*e^3 + 210*a*c^3*d^4*e^3 - 100*b^3*c*d^3*e ^4 - 300*a*b*c^2*d^3*e^4 + 9*b^4*d^2*e^5 + 108*a*b^2*c*d^2*e^5 + 54*a^2*c^ 2*d^2*e^5 - 6*a*b^3*d*e^6 - 18*a^2*b*c*d*e^6 - 3*a^2*b^2*e^7 - 2*a^3*c*e^7 )*x)/((e*x + d)^3*e^8) + 1/6*(3*c^4*e^12*x^4 - 16*c^4*d*e^11*x^3 + 14*b*c^ 3*e^12*x^3 + 60*c^4*d^2*e^10*x^2 - 84*b*c^3*d*e^11*x^2 + 27*b^2*c^2*e^12*x ^2 + 18*a*c^3*e^12*x^2 - 240*c^4*d^3*e^9*x + 420*b*c^3*d^2*e^10*x - 216*b^ 2*c^2*d*e^11*x - 144*a*c^3*d*e^11*x + 30*b^3*c*e^12*x + 90*a*b*c^2*e^12*x) /e^16
Time = 10.77 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.04 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=x^3\,\left (\frac {7\,b\,c^3}{3\,e^4}-\frac {8\,c^4\,d}{3\,e^5}\right )-\frac {x\,\left (a^3\,c\,e^6+\frac {3\,a^2\,b^2\,e^6}{2}+9\,a^2\,b\,c\,d\,e^5-27\,a^2\,c^2\,d^2\,e^4+3\,a\,b^3\,d\,e^5-54\,a\,b^2\,c\,d^2\,e^4+150\,a\,b\,c^2\,d^3\,e^3-105\,a\,c^3\,d^4\,e^2-\frac {9\,b^4\,d^2\,e^4}{2}+50\,b^3\,c\,d^3\,e^3-\frac {315\,b^2\,c^2\,d^4\,e^2}{2}+189\,b\,c^3\,d^5\,e-77\,c^4\,d^6\right )-x^2\,\left (-9\,a^2\,b\,c\,e^6+18\,a^2\,c^2\,d\,e^5-3\,a\,b^3\,e^6+36\,a\,b^2\,c\,d\,e^5-90\,a\,b\,c^2\,d^2\,e^4+60\,a\,c^3\,d^3\,e^3+3\,b^4\,d\,e^5-30\,b^3\,c\,d^2\,e^4+90\,b^2\,c^2\,d^3\,e^3-105\,b\,c^3\,d^4\,e^2+42\,c^4\,d^5\,e\right )+\frac {2\,a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6+18\,a^2\,b\,c\,d^2\,e^5-66\,a^2\,c^2\,d^3\,e^4+6\,a\,b^3\,d^2\,e^5-132\,a\,b^2\,c\,d^3\,e^4+390\,a\,b\,c^2\,d^4\,e^3-282\,a\,c^3\,d^5\,e^2-11\,b^4\,d^3\,e^4+130\,b^3\,c\,d^4\,e^3-423\,b^2\,c^2\,d^5\,e^2+518\,b\,c^3\,d^6\,e-214\,c^4\,d^7}{6\,e}}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}-x^2\,\left (\frac {2\,d\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{2\,e^4}+\frac {6\,c^4\,d^2}{e^6}\right )-x\,\left (\frac {8\,c^4\,d^3}{e^7}+\frac {6\,d^2\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {7\,b\,c^3}{e^4}-\frac {8\,c^4\,d}{e^5}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^4}+\frac {12\,c^4\,d^2}{e^6}\right )}{e}-\frac {5\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )+\frac {c^4\,x^4}{2\,e^4}+\frac {\ln \left (d+e\,x\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{e^8} \]
x^3*((7*b*c^3)/(3*e^4) - (8*c^4*d)/(3*e^5)) - (x*(a^3*c*e^6 - 77*c^4*d^6 + (3*a^2*b^2*e^6)/2 - (9*b^4*d^2*e^4)/2 - 105*a*c^3*d^4*e^2 + 50*b^3*c*d^3* e^3 - 27*a^2*c^2*d^2*e^4 - (315*b^2*c^2*d^4*e^2)/2 + 3*a*b^3*d*e^5 + 189*b *c^3*d^5*e + 9*a^2*b*c*d*e^5 + 150*a*b*c^2*d^3*e^3 - 54*a*b^2*c*d^2*e^4) - x^2*(3*b^4*d*e^5 - 3*a*b^3*e^6 + 42*c^4*d^5*e + 60*a*c^3*d^3*e^3 + 18*a^2 *c^2*d*e^5 - 105*b*c^3*d^4*e^2 - 30*b^3*c*d^2*e^4 + 90*b^2*c^2*d^3*e^3 - 9 *a^2*b*c*e^6 + 36*a*b^2*c*d*e^5 - 90*a*b*c^2*d^2*e^4) + (2*a^3*b*e^7 - 214 *c^4*d^7 - 11*b^4*d^3*e^4 + 6*a*b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 - 282*a*c^3* d^5*e^2 + 130*b^3*c*d^4*e^3 - 66*a^2*c^2*d^3*e^4 - 423*b^2*c^2*d^5*e^2 + 2 *a^3*c*d*e^6 + 518*b*c^3*d^6*e + 390*a*b*c^2*d^4*e^3 - 132*a*b^2*c*d^3*e^4 + 18*a^2*b*c*d^2*e^5)/(6*e))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9* x^2) - x^2*((2*d*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e - (6*a*c^3 + 9*b^2*c^2 )/(2*e^4) + (6*c^4*d^2)/e^6) - x*((8*c^4*d^3)/e^7 + (6*d^2*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e^2 - (4*d*((4*d*((7*b*c^3)/e^4 - (8*c^4*d)/e^5))/e - (6 *a*c^3 + 9*b^2*c^2)/e^4 + (12*c^4*d^2)/e^6))/e - (5*b*c*(3*a*c + b^2))/e^4 ) + (c^4*x^4)/(2*e^4) + (log(d + e*x)*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^ 4 + 60*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3 *e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/e^8