\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^{10}} \, dx\) [458]

Optimal result

Integrand size = 20, antiderivative size = 436 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{2 e^9 (d+e x)^8}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{7 e^9 (d+e x)^7}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^6}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{5 e^9 (d+e x)^5}+\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^4}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac {c^4}{e^9 (d+e x)} \] Output:

-1/9*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^9+1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d 
^2)^3/e^9/(e*x+d)^8-2/7*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e* 
(-a*e+7*b*d))/e^9/(e*x+d)^7+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^9/(e*x+d)^6-1/5*(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^9/(e*x+d)^5+c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3* 
a*e+7*b*d))/e^9/(e*x+d)^4-2/3*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d) 
)/e^9/(e*x+d)^3+2*c^3*(-b*e+2*c*d)/e^9/(e*x+d)^2-c^4/e^9/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {70 c^4 \left (d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right )+e^4 \left (70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+c e^3 \left (10 a^3 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+15 a^2 b e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 a b^2 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b^3 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 a b e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+5 b^2 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+5 c^3 e \left (2 a e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 b \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{630 e^9 (d+e x)^9} \] Input:

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^10,x]
 

Output:

-1/630*(70*c^4*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^ 
4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8) + 
 e^4*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e* 
x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) 
+ b^4*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + c 
*e^3*(10*a^3*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2) + 15*a^2*b*e^2*(d^3 + 9*d^2* 
e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*a*b^2*e*(d^4 + 9*d^3*e*x + 36*d^2*e^ 
2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b^3*(d^5 + 9*d^4*e*x + 36*d^3*e^2* 
x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 3*c^2*e^2*(2*a^2*e^ 
2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*a*b* 
e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126 
*e^5*x^5) + 5*b^2*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126 
*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 5*c^3*e*(2*a*e*(d^6 + 9*d^5* 
e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 
84*e^6*x^6) + 7*b*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126 
*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^9*(d + e* 
x)^9)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)^6}+\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^4}+\frac {4 c (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8 (d+e x)^5}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^7}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^8}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^9}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^{10}}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^3}+\frac {c^4}{e^8 (d+e x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{5 e^9 (d+e x)^5}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac {c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9 (d+e x)^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^9 (d+e x)^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{9 e^9 (d+e x)^9}+\frac {2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac {c^4}{e^9 (d+e x)}\)

Input:

Int[(a + b*x + c*x^2)^4/(d + e*x)^10,x]
 

Output:

-1/9*(c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)^9) + ((2*c*d - b*e)*(c*d^2 - 
 b*d*e + a*e^2)^3)/(2*e^9*(d + e*x)^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)))/(7*e^9*(d + e*x)^7) + (2*(2*c* 
d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e 
)))/(3*e^9*(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))/(5*e^9*(d + e*x)^5) + (c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c* 
e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)^4) - (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2 
*c*e*(7*b*d - a*e)))/(3*e^9*(d + e*x)^3) + (2*c^3*(2*c*d - b*e))/(e^9*(d + 
 e*x)^2) - c^4/(e^9*(d + e*x))
 

Defintions of rubi rules used

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(879\) vs. \(2(424)=848\).

Time = 0.91 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.02

method	result	size
risch	\(\frac {-\frac {c^{4} x^{8}}{e}-\frac {2 c^{3} \left (b e +2 c d \right ) x^{7}}{e^{2}}-\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}+7 b c d e +14 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}-\frac {c \left (3 a b c \,e^{3}+2 d \,e^{2} a \,c^{2}+b^{3} e^{3}+3 d \,e^{2} b^{2} c +7 d^{2} e b \,c^{2}+14 d^{3} c^{3}\right ) x^{5}}{e^{4}}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+15 a b \,c^{2} d \,e^{3}+10 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+5 d \,e^{3} b^{3} c +15 d^{2} e^{2} b^{2} c^{2}+35 d^{3} e b \,c^{3}+70 d^{4} c^{4}\right ) x^{4}}{5 e^{5}}-\frac {2 \left (15 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+5 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}+10 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}+15 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e +70 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {2 \left (10 e^{6} c \,a^{3}+15 a^{2} b^{2} e^{6}+15 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+5 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+15 a b \,c^{2} d^{3} e^{3}+10 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+5 b^{3} c \,d^{3} e^{3}+15 b^{2} c^{2} d^{4} e^{2}+35 b \,c^{3} d^{5} e +70 d^{6} c^{4}\right ) x^{2}}{35 e^{7}}-\frac {\left (35 a^{3} b \,e^{7}+10 d \,e^{6} c \,a^{3}+15 a^{2} b^{2} d \,e^{6}+15 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+5 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}+10 d^{5} e^{2} a \,c^{3}+b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}+15 b^{2} c^{2} d^{5} e^{2}+35 b \,c^{3} d^{6} e +70 d^{7} c^{4}\right ) x}{70 e^{8}}-\frac {70 a^{4} e^{8}+35 a^{3} b d \,e^{7}+10 a^{3} c \,d^{2} e^{6}+15 a^{2} b^{2} d^{2} e^{6}+15 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+5 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+15 a b \,c^{2} d^{5} e^{3}+10 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+5 b^{3} c \,d^{5} e^{3}+15 b^{2} c^{2} d^{6} e^{2}+35 b \,c^{3} d^{7} e +70 c^{4} d^{8}}{630 e^{9}}}{\left (e x +d \right )^{9}}\)	\(880\)
norman	\(\frac {-\frac {c^{4} x^{8}}{e}-\frac {2 \left (b e \,c^{3}+2 c^{4} d \right ) x^{7}}{e^{2}}-\frac {2 \left (2 e^{2} a \,c^{3}+3 b^{2} c^{2} e^{2}+7 c^{3} d e b +14 c^{4} d^{2}\right ) x^{6}}{3 e^{3}}-\frac {\left (3 a b \,c^{2} e^{3}+2 a \,c^{3} d \,e^{2}+b^{3} c \,e^{3}+3 d \,e^{2} b^{2} c^{2}+7 b \,c^{3} d^{2} e +14 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+15 a b \,c^{2} d \,e^{3}+10 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+5 d \,e^{3} b^{3} c +15 d^{2} e^{2} b^{2} c^{2}+35 d^{3} e b \,c^{3}+70 d^{4} c^{4}\right ) x^{4}}{5 e^{5}}-\frac {2 \left (15 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+5 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}+10 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}+15 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e +70 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {2 \left (10 e^{6} c \,a^{3}+15 a^{2} b^{2} e^{6}+15 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+5 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+15 a b \,c^{2} d^{3} e^{3}+10 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+5 b^{3} c \,d^{3} e^{3}+15 b^{2} c^{2} d^{4} e^{2}+35 b \,c^{3} d^{5} e +70 d^{6} c^{4}\right ) x^{2}}{35 e^{7}}-\frac {\left (35 a^{3} b \,e^{7}+10 d \,e^{6} c \,a^{3}+15 a^{2} b^{2} d \,e^{6}+15 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+5 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}+10 d^{5} e^{2} a \,c^{3}+b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}+15 b^{2} c^{2} d^{5} e^{2}+35 b \,c^{3} d^{6} e +70 d^{7} c^{4}\right ) x}{70 e^{8}}-\frac {70 a^{4} e^{8}+35 a^{3} b d \,e^{7}+10 a^{3} c \,d^{2} e^{6}+15 a^{2} b^{2} d^{2} e^{6}+15 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+5 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+15 a b \,c^{2} d^{5} e^{3}+10 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+5 b^{3} c \,d^{5} e^{3}+15 b^{2} c^{2} d^{6} e^{2}+35 b \,c^{3} d^{7} e +70 c^{4} d^{8}}{630 e^{9}}}{\left (e x +d \right )^{9}}\)	\(890\)
default	\(-\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{3 e^{9} \left (e x +d \right )^{3}}-\frac {c \left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-9 d \,e^{2} b^{2} c +21 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right )}{e^{9} \left (e x +d \right )^{4}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{9 e^{9} \left (e x +d \right )^{9}}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-20 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}-140 d^{3} e b \,c^{3}+70 d^{4} c^{4}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {c^{4}}{e^{9} \left (e x +d \right )}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} a \,c^{3}-4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {2 c^{3} \left (b e -2 c d \right )}{e^{9} \left (e x +d \right )^{2}}-\frac {12 a^{2} b c \,e^{5}-24 a^{2} c^{2} d \,e^{4}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 a \,c^{3} d^{3} e^{2}-4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(914\)
gosper	\(\text {Expression too large to display}\)	\(1015\)
orering	\(\text {Expression too large to display}\)	\(1015\)
parallelrisch	\(\text {Expression too large to display}\)	\(1016\)

Input:

int((c*x^2+b*x+a)^4/(e*x+d)^10,x,method=_RETURNVERBOSE)
 

Output:

(-c^4/e*x^8-2*c^3*(b*e+2*c*d)/e^2*x^7-2/3*c^2*(2*a*c*e^2+3*b^2*e^2+7*b*c*d 
*e+14*c^2*d^2)/e^3*x^6-c*(3*a*b*c*e^3+2*a*c^2*d*e^2+b^3*e^3+3*b^2*c*d*e^2+ 
7*b*c^2*d^2*e+14*c^3*d^3)/e^4*x^5-1/5*(6*a^2*c^2*e^4+12*a*b^2*c*e^4+15*a*b 
*c^2*d*e^3+10*a*c^3*d^2*e^2+b^4*e^4+5*b^3*c*d*e^3+15*b^2*c^2*d^2*e^2+35*b* 
c^3*d^3*e+70*c^4*d^4)/e^5*x^4-2/15*(15*a^2*b*c*e^5+6*a^2*c^2*d*e^4+5*a*b^3 
*e^5+12*a*b^2*c*d*e^4+15*a*b*c^2*d^2*e^3+10*a*c^3*d^3*e^2+b^4*d*e^4+5*b^3* 
c*d^2*e^3+15*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e+70*c^4*d^5)/e^6*x^3-2/35*(10*a 
^3*c*e^6+15*a^2*b^2*e^6+15*a^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4+5*a*b^3*d*e^5+1 
2*a*b^2*c*d^2*e^4+15*a*b*c^2*d^3*e^3+10*a*c^3*d^4*e^2+b^4*d^2*e^4+5*b^3*c* 
d^3*e^3+15*b^2*c^2*d^4*e^2+35*b*c^3*d^5*e+70*c^4*d^6)/e^7*x^2-1/70*(35*a^3 
*b*e^7+10*a^3*c*d*e^6+15*a^2*b^2*d*e^6+15*a^2*b*c*d^2*e^5+6*a^2*c^2*d^3*e^ 
4+5*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+10*a*c^3*d^5*e^2+b 
^4*d^3*e^4+5*b^3*c*d^4*e^3+15*b^2*c^2*d^5*e^2+35*b*c^3*d^6*e+70*c^4*d^7)/e 
^8*x-1/630*(70*a^4*e^8+35*a^3*b*d*e^7+10*a^3*c*d^2*e^6+15*a^2*b^2*d^2*e^6+ 
15*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4+5*a*b^3*d^3*e^5+12*a*b^2*c*d^4*e^4+15 
*a*b*c^2*d^5*e^3+10*a*c^3*d^6*e^2+b^4*d^4*e^4+5*b^3*c*d^5*e^3+15*b^2*c^2*d 
^6*e^2+35*b*c^3*d^7*e+70*c^4*d^8)/e^9)/(e*x+d)^9
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (424) = 848\).

Time = 0.08 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="fricas")
 

Output:

-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 7 
0*a^4*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 5*(b^3*c + 3*a*b*c^2)*d^5*e^ 
3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e^5 
 + 5*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 
420*(14*c^4*d^2*e^6 + 7*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630 
*(14*c^4*d^3*e^5 + 7*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c 
+ 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(3*b^2* 
c^2 + 2*a*c^3)*d^2*e^6 + 5*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 
 6*a^2*c^2)*e^8)*x^4 + 84*(70*c^4*d^5*e^3 + 35*b*c^3*d^4*e^4 + 5*(3*b^2*c^ 
2 + 2*a*c^3)*d^3*e^5 + 5*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 
 6*a^2*c^2)*d*e^7 + 5*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 36*(70*c^4*d^6*e^2 + 
35*b*c^3*d^5*e^3 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 5*(b^3*c + 3*a*b*c^2) 
*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 5*(a*b^3 + 3*a^2*b*c)* 
d*e^7 + 5*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6* 
e^2 + 35*a^3*b*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 5*(b^3*c + 3*a*b*c^ 
2)*d^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 5*(a*b^3 + 3*a^2*b*c 
)*d^2*e^6 + 5*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 3 
6*d^2*e^16*x^7 + 84*d^3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*e^13*x^4 + 8 
4*d^6*e^12*x^3 + 36*d^7*e^11*x^2 + 9*d^8*e^10*x + d^9*e^9)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**4/(e*x+d)**10,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (424) = 848\).

Time = 0.07 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="maxima")
 

Output:

-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 7 
0*a^4*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 5*(b^3*c + 3*a*b*c^2)*d^5*e^ 
3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e^5 
 + 5*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 
420*(14*c^4*d^2*e^6 + 7*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630 
*(14*c^4*d^3*e^5 + 7*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c 
+ 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(3*b^2* 
c^2 + 2*a*c^3)*d^2*e^6 + 5*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 
 6*a^2*c^2)*e^8)*x^4 + 84*(70*c^4*d^5*e^3 + 35*b*c^3*d^4*e^4 + 5*(3*b^2*c^ 
2 + 2*a*c^3)*d^3*e^5 + 5*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 
 6*a^2*c^2)*d*e^7 + 5*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 36*(70*c^4*d^6*e^2 + 
35*b*c^3*d^5*e^3 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 5*(b^3*c + 3*a*b*c^2) 
*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 5*(a*b^3 + 3*a^2*b*c)* 
d*e^7 + 5*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6* 
e^2 + 35*a^3*b*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 5*(b^3*c + 3*a*b*c^ 
2)*d^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 5*(a*b^3 + 3*a^2*b*c 
)*d^2*e^6 + 5*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 3 
6*d^2*e^16*x^7 + 84*d^3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*e^13*x^4 + 8 
4*d^6*e^12*x^3 + 36*d^7*e^11*x^2 + 9*d^8*e^10*x + d^9*e^9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (424) = 848\).

Time = 0.36 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.33 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="giac")
 

Output:

-1/630*(630*c^4*e^8*x^8 + 2520*c^4*d*e^7*x^7 + 1260*b*c^3*e^8*x^7 + 5880*c 
^4*d^2*e^6*x^6 + 2940*b*c^3*d*e^7*x^6 + 1260*b^2*c^2*e^8*x^6 + 840*a*c^3*e 
^8*x^6 + 8820*c^4*d^3*e^5*x^5 + 4410*b*c^3*d^2*e^6*x^5 + 1890*b^2*c^2*d*e^ 
7*x^5 + 1260*a*c^3*d*e^7*x^5 + 630*b^3*c*e^8*x^5 + 1890*a*b*c^2*e^8*x^5 + 
8820*c^4*d^4*e^4*x^4 + 4410*b*c^3*d^3*e^5*x^4 + 1890*b^2*c^2*d^2*e^6*x^4 + 
 1260*a*c^3*d^2*e^6*x^4 + 630*b^3*c*d*e^7*x^4 + 1890*a*b*c^2*d*e^7*x^4 + 1 
26*b^4*e^8*x^4 + 1512*a*b^2*c*e^8*x^4 + 756*a^2*c^2*e^8*x^4 + 5880*c^4*d^5 
*e^3*x^3 + 2940*b*c^3*d^4*e^4*x^3 + 1260*b^2*c^2*d^3*e^5*x^3 + 840*a*c^3*d 
^3*e^5*x^3 + 420*b^3*c*d^2*e^6*x^3 + 1260*a*b*c^2*d^2*e^6*x^3 + 84*b^4*d*e 
^7*x^3 + 1008*a*b^2*c*d*e^7*x^3 + 504*a^2*c^2*d*e^7*x^3 + 420*a*b^3*e^8*x^ 
3 + 1260*a^2*b*c*e^8*x^3 + 2520*c^4*d^6*e^2*x^2 + 1260*b*c^3*d^5*e^3*x^2 + 
 540*b^2*c^2*d^4*e^4*x^2 + 360*a*c^3*d^4*e^4*x^2 + 180*b^3*c*d^3*e^5*x^2 + 
 540*a*b*c^2*d^3*e^5*x^2 + 36*b^4*d^2*e^6*x^2 + 432*a*b^2*c*d^2*e^6*x^2 + 
216*a^2*c^2*d^2*e^6*x^2 + 180*a*b^3*d*e^7*x^2 + 540*a^2*b*c*d*e^7*x^2 + 54 
0*a^2*b^2*e^8*x^2 + 360*a^3*c*e^8*x^2 + 630*c^4*d^7*e*x + 315*b*c^3*d^6*e^ 
2*x + 135*b^2*c^2*d^5*e^3*x + 90*a*c^3*d^5*e^3*x + 45*b^3*c*d^4*e^4*x + 13 
5*a*b*c^2*d^4*e^4*x + 9*b^4*d^3*e^5*x + 108*a*b^2*c*d^3*e^5*x + 54*a^2*c^2 
*d^3*e^5*x + 45*a*b^3*d^2*e^6*x + 135*a^2*b*c*d^2*e^6*x + 135*a^2*b^2*d*e^ 
7*x + 90*a^3*c*d*e^7*x + 315*a^3*b*e^8*x + 70*c^4*d^8 + 35*b*c^3*d^7*e + 1 
5*b^2*c^2*d^6*e^2 + 10*a*c^3*d^6*e^2 + 5*b^3*c*d^5*e^3 + 15*a*b*c^2*d^5...
 

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {\frac {70\,a^4\,e^8+35\,a^3\,b\,d\,e^7+10\,a^3\,c\,d^2\,e^6+15\,a^2\,b^2\,d^2\,e^6+15\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+5\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+15\,a\,b\,c^2\,d^5\,e^3+10\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+5\,b^3\,c\,d^5\,e^3+15\,b^2\,c^2\,d^6\,e^2+35\,b\,c^3\,d^7\,e+70\,c^4\,d^8}{630\,e^9}+\frac {2\,x^3\,\left (15\,a^2\,b\,c\,e^5+6\,a^2\,c^2\,d\,e^4+5\,a\,b^3\,e^5+12\,a\,b^2\,c\,d\,e^4+15\,a\,b\,c^2\,d^2\,e^3+10\,a\,c^3\,d^3\,e^2+b^4\,d\,e^4+5\,b^3\,c\,d^2\,e^3+15\,b^2\,c^2\,d^3\,e^2+35\,b\,c^3\,d^4\,e+70\,c^4\,d^5\right )}{15\,e^6}+\frac {x^4\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4+15\,a\,b\,c^2\,d\,e^3+10\,a\,c^3\,d^2\,e^2+b^4\,e^4+5\,b^3\,c\,d\,e^3+15\,b^2\,c^2\,d^2\,e^2+35\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{5\,e^5}+\frac {x\,\left (35\,a^3\,b\,e^7+10\,a^3\,c\,d\,e^6+15\,a^2\,b^2\,d\,e^6+15\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+5\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4+15\,a\,b\,c^2\,d^4\,e^3+10\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4+5\,b^3\,c\,d^4\,e^3+15\,b^2\,c^2\,d^5\,e^2+35\,b\,c^3\,d^6\,e+70\,c^4\,d^7\right )}{70\,e^8}+\frac {c^4\,x^8}{e}+\frac {2\,x^2\,\left (10\,a^3\,c\,e^6+15\,a^2\,b^2\,e^6+15\,a^2\,b\,c\,d\,e^5+6\,a^2\,c^2\,d^2\,e^4+5\,a\,b^3\,d\,e^5+12\,a\,b^2\,c\,d^2\,e^4+15\,a\,b\,c^2\,d^3\,e^3+10\,a\,c^3\,d^4\,e^2+b^4\,d^2\,e^4+5\,b^3\,c\,d^3\,e^3+15\,b^2\,c^2\,d^4\,e^2+35\,b\,c^3\,d^5\,e+70\,c^4\,d^6\right )}{35\,e^7}+\frac {2\,c^3\,x^7\,\left (b\,e+2\,c\,d\right )}{e^2}+\frac {2\,c^2\,x^6\,\left (3\,b^2\,e^2+7\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^5\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+7\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3+14\,c^3\,d^3+2\,a\,c^2\,d\,e^2\right )}{e^4}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \] Input:

int((a + b*x + c*x^2)^4/(d + e*x)^10,x)
 

Output:

-((70*a^4*e^8 + 70*c^4*d^8 + b^4*d^4*e^4 + 5*a*b^3*d^3*e^5 + 10*a*c^3*d^6* 
e^2 + 10*a^3*c*d^2*e^6 + 5*b^3*c*d^5*e^3 + 15*a^2*b^2*d^2*e^6 + 6*a^2*c^2* 
d^4*e^4 + 15*b^2*c^2*d^6*e^2 + 35*a^3*b*d*e^7 + 35*b*c^3*d^7*e + 15*a*b*c^ 
2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 + 15*a^2*b*c*d^3*e^5)/(630*e^9) + (2*x^3*(7 
0*c^4*d^5 + 5*a*b^3*e^5 + b^4*d*e^4 + 10*a*c^3*d^3*e^2 + 6*a^2*c^2*d*e^4 + 
 5*b^3*c*d^2*e^3 + 15*b^2*c^2*d^3*e^2 + 15*a^2*b*c*e^5 + 35*b*c^3*d^4*e + 
12*a*b^2*c*d*e^4 + 15*a*b*c^2*d^2*e^3))/(15*e^6) + (x^4*(b^4*e^4 + 70*c^4* 
d^4 + 6*a^2*c^2*e^4 + 10*a*c^3*d^2*e^2 + 15*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e 
^4 + 35*b*c^3*d^3*e + 5*b^3*c*d*e^3 + 15*a*b*c^2*d*e^3))/(5*e^5) + (x*(70* 
c^4*d^7 + 35*a^3*b*e^7 + b^4*d^3*e^4 + 5*a*b^3*d^2*e^5 + 15*a^2*b^2*d*e^6 
+ 10*a*c^3*d^5*e^2 + 5*b^3*c*d^4*e^3 + 6*a^2*c^2*d^3*e^4 + 15*b^2*c^2*d^5* 
e^2 + 10*a^3*c*d*e^6 + 35*b*c^3*d^6*e + 15*a*b*c^2*d^4*e^3 + 12*a*b^2*c*d^ 
3*e^4 + 15*a^2*b*c*d^2*e^5))/(70*e^8) + (c^4*x^8)/e + (2*x^2*(70*c^4*d^6 + 
 10*a^3*c*e^6 + 15*a^2*b^2*e^6 + b^4*d^2*e^4 + 10*a*c^3*d^4*e^2 + 5*b^3*c* 
d^3*e^3 + 6*a^2*c^2*d^2*e^4 + 15*b^2*c^2*d^4*e^2 + 5*a*b^3*d*e^5 + 35*b*c^ 
3*d^5*e + 15*a^2*b*c*d*e^5 + 15*a*b*c^2*d^3*e^3 + 12*a*b^2*c*d^2*e^4))/(35 
*e^7) + (2*c^3*x^7*(b*e + 2*c*d))/e^2 + (2*c^2*x^6*(3*b^2*e^2 + 14*c^2*d^2 
 + 2*a*c*e^2 + 7*b*c*d*e))/(3*e^3) + (c*x^5*(b^3*e^3 + 14*c^3*d^3 + 3*a*b* 
c*e^3 + 2*a*c^2*d*e^2 + 7*b*c^2*d^2*e + 3*b^2*c*d*e^2))/e^4)/(d^9 + e^9*x^ 
9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 1...
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1039, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^4/(e*x+d)^10,x)
 

Output:

( - 70*a**4*d*e**7 - 35*a**3*b*d**2*e**6 - 315*a**3*b*d*e**7*x - 10*a**3*c 
*d**3*e**5 - 90*a**3*c*d**2*e**6*x - 360*a**3*c*d*e**7*x**2 - 15*a**2*b**2 
*d**3*e**5 - 135*a**2*b**2*d**2*e**6*x - 540*a**2*b**2*d*e**7*x**2 - 15*a* 
*2*b*c*d**4*e**4 - 135*a**2*b*c*d**3*e**5*x - 540*a**2*b*c*d**2*e**6*x**2 
- 1260*a**2*b*c*d*e**7*x**3 - 6*a**2*c**2*d**5*e**3 - 54*a**2*c**2*d**4*e* 
*4*x - 216*a**2*c**2*d**3*e**5*x**2 - 504*a**2*c**2*d**2*e**6*x**3 - 756*a 
**2*c**2*d*e**7*x**4 - 5*a*b**3*d**4*e**4 - 45*a*b**3*d**3*e**5*x - 180*a* 
b**3*d**2*e**6*x**2 - 420*a*b**3*d*e**7*x**3 - 12*a*b**2*c*d**5*e**3 - 108 
*a*b**2*c*d**4*e**4*x - 432*a*b**2*c*d**3*e**5*x**2 - 1008*a*b**2*c*d**2*e 
**6*x**3 - 1512*a*b**2*c*d*e**7*x**4 - 15*a*b*c**2*d**6*e**2 - 135*a*b*c** 
2*d**5*e**3*x - 540*a*b*c**2*d**4*e**4*x**2 - 1260*a*b*c**2*d**3*e**5*x**3 
 - 1890*a*b*c**2*d**2*e**6*x**4 - 1890*a*b*c**2*d*e**7*x**5 - 10*a*c**3*d* 
*7*e - 90*a*c**3*d**6*e**2*x - 360*a*c**3*d**5*e**3*x**2 - 840*a*c**3*d**4 
*e**4*x**3 - 1260*a*c**3*d**3*e**5*x**4 - 1260*a*c**3*d**2*e**6*x**5 - 840 
*a*c**3*d*e**7*x**6 - b**4*d**5*e**3 - 9*b**4*d**4*e**4*x - 36*b**4*d**3*e 
**5*x**2 - 84*b**4*d**2*e**6*x**3 - 126*b**4*d*e**7*x**4 - 5*b**3*c*d**6*e 
**2 - 45*b**3*c*d**5*e**3*x - 180*b**3*c*d**4*e**4*x**2 - 420*b**3*c*d**3* 
e**5*x**3 - 630*b**3*c*d**2*e**6*x**4 - 630*b**3*c*d*e**7*x**5 - 15*b**2*c 
**2*d**7*e - 135*b**2*c**2*d**6*e**2*x - 540*b**2*c**2*d**5*e**3*x**2 - 12 
60*b**2*c**2*d**4*e**4*x**3 - 1890*b**2*c**2*d**3*e**5*x**4 - 1890*b**2...