\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^{11}} \, dx\) [459]

Optimal result

Integrand size = 20, antiderivative size = 443 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{9 e^9 (d+e x)^9}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{4 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{7 e^9 (d+e x)^7}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{6 e^9 (d+e x)^6}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{5 e^9 (d+e x)^5}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{2 e^9 (d+e x)^4}+\frac {4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac {c^4}{2 e^9 (d+e x)^2} \] Output:

-1/10*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^10+4/9*(-b*e+2*c*d)*(a*e^2-b*d*e+c 
*d^2)^3/e^9/(e*x+d)^9-1/4*(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)^8+4/7*(-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^9/(e*x+d)^7-1/6*(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)^6+4/5*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)^5-1/2*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)^4+4/3*c^3*(-b*e+2*c*d)/e^9/(e*x+d)^3-1/2*c^4/e^9/(e*x+ 
d)^2
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {14 c^4 \left (d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right )+e^4 \left (126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+2 c e^3 \left (7 a^3 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a b^2 e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+2 b^3 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 b^2 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+2 c^3 e \left (3 a e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 b \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{1260 e^9 (d+e x)^{10}} \] Input:

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^11,x]
 

Output:

-1/1260*(14*c^4*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210 
*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8* 
x^8) + e^4*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 
+ 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 12 
0*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210* 
e^4*x^4)) + 2*c*e^3*(7*a^3*e^3*(d^2 + 10*d*e*x + 45*e^2*x^2) + 9*a^2*b*e^2 
*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 6*a*b^2*e*(d^4 + 10*d^3 
*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 2*b^3*(d^5 + 10*d^4 
*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)) + 
3*c^2*e^2*(2*a^2*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 
210*e^4*x^4) + 4*a*b*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^ 
3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*b^2*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^ 
2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + 2* 
c^3*e*(3*a*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^ 
2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6) + 7*b*(d^7 + 10*d^6*e*x + 45*d^5* 
e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6* 
x^6 + 120*e^7*x^7)))/(e^9*(d + e*x)^10)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 443, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)^7}+\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^5}+\frac {4 c (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8 (d+e x)^6}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^8}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^9}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^{10}}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^{11}}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^4}+\frac {c^4}{e^8 (d+e x)^3}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x + c*x^2)^4/(d + e*x)^11,x]
 

Output:

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

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. \(2(425)=850\).

Time = 0.92 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.99

method	result	size
risch	\(\frac {-\frac {c^{4} x^{8}}{2 e}-\frac {4 c^{3} \left (b e +c d \right ) x^{7}}{3 e^{2}}-\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}+14 b c d e +14 c^{2} d^{2}\right ) x^{6}}{6 e^{3}}-\frac {c \left (12 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+4 b^{3} e^{3}+9 d \,e^{2} b^{2} c +14 d^{2} e b \,c^{2}+14 d^{3} c^{3}\right ) x^{5}}{5 e^{4}}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}+6 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+4 d \,e^{3} b^{3} c +9 d^{2} e^{2} b^{2} c^{2}+14 d^{3} e b \,c^{3}+14 d^{4} c^{4}\right ) x^{4}}{6 e^{5}}-\frac {2 \left (18 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+6 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+12 a b \,c^{2} d^{2} e^{3}+6 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+4 b^{3} c \,d^{2} e^{3}+9 b^{2} c^{2} d^{3} e^{2}+14 b \,c^{3} d^{4} e +14 c^{4} d^{5}\right ) x^{3}}{21 e^{6}}-\frac {\left (14 e^{6} c \,a^{3}+21 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+12 a b \,c^{2} d^{3} e^{3}+6 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+4 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}+14 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) x^{2}}{28 e^{7}}-\frac {\left (56 a^{3} b \,e^{7}+14 d \,e^{6} c \,a^{3}+21 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+6 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+12 a b \,c^{2} d^{4} e^{3}+6 d^{5} e^{2} a \,c^{3}+b^{4} d^{3} e^{4}+4 b^{3} c \,d^{4} e^{3}+9 b^{2} c^{2} d^{5} e^{2}+14 b \,c^{3} d^{6} e +14 d^{7} c^{4}\right ) x}{126 e^{8}}-\frac {126 a^{4} e^{8}+56 a^{3} b d \,e^{7}+14 a^{3} c \,d^{2} e^{6}+21 a^{2} b^{2} d^{2} e^{6}+18 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+6 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+12 a b \,c^{2} d^{5} e^{3}+6 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+4 b^{3} c \,d^{5} e^{3}+9 b^{2} c^{2} d^{6} e^{2}+14 b \,c^{3} d^{7} e +14 c^{4} d^{8}}{1260 e^{9}}}{\left (e x +d \right )^{10}}\)	\(880\)
default	\(-\frac {4 c^{3} \left (b e -2 c d \right )}{3 e^{9} \left (e x +d \right )^{3}}-\frac {c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{2 e^{9} \left (e x +d \right )^{4}}-\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}}{9 e^{9} \left (e x +d \right )^{9}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {4 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 )}{5 e^{9} \left (e x +d \right )^{5}}-\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}}{8 e^{9} \left (e x +d \right )^{8}}-\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}}{10 e^{9} \left (e x +d \right )^{10}}-\frac {c^{4}}{2 e^{9} \left (e x +d \right )^{2}}-\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}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(914\)
norman	\(\frac {-\frac {c^{4} x^{8}}{2 e}-\frac {4 \left (e^{2} b \,c^{3}+d e \,c^{4}\right ) x^{7}}{3 e^{3}}-\frac {\left (6 a \,c^{3} e^{3}+9 b^{2} c^{2} e^{3}+14 d \,e^{2} b \,c^{3}+14 d^{2} e \,c^{4}\right ) x^{6}}{6 e^{4}}-\frac {\left (12 a b \,c^{2} e^{4}+6 a \,c^{3} d \,e^{3}+4 b^{3} c \,e^{4}+9 b^{2} c^{2} d \,e^{3}+14 b \,c^{3} d^{2} e^{2}+14 c^{4} d^{3} e \right ) x^{5}}{5 e^{5}}-\frac {\left (6 a^{2} c^{2} e^{5}+12 a \,b^{2} c \,e^{5}+12 a b \,c^{2} d \,e^{4}+6 a \,c^{3} d^{2} e^{3}+b^{4} e^{5}+4 b^{3} c d \,e^{4}+9 b^{2} c^{2} d^{2} e^{3}+14 b \,c^{3} d^{3} e^{2}+14 c^{4} d^{4} e \right ) x^{4}}{6 e^{6}}-\frac {2 \left (18 c \,a^{2} b \,e^{6}+6 a^{2} c^{2} d \,e^{5}+6 a \,b^{3} e^{6}+12 a \,b^{2} c d \,e^{5}+12 a b \,c^{2} d^{2} e^{4}+6 a \,c^{3} d^{3} e^{3}+b^{4} d \,e^{5}+4 b^{3} c \,d^{2} e^{4}+9 b^{2} c^{2} d^{3} e^{3}+14 b \,c^{3} d^{4} e^{2}+14 c^{4} d^{5} e \right ) x^{3}}{21 e^{7}}-\frac {\left (14 e^{7} c \,a^{3}+21 a^{2} b^{2} e^{7}+18 a^{2} b c d \,e^{6}+6 d^{2} e^{5} a^{2} c^{2}+6 a \,b^{3} d \,e^{6}+12 a \,b^{2} c \,d^{2} e^{5}+12 a b \,c^{2} d^{3} e^{4}+6 d^{4} e^{3} a \,c^{3}+b^{4} d^{2} e^{5}+4 b^{3} c \,d^{3} e^{4}+9 b^{2} c^{2} d^{4} e^{3}+14 b \,c^{3} d^{5} e^{2}+14 c^{4} d^{6} e \right ) x^{2}}{28 e^{8}}-\frac {\left (56 a^{3} b \,e^{8}+14 a^{3} e^{7} d c +21 a^{2} b^{2} d \,e^{7}+18 a^{2} b c \,d^{2} e^{6}+6 a^{2} c^{2} d^{3} e^{5}+6 a \,b^{3} d^{2} e^{6}+12 a \,b^{2} c \,d^{3} e^{5}+12 a b \,c^{2} d^{4} e^{4}+6 a \,c^{3} d^{5} e^{3}+b^{4} d^{3} e^{5}+4 b^{3} c \,d^{4} e^{4}+9 b^{2} c^{2} d^{5} e^{3}+14 b \,c^{3} d^{6} e^{2}+14 c^{4} d^{7} e \right ) x}{126 e^{9}}-\frac {126 a^{4} e^{9}+56 a^{3} b d \,e^{8}+14 a^{3} c \,d^{2} e^{7}+21 a^{2} b^{2} d^{2} e^{7}+18 a^{2} b c \,d^{3} e^{6}+6 a^{2} c^{2} d^{4} e^{5}+6 a \,b^{3} d^{3} e^{6}+12 a \,b^{2} c \,d^{4} e^{5}+12 a b \,c^{2} d^{5} e^{4}+6 a \,c^{3} d^{6} e^{3}+b^{4} d^{4} e^{5}+4 b^{3} c \,d^{5} e^{4}+9 b^{2} c^{2} d^{6} e^{3}+14 b \,c^{3} d^{7} e^{2}+14 c^{4} d^{8} e}{1260 e^{10}}}{\left (e x +d \right )^{10}}\)	\(914\)
gosper	\(\text {Expression too large to display}\)	\(1015\)
orering	\(\text {Expression too large to display}\)	\(1015\)
parallelrisch	\(\text {Expression too large to display}\)	\(1021\)

Input:

int((c*x^2+b*x+a)^4/(e*x+d)^11,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*c^4/e*x^8-4/3*c^3/e^2*(b*e+c*d)*x^7-1/6/e^3*c^2*(6*a*c*e^2+9*b^2*e^2 
+14*b*c*d*e+14*c^2*d^2)*x^6-1/5*c/e^4*(12*a*b*c*e^3+6*a*c^2*d*e^2+4*b^3*e^ 
3+9*b^2*c*d*e^2+14*b*c^2*d^2*e+14*c^3*d^3)*x^5-1/6/e^5*(6*a^2*c^2*e^4+12*a 
*b^2*c*e^4+12*a*b*c^2*d*e^3+6*a*c^3*d^2*e^2+b^4*e^4+4*b^3*c*d*e^3+9*b^2*c^ 
2*d^2*e^2+14*b*c^3*d^3*e+14*c^4*d^4)*x^4-2/21/e^6*(18*a^2*b*c*e^5+6*a^2*c^ 
2*d*e^4+6*a*b^3*e^5+12*a*b^2*c*d*e^4+12*a*b*c^2*d^2*e^3+6*a*c^3*d^3*e^2+b^ 
4*d*e^4+4*b^3*c*d^2*e^3+9*b^2*c^2*d^3*e^2+14*b*c^3*d^4*e+14*c^4*d^5)*x^3-1 
/28/e^7*(14*a^3*c*e^6+21*a^2*b^2*e^6+18*a^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4+6* 
a*b^3*d*e^5+12*a*b^2*c*d^2*e^4+12*a*b*c^2*d^3*e^3+6*a*c^3*d^4*e^2+b^4*d^2* 
e^4+4*b^3*c*d^3*e^3+9*b^2*c^2*d^4*e^2+14*b*c^3*d^5*e+14*c^4*d^6)*x^2-1/126 
/e^8*(56*a^3*b*e^7+14*a^3*c*d*e^6+21*a^2*b^2*d*e^6+18*a^2*b*c*d^2*e^5+6*a^ 
2*c^2*d^3*e^4+6*a*b^3*d^2*e^5+12*a*b^2*c*d^3*e^4+12*a*b*c^2*d^4*e^3+6*a*c^ 
3*d^5*e^2+b^4*d^3*e^4+4*b^3*c*d^4*e^3+9*b^2*c^2*d^5*e^2+14*b*c^3*d^6*e+14* 
c^4*d^7)*x-1/1260/e^9*(126*a^4*e^8+56*a^3*b*d*e^7+14*a^3*c*d^2*e^6+21*a^2* 
b^2*d^2*e^6+18*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4+6*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+6*a*c^3*d^6*e^2+b^4*d^4*e^4+4*b^3*c*d^5*e^3+9 
*b^2*c^2*d^6*e^2+14*b*c^3*d^7*e+14*c^4*d^8))/(e*x+d)^10
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (425) = 850\).

Time = 0.09 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^11,x, algorithm="fricas")
 

Output:

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 
126*a^4*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e 
^5 + 7*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 
210*(14*c^4*d^2*e^6 + 14*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 
252*(14*c^4*d^3*e^5 + 14*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4 
*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3 
*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 4*(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 + 120*(14*c^4*d^5*e^3 + 14*b*c^3*d^4*e^4 + 3* 
(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 45*(14*c^4*d 
^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 + 3* 
a^2*b*c)*d*e^7 + 7*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14* 
b*c^3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4*(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 + 6*(a*b^3 + 
 3*a^2*b*c)*d^2*e^6 + 7*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d* 
e^18*x^9 + 45*d^2*e^17*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x^6 + 252*d^5 
*e^14*x^5 + 210*d^6*e^13*x^4 + 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9 
*e^10*x + d^10*e^9)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**4/(e*x+d)**11,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (425) = 850\).

Time = 0.08 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^11,x, algorithm="maxima")
 

Output:

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 
126*a^4*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e 
^5 + 7*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 
210*(14*c^4*d^2*e^6 + 14*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 
252*(14*c^4*d^3*e^5 + 14*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4 
*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3 
*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 4*(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 + 120*(14*c^4*d^5*e^3 + 14*b*c^3*d^4*e^4 + 3* 
(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 45*(14*c^4*d 
^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 + 3* 
a^2*b*c)*d*e^7 + 7*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14* 
b*c^3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4*(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 + 6*(a*b^3 + 
 3*a^2*b*c)*d^2*e^6 + 7*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d* 
e^18*x^9 + 45*d^2*e^17*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x^6 + 252*d^5 
*e^14*x^5 + 210*d^6*e^13*x^4 + 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9 
*e^10*x + d^10*e^9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (425) = 850\).

Time = 0.39 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^11,x, algorithm="giac")
 

Output:

-1/1260*(630*c^4*e^8*x^8 + 1680*c^4*d*e^7*x^7 + 1680*b*c^3*e^8*x^7 + 2940* 
c^4*d^2*e^6*x^6 + 2940*b*c^3*d*e^7*x^6 + 1890*b^2*c^2*e^8*x^6 + 1260*a*c^3 
*e^8*x^6 + 3528*c^4*d^3*e^5*x^5 + 3528*b*c^3*d^2*e^6*x^5 + 2268*b^2*c^2*d* 
e^7*x^5 + 1512*a*c^3*d*e^7*x^5 + 1008*b^3*c*e^8*x^5 + 3024*a*b*c^2*e^8*x^5 
 + 2940*c^4*d^4*e^4*x^4 + 2940*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 + 840*b^3*c*d*e^7*x^4 + 2520*a*b*c^2*d*e^7*x^4 
+ 210*b^4*e^8*x^4 + 2520*a*b^2*c*e^8*x^4 + 1260*a^2*c^2*e^8*x^4 + 1680*c^4 
*d^5*e^3*x^3 + 1680*b*c^3*d^4*e^4*x^3 + 1080*b^2*c^2*d^3*e^5*x^3 + 720*a*c 
^3*d^3*e^5*x^3 + 480*b^3*c*d^2*e^6*x^3 + 1440*a*b*c^2*d^2*e^6*x^3 + 120*b^ 
4*d*e^7*x^3 + 1440*a*b^2*c*d*e^7*x^3 + 720*a^2*c^2*d*e^7*x^3 + 720*a*b^3*e 
^8*x^3 + 2160*a^2*b*c*e^8*x^3 + 630*c^4*d^6*e^2*x^2 + 630*b*c^3*d^5*e^3*x^ 
2 + 405*b^2*c^2*d^4*e^4*x^2 + 270*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 + 45*b^4*d^2*e^6*x^2 + 540*a*b^2*c*d^2*e^6*x^2 
 + 270*a^2*c^2*d^2*e^6*x^2 + 270*a*b^3*d*e^7*x^2 + 810*a^2*b*c*d*e^7*x^2 + 
 945*a^2*b^2*e^8*x^2 + 630*a^3*c*e^8*x^2 + 140*c^4*d^7*e*x + 140*b*c^3*d^6 
*e^2*x + 90*b^2*c^2*d^5*e^3*x + 60*a*c^3*d^5*e^3*x + 40*b^3*c*d^4*e^4*x + 
120*a*b*c^2*d^4*e^4*x + 10*b^4*d^3*e^5*x + 120*a*b^2*c*d^3*e^5*x + 60*a^2* 
c^2*d^3*e^5*x + 60*a*b^3*d^2*e^6*x + 180*a^2*b*c*d^2*e^6*x + 210*a^2*b^2*d 
*e^7*x + 140*a^3*c*d*e^7*x + 560*a^3*b*e^8*x + 14*c^4*d^8 + 14*b*c^3*d^7*e 
 + 9*b^2*c^2*d^6*e^2 + 6*a*c^3*d^6*e^2 + 4*b^3*c*d^5*e^3 + 12*a*b*c^2*d...
 

Mupad [B] (verification not implemented)

Time = 5.53 (sec) , antiderivative size = 979, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx=-\frac {\frac {126\,a^4\,e^8+56\,a^3\,b\,d\,e^7+14\,a^3\,c\,d^2\,e^6+21\,a^2\,b^2\,d^2\,e^6+18\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+6\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+12\,a\,b\,c^2\,d^5\,e^3+6\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+4\,b^3\,c\,d^5\,e^3+9\,b^2\,c^2\,d^6\,e^2+14\,b\,c^3\,d^7\,e+14\,c^4\,d^8}{1260\,e^9}+\frac {2\,x^3\,\left (18\,a^2\,b\,c\,e^5+6\,a^2\,c^2\,d\,e^4+6\,a\,b^3\,e^5+12\,a\,b^2\,c\,d\,e^4+12\,a\,b\,c^2\,d^2\,e^3+6\,a\,c^3\,d^3\,e^2+b^4\,d\,e^4+4\,b^3\,c\,d^2\,e^3+9\,b^2\,c^2\,d^3\,e^2+14\,b\,c^3\,d^4\,e+14\,c^4\,d^5\right )}{21\,e^6}+\frac {x^4\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3+6\,a\,c^3\,d^2\,e^2+b^4\,e^4+4\,b^3\,c\,d\,e^3+9\,b^2\,c^2\,d^2\,e^2+14\,b\,c^3\,d^3\,e+14\,c^4\,d^4\right )}{6\,e^5}+\frac {x\,\left (56\,a^3\,b\,e^7+14\,a^3\,c\,d\,e^6+21\,a^2\,b^2\,d\,e^6+18\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+6\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4+12\,a\,b\,c^2\,d^4\,e^3+6\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4+4\,b^3\,c\,d^4\,e^3+9\,b^2\,c^2\,d^5\,e^2+14\,b\,c^3\,d^6\,e+14\,c^4\,d^7\right )}{126\,e^8}+\frac {c^4\,x^8}{2\,e}+\frac {x^2\,\left (14\,a^3\,c\,e^6+21\,a^2\,b^2\,e^6+18\,a^2\,b\,c\,d\,e^5+6\,a^2\,c^2\,d^2\,e^4+6\,a\,b^3\,d\,e^5+12\,a\,b^2\,c\,d^2\,e^4+12\,a\,b\,c^2\,d^3\,e^3+6\,a\,c^3\,d^4\,e^2+b^4\,d^2\,e^4+4\,b^3\,c\,d^3\,e^3+9\,b^2\,c^2\,d^4\,e^2+14\,b\,c^3\,d^5\,e+14\,c^4\,d^6\right )}{28\,e^7}+\frac {4\,c^3\,x^7\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {c^2\,x^6\,\left (9\,b^2\,e^2+14\,b\,c\,d\,e+14\,c^2\,d^2+6\,a\,c\,e^2\right )}{6\,e^3}+\frac {c\,x^5\,\left (4\,b^3\,e^3+9\,b^2\,c\,d\,e^2+14\,b\,c^2\,d^2\,e+12\,a\,b\,c\,e^3+14\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{5\,e^4}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \] Input:

int((a + b*x + c*x^2)^4/(d + e*x)^11,x)
 

Output:

-((126*a^4*e^8 + 14*c^4*d^8 + b^4*d^4*e^4 + 6*a*b^3*d^3*e^5 + 6*a*c^3*d^6* 
e^2 + 14*a^3*c*d^2*e^6 + 4*b^3*c*d^5*e^3 + 21*a^2*b^2*d^2*e^6 + 6*a^2*c^2* 
d^4*e^4 + 9*b^2*c^2*d^6*e^2 + 56*a^3*b*d*e^7 + 14*b*c^3*d^7*e + 12*a*b*c^2 
*d^5*e^3 + 12*a*b^2*c*d^4*e^4 + 18*a^2*b*c*d^3*e^5)/(1260*e^9) + (2*x^3*(1 
4*c^4*d^5 + 6*a*b^3*e^5 + b^4*d*e^4 + 6*a*c^3*d^3*e^2 + 6*a^2*c^2*d*e^4 + 
4*b^3*c*d^2*e^3 + 9*b^2*c^2*d^3*e^2 + 18*a^2*b*c*e^5 + 14*b*c^3*d^4*e + 12 
*a*b^2*c*d*e^4 + 12*a*b*c^2*d^2*e^3))/(21*e^6) + (x^4*(b^4*e^4 + 14*c^4*d^ 
4 + 6*a^2*c^2*e^4 + 6*a*c^3*d^2*e^2 + 9*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 + 
 14*b*c^3*d^3*e + 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(6*e^5) + (x*(14*c^4* 
d^7 + 56*a^3*b*e^7 + b^4*d^3*e^4 + 6*a*b^3*d^2*e^5 + 21*a^2*b^2*d*e^6 + 6* 
a*c^3*d^5*e^2 + 4*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 + 
14*a^3*c*d*e^6 + 14*b*c^3*d^6*e + 12*a*b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 
+ 18*a^2*b*c*d^2*e^5))/(126*e^8) + (c^4*x^8)/(2*e) + (x^2*(14*c^4*d^6 + 14 
*a^3*c*e^6 + 21*a^2*b^2*e^6 + b^4*d^2*e^4 + 6*a*c^3*d^4*e^2 + 4*b^3*c*d^3* 
e^3 + 6*a^2*c^2*d^2*e^4 + 9*b^2*c^2*d^4*e^2 + 6*a*b^3*d*e^5 + 14*b*c^3*d^5 
*e + 18*a^2*b*c*d*e^5 + 12*a*b*c^2*d^3*e^3 + 12*a*b^2*c*d^2*e^4))/(28*e^7) 
 + (4*c^3*x^7*(b*e + c*d))/(3*e^2) + (c^2*x^6*(9*b^2*e^2 + 14*c^2*d^2 + 6* 
a*c*e^2 + 14*b*c*d*e))/(6*e^3) + (c*x^5*(4*b^3*e^3 + 14*c^3*d^3 + 12*a*b*c 
*e^3 + 6*a*c^2*d*e^2 + 14*b*c^2*d^2*e + 9*b^2*c*d*e^2))/(5*e^4))/(d^10 + e 
^10*x^10 + 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1114, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^4/(e*x+d)^11,x)
 

Output:

( - 126*a**4*e**8 - 56*a**3*b*d*e**7 - 560*a**3*b*e**8*x - 14*a**3*c*d**2* 
e**6 - 140*a**3*c*d*e**7*x - 630*a**3*c*e**8*x**2 - 21*a**2*b**2*d**2*e**6 
 - 210*a**2*b**2*d*e**7*x - 945*a**2*b**2*e**8*x**2 - 18*a**2*b*c*d**3*e** 
5 - 180*a**2*b*c*d**2*e**6*x - 810*a**2*b*c*d*e**7*x**2 - 2160*a**2*b*c*e* 
*8*x**3 - 6*a**2*c**2*d**4*e**4 - 60*a**2*c**2*d**3*e**5*x - 270*a**2*c**2 
*d**2*e**6*x**2 - 720*a**2*c**2*d*e**7*x**3 - 1260*a**2*c**2*e**8*x**4 - 6 
*a*b**3*d**3*e**5 - 60*a*b**3*d**2*e**6*x - 270*a*b**3*d*e**7*x**2 - 720*a 
*b**3*e**8*x**3 - 12*a*b**2*c*d**4*e**4 - 120*a*b**2*c*d**3*e**5*x - 540*a 
*b**2*c*d**2*e**6*x**2 - 1440*a*b**2*c*d*e**7*x**3 - 2520*a*b**2*c*e**8*x* 
*4 - 12*a*b*c**2*d**5*e**3 - 120*a*b*c**2*d**4*e**4*x - 540*a*b*c**2*d**3* 
e**5*x**2 - 1440*a*b*c**2*d**2*e**6*x**3 - 2520*a*b*c**2*d*e**7*x**4 - 302 
4*a*b*c**2*e**8*x**5 - 6*a*c**3*d**6*e**2 - 60*a*c**3*d**5*e**3*x - 270*a* 
c**3*d**4*e**4*x**2 - 720*a*c**3*d**3*e**5*x**3 - 1260*a*c**3*d**2*e**6*x* 
*4 - 1512*a*c**3*d*e**7*x**5 - 1260*a*c**3*e**8*x**6 - b**4*d**4*e**4 - 10 
*b**4*d**3*e**5*x - 45*b**4*d**2*e**6*x**2 - 120*b**4*d*e**7*x**3 - 210*b* 
*4*e**8*x**4 - 4*b**3*c*d**5*e**3 - 40*b**3*c*d**4*e**4*x - 180*b**3*c*d** 
3*e**5*x**2 - 480*b**3*c*d**2*e**6*x**3 - 840*b**3*c*d*e**7*x**4 - 1008*b* 
*3*c*e**8*x**5 - 9*b**2*c**2*d**6*e**2 - 90*b**2*c**2*d**5*e**3*x - 405*b* 
*2*c**2*d**4*e**4*x**2 - 1080*b**2*c**2*d**3*e**5*x**3 - 1890*b**2*c**2*d* 
*2*e**6*x**4 - 2268*b**2*c**2*d*e**7*x**5 - 1890*b**2*c**2*e**8*x**6 - ...