\(\int (d+e x)^4 (a+b x+c x^2)^3 \, dx\) [429]

Optimal result

Integrand size = 20, antiderivative size = 272 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^7}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}{2 e^7}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^8}{8 e^7}+\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^9}{3 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^{10}}{10 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7} \] Output:

1/5*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^7-1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 
2)^2*(e*x+d)^6/e^7+3/7*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5* 
b*d))*(e*x+d)^7/e^7-1/8*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b 
*d))*(e*x+d)^8/e^7+1/3*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^9/e^ 
7-3/10*c^2*(-b*e+2*c*d)*(e*x+d)^10/e^7+1/11*c^3*(e*x+d)^11/e^7
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.83 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^3 d^4 x+\frac {1}{2} a^2 d^3 (3 b d+4 a e) x^2+a d^2 \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right ) x^3+\frac {1}{4} d \left (b^3 d^3+12 a b^2 d^2 e+4 a^2 e \left (3 c d^2+a e^2\right )+6 a b d \left (c d^2+3 a e^2\right )\right ) x^4+\frac {1}{5} \left (4 b^3 d^3 e+12 a b d e \left (2 c d^2+a e^2\right )+3 b^2 \left (c d^4+6 a d^2 e^2\right )+a \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{2} \left (2 b^3 d^2 e^2+4 a c d e \left (c d^2+a e^2\right )+4 b^2 \left (c d^3 e+a d e^3\right )+b \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} \left (c^3 d^4+6 c^2 d^2 e (2 b d+3 a e)+b^2 e^3 (4 b d+3 a e)+3 c e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^7+\frac {1}{8} e \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right ) x^8+\frac {1}{3} c e^2 \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right ) x^9+\frac {1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac {1}{11} c^3 e^4 x^{11} \] Input:

Integrate[(d + e*x)^4*(a + b*x + c*x^2)^3,x]
 

Output:

a^3*d^4*x + (a^2*d^3*(3*b*d + 4*a*e)*x^2)/2 + a*d^2*(b^2*d^2 + 4*a*b*d*e + 
 a*(c*d^2 + 2*a*e^2))*x^3 + (d*(b^3*d^3 + 12*a*b^2*d^2*e + 4*a^2*e*(3*c*d^ 
2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2))*x^4)/4 + ((4*b^3*d^3*e + 12*a*b*d* 
e*(2*c*d^2 + a*e^2) + 3*b^2*(c*d^4 + 6*a*d^2*e^2) + a*(3*c^2*d^4 + 18*a*c* 
d^2*e^2 + a^2*e^4))*x^5)/5 + ((2*b^3*d^2*e^2 + 4*a*c*d*e*(c*d^2 + a*e^2) + 
 4*b^2*(c*d^3*e + a*d*e^3) + b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x^6)/ 
2 + ((c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3* 
c*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^7)/7 + (e*(4*c^3*d^3 + b^3*e^3 
+ 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e))*x^8)/8 + (c*e^2*(2* 
c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e))*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x 
^10)/10 + (c^3*e^4*x^11)/11
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 272, 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.

\(\displaystyle \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx\)

\(\Big \downarrow \) 1140

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c (d+e x)^9 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {(d+e x)^8 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{8 e^7}+\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}\)

Input:

Int[(d + e*x)^4*(a + b*x + c*x^2)^3,x]
 

Output:

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e^7) - ((2*c*d - b*e)*(c*d^2 - 
b*d*e + a*e^2)^2*(d + e*x)^6)/(2*e^7) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2* 
d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^7)/(7*e^7) - ((2*c*d - b*e)*( 
10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^8)/(8*e^7) + (c*(5 
*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^9)/(3*e^7) - (3*c^2*(2*c 
*d - b*e)*(d + e*x)^10)/(10*e^7) + (c^3*(d + e*x)^11)/(11*e^7)
 

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. \(525\) vs. \(2(258)=516\).

Time = 0.82 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.93

method	result	size
norman	\(\frac {e^{4} c^{3} x^{11}}{11}+\left (\frac {3}{10} e^{4} b \,c^{2}+\frac {2}{5} d \,e^{3} c^{3}\right ) x^{10}+\left (\frac {1}{3} a \,c^{2} e^{4}+\frac {1}{3} b^{2} c \,e^{4}+\frac {4}{3} d \,e^{3} b \,c^{2}+\frac {2}{3} d^{2} e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{4} a b c \,e^{4}+\frac {3}{2} a \,c^{2} d \,e^{3}+\frac {1}{8} e^{4} b^{3}+\frac {3}{2} b^{2} c d \,e^{3}+\frac {9}{4} d^{2} e^{2} b \,c^{2}+\frac {1}{2} d^{3} e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} e^{4} a^{2} c +\frac {3}{7} a \,b^{2} e^{4}+\frac {24}{7} a b c d \,e^{3}+\frac {18}{7} d^{2} e^{2} a \,c^{2}+\frac {4}{7} b^{3} d \,e^{3}+\frac {18}{7} b^{2} c \,d^{2} e^{2}+\frac {12}{7} b \,c^{2} d^{3} e +\frac {1}{7} d^{4} c^{3}\right ) x^{7}+\left (\frac {1}{2} e^{4} a^{2} b +2 a^{2} c d \,e^{3}+2 a \,b^{2} d \,e^{3}+6 a b c \,d^{2} e^{2}+2 a \,c^{2} d^{3} e +b^{3} d^{2} e^{2}+2 b^{2} c \,d^{3} e +\frac {1}{2} b \,c^{2} d^{4}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{3}+\frac {12}{5} a^{2} d \,e^{3} b +\frac {18}{5} d^{2} e^{2} a^{2} c +\frac {18}{5} a \,b^{2} d^{2} e^{2}+\frac {24}{5} a b c \,d^{3} e +\frac {3}{5} d^{4} a \,c^{2}+\frac {4}{5} d^{3} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{4}\right ) x^{5}+\left (a^{3} d \,e^{3}+\frac {9}{2} d^{2} e^{2} a^{2} b +3 a^{2} c \,d^{3} e +3 a \,b^{2} d^{3} e +\frac {3}{2} a b c \,d^{4}+\frac {1}{4} b^{3} d^{4}\right ) x^{4}+\left (2 d^{2} e^{2} a^{3}+4 a^{2} b \,d^{3} e +a^{2} c \,d^{4}+a \,b^{2} d^{4}\right ) x^{3}+\left (2 d^{3} e \,a^{3}+\frac {3}{2} a^{2} b \,d^{4}\right ) x^{2}+d^{4} a^{3} x\)	\(526\)
gosper	\(\frac {4}{3} x^{9} d \,e^{3} b \,c^{2}+\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {24}{7} x^{7} a b c d \,e^{3}+\frac {9}{2} x^{4} d^{2} e^{2} a^{2} b +3 x^{4} a^{2} c \,d^{3} e +3 x^{4} a \,b^{2} d^{3} e +4 a^{2} b \,d^{3} e \,x^{3}+\frac {1}{2} x^{6} b \,c^{2} d^{4}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+2 d^{3} e \,a^{3} x^{2}+2 a^{3} d^{2} e^{2} x^{3}+\frac {3}{5} x^{5} d^{4} a \,c^{2}+\frac {18}{7} x^{7} d^{2} e^{2} a \,c^{2}+\frac {18}{7} x^{7} b^{2} c \,d^{2} e^{2}+\frac {12}{7} x^{7} b \,c^{2} d^{3} e +2 x^{6} a^{2} c d \,e^{3}+2 x^{6} a \,b^{2} d \,e^{3}+2 x^{6} a \,c^{2} d^{3} e +2 x^{6} b^{2} c \,d^{3} e +\frac {12}{5} x^{5} a^{2} d \,e^{3} b +\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {18}{5} x^{5} a \,b^{2} d^{2} e^{2}+\frac {24}{5} x^{5} a b c \,d^{3} e +\frac {9}{4} x^{8} d^{2} e^{2} b \,c^{2}+\frac {4}{7} x^{7} b^{3} d \,e^{3}+\frac {1}{3} x^{9} a \,c^{2} e^{4}+\frac {1}{3} x^{9} b^{2} c \,e^{4}+\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} c a b \,d^{4} x^{4}+\frac {3}{5} b^{2} c \,x^{5} d^{4}+6 x^{6} a b c \,d^{2} e^{2}+\frac {3}{2} x^{8} a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} b^{2} c d \,e^{3}+a^{2} c \,d^{4} x^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {3}{7} x^{7} a \,b^{2} e^{4}+d^{4} a^{3} x +\frac {3}{4} x^{8} a b c \,e^{4}+\frac {3}{10} x^{10} e^{4} b \,c^{2}+\frac {1}{11} e^{4} c^{3} x^{11}+\frac {1}{2} x^{6} e^{4} a^{2} b +x^{6} b^{3} d^{2} e^{2}+\frac {4}{5} x^{5} d^{3} e \,b^{3}+x^{4} a^{3} d \,e^{3}+\frac {3}{2} a^{2} b \,d^{4} x^{2}+\frac {1}{4} b^{3} d^{4} x^{4}+\frac {1}{7} x^{7} d^{4} c^{3}+\frac {1}{8} x^{8} e^{4} b^{3}+a \,b^{2} d^{4} x^{3}+\frac {1}{5} x^{5} e^{4} a^{3}\)	\(625\)
risch	\(\frac {4}{3} x^{9} d \,e^{3} b \,c^{2}+\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {24}{7} x^{7} a b c d \,e^{3}+\frac {9}{2} x^{4} d^{2} e^{2} a^{2} b +3 x^{4} a^{2} c \,d^{3} e +3 x^{4} a \,b^{2} d^{3} e +4 a^{2} b \,d^{3} e \,x^{3}+\frac {1}{2} x^{6} b \,c^{2} d^{4}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+2 d^{3} e \,a^{3} x^{2}+2 a^{3} d^{2} e^{2} x^{3}+\frac {3}{5} x^{5} d^{4} a \,c^{2}+\frac {18}{7} x^{7} d^{2} e^{2} a \,c^{2}+\frac {18}{7} x^{7} b^{2} c \,d^{2} e^{2}+\frac {12}{7} x^{7} b \,c^{2} d^{3} e +2 x^{6} a^{2} c d \,e^{3}+2 x^{6} a \,b^{2} d \,e^{3}+2 x^{6} a \,c^{2} d^{3} e +2 x^{6} b^{2} c \,d^{3} e +\frac {12}{5} x^{5} a^{2} d \,e^{3} b +\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {18}{5} x^{5} a \,b^{2} d^{2} e^{2}+\frac {24}{5} x^{5} a b c \,d^{3} e +\frac {9}{4} x^{8} d^{2} e^{2} b \,c^{2}+\frac {4}{7} x^{7} b^{3} d \,e^{3}+\frac {1}{3} x^{9} a \,c^{2} e^{4}+\frac {1}{3} x^{9} b^{2} c \,e^{4}+\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} c a b \,d^{4} x^{4}+\frac {3}{5} b^{2} c \,x^{5} d^{4}+6 x^{6} a b c \,d^{2} e^{2}+\frac {3}{2} x^{8} a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} b^{2} c d \,e^{3}+a^{2} c \,d^{4} x^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {3}{7} x^{7} a \,b^{2} e^{4}+d^{4} a^{3} x +\frac {3}{4} x^{8} a b c \,e^{4}+\frac {3}{10} x^{10} e^{4} b \,c^{2}+\frac {1}{11} e^{4} c^{3} x^{11}+\frac {1}{2} x^{6} e^{4} a^{2} b +x^{6} b^{3} d^{2} e^{2}+\frac {4}{5} x^{5} d^{3} e \,b^{3}+x^{4} a^{3} d \,e^{3}+\frac {3}{2} a^{2} b \,d^{4} x^{2}+\frac {1}{4} b^{3} d^{4} x^{4}+\frac {1}{7} x^{7} d^{4} c^{3}+\frac {1}{8} x^{8} e^{4} b^{3}+a \,b^{2} d^{4} x^{3}+\frac {1}{5} x^{5} e^{4} a^{3}\)	\(625\)
parallelrisch	\(\frac {4}{3} x^{9} d \,e^{3} b \,c^{2}+\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {24}{7} x^{7} a b c d \,e^{3}+\frac {9}{2} x^{4} d^{2} e^{2} a^{2} b +3 x^{4} a^{2} c \,d^{3} e +3 x^{4} a \,b^{2} d^{3} e +4 a^{2} b \,d^{3} e \,x^{3}+\frac {1}{2} x^{6} b \,c^{2} d^{4}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+2 d^{3} e \,a^{3} x^{2}+2 a^{3} d^{2} e^{2} x^{3}+\frac {3}{5} x^{5} d^{4} a \,c^{2}+\frac {18}{7} x^{7} d^{2} e^{2} a \,c^{2}+\frac {18}{7} x^{7} b^{2} c \,d^{2} e^{2}+\frac {12}{7} x^{7} b \,c^{2} d^{3} e +2 x^{6} a^{2} c d \,e^{3}+2 x^{6} a \,b^{2} d \,e^{3}+2 x^{6} a \,c^{2} d^{3} e +2 x^{6} b^{2} c \,d^{3} e +\frac {12}{5} x^{5} a^{2} d \,e^{3} b +\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {18}{5} x^{5} a \,b^{2} d^{2} e^{2}+\frac {24}{5} x^{5} a b c \,d^{3} e +\frac {9}{4} x^{8} d^{2} e^{2} b \,c^{2}+\frac {4}{7} x^{7} b^{3} d \,e^{3}+\frac {1}{3} x^{9} a \,c^{2} e^{4}+\frac {1}{3} x^{9} b^{2} c \,e^{4}+\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} c a b \,d^{4} x^{4}+\frac {3}{5} b^{2} c \,x^{5} d^{4}+6 x^{6} a b c \,d^{2} e^{2}+\frac {3}{2} x^{8} a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} b^{2} c d \,e^{3}+a^{2} c \,d^{4} x^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {3}{7} x^{7} a \,b^{2} e^{4}+d^{4} a^{3} x +\frac {3}{4} x^{8} a b c \,e^{4}+\frac {3}{10} x^{10} e^{4} b \,c^{2}+\frac {1}{11} e^{4} c^{3} x^{11}+\frac {1}{2} x^{6} e^{4} a^{2} b +x^{6} b^{3} d^{2} e^{2}+\frac {4}{5} x^{5} d^{3} e \,b^{3}+x^{4} a^{3} d \,e^{3}+\frac {3}{2} a^{2} b \,d^{4} x^{2}+\frac {1}{4} b^{3} d^{4} x^{4}+\frac {1}{7} x^{7} d^{4} c^{3}+\frac {1}{8} x^{8} e^{4} b^{3}+a \,b^{2} d^{4} x^{3}+\frac {1}{5} x^{5} e^{4} a^{3}\)	\(625\)
orering	\(\frac {x \left (840 e^{4} c^{3} x^{10}+2772 b \,c^{2} e^{4} x^{9}+3696 c^{3} d \,e^{3} x^{9}+3080 a \,c^{2} e^{4} x^{8}+3080 b^{2} c \,e^{4} x^{8}+12320 b \,c^{2} d \,e^{3} x^{8}+6160 c^{3} d^{2} e^{2} x^{8}+6930 a b c \,e^{4} x^{7}+13860 a \,c^{2} d \,e^{3} x^{7}+1155 e^{4} b^{3} x^{7}+13860 b^{2} c d \,e^{3} x^{7}+20790 b \,c^{2} d^{2} e^{2} x^{7}+4620 c^{3} d^{3} e \,x^{7}+3960 a^{2} c \,e^{4} x^{6}+3960 x^{6} e^{4} a \,b^{2}+31680 a b c d \,e^{3} x^{6}+23760 a \,c^{2} d^{2} e^{2} x^{6}+5280 x^{6} d \,e^{3} b^{3}+23760 b^{2} c \,d^{2} e^{2} x^{6}+15840 b \,c^{2} d^{3} e \,x^{6}+1320 c^{3} d^{4} x^{6}+4620 x^{5} e^{4} a^{2} b +18480 a^{2} c d \,e^{3} x^{5}+18480 x^{5} d \,e^{3} a \,b^{2}+55440 a b c \,d^{2} e^{2} x^{5}+18480 a \,c^{2} d^{3} e \,x^{5}+9240 x^{5} d^{2} e^{2} b^{3}+18480 b^{2} c \,d^{3} e \,x^{5}+4620 b \,c^{2} d^{4} x^{5}+1848 x^{4} e^{4} a^{3}+22176 x^{4} d \,e^{3} a^{2} b +33264 a^{2} c \,d^{2} e^{2} x^{4}+33264 x^{4} d^{2} e^{2} a \,b^{2}+44352 a b c \,d^{3} e \,x^{4}+5544 a \,c^{2} d^{4} x^{4}+7392 x^{4} d^{3} e \,b^{3}+5544 b^{2} c \,x^{4} d^{4}+9240 x^{3} a^{3} d \,e^{3}+41580 x^{3} d^{2} e^{2} a^{2} b +27720 a^{2} c \,d^{3} e \,x^{3}+27720 x^{3} d^{3} e a \,b^{2}+13860 a b c \,d^{4} x^{3}+2310 x^{3} d^{4} b^{3}+18480 a^{3} d^{2} e^{2} x^{2}+36960 a^{2} b \,d^{3} e \,x^{2}+9240 a^{2} c \,d^{4} x^{2}+9240 a \,b^{2} d^{4} x^{2}+18480 x \,d^{3} e \,a^{3}+13860 x \,a^{2} b \,d^{4}+9240 a^{3} d^{4}\right )}{9240}\)	\(628\)
default	\(\frac {e^{4} c^{3} x^{11}}{11}+\frac {\left (3 e^{4} b \,c^{2}+4 d \,e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (6 d^{2} e^{2} c^{3}+12 d \,e^{3} b \,c^{2}+e^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (4 d^{3} e \,c^{3}+18 d^{2} e^{2} b \,c^{2}+4 d \,e^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (d^{4} c^{3}+12 b \,c^{2} d^{3} e +6 d^{2} e^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 d \,e^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{7}}{7}+\frac {\left (3 b \,c^{2} d^{4}+4 d^{3} e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+6 d^{2} e^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 d \,e^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 e^{4} a^{2} b \right ) x^{6}}{6}+\frac {\left (d^{4} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+4 d^{3} e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+6 d^{2} e^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 a^{2} d \,e^{3} b +e^{4} a^{3}\right ) x^{5}}{5}+\frac {\left (d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+4 d^{3} e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+18 d^{2} e^{2} a^{2} b +4 a^{3} d \,e^{3}\right ) x^{4}}{4}+\frac {\left (d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+12 a^{2} b \,d^{3} e +6 d^{2} e^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{3}+3 a^{2} b \,d^{4}\right ) x^{2}}{2}+d^{4} a^{3} x\)	\(631\)

Input:

int((e*x+d)^4*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/11*e^4*c^3*x^11+(3/10*e^4*b*c^2+2/5*d*e^3*c^3)*x^10+(1/3*a*c^2*e^4+1/3*b 
^2*c*e^4+4/3*d*e^3*b*c^2+2/3*d^2*e^2*c^3)*x^9+(3/4*a*b*c*e^4+3/2*a*c^2*d*e 
^3+1/8*e^4*b^3+3/2*b^2*c*d*e^3+9/4*d^2*e^2*b*c^2+1/2*d^3*e*c^3)*x^8+(3/7*e 
^4*a^2*c+3/7*a*b^2*e^4+24/7*a*b*c*d*e^3+18/7*d^2*e^2*a*c^2+4/7*b^3*d*e^3+1 
8/7*b^2*c*d^2*e^2+12/7*b*c^2*d^3*e+1/7*d^4*c^3)*x^7+(1/2*e^4*a^2*b+2*a^2*c 
*d*e^3+2*a*b^2*d*e^3+6*a*b*c*d^2*e^2+2*a*c^2*d^3*e+b^3*d^2*e^2+2*b^2*c*d^3 
*e+1/2*b*c^2*d^4)*x^6+(1/5*e^4*a^3+12/5*a^2*d*e^3*b+18/5*d^2*e^2*a^2*c+18/ 
5*a*b^2*d^2*e^2+24/5*a*b*c*d^3*e+3/5*d^4*a*c^2+4/5*d^3*e*b^3+3/5*b^2*c*d^4 
)*x^5+(a^3*d*e^3+9/2*d^2*e^2*a^2*b+3*a^2*c*d^3*e+3*a*b^2*d^3*e+3/2*a*b*c*d 
^4+1/4*b^3*d^4)*x^4+(2*a^3*d^2*e^2+4*a^2*b*d^3*e+a^2*c*d^4+a*b^2*d^4)*x^3+ 
(2*d^3*e*a^3+3/2*a^2*b*d^4)*x^2+d^4*a^3*x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + a^{2} b e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (12 \, a^{2} b d e^{3} + a^{3} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 12 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{4} + {\left (4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \] Input:

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

1/11*c^3*e^4*x^11 + 1/10*(4*c^3*d*e^3 + 3*b*c^2*e^4)*x^10 + 1/3*(2*c^3*d^2 
*e^2 + 4*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^9 + 1/8*(4*c^3*d^3*e + 18*b* 
c^2*d^2*e^2 + 12*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)*x^8 + a^3*d^ 
4*x + 1/7*(c^3*d^4 + 12*b*c^2*d^3*e + 18*(b^2*c + a*c^2)*d^2*e^2 + 4*(b^3 
+ 6*a*b*c)*d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*x^7 + 1/2*(b*c^2*d^4 + a^2*b*e^4 
 + 4*(b^2*c + a*c^2)*d^3*e + 2*(b^3 + 6*a*b*c)*d^2*e^2 + 4*(a*b^2 + a^2*c) 
*d*e^3)*x^6 + 1/5*(12*a^2*b*d*e^3 + a^3*e^4 + 3*(b^2*c + a*c^2)*d^4 + 4*(b 
^3 + 6*a*b*c)*d^3*e + 18*(a*b^2 + a^2*c)*d^2*e^2)*x^5 + 1/4*(18*a^2*b*d^2* 
e^2 + 4*a^3*d*e^3 + (b^3 + 6*a*b*c)*d^4 + 12*(a*b^2 + a^2*c)*d^3*e)*x^4 + 
(4*a^2*b*d^3*e + 2*a^3*d^2*e^2 + (a*b^2 + a^2*c)*d^4)*x^3 + 1/2*(3*a^2*b*d 
^4 + 4*a^3*d^3*e)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (258) = 516\).

Time = 0.06 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.28 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^{3} d^{4} x + \frac {c^{3} e^{4} x^{11}}{11} + x^{10} \cdot \left (\frac {3 b c^{2} e^{4}}{10} + \frac {2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac {a c^{2} e^{4}}{3} + \frac {b^{2} c e^{4}}{3} + \frac {4 b c^{2} d e^{3}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a b c e^{4}}{4} + \frac {3 a c^{2} d e^{3}}{2} + \frac {b^{3} e^{4}}{8} + \frac {3 b^{2} c d e^{3}}{2} + \frac {9 b c^{2} d^{2} e^{2}}{4} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c e^{4}}{7} + \frac {3 a b^{2} e^{4}}{7} + \frac {24 a b c d e^{3}}{7} + \frac {18 a c^{2} d^{2} e^{2}}{7} + \frac {4 b^{3} d e^{3}}{7} + \frac {18 b^{2} c d^{2} e^{2}}{7} + \frac {12 b c^{2} d^{3} e}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \left (\frac {a^{2} b e^{4}}{2} + 2 a^{2} c d e^{3} + 2 a b^{2} d e^{3} + 6 a b c d^{2} e^{2} + 2 a c^{2} d^{3} e + b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac {b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {a^{3} e^{4}}{5} + \frac {12 a^{2} b d e^{3}}{5} + \frac {18 a^{2} c d^{2} e^{2}}{5} + \frac {18 a b^{2} d^{2} e^{2}}{5} + \frac {24 a b c d^{3} e}{5} + \frac {3 a c^{2} d^{4}}{5} + \frac {4 b^{3} d^{3} e}{5} + \frac {3 b^{2} c d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + \frac {9 a^{2} b d^{2} e^{2}}{2} + 3 a^{2} c d^{3} e + 3 a b^{2} d^{3} e + \frac {3 a b c d^{4}}{2} + \frac {b^{3} d^{4}}{4}\right ) + x^{3} \cdot \left (2 a^{3} d^{2} e^{2} + 4 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) + x^{2} \cdot \left (2 a^{3} d^{3} e + \frac {3 a^{2} b d^{4}}{2}\right ) \] Input:

integrate((e*x+d)**4*(c*x**2+b*x+a)**3,x)
 

Output:

a**3*d**4*x + c**3*e**4*x**11/11 + x**10*(3*b*c**2*e**4/10 + 2*c**3*d*e**3 
/5) + x**9*(a*c**2*e**4/3 + b**2*c*e**4/3 + 4*b*c**2*d*e**3/3 + 2*c**3*d** 
2*e**2/3) + x**8*(3*a*b*c*e**4/4 + 3*a*c**2*d*e**3/2 + b**3*e**4/8 + 3*b** 
2*c*d*e**3/2 + 9*b*c**2*d**2*e**2/4 + c**3*d**3*e/2) + x**7*(3*a**2*c*e**4 
/7 + 3*a*b**2*e**4/7 + 24*a*b*c*d*e**3/7 + 18*a*c**2*d**2*e**2/7 + 4*b**3* 
d*e**3/7 + 18*b**2*c*d**2*e**2/7 + 12*b*c**2*d**3*e/7 + c**3*d**4/7) + x** 
6*(a**2*b*e**4/2 + 2*a**2*c*d*e**3 + 2*a*b**2*d*e**3 + 6*a*b*c*d**2*e**2 + 
 2*a*c**2*d**3*e + b**3*d**2*e**2 + 2*b**2*c*d**3*e + b*c**2*d**4/2) + x** 
5*(a**3*e**4/5 + 12*a**2*b*d*e**3/5 + 18*a**2*c*d**2*e**2/5 + 18*a*b**2*d* 
*2*e**2/5 + 24*a*b*c*d**3*e/5 + 3*a*c**2*d**4/5 + 4*b**3*d**3*e/5 + 3*b**2 
*c*d**4/5) + x**4*(a**3*d*e**3 + 9*a**2*b*d**2*e**2/2 + 3*a**2*c*d**3*e + 
3*a*b**2*d**3*e + 3*a*b*c*d**4/2 + b**3*d**4/4) + x**3*(2*a**3*d**2*e**2 + 
 4*a**2*b*d**3*e + a**2*c*d**4 + a*b**2*d**4) + x**2*(2*a**3*d**3*e + 3*a* 
*2*b*d**4/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + a^{2} b e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (12 \, a^{2} b d e^{3} + a^{3} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 12 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{4} + {\left (4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \] Input:

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

1/11*c^3*e^4*x^11 + 1/10*(4*c^3*d*e^3 + 3*b*c^2*e^4)*x^10 + 1/3*(2*c^3*d^2 
*e^2 + 4*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^9 + 1/8*(4*c^3*d^3*e + 18*b* 
c^2*d^2*e^2 + 12*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)*x^8 + a^3*d^ 
4*x + 1/7*(c^3*d^4 + 12*b*c^2*d^3*e + 18*(b^2*c + a*c^2)*d^2*e^2 + 4*(b^3 
+ 6*a*b*c)*d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*x^7 + 1/2*(b*c^2*d^4 + a^2*b*e^4 
 + 4*(b^2*c + a*c^2)*d^3*e + 2*(b^3 + 6*a*b*c)*d^2*e^2 + 4*(a*b^2 + a^2*c) 
*d*e^3)*x^6 + 1/5*(12*a^2*b*d*e^3 + a^3*e^4 + 3*(b^2*c + a*c^2)*d^4 + 4*(b 
^3 + 6*a*b*c)*d^3*e + 18*(a*b^2 + a^2*c)*d^2*e^2)*x^5 + 1/4*(18*a^2*b*d^2* 
e^2 + 4*a^3*d*e^3 + (b^3 + 6*a*b*c)*d^4 + 12*(a*b^2 + a^2*c)*d^3*e)*x^4 + 
(4*a^2*b*d^3*e + 2*a^3*d^2*e^2 + (a*b^2 + a^2*c)*d^4)*x^3 + 1/2*(3*a^2*b*d 
^4 + 4*a^3*d^3*e)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (258) = 516\).

Time = 0.35 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.29 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {3}{10} \, b c^{2} e^{4} x^{10} + \frac {2}{3} \, c^{3} d^{2} e^{2} x^{9} + \frac {4}{3} \, b c^{2} d e^{3} x^{9} + \frac {1}{3} \, b^{2} c e^{4} x^{9} + \frac {1}{3} \, a c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{3} d^{3} e x^{8} + \frac {9}{4} \, b c^{2} d^{2} e^{2} x^{8} + \frac {3}{2} \, b^{2} c d e^{3} x^{8} + \frac {3}{2} \, a c^{2} d e^{3} x^{8} + \frac {1}{8} \, b^{3} e^{4} x^{8} + \frac {3}{4} \, a b c e^{4} x^{8} + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {12}{7} \, b c^{2} d^{3} e x^{7} + \frac {18}{7} \, b^{2} c d^{2} e^{2} x^{7} + \frac {18}{7} \, a c^{2} d^{2} e^{2} x^{7} + \frac {4}{7} \, b^{3} d e^{3} x^{7} + \frac {24}{7} \, a b c d e^{3} x^{7} + \frac {3}{7} \, a b^{2} e^{4} x^{7} + \frac {3}{7} \, a^{2} c e^{4} x^{7} + \frac {1}{2} \, b c^{2} d^{4} x^{6} + 2 \, b^{2} c d^{3} e x^{6} + 2 \, a c^{2} d^{3} e x^{6} + b^{3} d^{2} e^{2} x^{6} + 6 \, a b c d^{2} e^{2} x^{6} + 2 \, a b^{2} d e^{3} x^{6} + 2 \, a^{2} c d e^{3} x^{6} + \frac {1}{2} \, a^{2} b e^{4} x^{6} + \frac {3}{5} \, b^{2} c d^{4} x^{5} + \frac {3}{5} \, a c^{2} d^{4} x^{5} + \frac {4}{5} \, b^{3} d^{3} e x^{5} + \frac {24}{5} \, a b c d^{3} e x^{5} + \frac {18}{5} \, a b^{2} d^{2} e^{2} x^{5} + \frac {18}{5} \, a^{2} c d^{2} e^{2} x^{5} + \frac {12}{5} \, a^{2} b d e^{3} x^{5} + \frac {1}{5} \, a^{3} e^{4} x^{5} + \frac {1}{4} \, b^{3} d^{4} x^{4} + \frac {3}{2} \, a b c d^{4} x^{4} + 3 \, a b^{2} d^{3} e x^{4} + 3 \, a^{2} c d^{3} e x^{4} + \frac {9}{2} \, a^{2} b d^{2} e^{2} x^{4} + a^{3} d e^{3} x^{4} + a b^{2} d^{4} x^{3} + a^{2} c d^{4} x^{3} + 4 \, a^{2} b d^{3} e x^{3} + 2 \, a^{3} d^{2} e^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{4} x^{2} + 2 \, a^{3} d^{3} e x^{2} + a^{3} d^{4} x \] Input:

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 3/10*b*c^2*e^4*x^10 + 2/3*c^3*d^2 
*e^2*x^9 + 4/3*b*c^2*d*e^3*x^9 + 1/3*b^2*c*e^4*x^9 + 1/3*a*c^2*e^4*x^9 + 1 
/2*c^3*d^3*e*x^8 + 9/4*b*c^2*d^2*e^2*x^8 + 3/2*b^2*c*d*e^3*x^8 + 3/2*a*c^2 
*d*e^3*x^8 + 1/8*b^3*e^4*x^8 + 3/4*a*b*c*e^4*x^8 + 1/7*c^3*d^4*x^7 + 12/7* 
b*c^2*d^3*e*x^7 + 18/7*b^2*c*d^2*e^2*x^7 + 18/7*a*c^2*d^2*e^2*x^7 + 4/7*b^ 
3*d*e^3*x^7 + 24/7*a*b*c*d*e^3*x^7 + 3/7*a*b^2*e^4*x^7 + 3/7*a^2*c*e^4*x^7 
 + 1/2*b*c^2*d^4*x^6 + 2*b^2*c*d^3*e*x^6 + 2*a*c^2*d^3*e*x^6 + b^3*d^2*e^2 
*x^6 + 6*a*b*c*d^2*e^2*x^6 + 2*a*b^2*d*e^3*x^6 + 2*a^2*c*d*e^3*x^6 + 1/2*a 
^2*b*e^4*x^6 + 3/5*b^2*c*d^4*x^5 + 3/5*a*c^2*d^4*x^5 + 4/5*b^3*d^3*e*x^5 + 
 24/5*a*b*c*d^3*e*x^5 + 18/5*a*b^2*d^2*e^2*x^5 + 18/5*a^2*c*d^2*e^2*x^5 + 
12/5*a^2*b*d*e^3*x^5 + 1/5*a^3*e^4*x^5 + 1/4*b^3*d^4*x^4 + 3/2*a*b*c*d^4*x 
^4 + 3*a*b^2*d^3*e*x^4 + 3*a^2*c*d^3*e*x^4 + 9/2*a^2*b*d^2*e^2*x^4 + a^3*d 
*e^3*x^4 + a*b^2*d^4*x^3 + a^2*c*d^4*x^3 + 4*a^2*b*d^3*e*x^3 + 2*a^3*d^2*e 
^2*x^3 + 3/2*a^2*b*d^4*x^2 + 2*a^3*d^3*e*x^2 + a^3*d^4*x
 

Mupad [B] (verification not implemented)

Time = 5.45 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.86 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=x^4\,\left (a^3\,d\,e^3+\frac {9\,a^2\,b\,d^2\,e^2}{2}+3\,c\,a^2\,d^3\,e+3\,a\,b^2\,d^3\,e+\frac {3\,c\,a\,b\,d^4}{2}+\frac {b^3\,d^4}{4}\right )+x^8\,\left (\frac {b^3\,e^4}{8}+\frac {3\,b^2\,c\,d\,e^3}{2}+\frac {9\,b\,c^2\,d^2\,e^2}{4}+\frac {3\,a\,b\,c\,e^4}{4}+\frac {c^3\,d^3\,e}{2}+\frac {3\,a\,c^2\,d\,e^3}{2}\right )+x^6\,\left (\frac {a^2\,b\,e^4}{2}+2\,a^2\,c\,d\,e^3+2\,a\,b^2\,d\,e^3+6\,a\,b\,c\,d^2\,e^2+2\,a\,c^2\,d^3\,e+b^3\,d^2\,e^2+2\,b^2\,c\,d^3\,e+\frac {b\,c^2\,d^4}{2}\right )+x^5\,\left (\frac {a^3\,e^4}{5}+\frac {12\,a^2\,b\,d\,e^3}{5}+\frac {18\,a^2\,c\,d^2\,e^2}{5}+\frac {18\,a\,b^2\,d^2\,e^2}{5}+\frac {24\,a\,b\,c\,d^3\,e}{5}+\frac {3\,a\,c^2\,d^4}{5}+\frac {4\,b^3\,d^3\,e}{5}+\frac {3\,b^2\,c\,d^4}{5}\right )+x^7\,\left (\frac {3\,a^2\,c\,e^4}{7}+\frac {3\,a\,b^2\,e^4}{7}+\frac {24\,a\,b\,c\,d\,e^3}{7}+\frac {18\,a\,c^2\,d^2\,e^2}{7}+\frac {4\,b^3\,d\,e^3}{7}+\frac {18\,b^2\,c\,d^2\,e^2}{7}+\frac {12\,b\,c^2\,d^3\,e}{7}+\frac {c^3\,d^4}{7}\right )+a^3\,d^4\,x+\frac {c^3\,e^4\,x^{11}}{11}+a\,d^2\,x^3\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {c\,e^2\,x^9\,\left (b^2\,e^2+4\,b\,c\,d\,e+2\,c^2\,d^2+a\,c\,e^2\right )}{3}+\frac {a^2\,d^3\,x^2\,\left (4\,a\,e+3\,b\,d\right )}{2}+\frac {c^2\,e^3\,x^{10}\,\left (3\,b\,e+4\,c\,d\right )}{10} \] Input:

int((d + e*x)^4*(a + b*x + c*x^2)^3,x)
 

Output:

x^4*((b^3*d^4)/4 + a^3*d*e^3 + (9*a^2*b*d^2*e^2)/2 + (3*a*b*c*d^4)/2 + 3*a 
*b^2*d^3*e + 3*a^2*c*d^3*e) + x^8*((b^3*e^4)/8 + (c^3*d^3*e)/2 + (9*b*c^2* 
d^2*e^2)/4 + (3*a*b*c*e^4)/4 + (3*a*c^2*d*e^3)/2 + (3*b^2*c*d*e^3)/2) + x^ 
6*((a^2*b*e^4)/2 + (b*c^2*d^4)/2 + b^3*d^2*e^2 + 2*a*b^2*d*e^3 + 2*a*c^2*d 
^3*e + 2*a^2*c*d*e^3 + 2*b^2*c*d^3*e + 6*a*b*c*d^2*e^2) + x^5*((a^3*e^4)/5 
 + (3*a*c^2*d^4)/5 + (3*b^2*c*d^4)/5 + (4*b^3*d^3*e)/5 + (18*a*b^2*d^2*e^2 
)/5 + (18*a^2*c*d^2*e^2)/5 + (12*a^2*b*d*e^3)/5 + (24*a*b*c*d^3*e)/5) + x^ 
7*((c^3*d^4)/7 + (3*a*b^2*e^4)/7 + (3*a^2*c*e^4)/7 + (4*b^3*d*e^3)/7 + (18 
*a*c^2*d^2*e^2)/7 + (18*b^2*c*d^2*e^2)/7 + (12*b*c^2*d^3*e)/7 + (24*a*b*c* 
d*e^3)/7) + a^3*d^4*x + (c^3*e^4*x^11)/11 + a*d^2*x^3*(2*a^2*e^2 + b^2*d^2 
 + a*c*d^2 + 4*a*b*d*e) + (c*e^2*x^9*(b^2*e^2 + 2*c^2*d^2 + a*c*e^2 + 4*b* 
c*d*e))/3 + (a^2*d^3*x^2*(4*a*e + 3*b*d))/2 + (c^2*e^3*x^10*(3*b*e + 4*c*d 
))/10
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.31 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {x \left (840 c^{3} e^{4} x^{10}+2772 b \,c^{2} e^{4} x^{9}+3696 c^{3} d \,e^{3} x^{9}+3080 a \,c^{2} e^{4} x^{8}+3080 b^{2} c \,e^{4} x^{8}+12320 b \,c^{2} d \,e^{3} x^{8}+6160 c^{3} d^{2} e^{2} x^{8}+6930 a b c \,e^{4} x^{7}+13860 a \,c^{2} d \,e^{3} x^{7}+1155 b^{3} e^{4} x^{7}+13860 b^{2} c d \,e^{3} x^{7}+20790 b \,c^{2} d^{2} e^{2} x^{7}+4620 c^{3} d^{3} e \,x^{7}+3960 a^{2} c \,e^{4} x^{6}+3960 a \,b^{2} e^{4} x^{6}+31680 a b c d \,e^{3} x^{6}+23760 a \,c^{2} d^{2} e^{2} x^{6}+5280 b^{3} d \,e^{3} x^{6}+23760 b^{2} c \,d^{2} e^{2} x^{6}+15840 b \,c^{2} d^{3} e \,x^{6}+1320 c^{3} d^{4} x^{6}+4620 a^{2} b \,e^{4} x^{5}+18480 a^{2} c d \,e^{3} x^{5}+18480 a \,b^{2} d \,e^{3} x^{5}+55440 a b c \,d^{2} e^{2} x^{5}+18480 a \,c^{2} d^{3} e \,x^{5}+9240 b^{3} d^{2} e^{2} x^{5}+18480 b^{2} c \,d^{3} e \,x^{5}+4620 b \,c^{2} d^{4} x^{5}+1848 a^{3} e^{4} x^{4}+22176 a^{2} b d \,e^{3} x^{4}+33264 a^{2} c \,d^{2} e^{2} x^{4}+33264 a \,b^{2} d^{2} e^{2} x^{4}+44352 a b c \,d^{3} e \,x^{4}+5544 a \,c^{2} d^{4} x^{4}+7392 b^{3} d^{3} e \,x^{4}+5544 b^{2} c \,d^{4} x^{4}+9240 a^{3} d \,e^{3} x^{3}+41580 a^{2} b \,d^{2} e^{2} x^{3}+27720 a^{2} c \,d^{3} e \,x^{3}+27720 a \,b^{2} d^{3} e \,x^{3}+13860 a b c \,d^{4} x^{3}+2310 b^{3} d^{4} x^{3}+18480 a^{3} d^{2} e^{2} x^{2}+36960 a^{2} b \,d^{3} e \,x^{2}+9240 a^{2} c \,d^{4} x^{2}+9240 a \,b^{2} d^{4} x^{2}+18480 a^{3} d^{3} e x +13860 a^{2} b \,d^{4} x +9240 a^{3} d^{4}\right )}{9240} \] Input:

int((e*x+d)^4*(c*x^2+b*x+a)^3,x)
 

Output:

(x*(9240*a**3*d**4 + 18480*a**3*d**3*e*x + 18480*a**3*d**2*e**2*x**2 + 924 
0*a**3*d*e**3*x**3 + 1848*a**3*e**4*x**4 + 13860*a**2*b*d**4*x + 36960*a** 
2*b*d**3*e*x**2 + 41580*a**2*b*d**2*e**2*x**3 + 22176*a**2*b*d*e**3*x**4 + 
 4620*a**2*b*e**4*x**5 + 9240*a**2*c*d**4*x**2 + 27720*a**2*c*d**3*e*x**3 
+ 33264*a**2*c*d**2*e**2*x**4 + 18480*a**2*c*d*e**3*x**5 + 3960*a**2*c*e** 
4*x**6 + 9240*a*b**2*d**4*x**2 + 27720*a*b**2*d**3*e*x**3 + 33264*a*b**2*d 
**2*e**2*x**4 + 18480*a*b**2*d*e**3*x**5 + 3960*a*b**2*e**4*x**6 + 13860*a 
*b*c*d**4*x**3 + 44352*a*b*c*d**3*e*x**4 + 55440*a*b*c*d**2*e**2*x**5 + 31 
680*a*b*c*d*e**3*x**6 + 6930*a*b*c*e**4*x**7 + 5544*a*c**2*d**4*x**4 + 184 
80*a*c**2*d**3*e*x**5 + 23760*a*c**2*d**2*e**2*x**6 + 13860*a*c**2*d*e**3* 
x**7 + 3080*a*c**2*e**4*x**8 + 2310*b**3*d**4*x**3 + 7392*b**3*d**3*e*x**4 
 + 9240*b**3*d**2*e**2*x**5 + 5280*b**3*d*e**3*x**6 + 1155*b**3*e**4*x**7 
+ 5544*b**2*c*d**4*x**4 + 18480*b**2*c*d**3*e*x**5 + 23760*b**2*c*d**2*e** 
2*x**6 + 13860*b**2*c*d*e**3*x**7 + 3080*b**2*c*e**4*x**8 + 4620*b*c**2*d* 
*4*x**5 + 15840*b*c**2*d**3*e*x**6 + 20790*b*c**2*d**2*e**2*x**7 + 12320*b 
*c**2*d*e**3*x**8 + 2772*b*c**2*e**4*x**9 + 1320*c**3*d**4*x**6 + 4620*c** 
3*d**3*e*x**7 + 6160*c**3*d**2*e**2*x**8 + 3696*c**3*d*e**3*x**9 + 840*c** 
3*e**4*x**10))/9240