\(\int (b+2 c x) (d+e x)^4 (a+b x+c x^2)^2 \, dx\) [525]

Optimal result

Integrand size = 26, antiderivative size = 240 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^6}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{3 e^6}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^7}{7 e^6}+\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{2 e^6}-\frac {5 c^2 (2 c d-b e) (d+e x)^9}{9 e^6}+\frac {c^3 (d+e x)^{10}}{5 e^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.80 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^2 b d^4 x+a d^3 \left (b^2 d+a c d+2 a b e\right ) x^2+\frac {1}{3} d^2 \left (b^3 d^2+8 a b^2 d e+8 a^2 c d e+6 a b \left (c d^2+a e^2\right )\right ) x^3+d \left (b^3 d^2 e+a b e \left (6 c d^2+a e^2\right )+a c d \left (c d^2+3 a e^2\right )+b^2 \left (c d^3+3 a d e^2\right )\right ) x^4+\frac {1}{5} \left (6 b^3 d^2 e^2+8 a c d e \left (2 c d^2+a e^2\right )+8 b^2 \left (2 c d^3 e+a d e^3\right )+b \left (5 c^2 d^4+36 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{3} \left (c^3 d^4+b^2 e^3 (2 b d+a e)+2 c^2 d^2 e (5 b d+6 a e)+c e^2 \left (12 b^2 d^2+12 a b d e+a^2 e^2\right )\right ) x^6+\frac {1}{7} e \left (8 c^3 d^3+b^3 e^3+2 b c e^2 (8 b d+3 a e)+2 c^2 d e (15 b d+8 a e)\right ) x^7+\frac {1}{2} c e^2 \left (3 c^2 d^2+b^2 e^2+c e (5 b d+a e)\right ) x^8+\frac {1}{9} c^2 e^3 (8 c d+5 b e) x^9+\frac {1}{5} c^3 e^4 x^{10} \] Input:

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

Output:

a^2*b*d^4*x + a*d^3*(b^2*d + a*c*d + 2*a*b*e)*x^2 + (d^2*(b^3*d^2 + 8*a*b^ 
2*d*e + 8*a^2*c*d*e + 6*a*b*(c*d^2 + a*e^2))*x^3)/3 + d*(b^3*d^2*e + a*b*e 
*(6*c*d^2 + a*e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*x^ 
4 + ((6*b^3*d^2*e^2 + 8*a*c*d*e*(2*c*d^2 + a*e^2) + 8*b^2*(2*c*d^3*e + a*d 
*e^3) + b*(5*c^2*d^4 + 36*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + ((c^3*d^4 + b^2 
*e^3*(2*b*d + a*e) + 2*c^2*d^2*e*(5*b*d + 6*a*e) + c*e^2*(12*b^2*d^2 + 12* 
a*b*d*e + a^2*e^2))*x^6)/3 + (e*(8*c^3*d^3 + b^3*e^3 + 2*b*c*e^2*(8*b*d + 
3*a*e) + 2*c^2*d*e*(15*b*d + 8*a*e))*x^7)/7 + (c*e^2*(3*c^2*d^2 + b^2*e^2 
+ c*e*(5*b*d + a*e))*x^8)/2 + (c^2*e^3*(8*c*d + 5*b*e)*x^9)/9 + (c^3*e^4*x 
^10)/5
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 240, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1195

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]
 

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. \(472\) vs. \(2(228)=456\).

Time = 1.31 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.97

method	result	size
norman	\(\frac {c^{3} e^{4} x^{10}}{5}+\left (\frac {5}{9} c^{2} e^{4} b +\frac {8}{9} c^{3} d \,e^{3}\right ) x^{9}+\left (\frac {1}{2} a \,c^{2} e^{4}+\frac {1}{2} b^{2} c \,e^{4}+\frac {5}{2} d \,e^{3} b \,c^{2}+\frac {3}{2} d^{2} e^{2} c^{3}\right ) x^{8}+\left (\frac {6}{7} a b c \,e^{4}+\frac {16}{7} a \,c^{2} d \,e^{3}+\frac {1}{7} e^{4} b^{3}+\frac {16}{7} b^{2} c d \,e^{3}+\frac {30}{7} d^{2} e^{2} b \,c^{2}+\frac {8}{7} d^{3} e \,c^{3}\right ) x^{7}+\left (\frac {1}{3} e^{4} a^{2} c +\frac {1}{3} a \,b^{2} e^{4}+4 a b c d \,e^{3}+4 d^{2} e^{2} a \,c^{2}+\frac {2}{3} b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+\frac {10}{3} b \,c^{2} d^{3} e +\frac {1}{3} d^{4} c^{3}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{2} b +\frac {8}{5} a^{2} c d \,e^{3}+\frac {8}{5} a \,b^{2} d \,e^{3}+\frac {36}{5} a b c \,d^{2} e^{2}+\frac {16}{5} a \,c^{2} d^{3} e +\frac {6}{5} d^{2} e^{2} b^{3}+\frac {16}{5} b^{2} c \,d^{3} e +b \,c^{2} d^{4}\right ) x^{5}+\left (a^{2} d \,e^{3} b +3 d^{2} e^{2} a^{2} c +3 a \,b^{2} d^{2} e^{2}+6 a b c \,d^{3} e +d^{4} a \,c^{2}+d^{3} e \,b^{3}+b^{2} c \,d^{4}\right ) x^{4}+\left (2 d^{2} e^{2} a^{2} b +\frac {8}{3} a^{2} c \,d^{3} e +\frac {8}{3} a \,b^{2} d^{3} e +2 a b c \,d^{4}+\frac {1}{3} b^{3} d^{4}\right ) x^{3}+\left (2 a^{2} b \,d^{3} e +a^{2} c \,d^{4}+d^{4} a \,b^{2}\right ) x^{2}+x \,a^{2} b \,d^{4}\)	\(473\)
default	\(\frac {c^{3} e^{4} x^{10}}{5}+\frac {\left (\left (b \,e^{4}+8 c d \,e^{3}\right ) c^{2}+4 c^{2} e^{4} b \right ) x^{9}}{9}+\frac {\left (\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) c^{2}+2 \left (b \,e^{4}+8 c d \,e^{3}\right ) b c +2 c \,e^{4} \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c^{2}+2 \left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) b c +\left (b \,e^{4}+8 c d \,e^{3}\right ) \left (2 a c +b^{2}\right )+4 a b c \,e^{4}\right ) x^{7}}{7}+\frac {\left (\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c^{2}+2 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b c +\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) \left (2 a c +b^{2}\right )+2 \left (b \,e^{4}+8 c d \,e^{3}\right ) a b +2 e^{4} a^{2} c \right ) x^{6}}{6}+\frac {\left (b \,c^{2} d^{4}+2 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) b c +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) \left (2 a c +b^{2}\right )+2 \left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) a b +\left (b \,e^{4}+8 c d \,e^{3}\right ) a^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{2} c \,d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) \left (2 a c +b^{2}\right )+2 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a b +\left (4 b d \,e^{3}+12 d^{2} e^{2} c \right ) a^{2}\right ) x^{4}}{4}+\frac {\left (b \,d^{4} \left (2 a c +b^{2}\right )+2 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) a b +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 d^{4} a \,b^{2}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a^{2}\right ) x^{2}}{2}+x \,a^{2} b \,d^{4}\)	\(554\)
gosper	\(\frac {6}{5} x^{5} d^{2} e^{2} b^{3}+x^{4} d^{3} e \,b^{3}+2 a^{2} b \,d^{3} e \,x^{2}+\frac {8}{3} x^{3} d^{3} e a \,b^{2}+3 x^{4} d^{2} e^{2} a \,b^{2}+2 x^{3} d^{2} e^{2} a^{2} b +\frac {8}{5} x^{5} d \,e^{3} a \,b^{2}+x^{4} d \,e^{3} a^{2} b +\frac {8}{7} x^{7} d^{3} e \,c^{3}+2 a b c \,d^{4} x^{3}+\frac {1}{2} x^{8} b^{2} c \,e^{4}+a \,c^{2} d^{4} x^{4}+\frac {1}{3} x^{6} e^{4} a \,b^{2}+\frac {5}{2} x^{8} d \,e^{3} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{4}+\frac {16}{7} x^{7} a \,c^{2} d \,e^{3}+\frac {16}{7} x^{7} b^{2} c d \,e^{3}+x^{5} b \,c^{2} d^{4}+\frac {30}{7} x^{7} d^{2} e^{2} b \,c^{2}+4 x^{6} d^{2} e^{2} a \,c^{2}+4 x^{6} b^{2} c \,d^{2} e^{2}+\frac {10}{3} x^{6} b \,c^{2} d^{3} e +\frac {8}{5} x^{5} a^{2} c d \,e^{3}+\frac {16}{5} x^{5} a \,c^{2} d^{3} e +\frac {16}{5} x^{5} b^{2} c \,d^{3} e +\frac {8}{3} x^{3} a^{2} c \,d^{3} e +a \,b^{2} d^{4} x^{2}+\frac {1}{2} x^{8} a \,c^{2} e^{4}+4 x^{6} a b c d \,e^{3}+\frac {8}{9} x^{9} c^{3} d \,e^{3}+\frac {5}{9} x^{9} c^{2} e^{4} b +a^{2} c \,d^{4} x^{2}+6 a b c \,d^{3} e \,x^{4}+x \,a^{2} b \,d^{4}+\frac {3}{2} x^{8} d^{2} e^{2} c^{3}+\frac {36}{5} x^{5} a b c \,d^{2} e^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{3}+\frac {1}{5} x^{5} e^{4} a^{2} b +3 a^{2} c \,d^{2} e^{2} x^{4}+\frac {1}{3} x^{6} e^{4} a^{2} c +\frac {1}{7} e^{4} b^{3} x^{7}+\frac {1}{3} x^{3} d^{4} b^{3}+\frac {1}{5} c^{3} e^{4} x^{10}+b^{2} c \,x^{4} d^{4}+\frac {1}{3} x^{6} d^{4} c^{3}\)	\(562\)
risch	\(\frac {6}{5} x^{5} d^{2} e^{2} b^{3}+x^{4} d^{3} e \,b^{3}+2 a^{2} b \,d^{3} e \,x^{2}+\frac {8}{3} x^{3} d^{3} e a \,b^{2}+3 x^{4} d^{2} e^{2} a \,b^{2}+2 x^{3} d^{2} e^{2} a^{2} b +\frac {8}{5} x^{5} d \,e^{3} a \,b^{2}+x^{4} d \,e^{3} a^{2} b +\frac {8}{7} x^{7} d^{3} e \,c^{3}+2 a b c \,d^{4} x^{3}+\frac {1}{2} x^{8} b^{2} c \,e^{4}+a \,c^{2} d^{4} x^{4}+\frac {1}{3} x^{6} e^{4} a \,b^{2}+\frac {5}{2} x^{8} d \,e^{3} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{4}+\frac {16}{7} x^{7} a \,c^{2} d \,e^{3}+\frac {16}{7} x^{7} b^{2} c d \,e^{3}+x^{5} b \,c^{2} d^{4}+\frac {30}{7} x^{7} d^{2} e^{2} b \,c^{2}+4 x^{6} d^{2} e^{2} a \,c^{2}+4 x^{6} b^{2} c \,d^{2} e^{2}+\frac {10}{3} x^{6} b \,c^{2} d^{3} e +\frac {8}{5} x^{5} a^{2} c d \,e^{3}+\frac {16}{5} x^{5} a \,c^{2} d^{3} e +\frac {16}{5} x^{5} b^{2} c \,d^{3} e +\frac {8}{3} x^{3} a^{2} c \,d^{3} e +a \,b^{2} d^{4} x^{2}+\frac {1}{2} x^{8} a \,c^{2} e^{4}+4 x^{6} a b c d \,e^{3}+\frac {8}{9} x^{9} c^{3} d \,e^{3}+\frac {5}{9} x^{9} c^{2} e^{4} b +a^{2} c \,d^{4} x^{2}+6 a b c \,d^{3} e \,x^{4}+x \,a^{2} b \,d^{4}+\frac {3}{2} x^{8} d^{2} e^{2} c^{3}+\frac {36}{5} x^{5} a b c \,d^{2} e^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{3}+\frac {1}{5} x^{5} e^{4} a^{2} b +3 a^{2} c \,d^{2} e^{2} x^{4}+\frac {1}{3} x^{6} e^{4} a^{2} c +\frac {1}{7} e^{4} b^{3} x^{7}+\frac {1}{3} x^{3} d^{4} b^{3}+\frac {1}{5} c^{3} e^{4} x^{10}+b^{2} c \,x^{4} d^{4}+\frac {1}{3} x^{6} d^{4} c^{3}\)	\(562\)
parallelrisch	\(\frac {6}{5} x^{5} d^{2} e^{2} b^{3}+x^{4} d^{3} e \,b^{3}+2 a^{2} b \,d^{3} e \,x^{2}+\frac {8}{3} x^{3} d^{3} e a \,b^{2}+3 x^{4} d^{2} e^{2} a \,b^{2}+2 x^{3} d^{2} e^{2} a^{2} b +\frac {8}{5} x^{5} d \,e^{3} a \,b^{2}+x^{4} d \,e^{3} a^{2} b +\frac {8}{7} x^{7} d^{3} e \,c^{3}+2 a b c \,d^{4} x^{3}+\frac {1}{2} x^{8} b^{2} c \,e^{4}+a \,c^{2} d^{4} x^{4}+\frac {1}{3} x^{6} e^{4} a \,b^{2}+\frac {5}{2} x^{8} d \,e^{3} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{4}+\frac {16}{7} x^{7} a \,c^{2} d \,e^{3}+\frac {16}{7} x^{7} b^{2} c d \,e^{3}+x^{5} b \,c^{2} d^{4}+\frac {30}{7} x^{7} d^{2} e^{2} b \,c^{2}+4 x^{6} d^{2} e^{2} a \,c^{2}+4 x^{6} b^{2} c \,d^{2} e^{2}+\frac {10}{3} x^{6} b \,c^{2} d^{3} e +\frac {8}{5} x^{5} a^{2} c d \,e^{3}+\frac {16}{5} x^{5} a \,c^{2} d^{3} e +\frac {16}{5} x^{5} b^{2} c \,d^{3} e +\frac {8}{3} x^{3} a^{2} c \,d^{3} e +a \,b^{2} d^{4} x^{2}+\frac {1}{2} x^{8} a \,c^{2} e^{4}+4 x^{6} a b c d \,e^{3}+\frac {8}{9} x^{9} c^{3} d \,e^{3}+\frac {5}{9} x^{9} c^{2} e^{4} b +a^{2} c \,d^{4} x^{2}+6 a b c \,d^{3} e \,x^{4}+x \,a^{2} b \,d^{4}+\frac {3}{2} x^{8} d^{2} e^{2} c^{3}+\frac {36}{5} x^{5} a b c \,d^{2} e^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{3}+\frac {1}{5} x^{5} e^{4} a^{2} b +3 a^{2} c \,d^{2} e^{2} x^{4}+\frac {1}{3} x^{6} e^{4} a^{2} c +\frac {1}{7} e^{4} b^{3} x^{7}+\frac {1}{3} x^{3} d^{4} b^{3}+\frac {1}{5} c^{3} e^{4} x^{10}+b^{2} c \,x^{4} d^{4}+\frac {1}{3} x^{6} d^{4} c^{3}\)	\(562\)
orering	\(\frac {x \left (126 c^{3} e^{4} x^{9}+350 b \,c^{2} e^{4} x^{8}+560 c^{3} d \,e^{3} x^{8}+315 a \,c^{2} e^{4} x^{7}+315 b^{2} c \,e^{4} x^{7}+1575 b \,c^{2} d \,e^{3} x^{7}+945 c^{3} d^{2} e^{2} x^{7}+540 a b c \,e^{4} x^{6}+1440 a \,c^{2} d \,e^{3} x^{6}+90 b^{3} e^{4} x^{6}+1440 b^{2} c d \,e^{3} x^{6}+2700 b \,c^{2} d^{2} e^{2} x^{6}+720 c^{3} d^{3} e \,x^{6}+210 a^{2} c \,e^{4} x^{5}+210 a \,b^{2} e^{4} x^{5}+2520 a b c d \,e^{3} x^{5}+2520 a \,c^{2} d^{2} e^{2} x^{5}+420 b^{3} d \,e^{3} x^{5}+2520 b^{2} c \,d^{2} e^{2} x^{5}+2100 b \,c^{2} d^{3} e \,x^{5}+210 c^{3} d^{4} x^{5}+126 a^{2} b \,e^{4} x^{4}+1008 a^{2} c d \,e^{3} x^{4}+1008 a \,b^{2} d \,e^{3} x^{4}+4536 a b c \,d^{2} e^{2} x^{4}+2016 a \,c^{2} d^{3} e \,x^{4}+756 b^{3} d^{2} e^{2} x^{4}+2016 b^{2} c \,d^{3} e \,x^{4}+630 b \,c^{2} d^{4} x^{4}+630 a^{2} b d \,e^{3} x^{3}+1890 a^{2} c \,d^{2} e^{2} x^{3}+1890 a \,b^{2} d^{2} e^{2} x^{3}+3780 a b c \,d^{3} e \,x^{3}+630 a \,c^{2} d^{4} x^{3}+630 b^{3} d^{3} e \,x^{3}+630 b^{2} c \,d^{4} x^{3}+1260 a^{2} b \,d^{2} e^{2} x^{2}+1680 a^{2} c \,d^{3} e \,x^{2}+1680 a \,b^{2} d^{3} e \,x^{2}+1260 a b c \,d^{4} x^{2}+210 b^{3} d^{4} x^{2}+1260 a^{2} b \,d^{3} e x +630 a^{2} c \,d^{4} x +630 a \,b^{2} d^{4} x +630 a^{2} b \,d^{4}\right )}{630}\)	\(566\)

Input:

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

Output:

1/5*c^3*e^4*x^10+(5/9*c^2*e^4*b+8/9*c^3*d*e^3)*x^9+(1/2*a*c^2*e^4+1/2*b^2* 
c*e^4+5/2*d*e^3*b*c^2+3/2*d^2*e^2*c^3)*x^8+(6/7*a*b*c*e^4+16/7*a*c^2*d*e^3 
+1/7*e^4*b^3+16/7*b^2*c*d*e^3+30/7*d^2*e^2*b*c^2+8/7*d^3*e*c^3)*x^7+(1/3*e 
^4*a^2*c+1/3*a*b^2*e^4+4*a*b*c*d*e^3+4*d^2*e^2*a*c^2+2/3*b^3*d*e^3+4*b^2*c 
*d^2*e^2+10/3*b*c^2*d^3*e+1/3*d^4*c^3)*x^6+(1/5*e^4*a^2*b+8/5*a^2*c*d*e^3+ 
8/5*a*b^2*d*e^3+36/5*a*b*c*d^2*e^2+16/5*a*c^2*d^3*e+6/5*d^2*e^2*b^3+16/5*b 
^2*c*d^3*e+b*c^2*d^4)*x^5+(a^2*b*d*e^3+3*a^2*c*d^2*e^2+3*a*b^2*d^2*e^2+6*a 
*b*c*d^3*e+a*c^2*d^4+b^3*d^3*e+b^2*c*d^4)*x^4+(2*d^2*e^2*a^2*b+8/3*a^2*c*d 
^3*e+8/3*a*b^2*d^3*e+2*a*b*c*d^4+1/3*b^3*d^4)*x^3+(2*a^2*b*d^3*e+a^2*c*d^4 
+a*b^2*d^4)*x^2+x*a^2*b*d^4
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.79 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{3} e^{4} x^{10} + \frac {1}{9} \, {\left (8 \, c^{3} d e^{3} + 5 \, b c^{2} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (3 \, c^{3} d^{2} e^{2} + 5 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{8} + a^{2} b d^{4} x + \frac {1}{7} \, {\left (8 \, c^{3} d^{3} e + 30 \, b c^{2} d^{2} e^{2} + 16 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{7} + \frac {1}{3} \, {\left (c^{3} d^{4} + 10 \, b c^{2} d^{3} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{4} + a^{2} b e^{4} + 16 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 6 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 8 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{5} + {\left (a^{2} b d e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{4} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d^{2} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 8 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{3} + {\left (2 \, a^{2} b d^{3} e + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{2} \] Input:

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

Output:

1/5*c^3*e^4*x^10 + 1/9*(8*c^3*d*e^3 + 5*b*c^2*e^4)*x^9 + 1/2*(3*c^3*d^2*e^ 
2 + 5*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^8 + a^2*b*d^4*x + 1/7*(8*c^3*d^ 
3*e + 30*b*c^2*d^2*e^2 + 16*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)*x 
^7 + 1/3*(c^3*d^4 + 10*b*c^2*d^3*e + 12*(b^2*c + a*c^2)*d^2*e^2 + 2*(b^3 + 
 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*x^6 + 1/5*(5*b*c^2*d^4 + a^2*b*e^4 
+ 16*(b^2*c + a*c^2)*d^3*e + 6*(b^3 + 6*a*b*c)*d^2*e^2 + 8*(a*b^2 + a^2*c) 
*d*e^3)*x^5 + (a^2*b*d*e^3 + (b^2*c + a*c^2)*d^4 + (b^3 + 6*a*b*c)*d^3*e + 
 3*(a*b^2 + a^2*c)*d^2*e^2)*x^4 + 1/3*(6*a^2*b*d^2*e^2 + (b^3 + 6*a*b*c)*d 
^4 + 8*(a*b^2 + a^2*c)*d^3*e)*x^3 + (2*a^2*b*d^3*e + (a*b^2 + a^2*c)*d^4)* 
x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (230) = 460\).

Time = 0.06 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.30 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b d^{4} x + \frac {c^{3} e^{4} x^{10}}{5} + x^{9} \cdot \left (\frac {5 b c^{2} e^{4}}{9} + \frac {8 c^{3} d e^{3}}{9}\right ) + x^{8} \left (\frac {a c^{2} e^{4}}{2} + \frac {b^{2} c e^{4}}{2} + \frac {5 b c^{2} d e^{3}}{2} + \frac {3 c^{3} d^{2} e^{2}}{2}\right ) + x^{7} \cdot \left (\frac {6 a b c e^{4}}{7} + \frac {16 a c^{2} d e^{3}}{7} + \frac {b^{3} e^{4}}{7} + \frac {16 b^{2} c d e^{3}}{7} + \frac {30 b c^{2} d^{2} e^{2}}{7} + \frac {8 c^{3} d^{3} e}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{4}}{3} + \frac {a b^{2} e^{4}}{3} + 4 a b c d e^{3} + 4 a c^{2} d^{2} e^{2} + \frac {2 b^{3} d e^{3}}{3} + 4 b^{2} c d^{2} e^{2} + \frac {10 b c^{2} d^{3} e}{3} + \frac {c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac {a^{2} b e^{4}}{5} + \frac {8 a^{2} c d e^{3}}{5} + \frac {8 a b^{2} d e^{3}}{5} + \frac {36 a b c d^{2} e^{2}}{5} + \frac {16 a c^{2} d^{3} e}{5} + \frac {6 b^{3} d^{2} e^{2}}{5} + \frac {16 b^{2} c d^{3} e}{5} + b c^{2} d^{4}\right ) + x^{4} \left (a^{2} b d e^{3} + 3 a^{2} c d^{2} e^{2} + 3 a b^{2} d^{2} e^{2} + 6 a b c d^{3} e + a c^{2} d^{4} + b^{3} d^{3} e + b^{2} c d^{4}\right ) + x^{3} \cdot \left (2 a^{2} b d^{2} e^{2} + \frac {8 a^{2} c d^{3} e}{3} + \frac {8 a b^{2} d^{3} e}{3} + 2 a b c d^{4} + \frac {b^{3} d^{4}}{3}\right ) + x^{2} \cdot \left (2 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) \] Input:

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

Output:

a**2*b*d**4*x + c**3*e**4*x**10/5 + x**9*(5*b*c**2*e**4/9 + 8*c**3*d*e**3/ 
9) + x**8*(a*c**2*e**4/2 + b**2*c*e**4/2 + 5*b*c**2*d*e**3/2 + 3*c**3*d**2 
*e**2/2) + x**7*(6*a*b*c*e**4/7 + 16*a*c**2*d*e**3/7 + b**3*e**4/7 + 16*b* 
*2*c*d*e**3/7 + 30*b*c**2*d**2*e**2/7 + 8*c**3*d**3*e/7) + x**6*(a**2*c*e* 
*4/3 + a*b**2*e**4/3 + 4*a*b*c*d*e**3 + 4*a*c**2*d**2*e**2 + 2*b**3*d*e**3 
/3 + 4*b**2*c*d**2*e**2 + 10*b*c**2*d**3*e/3 + c**3*d**4/3) + x**5*(a**2*b 
*e**4/5 + 8*a**2*c*d*e**3/5 + 8*a*b**2*d*e**3/5 + 36*a*b*c*d**2*e**2/5 + 1 
6*a*c**2*d**3*e/5 + 6*b**3*d**2*e**2/5 + 16*b**2*c*d**3*e/5 + b*c**2*d**4) 
 + x**4*(a**2*b*d*e**3 + 3*a**2*c*d**2*e**2 + 3*a*b**2*d**2*e**2 + 6*a*b*c 
*d**3*e + a*c**2*d**4 + b**3*d**3*e + b**2*c*d**4) + x**3*(2*a**2*b*d**2*e 
**2 + 8*a**2*c*d**3*e/3 + 8*a*b**2*d**3*e/3 + 2*a*b*c*d**4 + b**3*d**4/3) 
+ x**2*(2*a**2*b*d**3*e + a**2*c*d**4 + a*b**2*d**4)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.79 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{3} e^{4} x^{10} + \frac {1}{9} \, {\left (8 \, c^{3} d e^{3} + 5 \, b c^{2} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (3 \, c^{3} d^{2} e^{2} + 5 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{8} + a^{2} b d^{4} x + \frac {1}{7} \, {\left (8 \, c^{3} d^{3} e + 30 \, b c^{2} d^{2} e^{2} + 16 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{7} + \frac {1}{3} \, {\left (c^{3} d^{4} + 10 \, b c^{2} d^{3} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{4} + a^{2} b e^{4} + 16 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 6 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 8 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{5} + {\left (a^{2} b d e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{4} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d^{2} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 8 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{3} + {\left (2 \, a^{2} b d^{3} e + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{2} \] Input:

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

Output:

1/5*c^3*e^4*x^10 + 1/9*(8*c^3*d*e^3 + 5*b*c^2*e^4)*x^9 + 1/2*(3*c^3*d^2*e^ 
2 + 5*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^8 + a^2*b*d^4*x + 1/7*(8*c^3*d^ 
3*e + 30*b*c^2*d^2*e^2 + 16*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)*x 
^7 + 1/3*(c^3*d^4 + 10*b*c^2*d^3*e + 12*(b^2*c + a*c^2)*d^2*e^2 + 2*(b^3 + 
 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*x^6 + 1/5*(5*b*c^2*d^4 + a^2*b*e^4 
+ 16*(b^2*c + a*c^2)*d^3*e + 6*(b^3 + 6*a*b*c)*d^2*e^2 + 8*(a*b^2 + a^2*c) 
*d*e^3)*x^5 + (a^2*b*d*e^3 + (b^2*c + a*c^2)*d^4 + (b^3 + 6*a*b*c)*d^3*e + 
 3*(a*b^2 + a^2*c)*d^2*e^2)*x^4 + 1/3*(6*a^2*b*d^2*e^2 + (b^3 + 6*a*b*c)*d 
^4 + 8*(a*b^2 + a^2*c)*d^3*e)*x^3 + (2*a^2*b*d^3*e + (a*b^2 + a^2*c)*d^4)* 
x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (228) = 456\).

Time = 0.21 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.34 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{5} \, c^{3} e^{4} x^{10} + \frac {8}{9} \, c^{3} d e^{3} x^{9} + \frac {5}{9} \, b c^{2} e^{4} x^{9} + \frac {3}{2} \, c^{3} d^{2} e^{2} x^{8} + \frac {5}{2} \, b c^{2} d e^{3} x^{8} + \frac {1}{2} \, b^{2} c e^{4} x^{8} + \frac {1}{2} \, a c^{2} e^{4} x^{8} + \frac {8}{7} \, c^{3} d^{3} e x^{7} + \frac {30}{7} \, b c^{2} d^{2} e^{2} x^{7} + \frac {16}{7} \, b^{2} c d e^{3} x^{7} + \frac {16}{7} \, a c^{2} d e^{3} x^{7} + \frac {1}{7} \, b^{3} e^{4} x^{7} + \frac {6}{7} \, a b c e^{4} x^{7} + \frac {1}{3} \, c^{3} d^{4} x^{6} + \frac {10}{3} \, b c^{2} d^{3} e x^{6} + 4 \, b^{2} c d^{2} e^{2} x^{6} + 4 \, a c^{2} d^{2} e^{2} x^{6} + \frac {2}{3} \, b^{3} d e^{3} x^{6} + 4 \, a b c d e^{3} x^{6} + \frac {1}{3} \, a b^{2} e^{4} x^{6} + \frac {1}{3} \, a^{2} c e^{4} x^{6} + b c^{2} d^{4} x^{5} + \frac {16}{5} \, b^{2} c d^{3} e x^{5} + \frac {16}{5} \, a c^{2} d^{3} e x^{5} + \frac {6}{5} \, b^{3} d^{2} e^{2} x^{5} + \frac {36}{5} \, a b c d^{2} e^{2} x^{5} + \frac {8}{5} \, a b^{2} d e^{3} x^{5} + \frac {8}{5} \, a^{2} c d e^{3} x^{5} + \frac {1}{5} \, a^{2} b e^{4} x^{5} + b^{2} c d^{4} x^{4} + a c^{2} d^{4} x^{4} + b^{3} d^{3} e x^{4} + 6 \, a b c d^{3} e x^{4} + 3 \, a b^{2} d^{2} e^{2} x^{4} + 3 \, a^{2} c d^{2} e^{2} x^{4} + a^{2} b d e^{3} x^{4} + \frac {1}{3} \, b^{3} d^{4} x^{3} + 2 \, a b c d^{4} x^{3} + \frac {8}{3} \, a b^{2} d^{3} e x^{3} + \frac {8}{3} \, a^{2} c d^{3} e x^{3} + 2 \, a^{2} b d^{2} e^{2} x^{3} + a b^{2} d^{4} x^{2} + a^{2} c d^{4} x^{2} + 2 \, a^{2} b d^{3} e x^{2} + a^{2} b d^{4} x \] Input:

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

Output:

1/5*c^3*e^4*x^10 + 8/9*c^3*d*e^3*x^9 + 5/9*b*c^2*e^4*x^9 + 3/2*c^3*d^2*e^2 
*x^8 + 5/2*b*c^2*d*e^3*x^8 + 1/2*b^2*c*e^4*x^8 + 1/2*a*c^2*e^4*x^8 + 8/7*c 
^3*d^3*e*x^7 + 30/7*b*c^2*d^2*e^2*x^7 + 16/7*b^2*c*d*e^3*x^7 + 16/7*a*c^2* 
d*e^3*x^7 + 1/7*b^3*e^4*x^7 + 6/7*a*b*c*e^4*x^7 + 1/3*c^3*d^4*x^6 + 10/3*b 
*c^2*d^3*e*x^6 + 4*b^2*c*d^2*e^2*x^6 + 4*a*c^2*d^2*e^2*x^6 + 2/3*b^3*d*e^3 
*x^6 + 4*a*b*c*d*e^3*x^6 + 1/3*a*b^2*e^4*x^6 + 1/3*a^2*c*e^4*x^6 + b*c^2*d 
^4*x^5 + 16/5*b^2*c*d^3*e*x^5 + 16/5*a*c^2*d^3*e*x^5 + 6/5*b^3*d^2*e^2*x^5 
 + 36/5*a*b*c*d^2*e^2*x^5 + 8/5*a*b^2*d*e^3*x^5 + 8/5*a^2*c*d*e^3*x^5 + 1/ 
5*a^2*b*e^4*x^5 + b^2*c*d^4*x^4 + a*c^2*d^4*x^4 + b^3*d^3*e*x^4 + 6*a*b*c* 
d^3*e*x^4 + 3*a*b^2*d^2*e^2*x^4 + 3*a^2*c*d^2*e^2*x^4 + a^2*b*d*e^3*x^4 + 
1/3*b^3*d^4*x^3 + 2*a*b*c*d^4*x^3 + 8/3*a*b^2*d^3*e*x^3 + 8/3*a^2*c*d^3*e* 
x^3 + 2*a^2*b*d^2*e^2*x^3 + a*b^2*d^4*x^2 + a^2*c*d^4*x^2 + 2*a^2*b*d^3*e* 
x^2 + a^2*b*d^4*x
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.89 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (2\,a^2\,b\,d^2\,e^2+\frac {8\,c\,a^2\,d^3\,e}{3}+\frac {8\,a\,b^2\,d^3\,e}{3}+2\,c\,a\,b\,d^4+\frac {b^3\,d^4}{3}\right )+x^7\,\left (\frac {b^3\,e^4}{7}+\frac {16\,b^2\,c\,d\,e^3}{7}+\frac {30\,b\,c^2\,d^2\,e^2}{7}+\frac {6\,a\,b\,c\,e^4}{7}+\frac {8\,c^3\,d^3\,e}{7}+\frac {16\,a\,c^2\,d\,e^3}{7}\right )+x^5\,\left (\frac {a^2\,b\,e^4}{5}+\frac {8\,a^2\,c\,d\,e^3}{5}+\frac {8\,a\,b^2\,d\,e^3}{5}+\frac {36\,a\,b\,c\,d^2\,e^2}{5}+\frac {16\,a\,c^2\,d^3\,e}{5}+\frac {6\,b^3\,d^2\,e^2}{5}+\frac {16\,b^2\,c\,d^3\,e}{5}+b\,c^2\,d^4\right )+x^6\,\left (\frac {a^2\,c\,e^4}{3}+\frac {a\,b^2\,e^4}{3}+4\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+\frac {2\,b^3\,d\,e^3}{3}+4\,b^2\,c\,d^2\,e^2+\frac {10\,b\,c^2\,d^3\,e}{3}+\frac {c^3\,d^4}{3}\right )+x^4\,\left (a^2\,b\,d\,e^3+3\,a^2\,c\,d^2\,e^2+3\,a\,b^2\,d^2\,e^2+6\,a\,b\,c\,d^3\,e+a\,c^2\,d^4+b^3\,d^3\,e+b^2\,c\,d^4\right )+\frac {c^3\,e^4\,x^{10}}{5}+a\,d^3\,x^2\,\left (d\,b^2+2\,a\,e\,b+a\,c\,d\right )+\frac {c\,e^2\,x^8\,\left (b^2\,e^2+5\,b\,c\,d\,e+3\,c^2\,d^2+a\,c\,e^2\right )}{2}+\frac {c^2\,e^3\,x^9\,\left (5\,b\,e+8\,c\,d\right )}{9}+a^2\,b\,d^4\,x \] Input:

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

Output:

x^3*((b^3*d^4)/3 + 2*a^2*b*d^2*e^2 + 2*a*b*c*d^4 + (8*a*b^2*d^3*e)/3 + (8* 
a^2*c*d^3*e)/3) + x^7*((b^3*e^4)/7 + (8*c^3*d^3*e)/7 + (30*b*c^2*d^2*e^2)/ 
7 + (6*a*b*c*e^4)/7 + (16*a*c^2*d*e^3)/7 + (16*b^2*c*d*e^3)/7) + x^5*((a^2 
*b*e^4)/5 + b*c^2*d^4 + (6*b^3*d^2*e^2)/5 + (8*a*b^2*d*e^3)/5 + (16*a*c^2* 
d^3*e)/5 + (8*a^2*c*d*e^3)/5 + (16*b^2*c*d^3*e)/5 + (36*a*b*c*d^2*e^2)/5) 
+ x^6*((c^3*d^4)/3 + (a*b^2*e^4)/3 + (a^2*c*e^4)/3 + (2*b^3*d*e^3)/3 + 4*a 
*c^2*d^2*e^2 + 4*b^2*c*d^2*e^2 + (10*b*c^2*d^3*e)/3 + 4*a*b*c*d*e^3) + x^4 
*(a*c^2*d^4 + b^2*c*d^4 + b^3*d^3*e + 3*a*b^2*d^2*e^2 + 3*a^2*c*d^2*e^2 + 
a^2*b*d*e^3 + 6*a*b*c*d^3*e) + (c^3*e^4*x^10)/5 + a*d^3*x^2*(b^2*d + 2*a*b 
*e + a*c*d) + (c*e^2*x^8*(b^2*e^2 + 3*c^2*d^2 + a*c*e^2 + 5*b*c*d*e))/2 + 
(c^2*e^3*x^9*(5*b*e + 8*c*d))/9 + a^2*b*d^4*x
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.35 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {x \left (126 c^{3} e^{4} x^{9}+350 b \,c^{2} e^{4} x^{8}+560 c^{3} d \,e^{3} x^{8}+315 a \,c^{2} e^{4} x^{7}+315 b^{2} c \,e^{4} x^{7}+1575 b \,c^{2} d \,e^{3} x^{7}+945 c^{3} d^{2} e^{2} x^{7}+540 a b c \,e^{4} x^{6}+1440 a \,c^{2} d \,e^{3} x^{6}+90 b^{3} e^{4} x^{6}+1440 b^{2} c d \,e^{3} x^{6}+2700 b \,c^{2} d^{2} e^{2} x^{6}+720 c^{3} d^{3} e \,x^{6}+210 a^{2} c \,e^{4} x^{5}+210 a \,b^{2} e^{4} x^{5}+2520 a b c d \,e^{3} x^{5}+2520 a \,c^{2} d^{2} e^{2} x^{5}+420 b^{3} d \,e^{3} x^{5}+2520 b^{2} c \,d^{2} e^{2} x^{5}+2100 b \,c^{2} d^{3} e \,x^{5}+210 c^{3} d^{4} x^{5}+126 a^{2} b \,e^{4} x^{4}+1008 a^{2} c d \,e^{3} x^{4}+1008 a \,b^{2} d \,e^{3} x^{4}+4536 a b c \,d^{2} e^{2} x^{4}+2016 a \,c^{2} d^{3} e \,x^{4}+756 b^{3} d^{2} e^{2} x^{4}+2016 b^{2} c \,d^{3} e \,x^{4}+630 b \,c^{2} d^{4} x^{4}+630 a^{2} b d \,e^{3} x^{3}+1890 a^{2} c \,d^{2} e^{2} x^{3}+1890 a \,b^{2} d^{2} e^{2} x^{3}+3780 a b c \,d^{3} e \,x^{3}+630 a \,c^{2} d^{4} x^{3}+630 b^{3} d^{3} e \,x^{3}+630 b^{2} c \,d^{4} x^{3}+1260 a^{2} b \,d^{2} e^{2} x^{2}+1680 a^{2} c \,d^{3} e \,x^{2}+1680 a \,b^{2} d^{3} e \,x^{2}+1260 a b c \,d^{4} x^{2}+210 b^{3} d^{4} x^{2}+1260 a^{2} b \,d^{3} e x +630 a^{2} c \,d^{4} x +630 a \,b^{2} d^{4} x +630 a^{2} b \,d^{4}\right )}{630} \] Input:

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

Output:

(x*(630*a**2*b*d**4 + 1260*a**2*b*d**3*e*x + 1260*a**2*b*d**2*e**2*x**2 + 
630*a**2*b*d*e**3*x**3 + 126*a**2*b*e**4*x**4 + 630*a**2*c*d**4*x + 1680*a 
**2*c*d**3*e*x**2 + 1890*a**2*c*d**2*e**2*x**3 + 1008*a**2*c*d*e**3*x**4 + 
 210*a**2*c*e**4*x**5 + 630*a*b**2*d**4*x + 1680*a*b**2*d**3*e*x**2 + 1890 
*a*b**2*d**2*e**2*x**3 + 1008*a*b**2*d*e**3*x**4 + 210*a*b**2*e**4*x**5 + 
1260*a*b*c*d**4*x**2 + 3780*a*b*c*d**3*e*x**3 + 4536*a*b*c*d**2*e**2*x**4 
+ 2520*a*b*c*d*e**3*x**5 + 540*a*b*c*e**4*x**6 + 630*a*c**2*d**4*x**3 + 20 
16*a*c**2*d**3*e*x**4 + 2520*a*c**2*d**2*e**2*x**5 + 1440*a*c**2*d*e**3*x* 
*6 + 315*a*c**2*e**4*x**7 + 210*b**3*d**4*x**2 + 630*b**3*d**3*e*x**3 + 75 
6*b**3*d**2*e**2*x**4 + 420*b**3*d*e**3*x**5 + 90*b**3*e**4*x**6 + 630*b** 
2*c*d**4*x**3 + 2016*b**2*c*d**3*e*x**4 + 2520*b**2*c*d**2*e**2*x**5 + 144 
0*b**2*c*d*e**3*x**6 + 315*b**2*c*e**4*x**7 + 630*b*c**2*d**4*x**4 + 2100* 
b*c**2*d**3*e*x**5 + 2700*b*c**2*d**2*e**2*x**6 + 1575*b*c**2*d*e**3*x**7 
+ 350*b*c**2*e**4*x**8 + 210*c**3*d**4*x**5 + 720*c**3*d**3*e*x**6 + 945*c 
**3*d**2*e**2*x**7 + 560*c**3*d*e**3*x**8 + 126*c**3*e**4*x**9))/630