\(\int (d+e x)^3 (a+b x^2+c x^4)^2 \, dx\) [229]

Optimal result

Integrand size = 22, antiderivative size = 235 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{3} a b d^3 x^3+\frac {3}{2} a b d^2 e x^4+\frac {1}{5} d \left (b^2 d^2+2 a c d^2+6 a b e^2\right ) x^5+\frac {1}{6} e \left (3 b^2 d^2+6 a c d^2+2 a b e^2\right ) x^6+\frac {1}{7} d \left (2 b c d^2+3 b^2 e^2+6 a c e^2\right ) x^7+\frac {1}{8} e \left (6 b c d^2+b^2 e^2+2 a c e^2\right ) x^8+\frac {1}{9} c d \left (c d^2+6 b e^2\right ) x^9+\frac {1}{10} c e \left (3 c d^2+2 b e^2\right ) x^{10}+\frac {3}{11} c^2 d e^2 x^{11}+\frac {1}{12} c^2 e^3 x^{12}+\frac {a^2 (d+e x)^4}{4 e} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.10 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^3 x+\frac {3}{2} a^2 d^2 e x^2+\frac {1}{3} a d \left (2 b d^2+3 a e^2\right ) x^3+\frac {1}{4} a e \left (6 b d^2+a e^2\right ) x^4+\frac {1}{5} d \left (b^2 d^2+2 a c d^2+6 a b e^2\right ) x^5+\frac {1}{6} e \left (3 b^2 d^2+6 a c d^2+2 a b e^2\right ) x^6+\frac {1}{7} d \left (2 b c d^2+3 b^2 e^2+6 a c e^2\right ) x^7+\frac {1}{8} e \left (6 b c d^2+b^2 e^2+2 a c e^2\right ) x^8+\frac {1}{9} c d \left (c d^2+6 b e^2\right ) x^9+\frac {1}{10} c e \left (3 c d^2+2 b e^2\right ) x^{10}+\frac {3}{11} c^2 d e^2 x^{11}+\frac {1}{12} c^2 e^3 x^{12} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2200, 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.

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

\(\Big \downarrow \) 2200

\(\displaystyle \int \left (\frac {(d+e x)^7 \left (2 a c e^4+b^2 e^4+30 b c d^2 e^2+70 c^2 d^4\right )}{e^8}-\frac {4 d (d+e x)^6 \left (2 a c e^4+b^2 e^4+10 b c d^2 e^2+14 c^2 d^4\right )}{e^8}+\frac {2 (d+e x)^5 \left (a b e^6+6 a c d^2 e^4+3 b^2 d^2 e^4+15 b c d^4 e^2+14 c^2 d^6\right )}{e^8}-\frac {4 d (d+e x)^4 \left (b e^2+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8}+\frac {(d+e x)^3 \left (a e^4+b d^2 e^2+c d^4\right )^2}{e^8}+\frac {2 c (d+e x)^9 \left (b e^2+14 c d^2\right )}{e^8}-\frac {4 c d (d+e x)^8 \left (3 b e^2+14 c d^2\right )}{e^8}+\frac {c^2 (d+e x)^{11}}{e^8}-\frac {8 c^2 d (d+e x)^{10}}{e^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^8 \left (2 a c e^4+b^2 e^4+30 b c d^2 e^2+70 c^2 d^4\right )}{8 e^9}-\frac {4 d (d+e x)^7 \left (2 a c e^4+b^2 e^4+10 b c d^2 e^2+14 c^2 d^4\right )}{7 e^9}+\frac {(d+e x)^6 \left (a b e^6+6 a c d^2 e^4+3 b^2 d^2 e^4+15 b c d^4 e^2+14 c^2 d^6\right )}{3 e^9}-\frac {4 d (d+e x)^5 \left (b e^2+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{5 e^9}+\frac {(d+e x)^4 \left (a e^4+b d^2 e^2+c d^4\right )^2}{4 e^9}+\frac {c (d+e x)^{10} \left (b e^2+14 c d^2\right )}{5 e^9}-\frac {4 c d (d+e x)^9 \left (3 b e^2+14 c d^2\right )}{9 e^9}+\frac {c^2 (d+e x)^{12}}{12 e^9}-\frac {8 c^2 d (d+e x)^{11}}{11 e^9}\)

Input:

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

Output:

((c*d^4 + b*d^2*e^2 + a*e^4)^2*(d + e*x)^4)/(4*e^9) - (4*d*(2*c*d^2 + b*e^ 
2)*(c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x)^5)/(5*e^9) + ((14*c^2*d^6 + 15*b* 
c*d^4*e^2 + 3*b^2*d^2*e^4 + 6*a*c*d^2*e^4 + a*b*e^6)*(d + e*x)^6)/(3*e^9) 
- (4*d*(14*c^2*d^4 + 10*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*(d + e*x)^7)/(7 
*e^9) + ((70*c^2*d^4 + 30*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*(d + e*x)^8)/ 
(8*e^9) - (4*c*d*(14*c*d^2 + 3*b*e^2)*(d + e*x)^9)/(9*e^9) + (c*(14*c*d^2 
+ b*e^2)*(d + e*x)^10)/(5*e^9) - (8*c^2*d*(d + e*x)^11)/(11*e^9) + (c^2*(d 
 + e*x)^12)/(12*e^9)
 

Defintions of rubi rules used

rule 2009

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

rule 2200

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[Expa 
ndIntegrand[Px*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && Poly 
Q[Px, x] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {c^{2} e^{3} x^{12}}{12}+\frac {3 c^{2} d \,e^{2} x^{11}}{11}+\frac {\left (2 e^{3} b c +3 d^{2} e \,c^{2}\right ) x^{10}}{10}+\frac {\left (6 b c d \,e^{2}+c^{2} d^{3}\right ) x^{9}}{9}+\frac {\left (6 b c \,d^{2} e +e^{3} \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (2 b c \,d^{3}+3 d \,e^{2} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (3 d^{2} e \left (2 a c +b^{2}\right )+2 a b \,e^{3}\right ) x^{6}}{6}+\frac {\left (d^{3} \left (2 a c +b^{2}\right )+6 a b d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 d^{2} e a b \right ) x^{4}}{4}+\frac {\left (3 a^{2} d \,e^{2}+2 d^{3} a b \right ) x^{3}}{3}+\frac {3 a^{2} d^{2} e \,x^{2}}{2}+a^{2} d^{3} x\)	\(237\)
norman	\(\frac {c^{2} e^{3} x^{12}}{12}+\frac {3 c^{2} d \,e^{2} x^{11}}{11}+\left (\frac {1}{5} e^{3} b c +\frac {3}{10} d^{2} e \,c^{2}\right ) x^{10}+\left (\frac {2}{3} b c d \,e^{2}+\frac {1}{9} c^{2} d^{3}\right ) x^{9}+\left (\frac {1}{4} a c \,e^{3}+\frac {1}{8} b^{2} e^{3}+\frac {3}{4} b c \,d^{2} e \right ) x^{8}+\left (\frac {6}{7} a c d \,e^{2}+\frac {3}{7} b^{2} d \,e^{2}+\frac {2}{7} b c \,d^{3}\right ) x^{7}+\left (\frac {1}{3} a b \,e^{3}+a c \,d^{2} e +\frac {1}{2} b^{2} d^{2} e \right ) x^{6}+\left (\frac {6}{5} a b d \,e^{2}+\frac {2}{5} a c \,d^{3}+\frac {1}{5} b^{2} d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{2} e^{3}+\frac {3}{2} d^{2} e a b \right ) x^{4}+\left (a^{2} d \,e^{2}+\frac {2}{3} d^{3} a b \right ) x^{3}+\frac {3 a^{2} d^{2} e \,x^{2}}{2}+a^{2} d^{3} x\)	\(241\)
gosper	\(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {1}{5} x^{10} e^{3} b c +\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {2}{3} x^{9} b c d \,e^{2}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {1}{8} x^{8} b^{2} e^{3}+\frac {3}{4} x^{8} b c \,d^{2} e +\frac {6}{7} a c d \,e^{2} x^{7}+\frac {3}{7} x^{7} b^{2} d \,e^{2}+\frac {2}{7} x^{7} b c \,d^{3}+\frac {1}{3} x^{6} a b \,e^{3}+a c \,d^{2} e \,x^{6}+\frac {1}{2} x^{6} b^{2} d^{2} e +\frac {6}{5} x^{5} a b d \,e^{2}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{4} a^{2} e^{3} x^{4}+\frac {3}{2} a b \,d^{2} e \,x^{4}+a^{2} d \,e^{2} x^{3}+\frac {2}{3} a b \,d^{3} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\)	\(261\)
risch	\(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {1}{5} x^{10} e^{3} b c +\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {2}{3} x^{9} b c d \,e^{2}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {1}{8} x^{8} b^{2} e^{3}+\frac {3}{4} x^{8} b c \,d^{2} e +\frac {6}{7} a c d \,e^{2} x^{7}+\frac {3}{7} x^{7} b^{2} d \,e^{2}+\frac {2}{7} x^{7} b c \,d^{3}+\frac {1}{3} x^{6} a b \,e^{3}+a c \,d^{2} e \,x^{6}+\frac {1}{2} x^{6} b^{2} d^{2} e +\frac {6}{5} x^{5} a b d \,e^{2}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{4} a^{2} e^{3} x^{4}+\frac {3}{2} a b \,d^{2} e \,x^{4}+a^{2} d \,e^{2} x^{3}+\frac {2}{3} a b \,d^{3} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\)	\(261\)
parallelrisch	\(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {1}{5} x^{10} e^{3} b c +\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {2}{3} x^{9} b c d \,e^{2}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {1}{8} x^{8} b^{2} e^{3}+\frac {3}{4} x^{8} b c \,d^{2} e +\frac {6}{7} a c d \,e^{2} x^{7}+\frac {3}{7} x^{7} b^{2} d \,e^{2}+\frac {2}{7} x^{7} b c \,d^{3}+\frac {1}{3} x^{6} a b \,e^{3}+a c \,d^{2} e \,x^{6}+\frac {1}{2} x^{6} b^{2} d^{2} e +\frac {6}{5} x^{5} a b d \,e^{2}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{5} x^{5} b^{2} d^{3}+\frac {1}{4} a^{2} e^{3} x^{4}+\frac {3}{2} a b \,d^{2} e \,x^{4}+a^{2} d \,e^{2} x^{3}+\frac {2}{3} a b \,d^{3} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\)	\(261\)
orering	\(\frac {x \left (2310 e^{3} c^{2} x^{11}+7560 c^{2} d \,e^{2} x^{10}+5544 b c \,e^{3} x^{9}+8316 c^{2} d^{2} e \,x^{9}+18480 b c d \,e^{2} x^{8}+3080 c^{2} d^{3} x^{8}+6930 a c \,e^{3} x^{7}+3465 b^{2} e^{3} x^{7}+20790 b c \,d^{2} e \,x^{7}+23760 a c d \,e^{2} x^{6}+11880 b^{2} d \,e^{2} x^{6}+7920 b c \,d^{3} x^{6}+9240 a b \,e^{3} x^{5}+27720 a c \,d^{2} e \,x^{5}+13860 b^{2} d^{2} e \,x^{5}+33264 a b d \,e^{2} x^{4}+11088 a c \,d^{3} x^{4}+5544 b^{2} d^{3} x^{4}+6930 a^{2} e^{3} x^{3}+41580 a b \,d^{2} e \,x^{3}+27720 a^{2} d \,e^{2} x^{2}+18480 a b \,d^{3} x^{2}+41580 d^{2} e \,a^{2} x +27720 d^{3} a^{2}\right )}{27720}\)	\(264\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {1}{10} \, {\left (3 \, c^{2} d^{2} e + 2 \, b c e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, b c d e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, b c d^{2} e + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, b c d^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{7} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {1}{6} \, {\left (2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (6 \, a b d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (6 \, a b d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d e^{2}\right )} x^{3} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.18 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + \frac {3 c^{2} d e^{2} x^{11}}{11} + \frac {c^{2} e^{3} x^{12}}{12} + x^{10} \left (\frac {b c e^{3}}{5} + \frac {3 c^{2} d^{2} e}{10}\right ) + x^{9} \cdot \left (\frac {2 b c d e^{2}}{3} + \frac {c^{2} d^{3}}{9}\right ) + x^{8} \left (\frac {a c e^{3}}{4} + \frac {b^{2} e^{3}}{8} + \frac {3 b c d^{2} e}{4}\right ) + x^{7} \cdot \left (\frac {6 a c d e^{2}}{7} + \frac {3 b^{2} d e^{2}}{7} + \frac {2 b c d^{3}}{7}\right ) + x^{6} \left (\frac {a b e^{3}}{3} + a c d^{2} e + \frac {b^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {6 a b d e^{2}}{5} + \frac {2 a c d^{3}}{5} + \frac {b^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d^{2} e}{2}\right ) + x^{3} \left (a^{2} d e^{2} + \frac {2 a b d^{3}}{3}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {1}{10} \, {\left (3 \, c^{2} d^{2} e + 2 \, b c e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, b c d e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, b c d^{2} e + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, b c d^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{7} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {1}{6} \, {\left (2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (6 \, a b d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (6 \, a b d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d e^{2}\right )} x^{3} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.11 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {3}{10} \, c^{2} d^{2} e x^{10} + \frac {1}{5} \, b c e^{3} x^{10} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {2}{3} \, b c d e^{2} x^{9} + \frac {3}{4} \, b c d^{2} e x^{8} + \frac {1}{8} \, b^{2} e^{3} x^{8} + \frac {1}{4} \, a c e^{3} x^{8} + \frac {2}{7} \, b c d^{3} x^{7} + \frac {3}{7} \, b^{2} d e^{2} x^{7} + \frac {6}{7} \, a c d e^{2} x^{7} + \frac {1}{2} \, b^{2} d^{2} e x^{6} + a c d^{2} e x^{6} + \frac {1}{3} \, a b e^{3} x^{6} + \frac {1}{5} \, b^{2} d^{3} x^{5} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {6}{5} \, a b d e^{2} x^{5} + \frac {3}{2} \, a b d^{2} e x^{4} + \frac {1}{4} \, a^{2} e^{3} x^{4} + \frac {2}{3} \, a b d^{3} x^{3} + a^{2} d e^{2} x^{3} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + a^{2} d^{3} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.03 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {b^2\,d^3}{5}+\frac {6\,a\,b\,d\,e^2}{5}+\frac {2\,a\,c\,d^3}{5}\right )+x^8\,\left (\frac {b^2\,e^3}{8}+\frac {3\,c\,b\,d^2\,e}{4}+\frac {a\,c\,e^3}{4}\right )+x^3\,\left (a^2\,d\,e^2+\frac {2\,b\,a\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,b\,a\,d^2\,e}{2}\right )+x^9\,\left (\frac {c^2\,d^3}{9}+\frac {2\,b\,c\,d\,e^2}{3}\right )+x^{10}\,\left (\frac {3\,c^2\,d^2\,e}{10}+\frac {b\,c\,e^3}{5}\right )+a^2\,d^3\,x+\frac {e\,x^6\,\left (3\,b^2\,d^2+2\,a\,b\,e^2+6\,a\,c\,d^2\right )}{6}+\frac {d\,x^7\,\left (3\,b^2\,e^2+2\,c\,b\,d^2+6\,a\,c\,e^2\right )}{7}+\frac {c^2\,e^3\,x^{12}}{12}+\frac {3\,a^2\,d^2\,e\,x^2}{2}+\frac {3\,c^2\,d\,e^2\,x^{11}}{11} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {x \left (2310 c^{2} e^{3} x^{11}+7560 c^{2} d \,e^{2} x^{10}+5544 b c \,e^{3} x^{9}+8316 c^{2} d^{2} e \,x^{9}+18480 b c d \,e^{2} x^{8}+3080 c^{2} d^{3} x^{8}+6930 a c \,e^{3} x^{7}+3465 b^{2} e^{3} x^{7}+20790 b c \,d^{2} e \,x^{7}+23760 a c d \,e^{2} x^{6}+11880 b^{2} d \,e^{2} x^{6}+7920 b c \,d^{3} x^{6}+9240 a b \,e^{3} x^{5}+27720 a c \,d^{2} e \,x^{5}+13860 b^{2} d^{2} e \,x^{5}+33264 a b d \,e^{2} x^{4}+11088 a c \,d^{3} x^{4}+5544 b^{2} d^{3} x^{4}+6930 a^{2} e^{3} x^{3}+41580 a b \,d^{2} e \,x^{3}+27720 a^{2} d \,e^{2} x^{2}+18480 a b \,d^{3} x^{2}+41580 a^{2} d^{2} e x +27720 a^{2} d^{3}\right )}{27720} \] Input:

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

Output:

(x*(27720*a**2*d**3 + 41580*a**2*d**2*e*x + 27720*a**2*d*e**2*x**2 + 6930* 
a**2*e**3*x**3 + 18480*a*b*d**3*x**2 + 41580*a*b*d**2*e*x**3 + 33264*a*b*d 
*e**2*x**4 + 9240*a*b*e**3*x**5 + 11088*a*c*d**3*x**4 + 27720*a*c*d**2*e*x 
**5 + 23760*a*c*d*e**2*x**6 + 6930*a*c*e**3*x**7 + 5544*b**2*d**3*x**4 + 1 
3860*b**2*d**2*e*x**5 + 11880*b**2*d*e**2*x**6 + 3465*b**2*e**3*x**7 + 792 
0*b*c*d**3*x**6 + 20790*b*c*d**2*e*x**7 + 18480*b*c*d*e**2*x**8 + 5544*b*c 
*e**3*x**9 + 3080*c**2*d**3*x**8 + 8316*c**2*d**2*e*x**9 + 7560*c**2*d*e** 
2*x**10 + 2310*c**2*e**3*x**11))/27720