\(\int (d+e x)^2 (a+b x+c x^2)^4 \, dx\) [2148]

Optimal result

Integrand size = 20, antiderivative size = 441 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^3}{3 e^9}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^6}{3 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^7}{7 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{9 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^{10}}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \]

[Out]

1/3*(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^3/e^9-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^4/e^9+2/5*(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*x+d)^5/e^9-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b
^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^6/e^9+1/7*(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*x+d)^7/e^9-1/2*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(
-3*a*e+7*b*d))*(e*x+d)^8/e^9+2/9*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^9/e^9-2/5*c^3*(-b*e+2*c
*d)*(e*x+d)^10/e^9+1/11*c^4*(e*x+d)^11/e^9

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=\frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^9}+\frac {2 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9}-\frac {c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}-\frac {2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^4}{3 e^9}-\frac {2 c^3 (d+e x)^{10} (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \]

[In]

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

[Out]

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

Rule 712

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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}{e^8}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^4}{e^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+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^5}{e^8}+\frac {\left (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 )\right ) (d+e x)^6}{e^8}+\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 ) (d+e x)^7}{e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^8}{e^8}-\frac {4 c^3 (2 c d-b e) (d+e x)^9}{e^8}+\frac {c^4 (d+e x)^{10}}{e^8}\right ) \, dx \\ & = \frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^3}{3 e^9}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^6}{3 e^9}+\frac {\left (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 )\right ) (d+e x)^7}{7 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{9 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^{10}}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.97 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=a^4 d^2 x+a^3 d (2 b d+a e) x^2+\frac {1}{3} a^2 \left (6 b^2 d^2+8 a b d e+a \left (4 c d^2+a e^2\right )\right ) x^3+a \left (b^3 d^2+3 a b^2 d e+2 a^2 c d e+a b \left (3 c d^2+a e^2\right )\right ) x^4+\frac {1}{5} \left (b^4 d^2+8 a b^3 d e+24 a^2 b c d e+6 a b^2 \left (2 c d^2+a e^2\right )+2 a^2 c \left (3 c d^2+2 a e^2\right )\right ) x^5+\frac {1}{3} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e+2 b^3 \left (c d^2+a e^2\right )+6 a b c \left (c d^2+a e^2\right )\right ) x^6+\frac {1}{7} \left (8 b^3 c d e+24 a b c^2 d e+b^4 e^2+6 b^2 c \left (c d^2+2 a e^2\right )+2 a c^2 \left (2 c d^2+3 a e^2\right )\right ) x^7+\frac {1}{2} c \left (3 b^2 c d e+2 a c^2 d e+b^3 e^2+b c \left (c d^2+3 a e^2\right )\right ) x^8+\frac {1}{9} c^2 \left (c^2 d^2+6 b^2 e^2+4 c e (2 b d+a e)\right ) x^9+\frac {1}{5} c^3 e (c d+2 b e) x^{10}+\frac {1}{11} c^4 e^2 x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.86 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.03

method	result	size
norman	\(\frac {c^{4} e^{2} x^{11}}{11}+\left (\frac {2}{5} e^{2} c^{3} b +\frac {1}{5} d e \,c^{4}\right ) x^{10}+\left (\frac {4}{9} a \,c^{3} e^{2}+\frac {2}{3} b^{2} c^{2} e^{2}+\frac {8}{9} d e \,c^{3} b +\frac {1}{9} c^{4} d^{2}\right ) x^{9}+\left (\frac {3}{2} a b \,c^{2} e^{2}+a \,c^{3} d e +\frac {1}{2} c \,e^{2} b^{3}+\frac {3}{2} b^{2} c^{2} d e +\frac {1}{2} b \,c^{3} d^{2}\right ) x^{8}+\left (\frac {6}{7} a^{2} c^{2} e^{2}+\frac {12}{7} a \,b^{2} c \,e^{2}+\frac {24}{7} a b \,c^{2} d e +\frac {4}{7} a \,c^{3} d^{2}+\frac {1}{7} b^{4} e^{2}+\frac {8}{7} b^{3} c d e +\frac {6}{7} b^{2} c^{2} d^{2}\right ) x^{7}+\left (2 a^{2} b c \,e^{2}+2 a^{2} c^{2} d e +\frac {2}{3} a \,b^{3} e^{2}+4 a \,b^{2} c d e +2 a b \,c^{2} d^{2}+\frac {1}{3} b^{4} d e +\frac {2}{3} b^{3} c \,d^{2}\right ) x^{6}+\left (\frac {4}{5} a^{3} c \,e^{2}+\frac {6}{5} a^{2} b^{2} e^{2}+\frac {24}{5} a^{2} b c d e +\frac {6}{5} a^{2} c^{2} d^{2}+\frac {8}{5} a \,b^{3} d e +\frac {12}{5} a \,b^{2} c \,d^{2}+\frac {1}{5} b^{4} d^{2}\right ) x^{5}+\left (e^{2} a^{3} b +2 a^{3} c d e +3 a^{2} b^{2} d e +3 a^{2} b c \,d^{2}+a \,b^{3} d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{4} e^{2}+\frac {8}{3} d e \,a^{3} b +\frac {4}{3} a^{3} c \,d^{2}+2 b^{2} d^{2} a^{2}\right ) x^{3}+\left (d e \,a^{4}+2 a^{3} b \,d^{2}\right ) x^{2}+a^{4} d^{2} x\)	\(454\)
gosper	\(\frac {4}{5} x^{5} a^{3} c \,e^{2}+2 d e c \,a^{3} x^{4}+\frac {3}{2} x^{8} a b \,c^{2} e^{2}+a^{4} d^{2} x +\frac {1}{5} d^{2} x^{5} b^{4}+\frac {12}{7} x^{7} a \,b^{2} c \,e^{2}+2 a^{3} b \,d^{2} x^{2}+a \,b^{3} d^{2} x^{4}+2 a^{2} b^{2} d^{2} x^{3}+2 x^{6} a b \,c^{2} d^{2}+\frac {3}{2} x^{8} b^{2} c^{2} d e +\frac {8}{5} x^{5} a \,b^{3} d e +3 a^{2} b^{2} d e \,x^{4}+\frac {8}{9} x^{9} d e \,c^{3} b +d e \,c^{3} a \,x^{8}+\frac {8}{7} x^{7} b^{3} c d e +2 x^{6} a^{2} b c \,e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} c^{2} d e \,x^{6}+\frac {1}{3} x^{6} b^{4} d e +\frac {2}{3} x^{6} b^{3} c \,d^{2}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {2}{3} x^{9} b^{2} c^{2} e^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+\frac {24}{5} x^{5} a^{2} b c d e +\frac {24}{7} x^{7} a b \,c^{2} d e +d e \,a^{4} x^{2}+4 x^{6} a \,b^{2} c d e +\frac {4}{9} x^{9} a \,c^{3} e^{2}+\frac {1}{2} x^{8} b \,c^{3} d^{2}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {2}{5} x^{10} e^{2} c^{3} b +\frac {1}{7} e^{2} b^{4} x^{7}+\frac {1}{9} x^{9} c^{4} d^{2}+\frac {12}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{2}+\frac {2}{3} x^{6} a \,b^{3} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+\frac {1}{11} c^{4} e^{2} x^{11}+3 a^{2} b c \,d^{2} x^{4}+\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+a^{3} b \,e^{2} x^{4}+\frac {1}{2} x^{8} c \,e^{2} b^{3}+\frac {1}{3} x^{3} a^{4} e^{2}\)	\(538\)
risch	\(\frac {4}{5} x^{5} a^{3} c \,e^{2}+2 d e c \,a^{3} x^{4}+\frac {3}{2} x^{8} a b \,c^{2} e^{2}+a^{4} d^{2} x +\frac {1}{5} d^{2} x^{5} b^{4}+\frac {12}{7} x^{7} a \,b^{2} c \,e^{2}+2 a^{3} b \,d^{2} x^{2}+a \,b^{3} d^{2} x^{4}+2 a^{2} b^{2} d^{2} x^{3}+2 x^{6} a b \,c^{2} d^{2}+\frac {3}{2} x^{8} b^{2} c^{2} d e +\frac {8}{5} x^{5} a \,b^{3} d e +3 a^{2} b^{2} d e \,x^{4}+\frac {8}{9} x^{9} d e \,c^{3} b +d e \,c^{3} a \,x^{8}+\frac {8}{7} x^{7} b^{3} c d e +2 x^{6} a^{2} b c \,e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} c^{2} d e \,x^{6}+\frac {1}{3} x^{6} b^{4} d e +\frac {2}{3} x^{6} b^{3} c \,d^{2}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {2}{3} x^{9} b^{2} c^{2} e^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+\frac {24}{5} x^{5} a^{2} b c d e +\frac {24}{7} x^{7} a b \,c^{2} d e +d e \,a^{4} x^{2}+4 x^{6} a \,b^{2} c d e +\frac {4}{9} x^{9} a \,c^{3} e^{2}+\frac {1}{2} x^{8} b \,c^{3} d^{2}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {2}{5} x^{10} e^{2} c^{3} b +\frac {1}{7} e^{2} b^{4} x^{7}+\frac {1}{9} x^{9} c^{4} d^{2}+\frac {12}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{2}+\frac {2}{3} x^{6} a \,b^{3} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+\frac {1}{11} c^{4} e^{2} x^{11}+3 a^{2} b c \,d^{2} x^{4}+\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+a^{3} b \,e^{2} x^{4}+\frac {1}{2} x^{8} c \,e^{2} b^{3}+\frac {1}{3} x^{3} a^{4} e^{2}\)	\(538\)
parallelrisch	\(\frac {4}{5} x^{5} a^{3} c \,e^{2}+2 d e c \,a^{3} x^{4}+\frac {3}{2} x^{8} a b \,c^{2} e^{2}+a^{4} d^{2} x +\frac {1}{5} d^{2} x^{5} b^{4}+\frac {12}{7} x^{7} a \,b^{2} c \,e^{2}+2 a^{3} b \,d^{2} x^{2}+a \,b^{3} d^{2} x^{4}+2 a^{2} b^{2} d^{2} x^{3}+2 x^{6} a b \,c^{2} d^{2}+\frac {3}{2} x^{8} b^{2} c^{2} d e +\frac {8}{5} x^{5} a \,b^{3} d e +3 a^{2} b^{2} d e \,x^{4}+\frac {8}{9} x^{9} d e \,c^{3} b +d e \,c^{3} a \,x^{8}+\frac {8}{7} x^{7} b^{3} c d e +2 x^{6} a^{2} b c \,e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} c^{2} d e \,x^{6}+\frac {1}{3} x^{6} b^{4} d e +\frac {2}{3} x^{6} b^{3} c \,d^{2}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {2}{3} x^{9} b^{2} c^{2} e^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+\frac {24}{5} x^{5} a^{2} b c d e +\frac {24}{7} x^{7} a b \,c^{2} d e +d e \,a^{4} x^{2}+4 x^{6} a \,b^{2} c d e +\frac {4}{9} x^{9} a \,c^{3} e^{2}+\frac {1}{2} x^{8} b \,c^{3} d^{2}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {2}{5} x^{10} e^{2} c^{3} b +\frac {1}{7} e^{2} b^{4} x^{7}+\frac {1}{9} x^{9} c^{4} d^{2}+\frac {12}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{2}+\frac {2}{3} x^{6} a \,b^{3} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+\frac {1}{11} c^{4} e^{2} x^{11}+3 a^{2} b c \,d^{2} x^{4}+\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+a^{3} b \,e^{2} x^{4}+\frac {1}{2} x^{8} c \,e^{2} b^{3}+\frac {1}{3} x^{3} a^{4} e^{2}\)	\(538\)
default	\(\frac {c^{4} e^{2} x^{11}}{11}+\frac {\left (4 e^{2} c^{3} b +2 d e \,c^{4}\right ) x^{10}}{10}+\frac {\left (c^{4} d^{2}+8 d e \,c^{3} b +e^{2} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )\right ) x^{9}}{9}+\frac {\left (4 b \,c^{3} d^{2}+2 d e \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+e^{2} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )\right ) x^{8}}{8}+\frac {\left (d^{2} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+2 d e \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+e^{2} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )\right ) x^{7}}{7}+\frac {\left (d^{2} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+2 d e \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+e^{2} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d^{2} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+2 d e \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+e^{2} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )\right ) x^{5}}{5}+\frac {\left (d^{2} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+2 d e \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+4 e^{2} a^{3} b \right ) x^{4}}{4}+\frac {\left (d^{2} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+8 d e \,a^{3} b +a^{4} e^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{4}+4 a^{3} b \,d^{2}\right ) x^{2}}{2}+a^{4} d^{2} x\)	\(545\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, {\left (c^{4} d e + 2 \, b c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{2} + 8 \, b c^{3} d e + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{2} \, {\left (b c^{3} d^{2} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{5} + {\left (a^{3} b e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a^{3} b d e + a^{4} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=a^{4} d^{2} x + \frac {c^{4} e^{2} x^{11}}{11} + x^{10} \cdot \left (\frac {2 b c^{3} e^{2}}{5} + \frac {c^{4} d e}{5}\right ) + x^{9} \cdot \left (\frac {4 a c^{3} e^{2}}{9} + \frac {2 b^{2} c^{2} e^{2}}{3} + \frac {8 b c^{3} d e}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{8} \cdot \left (\frac {3 a b c^{2} e^{2}}{2} + a c^{3} d e + \frac {b^{3} c e^{2}}{2} + \frac {3 b^{2} c^{2} d e}{2} + \frac {b c^{3} d^{2}}{2}\right ) + x^{7} \cdot \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {12 a b^{2} c e^{2}}{7} + \frac {24 a b c^{2} d e}{7} + \frac {4 a c^{3} d^{2}}{7} + \frac {b^{4} e^{2}}{7} + \frac {8 b^{3} c d e}{7} + \frac {6 b^{2} c^{2} d^{2}}{7}\right ) + x^{6} \cdot \left (2 a^{2} b c e^{2} + 2 a^{2} c^{2} d e + \frac {2 a b^{3} e^{2}}{3} + 4 a b^{2} c d e + 2 a b c^{2} d^{2} + \frac {b^{4} d e}{3} + \frac {2 b^{3} c d^{2}}{3}\right ) + x^{5} \cdot \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} b^{2} e^{2}}{5} + \frac {24 a^{2} b c d e}{5} + \frac {6 a^{2} c^{2} d^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {12 a b^{2} c d^{2}}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 2 a^{3} c d e + 3 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + \frac {4 a^{3} c d^{2}}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, {\left (c^{4} d e + 2 \, b c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{2} + 8 \, b c^{3} d e + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{2} \, {\left (b c^{3} d^{2} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{5} + {\left (a^{3} b e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a^{3} b d e + a^{4} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{11} \, c^{4} e^{2} x^{11} + \frac {1}{5} \, c^{4} d e x^{10} + \frac {2}{5} \, b c^{3} e^{2} x^{10} + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {8}{9} \, b c^{3} d e x^{9} + \frac {2}{3} \, b^{2} c^{2} e^{2} x^{9} + \frac {4}{9} \, a c^{3} e^{2} x^{9} + \frac {1}{2} \, b c^{3} d^{2} x^{8} + \frac {3}{2} \, b^{2} c^{2} d e x^{8} + a c^{3} d e x^{8} + \frac {1}{2} \, b^{3} c e^{2} x^{8} + \frac {3}{2} \, a b c^{2} e^{2} x^{8} + \frac {6}{7} \, b^{2} c^{2} d^{2} x^{7} + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {8}{7} \, b^{3} c d e x^{7} + \frac {24}{7} \, a b c^{2} d e x^{7} + \frac {1}{7} \, b^{4} e^{2} x^{7} + \frac {12}{7} \, a b^{2} c e^{2} x^{7} + \frac {6}{7} \, a^{2} c^{2} e^{2} x^{7} + \frac {2}{3} \, b^{3} c d^{2} x^{6} + 2 \, a b c^{2} d^{2} x^{6} + \frac {1}{3} \, b^{4} d e x^{6} + 4 \, a b^{2} c d e x^{6} + 2 \, a^{2} c^{2} d e x^{6} + \frac {2}{3} \, a b^{3} e^{2} x^{6} + 2 \, a^{2} b c e^{2} x^{6} + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {12}{5} \, a b^{2} c d^{2} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {8}{5} \, a b^{3} d e x^{5} + \frac {24}{5} \, a^{2} b c d e x^{5} + \frac {6}{5} \, a^{2} b^{2} e^{2} x^{5} + \frac {4}{5} \, a^{3} c e^{2} x^{5} + a b^{3} d^{2} x^{4} + 3 \, a^{2} b c d^{2} x^{4} + 3 \, a^{2} b^{2} d e x^{4} + 2 \, a^{3} c d e x^{4} + a^{3} b e^{2} x^{4} + 2 \, a^{2} b^{2} d^{2} x^{3} + \frac {4}{3} \, a^{3} c d^{2} x^{3} + \frac {8}{3} \, a^{3} b d e x^{3} + \frac {1}{3} \, a^{4} e^{2} x^{3} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d e x^{2} + a^{4} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx=x^5\,\left (\frac {4\,a^3\,c\,e^2}{5}+\frac {6\,a^2\,b^2\,e^2}{5}+\frac {24\,a^2\,b\,c\,d\,e}{5}+\frac {6\,a^2\,c^2\,d^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {12\,a\,b^2\,c\,d^2}{5}+\frac {b^4\,d^2}{5}\right )+x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+\frac {4\,c\,a^3\,d^2}{3}+2\,a^2\,b^2\,d^2\right )+x^7\,\left (\frac {6\,a^2\,c^2\,e^2}{7}+\frac {12\,a\,b^2\,c\,e^2}{7}+\frac {24\,a\,b\,c^2\,d\,e}{7}+\frac {4\,a\,c^3\,d^2}{7}+\frac {b^4\,e^2}{7}+\frac {8\,b^3\,c\,d\,e}{7}+\frac {6\,b^2\,c^2\,d^2}{7}\right )+x^9\,\left (\frac {2\,b^2\,c^2\,e^2}{3}+\frac {8\,b\,c^3\,d\,e}{9}+\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+x^6\,\left (2\,a^2\,b\,c\,e^2+2\,a^2\,c^2\,d\,e+\frac {2\,a\,b^3\,e^2}{3}+4\,a\,b^2\,c\,d\,e+2\,a\,b\,c^2\,d^2+\frac {b^4\,d\,e}{3}+\frac {2\,b^3\,c\,d^2}{3}\right )+x^4\,\left (a^3\,b\,e^2+2\,c\,a^3\,d\,e+3\,a^2\,b^2\,d\,e+3\,c\,a^2\,b\,d^2+a\,b^3\,d^2\right )+x^8\,\left (\frac {b^3\,c\,e^2}{2}+\frac {3\,b^2\,c^2\,d\,e}{2}+\frac {b\,c^3\,d^2}{2}+\frac {3\,a\,b\,c^2\,e^2}{2}+a\,c^3\,d\,e\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {c^3\,e\,x^{10}\,\left (2\,b\,e+c\,d\right )}{5} \]

[In]

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

[Out]

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