3.2336 \(\int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=310 \[ a^3 A d x+\frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.36218, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ a^3 A d x+\frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+b*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.324716, size = 310, normalized size = 1. \[ a^3 A d x+\frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 375, normalized size = 1.2 \[{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{3}+3\,Beb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\, \left ( Ae+Bd \right ) b{c}^{2}+Be \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Adb{c}^{2}+ \left ( Ae+Bd \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +Be \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( Ad \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( Ae+Bd \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +Be \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( Ad \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( Ae+Bd \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,Be{a}^{2}b \right ){x}^{4}}{4}}+{\frac{ \left ( Ad \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( Ae+Bd \right ){a}^{2}b+Be{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,Ad{a}^{2}b+ \left ( Ae+Bd \right ){a}^{3} \right ){x}^{2}}{2}}+{a}^{3}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)*(c*x^2+b*x+a)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.695703, size = 451, normalized size = 1.45 \[ \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} d +{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac{1}{5} \,{\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} d +{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d +{\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*(e*x + d),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.242697, size = 1, normalized size = 0. \[ \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{3}{8} x^{8} e c^{2} b B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} b B + \frac{3}{7} x^{7} e c b^{2} B + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \frac{3}{7} x^{7} e c^{2} b A + \frac{1}{2} x^{6} d c b^{2} B + \frac{1}{6} x^{6} e b^{3} B + \frac{1}{2} x^{6} d c^{2} a B + x^{6} e c b a B + \frac{1}{2} x^{6} d c^{2} b A + \frac{1}{2} x^{6} e c b^{2} A + \frac{1}{2} x^{6} e c^{2} a A + \frac{1}{5} x^{5} d b^{3} B + \frac{6}{5} x^{5} d c b a B + \frac{3}{5} x^{5} e b^{2} a B + \frac{3}{5} x^{5} e c a^{2} B + \frac{3}{5} x^{5} d c b^{2} A + \frac{1}{5} x^{5} e b^{3} A + \frac{3}{5} x^{5} d c^{2} a A + \frac{6}{5} x^{5} e c b a A + \frac{3}{4} x^{4} d b^{2} a B + \frac{3}{4} x^{4} d c a^{2} B + \frac{3}{4} x^{4} e b a^{2} B + \frac{1}{4} x^{4} d b^{3} A + \frac{3}{2} x^{4} d c b a A + \frac{3}{4} x^{4} e b^{2} a A + \frac{3}{4} x^{4} e c a^{2} A + x^{3} d b a^{2} B + \frac{1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} d c a^{2} A + x^{3} e b a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{3}{2} x^{2} d b a^{2} A + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*(e*x + d),x, algorithm="fricas")

[Out]

1/9*x^9*e*c^3*B + 1/8*x^8*d*c^3*B + 3/8*x^8*e*c^2*b*B + 1/8*x^8*e*c^3*A + 3/7*x^
7*d*c^2*b*B + 3/7*x^7*e*c*b^2*B + 3/7*x^7*e*c^2*a*B + 1/7*x^7*d*c^3*A + 3/7*x^7*
e*c^2*b*A + 1/2*x^6*d*c*b^2*B + 1/6*x^6*e*b^3*B + 1/2*x^6*d*c^2*a*B + x^6*e*c*b*
a*B + 1/2*x^6*d*c^2*b*A + 1/2*x^6*e*c*b^2*A + 1/2*x^6*e*c^2*a*A + 1/5*x^5*d*b^3*
B + 6/5*x^5*d*c*b*a*B + 3/5*x^5*e*b^2*a*B + 3/5*x^5*e*c*a^2*B + 3/5*x^5*d*c*b^2*
A + 1/5*x^5*e*b^3*A + 3/5*x^5*d*c^2*a*A + 6/5*x^5*e*c*b*a*A + 3/4*x^4*d*b^2*a*B
+ 3/4*x^4*d*c*a^2*B + 3/4*x^4*e*b*a^2*B + 1/4*x^4*d*b^3*A + 3/2*x^4*d*c*b*a*A +
3/4*x^4*e*b^2*a*A + 3/4*x^4*e*c*a^2*A + x^3*d*b*a^2*B + 1/3*x^3*e*a^3*B + x^3*d*
b^2*a*A + x^3*d*c*a^2*A + x^3*e*b*a^2*A + 1/2*x^2*d*a^3*B + 3/2*x^2*d*b*a^2*A +
1/2*x^2*e*a^3*A + x*d*a^3*A

_______________________________________________________________________________________

Sympy [A]  time = 0.290088, size = 435, normalized size = 1.4 \[ A a^{3} d x + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{3 B b c^{2} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{3 A b c^{2} e}{7} + \frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7} + \frac{3 B b^{2} c e}{7} + \frac{3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{A b^{2} c e}{2} + \frac{A b c^{2} d}{2} + B a b c e + \frac{B a c^{2} d}{2} + \frac{B b^{3} e}{6} + \frac{B b^{2} c d}{2}\right ) + x^{5} \left (\frac{6 A a b c e}{5} + \frac{3 A a c^{2} d}{5} + \frac{A b^{3} e}{5} + \frac{3 A b^{2} c d}{5} + \frac{3 B a^{2} c e}{5} + \frac{3 B a b^{2} e}{5} + \frac{6 B a b c d}{5} + \frac{B b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 A a b^{2} e}{4} + \frac{3 A a b c d}{2} + \frac{A b^{3} d}{4} + \frac{3 B a^{2} b e}{4} + \frac{3 B a^{2} c d}{4} + \frac{3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a^{2} c d + A a b^{2} d + \frac{B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{3 A a^{2} b d}{2} + \frac{B a^{3} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)*(c*x**2+b*x+a)**3,x)

[Out]

A*a**3*d*x + B*c**3*e*x**9/9 + x**8*(A*c**3*e/8 + 3*B*b*c**2*e/8 + B*c**3*d/8) +
x**7*(3*A*b*c**2*e/7 + A*c**3*d/7 + 3*B*a*c**2*e/7 + 3*B*b**2*c*e/7 + 3*B*b*c**
2*d/7) + x**6*(A*a*c**2*e/2 + A*b**2*c*e/2 + A*b*c**2*d/2 + B*a*b*c*e + B*a*c**2
*d/2 + B*b**3*e/6 + B*b**2*c*d/2) + x**5*(6*A*a*b*c*e/5 + 3*A*a*c**2*d/5 + A*b**
3*e/5 + 3*A*b**2*c*d/5 + 3*B*a**2*c*e/5 + 3*B*a*b**2*e/5 + 6*B*a*b*c*d/5 + B*b**
3*d/5) + x**4*(3*A*a**2*c*e/4 + 3*A*a*b**2*e/4 + 3*A*a*b*c*d/2 + A*b**3*d/4 + 3*
B*a**2*b*e/4 + 3*B*a**2*c*d/4 + 3*B*a*b**2*d/4) + x**3*(A*a**2*b*e + A*a**2*c*d
+ A*a*b**2*d + B*a**3*e/3 + B*a**2*b*d) + x**2*(A*a**3*e/2 + 3*A*a**2*b*d/2 + B*
a**3*d/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264707, size = 590, normalized size = 1.9 \[ \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, B b c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, B b c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B b^{2} c x^{7} e + \frac{3}{7} \, B a c^{2} x^{7} e + \frac{3}{7} \, A b c^{2} x^{7} e + \frac{1}{2} \, B b^{2} c d x^{6} + \frac{1}{2} \, B a c^{2} d x^{6} + \frac{1}{2} \, A b c^{2} d x^{6} + \frac{1}{6} \, B b^{3} x^{6} e + B a b c x^{6} e + \frac{1}{2} \, A b^{2} c x^{6} e + \frac{1}{2} \, A a c^{2} x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{6}{5} \, B a b c d x^{5} + \frac{3}{5} \, A b^{2} c d x^{5} + \frac{3}{5} \, A a c^{2} d x^{5} + \frac{3}{5} \, B a b^{2} x^{5} e + \frac{1}{5} \, A b^{3} x^{5} e + \frac{3}{5} \, B a^{2} c x^{5} e + \frac{6}{5} \, A a b c x^{5} e + \frac{3}{4} \, B a b^{2} d x^{4} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{3}{4} \, B a^{2} c d x^{4} + \frac{3}{2} \, A a b c d x^{4} + \frac{3}{4} \, B a^{2} b x^{4} e + \frac{3}{4} \, A a b^{2} x^{4} e + \frac{3}{4} \, A a^{2} c x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + A a^{2} c d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{3}{2} \, A a^{2} b d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*(e*x + d),x, algorithm="giac")

[Out]

1/9*B*c^3*x^9*e + 1/8*B*c^3*d*x^8 + 3/8*B*b*c^2*x^8*e + 1/8*A*c^3*x^8*e + 3/7*B*
b*c^2*d*x^7 + 1/7*A*c^3*d*x^7 + 3/7*B*b^2*c*x^7*e + 3/7*B*a*c^2*x^7*e + 3/7*A*b*
c^2*x^7*e + 1/2*B*b^2*c*d*x^6 + 1/2*B*a*c^2*d*x^6 + 1/2*A*b*c^2*d*x^6 + 1/6*B*b^
3*x^6*e + B*a*b*c*x^6*e + 1/2*A*b^2*c*x^6*e + 1/2*A*a*c^2*x^6*e + 1/5*B*b^3*d*x^
5 + 6/5*B*a*b*c*d*x^5 + 3/5*A*b^2*c*d*x^5 + 3/5*A*a*c^2*d*x^5 + 3/5*B*a*b^2*x^5*
e + 1/5*A*b^3*x^5*e + 3/5*B*a^2*c*x^5*e + 6/5*A*a*b*c*x^5*e + 3/4*B*a*b^2*d*x^4
+ 1/4*A*b^3*d*x^4 + 3/4*B*a^2*c*d*x^4 + 3/2*A*a*b*c*d*x^4 + 3/4*B*a^2*b*x^4*e +
3/4*A*a*b^2*x^4*e + 3/4*A*a^2*c*x^4*e + B*a^2*b*d*x^3 + A*a*b^2*d*x^3 + A*a^2*c*
d*x^3 + 1/3*B*a^3*x^3*e + A*a^2*b*x^3*e + 1/2*B*a^3*d*x^2 + 3/2*A*a^2*b*d*x^2 +
1/2*A*a^3*x^2*e + A*a^3*d*x