3.1659 \(\int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=120 \[ -\frac{b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac{(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac{b^2 B (d+e x)^9}{9 e^4} \]

[Out]

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

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Rubi [A]  time = 0.659344, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac{(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac{b^2 B (d+e x)^9}{9 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 94.2738, size = 112, normalized size = 0.93 \[ \frac{B b^{2} \left (d + e x\right )^{9}}{9 e^{4}} + \frac{b \left (d + e x\right )^{8} \left (A b e + 2 B a e - 3 B b d\right )}{8 e^{4}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{6 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [B]  time = 0.199483, size = 330, normalized size = 2.75 \[ \frac{1}{6} e^2 x^6 \left (a^2 e^2 (A e+5 B d)+10 a b d e (A e+2 B d)+10 b^2 d^2 (A e+B d)\right )+d e x^5 \left (a^2 e^2 (A e+2 B d)+4 a b d e (A e+B d)+b^2 d^2 (2 A e+B d)\right )+\frac{1}{4} d^2 x^4 \left (10 a^2 e^2 (A e+B d)+10 a b d e (2 A e+B d)+b^2 d^2 (5 A e+B d)\right )+\frac{1}{3} d^3 x^3 \left (A \left (10 a^2 e^2+10 a b d e+b^2 d^2\right )+a B d (5 a e+2 b d)\right )+\frac{1}{7} e^3 x^7 \left (a^2 B e^2+2 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+a^2 A d^5 x+\frac{1}{2} a d^4 x^2 (5 a A e+a B d+2 A b d)+\frac{1}{8} b e^4 x^8 (2 a B e+A b e+5 b B d)+\frac{1}{9} b^2 B e^5 x^9 \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Maple [B]  time = 0.003, size = 394, normalized size = 3.3 \[{\frac{B{e}^{5}{b}^{2}{x}^{9}}{9}}+{\frac{ \left ( \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){b}^{2}+2\,B{e}^{5}ab \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){b}^{2}+2\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) ab+B{e}^{5}{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){b}^{2}+2\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) ab+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){b}^{2}+2\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) ab+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){b}^{2}+2\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) ab+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{5}{b}^{2}+2\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) ab+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{5}ab+ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{5}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.712081, size = 498, normalized size = 4.15 \[ \frac{1}{9} \, B b^{2} e^{5} x^{9} + A a^{2} d^{5} x + \frac{1}{8} \,{\left (5 \, B b^{2} d e^{4} +{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{2} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, B b^{2} d^{3} e^{2} + A a^{2} e^{5} + 10 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} +{\left (B b^{2} d^{4} e + A a^{2} d e^{4} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{5} + 10 \, A a^{2} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{2} d^{3} e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{5} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{2} d^{4} e +{\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.259506, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{5} b^{2} B + \frac{5}{8} x^{8} e^{4} d b^{2} B + \frac{1}{4} x^{8} e^{5} b a B + \frac{1}{8} x^{8} e^{5} b^{2} A + \frac{10}{7} x^{7} e^{3} d^{2} b^{2} B + \frac{10}{7} x^{7} e^{4} d b a B + \frac{1}{7} x^{7} e^{5} a^{2} B + \frac{5}{7} x^{7} e^{4} d b^{2} A + \frac{2}{7} x^{7} e^{5} b a A + \frac{5}{3} x^{6} e^{2} d^{3} b^{2} B + \frac{10}{3} x^{6} e^{3} d^{2} b a B + \frac{5}{6} x^{6} e^{4} d a^{2} B + \frac{5}{3} x^{6} e^{3} d^{2} b^{2} A + \frac{5}{3} x^{6} e^{4} d b a A + \frac{1}{6} x^{6} e^{5} a^{2} A + x^{5} e d^{4} b^{2} B + 4 x^{5} e^{2} d^{3} b a B + 2 x^{5} e^{3} d^{2} a^{2} B + 2 x^{5} e^{2} d^{3} b^{2} A + 4 x^{5} e^{3} d^{2} b a A + x^{5} e^{4} d a^{2} A + \frac{1}{4} x^{4} d^{5} b^{2} B + \frac{5}{2} x^{4} e d^{4} b a B + \frac{5}{2} x^{4} e^{2} d^{3} a^{2} B + \frac{5}{4} x^{4} e d^{4} b^{2} A + 5 x^{4} e^{2} d^{3} b a A + \frac{5}{2} x^{4} e^{3} d^{2} a^{2} A + \frac{2}{3} x^{3} d^{5} b a B + \frac{5}{3} x^{3} e d^{4} a^{2} B + \frac{1}{3} x^{3} d^{5} b^{2} A + \frac{10}{3} x^{3} e d^{4} b a A + \frac{10}{3} x^{3} e^{2} d^{3} a^{2} A + \frac{1}{2} x^{2} d^{5} a^{2} B + x^{2} d^{5} b a A + \frac{5}{2} x^{2} e d^{4} a^{2} A + x d^{5} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^5,x, algorithm="fricas")

[Out]

1/9*x^9*e^5*b^2*B + 5/8*x^8*e^4*d*b^2*B + 1/4*x^8*e^5*b*a*B + 1/8*x^8*e^5*b^2*A
+ 10/7*x^7*e^3*d^2*b^2*B + 10/7*x^7*e^4*d*b*a*B + 1/7*x^7*e^5*a^2*B + 5/7*x^7*e^
4*d*b^2*A + 2/7*x^7*e^5*b*a*A + 5/3*x^6*e^2*d^3*b^2*B + 10/3*x^6*e^3*d^2*b*a*B +
5/6*x^6*e^4*d*a^2*B + 5/3*x^6*e^3*d^2*b^2*A + 5/3*x^6*e^4*d*b*a*A + 1/6*x^6*e^5
*a^2*A + x^5*e*d^4*b^2*B + 4*x^5*e^2*d^3*b*a*B + 2*x^5*e^3*d^2*a^2*B + 2*x^5*e^2
*d^3*b^2*A + 4*x^5*e^3*d^2*b*a*A + x^5*e^4*d*a^2*A + 1/4*x^4*d^5*b^2*B + 5/2*x^4
*e*d^4*b*a*B + 5/2*x^4*e^2*d^3*a^2*B + 5/4*x^4*e*d^4*b^2*A + 5*x^4*e^2*d^3*b*a*A
+ 5/2*x^4*e^3*d^2*a^2*A + 2/3*x^3*d^5*b*a*B + 5/3*x^3*e*d^4*a^2*B + 1/3*x^3*d^5
*b^2*A + 10/3*x^3*e*d^4*b*a*A + 10/3*x^3*e^2*d^3*a^2*A + 1/2*x^2*d^5*a^2*B + x^2
*d^5*b*a*A + 5/2*x^2*e*d^4*a^2*A + x*d^5*a^2*A

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Sympy [A]  time = 0.291514, size = 481, normalized size = 4.01 \[ A a^{2} d^{5} x + \frac{B b^{2} e^{5} x^{9}}{9} + x^{8} \left (\frac{A b^{2} e^{5}}{8} + \frac{B a b e^{5}}{4} + \frac{5 B b^{2} d e^{4}}{8}\right ) + x^{7} \left (\frac{2 A a b e^{5}}{7} + \frac{5 A b^{2} d e^{4}}{7} + \frac{B a^{2} e^{5}}{7} + \frac{10 B a b d e^{4}}{7} + \frac{10 B b^{2} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{A a^{2} e^{5}}{6} + \frac{5 A a b d e^{4}}{3} + \frac{5 A b^{2} d^{2} e^{3}}{3} + \frac{5 B a^{2} d e^{4}}{6} + \frac{10 B a b d^{2} e^{3}}{3} + \frac{5 B b^{2} d^{3} e^{2}}{3}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 2 A b^{2} d^{3} e^{2} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + B b^{2} d^{4} e\right ) + x^{4} \left (\frac{5 A a^{2} d^{2} e^{3}}{2} + 5 A a b d^{3} e^{2} + \frac{5 A b^{2} d^{4} e}{4} + \frac{5 B a^{2} d^{3} e^{2}}{2} + \frac{5 B a b d^{4} e}{2} + \frac{B b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac{10 A a^{2} d^{3} e^{2}}{3} + \frac{10 A a b d^{4} e}{3} + \frac{A b^{2} d^{5}}{3} + \frac{5 B a^{2} d^{4} e}{3} + \frac{2 B a b d^{5}}{3}\right ) + x^{2} \left (\frac{5 A a^{2} d^{4} e}{2} + A a b d^{5} + \frac{B a^{2} d^{5}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

A*a**2*d**5*x + B*b**2*e**5*x**9/9 + x**8*(A*b**2*e**5/8 + B*a*b*e**5/4 + 5*B*b*
*2*d*e**4/8) + x**7*(2*A*a*b*e**5/7 + 5*A*b**2*d*e**4/7 + B*a**2*e**5/7 + 10*B*a
*b*d*e**4/7 + 10*B*b**2*d**2*e**3/7) + x**6*(A*a**2*e**5/6 + 5*A*a*b*d*e**4/3 +
5*A*b**2*d**2*e**3/3 + 5*B*a**2*d*e**4/6 + 10*B*a*b*d**2*e**3/3 + 5*B*b**2*d**3*
e**2/3) + x**5*(A*a**2*d*e**4 + 4*A*a*b*d**2*e**3 + 2*A*b**2*d**3*e**2 + 2*B*a**
2*d**2*e**3 + 4*B*a*b*d**3*e**2 + B*b**2*d**4*e) + x**4*(5*A*a**2*d**2*e**3/2 +
5*A*a*b*d**3*e**2 + 5*A*b**2*d**4*e/4 + 5*B*a**2*d**3*e**2/2 + 5*B*a*b*d**4*e/2
+ B*b**2*d**5/4) + x**3*(10*A*a**2*d**3*e**2/3 + 10*A*a*b*d**4*e/3 + A*b**2*d**5
/3 + 5*B*a**2*d**4*e/3 + 2*B*a*b*d**5/3) + x**2*(5*A*a**2*d**4*e/2 + A*a*b*d**5
+ B*a**2*d**5/2)

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GIAC/XCAS [A]  time = 0.275623, size = 601, normalized size = 5.01 \[ \frac{1}{9} \, B b^{2} x^{9} e^{5} + \frac{5}{8} \, B b^{2} d x^{8} e^{4} + \frac{10}{7} \, B b^{2} d^{2} x^{7} e^{3} + \frac{5}{3} \, B b^{2} d^{3} x^{6} e^{2} + B b^{2} d^{4} x^{5} e + \frac{1}{4} \, B b^{2} d^{5} x^{4} + \frac{1}{4} \, B a b x^{8} e^{5} + \frac{1}{8} \, A b^{2} x^{8} e^{5} + \frac{10}{7} \, B a b d x^{7} e^{4} + \frac{5}{7} \, A b^{2} d x^{7} e^{4} + \frac{10}{3} \, B a b d^{2} x^{6} e^{3} + \frac{5}{3} \, A b^{2} d^{2} x^{6} e^{3} + 4 \, B a b d^{3} x^{5} e^{2} + 2 \, A b^{2} d^{3} x^{5} e^{2} + \frac{5}{2} \, B a b d^{4} x^{4} e + \frac{5}{4} \, A b^{2} d^{4} x^{4} e + \frac{2}{3} \, B a b d^{5} x^{3} + \frac{1}{3} \, A b^{2} d^{5} x^{3} + \frac{1}{7} \, B a^{2} x^{7} e^{5} + \frac{2}{7} \, A a b x^{7} e^{5} + \frac{5}{6} \, B a^{2} d x^{6} e^{4} + \frac{5}{3} \, A a b d x^{6} e^{4} + 2 \, B a^{2} d^{2} x^{5} e^{3} + 4 \, A a b d^{2} x^{5} e^{3} + \frac{5}{2} \, B a^{2} d^{3} x^{4} e^{2} + 5 \, A a b d^{3} x^{4} e^{2} + \frac{5}{3} \, B a^{2} d^{4} x^{3} e + \frac{10}{3} \, A a b d^{4} x^{3} e + \frac{1}{2} \, B a^{2} d^{5} x^{2} + A a b d^{5} x^{2} + \frac{1}{6} \, A a^{2} x^{6} e^{5} + A a^{2} d x^{5} e^{4} + \frac{5}{2} \, A a^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, A a^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, A a^{2} d^{4} x^{2} e + A a^{2} d^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^5,x, algorithm="giac")

[Out]

1/9*B*b^2*x^9*e^5 + 5/8*B*b^2*d*x^8*e^4 + 10/7*B*b^2*d^2*x^7*e^3 + 5/3*B*b^2*d^3
*x^6*e^2 + B*b^2*d^4*x^5*e + 1/4*B*b^2*d^5*x^4 + 1/4*B*a*b*x^8*e^5 + 1/8*A*b^2*x
^8*e^5 + 10/7*B*a*b*d*x^7*e^4 + 5/7*A*b^2*d*x^7*e^4 + 10/3*B*a*b*d^2*x^6*e^3 + 5
/3*A*b^2*d^2*x^6*e^3 + 4*B*a*b*d^3*x^5*e^2 + 2*A*b^2*d^3*x^5*e^2 + 5/2*B*a*b*d^4
*x^4*e + 5/4*A*b^2*d^4*x^4*e + 2/3*B*a*b*d^5*x^3 + 1/3*A*b^2*d^5*x^3 + 1/7*B*a^2
*x^7*e^5 + 2/7*A*a*b*x^7*e^5 + 5/6*B*a^2*d*x^6*e^4 + 5/3*A*a*b*d*x^6*e^4 + 2*B*a
^2*d^2*x^5*e^3 + 4*A*a*b*d^2*x^5*e^3 + 5/2*B*a^2*d^3*x^4*e^2 + 5*A*a*b*d^3*x^4*e
^2 + 5/3*B*a^2*d^4*x^3*e + 10/3*A*a*b*d^4*x^3*e + 1/2*B*a^2*d^5*x^2 + A*a*b*d^5*
x^2 + 1/6*A*a^2*x^6*e^5 + A*a^2*d*x^5*e^4 + 5/2*A*a^2*d^2*x^4*e^3 + 10/3*A*a^2*d
^3*x^3*e^2 + 5/2*A*a^2*d^4*x^2*e + A*a^2*d^5*x