3.1022 \(\int (a+b x)^2 (A+B x) (d+e x)^4 \, dx\)

Optimal. Leaf size=120 \[ -\frac {b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac {(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac {(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^4}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^5}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^6}{e^3}+\frac {b^2 B (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4}\\ \end {align*}

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Mathematica [B]  time = 0.10, size = 283, normalized size = 2.36 \[ \frac {1}{5} e x^5 \left (a^2 e^2 (A e+4 B d)+4 a b d e (2 A e+3 B d)+2 b^2 d^2 (3 A e+2 B d)\right )+\frac {1}{4} d x^4 \left (2 a^2 e^2 (2 A e+3 B d)+4 a b d e (3 A e+2 B d)+b^2 d^2 (4 A e+B d)\right )+\frac {1}{3} d^2 x^3 \left (A \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac {1}{6} e^2 x^6 \left (a^2 B e^2+2 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+a^2 A d^4 x+\frac {1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac {1}{7} b e^3 x^7 (2 a B e+A b e+4 b B d)+\frac {1}{8} b^2 B e^4 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.88, size = 374, normalized size = 3.12 \[ \frac {1}{8} x^{8} e^{4} b^{2} B + \frac {4}{7} x^{7} e^{3} d b^{2} B + \frac {2}{7} x^{7} e^{4} b a B + \frac {1}{7} x^{7} e^{4} b^{2} A + x^{6} e^{2} d^{2} b^{2} B + \frac {4}{3} x^{6} e^{3} d b a B + \frac {1}{6} x^{6} e^{4} a^{2} B + \frac {2}{3} x^{6} e^{3} d b^{2} A + \frac {1}{3} x^{6} e^{4} b a A + \frac {4}{5} x^{5} e d^{3} b^{2} B + \frac {12}{5} x^{5} e^{2} d^{2} b a B + \frac {4}{5} x^{5} e^{3} d a^{2} B + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac {8}{5} x^{5} e^{3} d b a A + \frac {1}{5} x^{5} e^{4} a^{2} A + \frac {1}{4} x^{4} d^{4} b^{2} B + 2 x^{4} e d^{3} b a B + \frac {3}{2} x^{4} e^{2} d^{2} a^{2} B + x^{4} e d^{3} b^{2} A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac {2}{3} x^{3} d^{4} b a B + \frac {4}{3} x^{3} e d^{3} a^{2} B + \frac {1}{3} x^{3} d^{4} b^{2} A + \frac {8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac {1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e^4*b^2*B + 4/7*x^7*e^3*d*b^2*B + 2/7*x^7*e^4*b*a*B + 1/7*x^7*e^4*b^2*A + x^6*e^2*d^2*b^2*B + 4/3*x^6*
e^3*d*b*a*B + 1/6*x^6*e^4*a^2*B + 2/3*x^6*e^3*d*b^2*A + 1/3*x^6*e^4*b*a*A + 4/5*x^5*e*d^3*b^2*B + 12/5*x^5*e^2
*d^2*b*a*B + 4/5*x^5*e^3*d*a^2*B + 6/5*x^5*e^2*d^2*b^2*A + 8/5*x^5*e^3*d*b*a*A + 1/5*x^5*e^4*a^2*A + 1/4*x^4*d
^4*b^2*B + 2*x^4*e*d^3*b*a*B + 3/2*x^4*e^2*d^2*a^2*B + x^4*e*d^3*b^2*A + 3*x^4*e^2*d^2*b*a*A + x^4*e^3*d*a^2*A
+ 2/3*x^3*d^4*b*a*B + 4/3*x^3*e*d^3*a^2*B + 1/3*x^3*d^4*b^2*A + 8/3*x^3*e*d^3*b*a*A + 2*x^3*e^2*d^2*a^2*A + 1
/2*x^2*d^4*a^2*B + x^2*d^4*b*a*A + 2*x^2*e*d^3*a^2*A + x*d^4*a^2*A

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giac [B]  time = 1.16, size = 362, normalized size = 3.02 \[ \frac {1}{8} \, B b^{2} x^{8} e^{4} + \frac {4}{7} \, B b^{2} d x^{7} e^{3} + B b^{2} d^{2} x^{6} e^{2} + \frac {4}{5} \, B b^{2} d^{3} x^{5} e + \frac {1}{4} \, B b^{2} d^{4} x^{4} + \frac {2}{7} \, B a b x^{7} e^{4} + \frac {1}{7} \, A b^{2} x^{7} e^{4} + \frac {4}{3} \, B a b d x^{6} e^{3} + \frac {2}{3} \, A b^{2} d x^{6} e^{3} + \frac {12}{5} \, B a b d^{2} x^{5} e^{2} + \frac {6}{5} \, A b^{2} d^{2} x^{5} e^{2} + 2 \, B a b d^{3} x^{4} e + A b^{2} d^{3} x^{4} e + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {1}{6} \, B a^{2} x^{6} e^{4} + \frac {1}{3} \, A a b x^{6} e^{4} + \frac {4}{5} \, B a^{2} d x^{5} e^{3} + \frac {8}{5} \, A a b d x^{5} e^{3} + \frac {3}{2} \, B a^{2} d^{2} x^{4} e^{2} + 3 \, A a b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{2} d^{3} x^{3} e + \frac {8}{3} \, A a b d^{3} x^{3} e + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + \frac {1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^2*x^8*e^4 + 4/7*B*b^2*d*x^7*e^3 + B*b^2*d^2*x^6*e^2 + 4/5*B*b^2*d^3*x^5*e + 1/4*B*b^2*d^4*x^4 + 2/7*B*
a*b*x^7*e^4 + 1/7*A*b^2*x^7*e^4 + 4/3*B*a*b*d*x^6*e^3 + 2/3*A*b^2*d*x^6*e^3 + 12/5*B*a*b*d^2*x^5*e^2 + 6/5*A*b
^2*d^2*x^5*e^2 + 2*B*a*b*d^3*x^4*e + A*b^2*d^3*x^4*e + 2/3*B*a*b*d^4*x^3 + 1/3*A*b^2*d^4*x^3 + 1/6*B*a^2*x^6*e
^4 + 1/3*A*a*b*x^6*e^4 + 4/5*B*a^2*d*x^5*e^3 + 8/5*A*a*b*d*x^5*e^3 + 3/2*B*a^2*d^2*x^4*e^2 + 3*A*a*b*d^2*x^4*e
^2 + 4/3*B*a^2*d^3*x^3*e + 8/3*A*a*b*d^3*x^3*e + 1/2*B*a^2*d^4*x^2 + A*a*b*d^4*x^2 + 1/5*A*a^2*x^5*e^4 + A*a^2
*d*x^4*e^3 + 2*A*a^2*d^2*x^3*e^2 + 2*A*a^2*d^3*x^2*e + A*a^2*d^4*x

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maple [B]  time = 0.00, size = 305, normalized size = 2.54 \[ \frac {B \,b^{2} e^{4} x^{8}}{8}+A \,a^{2} d^{4} x +\frac {\left (4 B \,b^{2} d \,e^{3}+\left (A \,b^{2}+2 B a b \right ) e^{4}\right ) x^{7}}{7}+\frac {\left (6 B \,b^{2} d^{2} e^{2}+4 \left (A \,b^{2}+2 B a b \right ) d \,e^{3}+\left (2 A a b +B \,a^{2}\right ) e^{4}\right ) x^{6}}{6}+\frac {\left (A \,a^{2} e^{4}+4 B \,b^{2} d^{3} e +6 \left (A \,b^{2}+2 B a b \right ) d^{2} e^{2}+4 \left (2 A a b +B \,a^{2}\right ) d \,e^{3}\right ) x^{5}}{5}+\frac {\left (4 A \,a^{2} d \,e^{3}+B \,b^{2} d^{4}+4 \left (A \,b^{2}+2 B a b \right ) d^{3} e +6 \left (2 A a b +B \,a^{2}\right ) d^{2} e^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} d^{2} e^{2}+\left (A \,b^{2}+2 B a b \right ) d^{4}+4 \left (2 A a b +B \,a^{2}\right ) d^{3} e \right ) x^{3}}{3}+\frac {\left (4 A \,a^{2} d^{3} e +\left (2 A a b +B \,a^{2}\right ) d^{4}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.48, size = 304, normalized size = 2.53 \[ \frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 305, normalized size = 2.54 \[ x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}\right )+x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^3\,x^7\,\left (A\,b\,e+2\,B\,a\,e+4\,B\,b\,d\right )}{7}+\frac {B\,b^2\,e^4\,x^8}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((B*b^2*d^4)/4 + A*a^2*d*e^3 + A*b^2*d^3*e + (3*B*a^2*d^2*e^2)/2 + 2*B*a*b*d^3*e + 3*A*a*b*d^2*e^2) + x^5*
((A*a^2*e^4)/5 + (4*B*a^2*d*e^3)/5 + (4*B*b^2*d^3*e)/5 + (6*A*b^2*d^2*e^2)/5 + (8*A*a*b*d*e^3)/5 + (12*B*a*b*d
^2*e^2)/5) + x^3*((A*b^2*d^4)/3 + (2*B*a*b*d^4)/3 + (4*B*a^2*d^3*e)/3 + 2*A*a^2*d^2*e^2 + (8*A*a*b*d^3*e)/3) +
x^6*((B*a^2*e^4)/6 + (A*a*b*e^4)/3 + (2*A*b^2*d*e^3)/3 + B*b^2*d^2*e^2 + (4*B*a*b*d*e^3)/3) + A*a^2*d^4*x + (
a*d^3*x^2*(4*A*a*e + 2*A*b*d + B*a*d))/2 + (b*e^3*x^7*(A*b*e + 2*B*a*e + 4*B*b*d))/7 + (B*b^2*e^4*x^8)/8

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sympy [B]  time = 0.12, size = 384, normalized size = 3.20 \[ A a^{2} d^{4} x + \frac {B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{2} e^{4}}{7} + \frac {2 B a b e^{4}}{7} + \frac {4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {2 A b^{2} d e^{3}}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*d**4*x + B*b**2*e**4*x**8/8 + x**7*(A*b**2*e**4/7 + 2*B*a*b*e**4/7 + 4*B*b**2*d*e**3/7) + x**6*(A*a*b*e
**4/3 + 2*A*b**2*d*e**3/3 + B*a**2*e**4/6 + 4*B*a*b*d*e**3/3 + B*b**2*d**2*e**2) + x**5*(A*a**2*e**4/5 + 8*A*a
*b*d*e**3/5 + 6*A*b**2*d**2*e**2/5 + 4*B*a**2*d*e**3/5 + 12*B*a*b*d**2*e**2/5 + 4*B*b**2*d**3*e/5) + x**4*(A*a
**2*d*e**3 + 3*A*a*b*d**2*e**2 + A*b**2*d**3*e + 3*B*a**2*d**2*e**2/2 + 2*B*a*b*d**3*e + B*b**2*d**4/4) + x**3
*(2*A*a**2*d**2*e**2 + 8*A*a*b*d**3*e/3 + A*b**2*d**4/3 + 4*B*a**2*d**3*e/3 + 2*B*a*b*d**4/3) + x**2*(2*A*a**2
*d**3*e + A*a*b*d**4 + B*a**2*d**4/2)

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