3.1678 \(\int (A+B x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=118 \[ \frac {e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac {(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac {B e^2 (a+b x)^8}{8 b^4} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \[ \frac {e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac {(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac {B e^2 (a+b x)^8}{8 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

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Mathematica [B]  time = 0.09, size = 288, normalized size = 2.44 \[ a^4 A d^2 x+\frac {1}{2} a^3 d x^2 (2 a A e+a B d+4 A b d)+\frac {1}{5} b x^5 \left (A b \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+4 a B \left (a^2 e^2+3 a b d e+b^2 d^2\right )\right )+\frac {1}{4} a x^4 \left (4 A b \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )\right )+\frac {1}{3} a^2 x^3 \left (A \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+2 a B d (a e+2 b d)\right )+\frac {1}{6} b^2 x^6 \left (6 a^2 B e^2+4 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+\frac {1}{7} b^3 e x^7 (4 a B e+A b e+2 b B d)+\frac {1}{8} b^4 B e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.68, size = 374, normalized size = 3.17 \[ \frac {1}{8} x^{8} e^{2} b^{4} B + \frac {2}{7} x^{7} e d b^{4} B + \frac {4}{7} x^{7} e^{2} b^{3} a B + \frac {1}{7} x^{7} e^{2} b^{4} A + \frac {1}{6} x^{6} d^{2} b^{4} B + \frac {4}{3} x^{6} e d b^{3} a B + x^{6} e^{2} b^{2} a^{2} B + \frac {1}{3} x^{6} e d b^{4} A + \frac {2}{3} x^{6} e^{2} b^{3} a A + \frac {4}{5} x^{5} d^{2} b^{3} a B + \frac {12}{5} x^{5} e d b^{2} a^{2} B + \frac {4}{5} x^{5} e^{2} b a^{3} B + \frac {1}{5} x^{5} d^{2} b^{4} A + \frac {8}{5} x^{5} e d b^{3} a A + \frac {6}{5} x^{5} e^{2} b^{2} a^{2} A + \frac {3}{2} x^{4} d^{2} b^{2} a^{2} B + 2 x^{4} e d b a^{3} B + \frac {1}{4} x^{4} e^{2} a^{4} B + x^{4} d^{2} b^{3} a A + 3 x^{4} e d b^{2} a^{2} A + x^{4} e^{2} b a^{3} A + \frac {4}{3} x^{3} d^{2} b a^{3} B + \frac {2}{3} x^{3} e d a^{4} B + 2 x^{3} d^{2} b^{2} a^{2} A + \frac {8}{3} x^{3} e d b a^{3} A + \frac {1}{3} x^{3} e^{2} a^{4} A + \frac {1}{2} x^{2} d^{2} a^{4} B + 2 x^{2} d^{2} b a^{3} A + x^{2} e d a^{4} A + x d^{2} a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 374, normalized size = 3.17 \[ \frac {1}{8} \, B b^{4} x^{8} e^{2} + \frac {2}{7} \, B b^{4} d x^{7} e + \frac {1}{6} \, B b^{4} d^{2} x^{6} + \frac {4}{7} \, B a b^{3} x^{7} e^{2} + \frac {1}{7} \, A b^{4} x^{7} e^{2} + \frac {4}{3} \, B a b^{3} d x^{6} e + \frac {1}{3} \, A b^{4} d x^{6} e + \frac {4}{5} \, B a b^{3} d^{2} x^{5} + \frac {1}{5} \, A b^{4} d^{2} x^{5} + B a^{2} b^{2} x^{6} e^{2} + \frac {2}{3} \, A a b^{3} x^{6} e^{2} + \frac {12}{5} \, B a^{2} b^{2} d x^{5} e + \frac {8}{5} \, A a b^{3} d x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d^{2} x^{4} + A a b^{3} d^{2} x^{4} + \frac {4}{5} \, B a^{3} b x^{5} e^{2} + \frac {6}{5} \, A a^{2} b^{2} x^{5} e^{2} + 2 \, B a^{3} b d x^{4} e + 3 \, A a^{2} b^{2} d x^{4} e + \frac {4}{3} \, B a^{3} b d^{2} x^{3} + 2 \, A a^{2} b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{4} x^{4} e^{2} + A a^{3} b x^{4} e^{2} + \frac {2}{3} \, B a^{4} d x^{3} e + \frac {8}{3} \, A a^{3} b d x^{3} e + \frac {1}{2} \, B a^{4} d^{2} x^{2} + 2 \, A a^{3} b d^{2} x^{2} + \frac {1}{3} \, A a^{4} x^{3} e^{2} + A a^{4} d x^{2} e + A a^{4} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 305, normalized size = 2.58 \[ \frac {B \,b^{4} e^{2} x^{8}}{8}+A \,a^{4} d^{2} x +\frac {\left (4 B a \,b^{3} e^{2}+\left (A \,e^{2}+2 B d e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (6 B \,a^{2} b^{2} e^{2}+4 \left (A \,e^{2}+2 B d e \right ) a \,b^{3}+\left (2 A d e +B \,d^{2}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{2}+4 B \,a^{3} b \,e^{2}+6 \left (A \,e^{2}+2 B d e \right ) a^{2} b^{2}+4 \left (2 A d e +B \,d^{2}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{2}+B \,a^{4} e^{2}+4 \left (A \,e^{2}+2 B d e \right ) a^{3} b +6 \left (2 A d e +B \,d^{2}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{2}+\left (A \,e^{2}+2 B d e \right ) a^{4}+4 \left (2 A d e +B \,d^{2}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{2}+\left (2 A d e +B \,d^{2}\right ) a^{4}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.52, size = 322, normalized size = 2.73 \[ \frac {1}{8} \, B b^{4} e^{2} x^{8} + A a^{4} d^{2} x + \frac {1}{7} \, {\left (2 \, B b^{4} d e + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{4} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{4} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.06, size = 305, normalized size = 2.58 \[ x^4\,\left (\frac {B\,a^4\,e^2}{4}+2\,B\,a^3\,b\,d\,e+A\,a^3\,b\,e^2+\frac {3\,B\,a^2\,b^2\,d^2}{2}+3\,A\,a^2\,b^2\,d\,e+A\,a\,b^3\,d^2\right )+x^5\,\left (\frac {4\,B\,a^3\,b\,e^2}{5}+\frac {12\,B\,a^2\,b^2\,d\,e}{5}+\frac {6\,A\,a^2\,b^2\,e^2}{5}+\frac {4\,B\,a\,b^3\,d^2}{5}+\frac {8\,A\,a\,b^3\,d\,e}{5}+\frac {A\,b^4\,d^2}{5}\right )+x^3\,\left (\frac {2\,B\,a^4\,d\,e}{3}+\frac {A\,a^4\,e^2}{3}+\frac {4\,B\,a^3\,b\,d^2}{3}+\frac {8\,A\,a^3\,b\,d\,e}{3}+2\,A\,a^2\,b^2\,d^2\right )+x^6\,\left (B\,a^2\,b^2\,e^2+\frac {4\,B\,a\,b^3\,d\,e}{3}+\frac {2\,A\,a\,b^3\,e^2}{3}+\frac {B\,b^4\,d^2}{6}+\frac {A\,b^4\,d\,e}{3}\right )+A\,a^4\,d^2\,x+\frac {a^3\,d\,x^2\,\left (2\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e\,x^7\,\left (A\,b\,e+4\,B\,a\,e+2\,B\,b\,d\right )}{7}+\frac {B\,b^4\,e^2\,x^8}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.12, size = 384, normalized size = 3.25 \[ A a^{4} d^{2} x + \frac {B b^{4} e^{2} x^{8}}{8} + x^{7} \left (\frac {A b^{4} e^{2}}{7} + \frac {4 B a b^{3} e^{2}}{7} + \frac {2 B b^{4} d e}{7}\right ) + x^{6} \left (\frac {2 A a b^{3} e^{2}}{3} + \frac {A b^{4} d e}{3} + B a^{2} b^{2} e^{2} + \frac {4 B a b^{3} d e}{3} + \frac {B b^{4} d^{2}}{6}\right ) + x^{5} \left (\frac {6 A a^{2} b^{2} e^{2}}{5} + \frac {8 A a b^{3} d e}{5} + \frac {A b^{4} d^{2}}{5} + \frac {4 B a^{3} b e^{2}}{5} + \frac {12 B a^{2} b^{2} d e}{5} + \frac {4 B a b^{3} d^{2}}{5}\right ) + x^{4} \left (A a^{3} b e^{2} + 3 A a^{2} b^{2} d e + A a b^{3} d^{2} + \frac {B a^{4} e^{2}}{4} + 2 B a^{3} b d e + \frac {3 B a^{2} b^{2} d^{2}}{2}\right ) + x^{3} \left (\frac {A a^{4} e^{2}}{3} + \frac {8 A a^{3} b d e}{3} + 2 A a^{2} b^{2} d^{2} + \frac {2 B a^{4} d e}{3} + \frac {4 B a^{3} b d^{2}}{3}\right ) + x^{2} \left (A a^{4} d e + 2 A a^{3} b d^{2} + \frac {B a^{4} d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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