3.11.50 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx\) [1050]

Optimal. Leaf size=163 \[ -\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{8 e^5 (d+e x)^8}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{7 e^5 (d+e x)^7}+\frac {b^2 (4 b B d-A b e-3 a B e)}{6 e^5 (d+e x)^6}-\frac {b^3 B}{5 e^5 (d+e x)^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 163, 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 = {78} \begin {gather*} \frac {b^2 (-3 a B e-A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (d+e x)^7}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (d+e x)^8}-\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}-\frac {b^3 B}{5 e^5 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

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

Rule 78

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 \frac {(a+b x)^3 (A+B x)}{(d+e x)^{10}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{10}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^9}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^8}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^7}+\frac {b^3 B}{e^4 (d+e x)^6}\right ) \, dx\\ &=-\frac {(b d-a e)^3 (B d-A e)}{9 e^5 (d+e x)^9}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{8 e^5 (d+e x)^8}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{7 e^5 (d+e x)^7}+\frac {b^2 (4 b B d-A b e-3 a B e)}{6 e^5 (d+e x)^6}-\frac {b^3 B}{5 e^5 (d+e x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 214, normalized size = 1.31 \begin {gather*} -\frac {35 a^3 e^3 (8 A e+B (d+9 e x))+15 a^2 b e^2 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a b^2 e \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+b^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )}{2520 e^5 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/2520*(35*a^3*e^3*(8*A*e + B*(d + 9*e*x)) + 15*a^2*b*e^2*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^
2)) + 15*a*b^2*e*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + b^3*
(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 +
126*e^4*x^4)))/(e^5*(d + e*x)^9)

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Maple [A]
time = 0.08, size = 281, normalized size = 1.72

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{5 e}-\frac {b^{2} \left (5 A b e +15 B a e +4 B b d \right ) x^{3}}{30 e^{2}}-\frac {b \left (30 A a b \,e^{2}+5 A \,b^{2} d e +30 B \,a^{2} e^{2}+15 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{70 e^{3}}-\frac {\left (105 A \,a^{2} b \,e^{3}+30 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +35 B \,a^{3} e^{3}+30 B \,a^{2} b d \,e^{2}+15 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{280 e^{4}}-\frac {280 a^{3} A \,e^{4}+105 A \,a^{2} b d \,e^{3}+30 A a \,b^{2} d^{2} e^{2}+5 A \,b^{3} d^{3} e +35 B \,a^{3} d \,e^{3}+30 B \,a^{2} b \,d^{2} e^{2}+15 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{2520 e^{5}}}{\left (e x +d \right )^{9}}\)	\(270\)
default	\(-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{8 e^{5} \left (e x +d \right )^{8}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{9 e^{5} \left (e x +d \right )^{9}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{6 e^{5} \left (e x +d \right )^{6}}-\frac {b^{3} B}{5 e^{5} \left (e x +d \right )^{5}}\)	\(281\)
gosper	\(-\frac {504 b^{3} B \,x^{4} e^{4}+420 A \,b^{3} e^{4} x^{3}+1260 B a \,b^{2} e^{4} x^{3}+336 B \,b^{3} d \,e^{3} x^{3}+1080 A a \,b^{2} e^{4} x^{2}+180 A \,b^{3} d \,e^{3} x^{2}+1080 B \,a^{2} b \,e^{4} x^{2}+540 B a \,b^{2} d \,e^{3} x^{2}+144 B \,b^{3} d^{2} e^{2} x^{2}+945 A \,a^{2} b \,e^{4} x +270 A a \,b^{2} d \,e^{3} x +45 A \,b^{3} d^{2} e^{2} x +315 B \,a^{3} e^{4} x +270 B \,a^{2} b d \,e^{3} x +135 B a \,b^{2} d^{2} e^{2} x +36 B \,b^{3} d^{3} e x +280 a^{3} A \,e^{4}+105 A \,a^{2} b d \,e^{3}+30 A a \,b^{2} d^{2} e^{2}+5 A \,b^{3} d^{3} e +35 B \,a^{3} d \,e^{3}+30 B \,a^{2} b \,d^{2} e^{2}+15 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{2520 e^{5} \left (e x +d \right )^{9}}\)	\(301\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{5 e}-\frac {\left (5 A \,b^{3} e^{5}+15 B a \,b^{2} e^{5}+4 b^{3} B d \,e^{4}\right ) x^{3}}{30 e^{6}}-\frac {\left (30 a \,b^{2} A \,e^{6}+5 A \,b^{3} d \,e^{5}+30 a^{2} b B \,e^{6}+15 B a \,b^{2} d \,e^{5}+4 B \,b^{3} d^{2} e^{4}\right ) x^{2}}{70 e^{7}}-\frac {\left (105 A \,a^{2} b \,e^{7}+30 A a \,b^{2} d \,e^{6}+5 A \,b^{3} d^{2} e^{5}+35 B \,a^{3} e^{7}+30 B \,a^{2} b d \,e^{6}+15 B a \,b^{2} d^{2} e^{5}+4 B \,b^{3} d^{3} e^{4}\right ) x}{280 e^{8}}-\frac {280 a^{3} A \,e^{8}+105 A \,a^{2} b d \,e^{7}+30 A a \,b^{2} d^{2} e^{6}+5 A \,b^{3} d^{3} e^{5}+35 B \,a^{3} d \,e^{7}+30 B \,a^{2} b \,d^{2} e^{6}+15 B a \,b^{2} d^{3} e^{5}+4 B \,b^{3} d^{4} e^{4}}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (163) = 326\).
time = 0.32, size = 347, normalized size = 2.13 \begin {gather*} -\frac {504 \, B b^{3} x^{4} e^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 84 \, {\left (4 \, B b^{3} d e^{3} + 15 \, B a b^{2} e^{4} + 5 \, A b^{3} e^{4}\right )} x^{3} + 30 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} + 36 \, {\left (4 \, B b^{3} d^{2} e^{2} + 30 \, B a^{2} b e^{4} + 30 \, A a b^{2} e^{4} + 5 \, {\left (3 \, B a b^{2} e^{3} + A b^{3} e^{3}\right )} d\right )} x^{2} + 35 \, {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d + 9 \, {\left (4 \, B b^{3} d^{3} e + 35 \, B a^{3} e^{4} + 105 \, A a^{2} b e^{4} + 5 \, {\left (3 \, B a b^{2} e^{2} + A b^{3} e^{2}\right )} d^{2} + 30 \, {\left (B a^{2} b e^{3} + A a b^{2} e^{3}\right )} d\right )} x}{2520 \, {\left (x^{9} e^{14} + 9 \, d x^{8} e^{13} + 36 \, d^{2} x^{7} e^{12} + 84 \, d^{3} x^{6} e^{11} + 126 \, d^{4} x^{5} e^{10} + 126 \, d^{5} x^{4} e^{9} + 84 \, d^{6} x^{3} e^{8} + 36 \, d^{7} x^{2} e^{7} + 9 \, d^{8} x e^{6} + d^{9} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*x^4*e^4 + 4*B*b^3*d^4 + 280*A*a^3*e^4 + 5*(3*B*a*b^2*e + A*b^3*e)*d^3 + 84*(4*B*b^3*d*e^3 +
15*B*a*b^2*e^4 + 5*A*b^3*e^4)*x^3 + 30*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 + 36*(4*B*b^3*d^2*e^2 + 30*B*a^2*b*e^4
+ 30*A*a*b^2*e^4 + 5*(3*B*a*b^2*e^3 + A*b^3*e^3)*d)*x^2 + 35*(B*a^3*e^3 + 3*A*a^2*b*e^3)*d + 9*(4*B*b^3*d^3*e
+ 35*B*a^3*e^4 + 105*A*a^2*b*e^4 + 5*(3*B*a*b^2*e^2 + A*b^3*e^2)*d^2 + 30*(B*a^2*b*e^3 + A*a*b^2*e^3)*d)*x)/(
x^9*e^14 + 9*d*x^8*e^13 + 36*d^2*x^7*e^12 + 84*d^3*x^6*e^11 + 126*d^4*x^5*e^10 + 126*d^5*x^4*e^9 + 84*d^6*x^3*
e^8 + 36*d^7*x^2*e^7 + 9*d^8*x*e^6 + d^9*e^5)

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Fricas [A]
time = 0.66, size = 326, normalized size = 2.00 \begin {gather*} -\frac {4 \, B b^{3} d^{4} + {\left (504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} + {\left (336 \, B b^{3} d x^{3} + 180 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} + 270 \, {\left (B a^{2} b + A a b^{2}\right )} d x + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 3 \, {\left (48 \, B b^{3} d^{2} x^{2} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + 10 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} + {\left (36 \, B b^{3} d^{3} x + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e}{2520 \, {\left (x^{9} e^{14} + 9 \, d x^{8} e^{13} + 36 \, d^{2} x^{7} e^{12} + 84 \, d^{3} x^{6} e^{11} + 126 \, d^{4} x^{5} e^{10} + 126 \, d^{5} x^{4} e^{9} + 84 \, d^{6} x^{3} e^{8} + 36 \, d^{7} x^{2} e^{7} + 9 \, d^{8} x e^{6} + d^{9} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(4*B*b^3*d^4 + (504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2
+ 315*(B*a^3 + 3*A*a^2*b)*x)*e^4 + (336*B*b^3*d*x^3 + 180*(3*B*a*b^2 + A*b^3)*d*x^2 + 270*(B*a^2*b + A*a*b^2)
*d*x + 35*(B*a^3 + 3*A*a^2*b)*d)*e^3 + 3*(48*B*b^3*d^2*x^2 + 15*(3*B*a*b^2 + A*b^3)*d^2*x + 10*(B*a^2*b + A*a*
b^2)*d^2)*e^2 + (36*B*b^3*d^3*x + 5*(3*B*a*b^2 + A*b^3)*d^3)*e)/(x^9*e^14 + 9*d*x^8*e^13 + 36*d^2*x^7*e^12 + 8
4*d^3*x^6*e^11 + 126*d^4*x^5*e^10 + 126*d^5*x^4*e^9 + 84*d^6*x^3*e^8 + 36*d^7*x^2*e^7 + 9*d^8*x*e^6 + d^9*e^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**10,x)

[Out]

Timed out

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Giac [A]
time = 1.83, size = 283, normalized size = 1.74 \begin {gather*} -\frac {{\left (504 \, B b^{3} x^{4} e^{4} + 336 \, B b^{3} d x^{3} e^{3} + 144 \, B b^{3} d^{2} x^{2} e^{2} + 36 \, B b^{3} d^{3} x e + 4 \, B b^{3} d^{4} + 1260 \, B a b^{2} x^{3} e^{4} + 420 \, A b^{3} x^{3} e^{4} + 540 \, B a b^{2} d x^{2} e^{3} + 180 \, A b^{3} d x^{2} e^{3} + 135 \, B a b^{2} d^{2} x e^{2} + 45 \, A b^{3} d^{2} x e^{2} + 15 \, B a b^{2} d^{3} e + 5 \, A b^{3} d^{3} e + 1080 \, B a^{2} b x^{2} e^{4} + 1080 \, A a b^{2} x^{2} e^{4} + 270 \, B a^{2} b d x e^{3} + 270 \, A a b^{2} d x e^{3} + 30 \, B a^{2} b d^{2} e^{2} + 30 \, A a b^{2} d^{2} e^{2} + 315 \, B a^{3} x e^{4} + 945 \, A a^{2} b x e^{4} + 35 \, B a^{3} d e^{3} + 105 \, A a^{2} b d e^{3} + 280 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{2520 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*x^4*e^4 + 336*B*b^3*d*x^3*e^3 + 144*B*b^3*d^2*x^2*e^2 + 36*B*b^3*d^3*x*e + 4*B*b^3*d^4 + 12
60*B*a*b^2*x^3*e^4 + 420*A*b^3*x^3*e^4 + 540*B*a*b^2*d*x^2*e^3 + 180*A*b^3*d*x^2*e^3 + 135*B*a*b^2*d^2*x*e^2 +
45*A*b^3*d^2*x*e^2 + 15*B*a*b^2*d^3*e + 5*A*b^3*d^3*e + 1080*B*a^2*b*x^2*e^4 + 1080*A*a*b^2*x^2*e^4 + 270*B*a
^2*b*d*x*e^3 + 270*A*a*b^2*d*x*e^3 + 30*B*a^2*b*d^2*e^2 + 30*A*a*b^2*d^2*e^2 + 315*B*a^3*x*e^4 + 945*A*a^2*b*x
*e^4 + 35*B*a^3*d*e^3 + 105*A*a^2*b*d*e^3 + 280*A*a^3*e^4)*e^(-5)/(x*e + d)^9

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Mupad [B]
time = 1.17, size = 358, normalized size = 2.20 \begin {gather*} -\frac {\frac {35\,B\,a^3\,d\,e^3+280\,A\,a^3\,e^4+30\,B\,a^2\,b\,d^2\,e^2+105\,A\,a^2\,b\,d\,e^3+15\,B\,a\,b^2\,d^3\,e+30\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+5\,A\,b^3\,d^3\,e}{2520\,e^5}+\frac {x\,\left (35\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+15\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{280\,e^4}+\frac {b^2\,x^3\,\left (5\,A\,b\,e+15\,B\,a\,e+4\,B\,b\,d\right )}{30\,e^2}+\frac {b\,x^2\,\left (30\,B\,a^2\,e^2+15\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{70\,e^3}+\frac {B\,b^3\,x^4}{5\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^10,x)

[Out]

-((280*A*a^3*e^4 + 4*B*b^3*d^4 + 5*A*b^3*d^3*e + 35*B*a^3*d*e^3 + 30*A*a*b^2*d^2*e^2 + 30*B*a^2*b*d^2*e^2 + 10
5*A*a^2*b*d*e^3 + 15*B*a*b^2*d^3*e)/(2520*e^5) + (x*(35*B*a^3*e^3 + 4*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^
2*e + 30*A*a*b^2*d*e^2 + 15*B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(280*e^4) + (b^2*x^3*(5*A*b*e + 15*B*a*e + 4*B*
b*d))/(30*e^2) + (b*x^2*(30*B*a^2*e^2 + 4*B*b^2*d^2 + 30*A*a*b*e^2 + 5*A*b^2*d*e + 15*B*a*b*d*e))/(70*e^3) + (
B*b^3*x^4)/(5*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e
^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)

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