3.1807 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 b^5 (d+e x)^{7/2} (-6 a B e-A b e+7 b B d)}{7 e^8}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{3/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{10 b^2 \sqrt{d+e x} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{6 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 \sqrt{d+e x}}-\frac{2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^{3/2}}+\frac{2 (b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac{2 b^6 B (d+e x)^{9/2}}{9 e^8} \]

[Out]

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

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Rubi [A]  time = 0.146419, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {27, 77} \[ -\frac{2 b^5 (d+e x)^{7/2} (-6 a B e-A b e+7 b B d)}{7 e^8}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{3/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{10 b^2 \sqrt{d+e x} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{6 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 \sqrt{d+e x}}-\frac{2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^{3/2}}+\frac{2 (b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac{2 b^6 B (d+e x)^{9/2}}{9 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

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

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

Mathematica [A]  time = 0.219486, size = 259, normalized size = 0.85 \[ \frac{2 \left (-45 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+189 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-525 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1575 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)+945 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)-105 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+63 (b d-a e)^6 (B d-A e)+35 b^6 B (d+e x)^7\right )}{315 e^8 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

(2*(63*(b*d - a*e)^6*(B*d - A*e) - 105*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) + 945*b*(b*d - a*e)
^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 1575*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^
3 - 525*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 189*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e -
5*a*B*e)*(d + e*x)^5 - 45*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 35*b^6*B*(d + e*x)^7))/(315*e^8*(d +
e*x)^(5/2))

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Maple [B]  time = 0.008, size = 913, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(7/2),x)

[Out]

-2/315*(-35*B*b^6*e^7*x^7-45*A*b^6*e^7*x^6-270*B*a*b^5*e^7*x^6+70*B*b^6*d*e^6*x^6-378*A*a*b^5*e^7*x^5+108*A*b^
6*d*e^6*x^5-945*B*a^2*b^4*e^7*x^5+648*B*a*b^5*d*e^6*x^5-168*B*b^6*d^2*e^5*x^5-1575*A*a^2*b^4*e^7*x^4+1260*A*a*
b^5*d*e^6*x^4-360*A*b^6*d^2*e^5*x^4-2100*B*a^3*b^3*e^7*x^4+3150*B*a^2*b^4*d*e^6*x^4-2160*B*a*b^5*d^2*e^5*x^4+5
60*B*b^6*d^3*e^4*x^4-6300*A*a^3*b^3*e^7*x^3+12600*A*a^2*b^4*d*e^6*x^3-10080*A*a*b^5*d^2*e^5*x^3+2880*A*b^6*d^3
*e^4*x^3-4725*B*a^4*b^2*e^7*x^3+16800*B*a^3*b^3*d*e^6*x^3-25200*B*a^2*b^4*d^2*e^5*x^3+17280*B*a*b^5*d^3*e^4*x^
3-4480*B*b^6*d^4*e^3*x^3+4725*A*a^4*b^2*e^7*x^2-37800*A*a^3*b^3*d*e^6*x^2+75600*A*a^2*b^4*d^2*e^5*x^2-60480*A*
a*b^5*d^3*e^4*x^2+17280*A*b^6*d^4*e^3*x^2+1890*B*a^5*b*e^7*x^2-28350*B*a^4*b^2*d*e^6*x^2+100800*B*a^3*b^3*d^2*
e^5*x^2-151200*B*a^2*b^4*d^3*e^4*x^2+103680*B*a*b^5*d^4*e^3*x^2-26880*B*b^6*d^5*e^2*x^2+630*A*a^5*b*e^7*x+6300
*A*a^4*b^2*d*e^6*x-50400*A*a^3*b^3*d^2*e^5*x+100800*A*a^2*b^4*d^3*e^4*x-80640*A*a*b^5*d^4*e^3*x+23040*A*b^6*d^
5*e^2*x+105*B*a^6*e^7*x+2520*B*a^5*b*d*e^6*x-37800*B*a^4*b^2*d^2*e^5*x+134400*B*a^3*b^3*d^3*e^4*x-201600*B*a^2
*b^4*d^4*e^3*x+138240*B*a*b^5*d^5*e^2*x-35840*B*b^6*d^6*e*x+63*A*a^6*e^7+252*A*a^5*b*d*e^6+2520*A*a^4*b^2*d^2*
e^5-20160*A*a^3*b^3*d^3*e^4+40320*A*a^2*b^4*d^4*e^3-32256*A*a*b^5*d^5*e^2+9216*A*b^6*d^6*e+42*B*a^6*d*e^6+1008
*B*a^5*b*d^2*e^5-15120*B*a^4*b^2*d^3*e^4+53760*B*a^3*b^3*d^4*e^3-80640*B*a^2*b^4*d^5*e^2+55296*B*a*b^5*d^6*e-1
4336*B*b^6*d^7)/(e*x+d)^(5/2)/e^8

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Maxima [B]  time = 1.01085, size = 1046, normalized size = 3.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*B*b^6 - 45*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(7/2) + 189*(7*B*b^6*d^2 -
2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(5/2) - 525*(7*B*b^6*d^3 - 3*(6*B*a*b^5
+ A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x + d)^(3/2) + 1575*(
7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*sqrt(e*x + d))/e^7 + 21*(3*B*b^6*d^7 - 3*A*a^6*e^7 - 3*(6*B*a*b^5 +
A*b^6)*d^6*e + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d^3*e^4 - 9*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 3*(B*a^6 + 6*A*a^5*b)*d*e^6 + 45*(7*B*b^6*d^5
- 5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3
+ 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^2 - 5*(7*B*b^6*d^6 - 6*(6*B*a
*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*
a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d))/((e*x
+ d)^(5/2)*e^7))/e

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Fricas [B]  time = 1.46242, size = 1752, normalized size = 5.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^6*e^7*x^7 + 14336*B*b^6*d^7 - 63*A*a^6*e^7 - 9216*(6*B*a*b^5 + A*b^6)*d^6*e + 16128*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^5*e^2 - 13440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5040*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 -
504*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 42*(B*a^6 + 6*A*a^5*b)*d*e^6 - 5*(14*B*b^6*d*e^6 - 9*(6*B*a*b^5 + A*b
^6)*e^7)*x^6 + 3*(56*B*b^6*d^2*e^5 - 36*(6*B*a*b^5 + A*b^6)*d*e^6 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 5*
(112*B*b^6*d^3*e^4 - 72*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 126*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 105*(4*B*a^3*b^3 +
3*A*a^2*b^4)*e^7)*x^4 + 5*(896*B*b^6*d^4*e^3 - 576*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 1008*(5*B*a^2*b^4 + 2*A*a*b^
5)*d^2*e^5 - 840*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 315*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 15*(1792*B*b^6
*d^5*e^2 - 1152*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 2016*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 1680*(4*B*a^3*b^3 + 3*A
*a^2*b^4)*d^2*e^5 + 630*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 63*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 5*(7168*B*
b^6*d^6*e - 4608*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 8064*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 6720*(4*B*a^3*b^3 + 3*
A*a^2*b^4)*d^3*e^4 + 2520*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 21*(B*a^
6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.20072, size = 1489, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^6*e^64 - 315*(x*e + d)^(7/2)*B*b^6*d*e^64 + 1323*(x*e + d)^(5/2)*B*b^6*d^2*e^64
- 3675*(x*e + d)^(3/2)*B*b^6*d^3*e^64 + 11025*sqrt(x*e + d)*B*b^6*d^4*e^64 + 270*(x*e + d)^(7/2)*B*a*b^5*e^65
+ 45*(x*e + d)^(7/2)*A*b^6*e^65 - 2268*(x*e + d)^(5/2)*B*a*b^5*d*e^65 - 378*(x*e + d)^(5/2)*A*b^6*d*e^65 + 945
0*(x*e + d)^(3/2)*B*a*b^5*d^2*e^65 + 1575*(x*e + d)^(3/2)*A*b^6*d^2*e^65 - 37800*sqrt(x*e + d)*B*a*b^5*d^3*e^6
5 - 6300*sqrt(x*e + d)*A*b^6*d^3*e^65 + 945*(x*e + d)^(5/2)*B*a^2*b^4*e^66 + 378*(x*e + d)^(5/2)*A*a*b^5*e^66
- 7875*(x*e + d)^(3/2)*B*a^2*b^4*d*e^66 - 3150*(x*e + d)^(3/2)*A*a*b^5*d*e^66 + 47250*sqrt(x*e + d)*B*a^2*b^4*
d^2*e^66 + 18900*sqrt(x*e + d)*A*a*b^5*d^2*e^66 + 2100*(x*e + d)^(3/2)*B*a^3*b^3*e^67 + 1575*(x*e + d)^(3/2)*A
*a^2*b^4*e^67 - 25200*sqrt(x*e + d)*B*a^3*b^3*d*e^67 - 18900*sqrt(x*e + d)*A*a^2*b^4*d*e^67 + 4725*sqrt(x*e +
d)*B*a^4*b^2*e^68 + 6300*sqrt(x*e + d)*A*a^3*b^3*e^68)*e^(-72) + 2/15*(315*(x*e + d)^2*B*b^6*d^5 - 35*(x*e + d
)*B*b^6*d^6 + 3*B*b^6*d^7 - 1350*(x*e + d)^2*B*a*b^5*d^4*e - 225*(x*e + d)^2*A*b^6*d^4*e + 180*(x*e + d)*B*a*b
^5*d^5*e + 30*(x*e + d)*A*b^6*d^5*e - 18*B*a*b^5*d^6*e - 3*A*b^6*d^6*e + 2250*(x*e + d)^2*B*a^2*b^4*d^3*e^2 +
900*(x*e + d)^2*A*a*b^5*d^3*e^2 - 375*(x*e + d)*B*a^2*b^4*d^4*e^2 - 150*(x*e + d)*A*a*b^5*d^4*e^2 + 45*B*a^2*b
^4*d^5*e^2 + 18*A*a*b^5*d^5*e^2 - 1800*(x*e + d)^2*B*a^3*b^3*d^2*e^3 - 1350*(x*e + d)^2*A*a^2*b^4*d^2*e^3 + 40
0*(x*e + d)*B*a^3*b^3*d^3*e^3 + 300*(x*e + d)*A*a^2*b^4*d^3*e^3 - 60*B*a^3*b^3*d^4*e^3 - 45*A*a^2*b^4*d^4*e^3
+ 675*(x*e + d)^2*B*a^4*b^2*d*e^4 + 900*(x*e + d)^2*A*a^3*b^3*d*e^4 - 225*(x*e + d)*B*a^4*b^2*d^2*e^4 - 300*(x
*e + d)*A*a^3*b^3*d^2*e^4 + 45*B*a^4*b^2*d^3*e^4 + 60*A*a^3*b^3*d^3*e^4 - 90*(x*e + d)^2*B*a^5*b*e^5 - 225*(x*
e + d)^2*A*a^4*b^2*e^5 + 60*(x*e + d)*B*a^5*b*d*e^5 + 150*(x*e + d)*A*a^4*b^2*d*e^5 - 18*B*a^5*b*d^2*e^5 - 45*
A*a^4*b^2*d^2*e^5 - 5*(x*e + d)*B*a^6*e^6 - 30*(x*e + d)*A*a^5*b*e^6 + 3*B*a^6*d*e^6 + 18*A*a^5*b*d*e^6 - 3*A*
a^6*e^7)*e^(-8)/(x*e + d)^(5/2)