3.1806 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{5/2}} \, dx\)
Optimal. Leaf size=302 \[ -\frac{2 b^5 (d+e x)^{9/2} (-6 a B e-A b e+7 b B d)}{9 e^8}+\frac{6 b^4 (d+e x)^{7/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8}-\frac{2 b^3 (d+e x)^{5/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac{6 b \sqrt{d+e x} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}-\frac{2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 \sqrt{d+e x}}+\frac{2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}+\frac{2 b^6 B (d+e x)^{11/2}}{11 e^8} \]
[Out]
(2*(b*d - a*e)^6*(B*d - A*e))/(3*e^8*(d + e*x)^(3/2)) - (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(e^8*Sqr
t[d + e*x]) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Sqrt[d + e*x])/e^8 + (10*b^2*(b*d - a*e)^3*(7*b
*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(3/2))/(3*e^8) - (2*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e
*x)^(5/2))/e^8 + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (2*b^5*(7*b*B*d -
A*b*e - 6*a*B*e)*(d + e*x)^(9/2))/(9*e^8) + (2*b^6*B*(d + e*x)^(11/2))/(11*e^8)
________________________________________________________________________________________
Rubi [A] time = 0.145187, antiderivative size = 302, 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} \[ -\frac{2 b^5 (d+e x)^{9/2} (-6 a B e-A b e+7 b B d)}{9 e^8}+\frac{6 b^4 (d+e x)^{7/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{7 e^8}-\frac{2 b^3 (d+e x)^{5/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8}-\frac{6 b \sqrt{d+e x} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}-\frac{2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 \sqrt{d+e x}}+\frac{2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}+\frac{2 b^6 B (d+e x)^{11/2}}{11 e^8} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]
[Out]
(2*(b*d - a*e)^6*(B*d - A*e))/(3*e^8*(d + e*x)^(3/2)) - (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(e^8*Sqr
t[d + e*x]) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Sqrt[d + e*x])/e^8 + (10*b^2*(b*d - a*e)^3*(7*b
*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(3/2))/(3*e^8) - (2*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e
*x)^(5/2))/e^8 + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (2*b^5*(7*b*B*d -
A*b*e - 6*a*B*e)*(d + e*x)^(9/2))/(9*e^8) + (2*b^6*B*(d + e*x)^(11/2))/(11*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)^{5/2}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{5/2}}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^{3/2}}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 \sqrt{d+e x}}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) \sqrt{d+e x}}{e^7}+\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^{3/2}}{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)^{5/2}}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{7/2}}{e^7}+\frac{b^6 B (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac{2 (b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^{3/2}}-\frac{2 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{e^8 \sqrt{d+e x}}-\frac{6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \sqrt{d+e x}}{e^8}+\frac{10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{3/2}}{3 e^8}-\frac{2 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{5/2}}{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)^{7/2}}{7 e^8}-\frac{2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{9/2}}{9 e^8}+\frac{2 b^6 B (d+e x)^{11/2}}{11 e^8}\\ \end{align*}
Mathematica [A] time = 0.20306, size = 259, normalized size = 0.86 \[ \frac{2 \left (-77 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+297 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-693 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1155 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-2079 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)-693 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+231 (b d-a e)^6 (B d-A e)+63 b^6 B (d+e x)^7\right )}{693 e^8 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]
[Out]
(2*(231*(b*d - a*e)^6*(B*d - A*e) - 693*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) - 2079*b*(b*d - a*
e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 1155*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x
)^3 - 693*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 297*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e
- 5*a*B*e)*(d + e*x)^5 - 77*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 63*b^6*B*(d + e*x)^7))/(693*e^8*(d
+ e*x)^(3/2))
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Maple [B] time = 0.011, 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)^(5/2),x)
[Out]
-2/693*(-63*B*b^6*e^7*x^7-77*A*b^6*e^7*x^6-462*B*a*b^5*e^7*x^6+98*B*b^6*d*e^6*x^6-594*A*a*b^5*e^7*x^5+132*A*b^
6*d*e^6*x^5-1485*B*a^2*b^4*e^7*x^5+792*B*a*b^5*d*e^6*x^5-168*B*b^6*d^2*e^5*x^5-2079*A*a^2*b^4*e^7*x^4+1188*A*a
*b^5*d*e^6*x^4-264*A*b^6*d^2*e^5*x^4-2772*B*a^3*b^3*e^7*x^4+2970*B*a^2*b^4*d*e^6*x^4-1584*B*a*b^5*d^2*e^5*x^4+
336*B*b^6*d^3*e^4*x^4-4620*A*a^3*b^3*e^7*x^3+5544*A*a^2*b^4*d*e^6*x^3-3168*A*a*b^5*d^2*e^5*x^3+704*A*b^6*d^3*e
^4*x^3-3465*B*a^4*b^2*e^7*x^3+7392*B*a^3*b^3*d*e^6*x^3-7920*B*a^2*b^4*d^2*e^5*x^3+4224*B*a*b^5*d^3*e^4*x^3-896
*B*b^6*d^4*e^3*x^3-10395*A*a^4*b^2*e^7*x^2+27720*A*a^3*b^3*d*e^6*x^2-33264*A*a^2*b^4*d^2*e^5*x^2+19008*A*a*b^5
*d^3*e^4*x^2-4224*A*b^6*d^4*e^3*x^2-4158*B*a^5*b*e^7*x^2+20790*B*a^4*b^2*d*e^6*x^2-44352*B*a^3*b^3*d^2*e^5*x^2
+47520*B*a^2*b^4*d^3*e^4*x^2-25344*B*a*b^5*d^4*e^3*x^2+5376*B*b^6*d^5*e^2*x^2+4158*A*a^5*b*e^7*x-41580*A*a^4*b
^2*d*e^6*x+110880*A*a^3*b^3*d^2*e^5*x-133056*A*a^2*b^4*d^3*e^4*x+76032*A*a*b^5*d^4*e^3*x-16896*A*b^6*d^5*e^2*x
+693*B*a^6*e^7*x-16632*B*a^5*b*d*e^6*x+83160*B*a^4*b^2*d^2*e^5*x-177408*B*a^3*b^3*d^3*e^4*x+190080*B*a^2*b^4*d
^4*e^3*x-101376*B*a*b^5*d^5*e^2*x+21504*B*b^6*d^6*e*x+231*A*a^6*e^7+2772*A*a^5*b*d*e^6-27720*A*a^4*b^2*d^2*e^5
+73920*A*a^3*b^3*d^3*e^4-88704*A*a^2*b^4*d^4*e^3+50688*A*a*b^5*d^5*e^2-11264*A*b^6*d^6*e+462*B*a^6*d*e^6-11088
*B*a^5*b*d^2*e^5+55440*B*a^4*b^2*d^3*e^4-118272*B*a^3*b^3*d^4*e^3+126720*B*a^2*b^4*d^5*e^2-67584*B*a*b^5*d^6*e
+14336*B*b^6*d^7)/(e*x+d)^(3/2)/e^8
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Maxima [B] time = 1.01418, size = 1044, normalized size = 3.46 \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)^(5/2),x, algorithm="maxima")
[Out]
2/693*((63*(e*x + d)^(11/2)*B*b^6 - 77*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(9/2) + 297*(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)^(7/2) - 693*(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)^(5/2) + 1155*
(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)*(e*x + d)^(3/2) - 2079*(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)*sqrt(e*x + d))/e^7 + 231*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)
*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 - 3*(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)^(3/2)*e^7))/e
________________________________________________________________________________________
Fricas [B] time = 1.43353, size = 1744, normalized size = 5.77 \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)^(5/2),x, algorithm="fricas")
[Out]
2/693*(63*B*b^6*e^7*x^7 - 14336*B*b^6*d^7 - 231*A*a^6*e^7 + 11264*(6*B*a*b^5 + A*b^6)*d^6*e - 25344*(5*B*a^2*b
^4 + 2*A*a*b^5)*d^5*e^2 + 29568*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 18480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^
4 + 5544*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 462*(B*a^6 + 6*A*a^5*b)*d*e^6 - 7*(14*B*b^6*d*e^6 - 11*(6*B*a*b^5
+ A*b^6)*e^7)*x^6 + 3*(56*B*b^6*d^2*e^5 - 44*(6*B*a*b^5 + A*b^6)*d*e^6 + 99*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^
5 - 3*(112*B*b^6*d^3*e^4 - 88*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 198*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 231*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + (896*B*b^6*d^4*e^3 - 704*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 1584*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^2*e^5 - 1848*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 1155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 3*(1792*
B*b^6*d^5*e^2 - 1408*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 3696*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^2*e^5 + 2310*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 693*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 - 3*(
7168*B*b^6*d^6*e - 5632*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 12672*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 14784*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 9240*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 2772*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6
+ 231*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)
________________________________________________________________________________________
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)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.20521, size = 1489, normalized size = 4.93 \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)^(5/2),x, algorithm="giac")
[Out]
2/693*(63*(x*e + d)^(11/2)*B*b^6*e^80 - 539*(x*e + d)^(9/2)*B*b^6*d*e^80 + 2079*(x*e + d)^(7/2)*B*b^6*d^2*e^80
- 4851*(x*e + d)^(5/2)*B*b^6*d^3*e^80 + 8085*(x*e + d)^(3/2)*B*b^6*d^4*e^80 - 14553*sqrt(x*e + d)*B*b^6*d^5*e
^80 + 462*(x*e + d)^(9/2)*B*a*b^5*e^81 + 77*(x*e + d)^(9/2)*A*b^6*e^81 - 3564*(x*e + d)^(7/2)*B*a*b^5*d*e^81 -
594*(x*e + d)^(7/2)*A*b^6*d*e^81 + 12474*(x*e + d)^(5/2)*B*a*b^5*d^2*e^81 + 2079*(x*e + d)^(5/2)*A*b^6*d^2*e^
81 - 27720*(x*e + d)^(3/2)*B*a*b^5*d^3*e^81 - 4620*(x*e + d)^(3/2)*A*b^6*d^3*e^81 + 62370*sqrt(x*e + d)*B*a*b^
5*d^4*e^81 + 10395*sqrt(x*e + d)*A*b^6*d^4*e^81 + 1485*(x*e + d)^(7/2)*B*a^2*b^4*e^82 + 594*(x*e + d)^(7/2)*A*
a*b^5*e^82 - 10395*(x*e + d)^(5/2)*B*a^2*b^4*d*e^82 - 4158*(x*e + d)^(5/2)*A*a*b^5*d*e^82 + 34650*(x*e + d)^(3
/2)*B*a^2*b^4*d^2*e^82 + 13860*(x*e + d)^(3/2)*A*a*b^5*d^2*e^82 - 103950*sqrt(x*e + d)*B*a^2*b^4*d^3*e^82 - 41
580*sqrt(x*e + d)*A*a*b^5*d^3*e^82 + 2772*(x*e + d)^(5/2)*B*a^3*b^3*e^83 + 2079*(x*e + d)^(5/2)*A*a^2*b^4*e^83
- 18480*(x*e + d)^(3/2)*B*a^3*b^3*d*e^83 - 13860*(x*e + d)^(3/2)*A*a^2*b^4*d*e^83 + 83160*sqrt(x*e + d)*B*a^3
*b^3*d^2*e^83 + 62370*sqrt(x*e + d)*A*a^2*b^4*d^2*e^83 + 3465*(x*e + d)^(3/2)*B*a^4*b^2*e^84 + 4620*(x*e + d)^
(3/2)*A*a^3*b^3*e^84 - 31185*sqrt(x*e + d)*B*a^4*b^2*d*e^84 - 41580*sqrt(x*e + d)*A*a^3*b^3*d*e^84 + 4158*sqrt
(x*e + d)*B*a^5*b*e^85 + 10395*sqrt(x*e + d)*A*a^4*b^2*e^85)*e^(-88) - 2/3*(21*(x*e + d)*B*b^6*d^6 - B*b^6*d^7
- 108*(x*e + d)*B*a*b^5*d^5*e - 18*(x*e + d)*A*b^6*d^5*e + 6*B*a*b^5*d^6*e + A*b^6*d^6*e + 225*(x*e + d)*B*a^
2*b^4*d^4*e^2 + 90*(x*e + d)*A*a*b^5*d^4*e^2 - 15*B*a^2*b^4*d^5*e^2 - 6*A*a*b^5*d^5*e^2 - 240*(x*e + d)*B*a^3*
b^3*d^3*e^3 - 180*(x*e + d)*A*a^2*b^4*d^3*e^3 + 20*B*a^3*b^3*d^4*e^3 + 15*A*a^2*b^4*d^4*e^3 + 135*(x*e + d)*B*
a^4*b^2*d^2*e^4 + 180*(x*e + d)*A*a^3*b^3*d^2*e^4 - 15*B*a^4*b^2*d^3*e^4 - 20*A*a^3*b^3*d^3*e^4 - 36*(x*e + d)
*B*a^5*b*d*e^5 - 90*(x*e + d)*A*a^4*b^2*d*e^5 + 6*B*a^5*b*d^2*e^5 + 15*A*a^4*b^2*d^2*e^5 + 3*(x*e + d)*B*a^6*e
^6 + 18*(x*e + d)*A*a^5*b*e^6 - B*a^6*d*e^6 - 6*A*a^5*b*d*e^6 + A*a^6*e^7)*e^(-8)/(x*e + d)^(3/2)