3.1805 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{3/2}} \, dx\)
Optimal. Leaf size=300 \[ -\frac{2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac{2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac{10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac{2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac{2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac{2 \sqrt{d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac{2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt{d+e x}}+\frac{2 b^6 B (d+e x)^{13/2}}{13 e^8} \]
[Out]
(2*(b*d - a*e)^6*(B*d - A*e))/(e^8*Sqrt[d + e*x]) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*Sqrt[d + e*x]
)/e^8 - (2*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(3/2))/e^8 + (2*b^2*(b*d - a*e)^3*(7*b*B*d
- 4*A*b*e - 3*a*B*e)*(d + e*x)^(5/2))/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)^(7/2
))/(7*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(9/2))/(3*e^8) - (2*b^5*(7*b*B*d - A*b
*e - 6*a*B*e)*(d + e*x)^(11/2))/(11*e^8) + (2*b^6*B*(d + e*x)^(13/2))/(13*e^8)
________________________________________________________________________________________
Rubi [A] time = 0.145125, antiderivative size = 300, 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} \[ -\frac{2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac{2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac{10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac{2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac{2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac{2 \sqrt{d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac{2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt{d+e x}}+\frac{2 b^6 B (d+e x)^{13/2}}{13 e^8} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(3/2),x]
[Out]
(2*(b*d - a*e)^6*(B*d - A*e))/(e^8*Sqrt[d + e*x]) + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*Sqrt[d + e*x]
)/e^8 - (2*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(3/2))/e^8 + (2*b^2*(b*d - a*e)^3*(7*b*B*d
- 4*A*b*e - 3*a*B*e)*(d + e*x)^(5/2))/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)^(7/2
))/(7*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(9/2))/(3*e^8) - (2*b^5*(7*b*B*d - A*b
*e - 6*a*B*e)*(d + e*x)^(11/2))/(11*e^8) + (2*b^6*B*(d + e*x)^(13/2))/(13*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)^{3/2}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^{3/2}}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 \sqrt{d+e x}}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) \sqrt{d+e x}}{e^7}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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)^{7/2}}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{9/2}}{e^7}+\frac{b^6 B (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac{2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt{d+e x}}+\frac{2 (b d-a e)^5 (7 b B d-6 A b e-a B e) \sqrt{d+e x}}{e^8}-\frac{2 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{3/2}}{e^8}+\frac{2 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{5/2}}{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)^{7/2}}{7 e^8}+\frac{2 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{9/2}}{3 e^8}-\frac{2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{11/2}}{11 e^8}+\frac{2 b^6 B (d+e x)^{13/2}}{13 e^8}\\ \end{align*}
Mathematica [A] time = 0.220065, size = 259, normalized size = 0.86 \[ \frac{2 \left (-273 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+1001 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-2145 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+3003 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-3003 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+3003 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+3003 (b d-a e)^6 (B d-A e)+231 b^6 B (d+e x)^7\right )}{3003 e^8 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(3/2),x]
[Out]
(2*(3003*(b*d - a*e)^6*(B*d - A*e) + 3003*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) - 3003*b*(b*d -
a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 3003*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e
*x)^3 - 2145*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 1001*b^4*(b*d - a*e)*(7*b*B*d - 2*A
*b*e - 5*a*B*e)*(d + e*x)^5 - 273*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 231*b^6*B*(d + e*x)^7))/(3003*
e^8*Sqrt[d + e*x])
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Maple [B] time = 0.009, 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)^(3/2),x)
[Out]
-2/3003*(-231*B*b^6*e^7*x^7-273*A*b^6*e^7*x^6-1638*B*a*b^5*e^7*x^6+294*B*b^6*d*e^6*x^6-2002*A*a*b^5*e^7*x^5+36
4*A*b^6*d*e^6*x^5-5005*B*a^2*b^4*e^7*x^5+2184*B*a*b^5*d*e^6*x^5-392*B*b^6*d^2*e^5*x^5-6435*A*a^2*b^4*e^7*x^4+2
860*A*a*b^5*d*e^6*x^4-520*A*b^6*d^2*e^5*x^4-8580*B*a^3*b^3*e^7*x^4+7150*B*a^2*b^4*d*e^6*x^4-3120*B*a*b^5*d^2*e
^5*x^4+560*B*b^6*d^3*e^4*x^4-12012*A*a^3*b^3*e^7*x^3+10296*A*a^2*b^4*d*e^6*x^3-4576*A*a*b^5*d^2*e^5*x^3+832*A*
b^6*d^3*e^4*x^3-9009*B*a^4*b^2*e^7*x^3+13728*B*a^3*b^3*d*e^6*x^3-11440*B*a^2*b^4*d^2*e^5*x^3+4992*B*a*b^5*d^3*
e^4*x^3-896*B*b^6*d^4*e^3*x^3-15015*A*a^4*b^2*e^7*x^2+24024*A*a^3*b^3*d*e^6*x^2-20592*A*a^2*b^4*d^2*e^5*x^2+91
52*A*a*b^5*d^3*e^4*x^2-1664*A*b^6*d^4*e^3*x^2-6006*B*a^5*b*e^7*x^2+18018*B*a^4*b^2*d*e^6*x^2-27456*B*a^3*b^3*d
^2*e^5*x^2+22880*B*a^2*b^4*d^3*e^4*x^2-9984*B*a*b^5*d^4*e^3*x^2+1792*B*b^6*d^5*e^2*x^2-18018*A*a^5*b*e^7*x+600
60*A*a^4*b^2*d*e^6*x-96096*A*a^3*b^3*d^2*e^5*x+82368*A*a^2*b^4*d^3*e^4*x-36608*A*a*b^5*d^4*e^3*x+6656*A*b^6*d^
5*e^2*x-3003*B*a^6*e^7*x+24024*B*a^5*b*d*e^6*x-72072*B*a^4*b^2*d^2*e^5*x+109824*B*a^3*b^3*d^3*e^4*x-91520*B*a^
2*b^4*d^4*e^3*x+39936*B*a*b^5*d^5*e^2*x-7168*B*b^6*d^6*e*x+3003*A*a^6*e^7-36036*A*a^5*b*d*e^6+120120*A*a^4*b^2
*d^2*e^5-192192*A*a^3*b^3*d^3*e^4+164736*A*a^2*b^4*d^4*e^3-73216*A*a*b^5*d^5*e^2+13312*A*b^6*d^6*e-6006*B*a^6*
d*e^6+48048*B*a^5*b*d^2*e^5-144144*B*a^4*b^2*d^3*e^4+219648*B*a^3*b^3*d^4*e^3-183040*B*a^2*b^4*d^5*e^2+79872*B
*a*b^5*d^6*e-14336*B*b^6*d^7)/(e*x+d)^(1/2)/e^8
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Maxima [B] time = 1.01508, size = 1046, normalized size = 3.49 \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)^(3/2),x, algorithm="maxima")
[Out]
2/3003*((231*(e*x + d)^(13/2)*B*b^6 - 273*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(11/2) + 1001*(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)^(9/2) - 2145*(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)^(7/2) +
3003*(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)^(5/2) - 3003*(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)^(3/2) + 3003*(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)*sqrt(e*x + d))/e^7 + 3003*(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)/(sqrt(e*x + d)*e^7))/e
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Fricas [B] time = 1.36841, size = 1732, 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)^(3/2),x, algorithm="fricas")
[Out]
2/3003*(231*B*b^6*e^7*x^7 + 14336*B*b^6*d^7 - 3003*A*a^6*e^7 - 13312*(6*B*a*b^5 + A*b^6)*d^6*e + 36608*(5*B*a^
2*b^4 + 2*A*a*b^5)*d^5*e^2 - 54912*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 48048*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3
*e^4 - 24024*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 6006*(B*a^6 + 6*A*a^5*b)*d*e^6 - 21*(14*B*b^6*d*e^6 - 13*(6*B
*a*b^5 + A*b^6)*e^7)*x^6 + 7*(56*B*b^6*d^2*e^5 - 52*(6*B*a*b^5 + A*b^6)*d*e^6 + 143*(5*B*a^2*b^4 + 2*A*a*b^5)*
e^7)*x^5 - 5*(112*B*b^6*d^3*e^4 - 104*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 286*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 429*
(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + (896*B*b^6*d^4*e^3 - 832*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 2288*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^2*e^5 - 3432*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 3003*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 -
(1792*B*b^6*d^5*e^2 - 1664*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 4576*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 6864*(4*B*a^
3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 6006*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 3003*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^
2 + (7168*B*b^6*d^6*e - 6656*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 18304*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 27456*(4*
B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 24024*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 12012*(2*B*a^5*b + 5*A*a^4*b^2)
*d*e^6 + 3003*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^9*x + d*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)**(3/2),x)
[Out]
Timed out
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Giac [B] time = 1.20847, size = 1527, normalized size = 5.09 \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)^(3/2),x, algorithm="giac")
[Out]
2/3003*(231*(x*e + d)^(13/2)*B*b^6*e^96 - 1911*(x*e + d)^(11/2)*B*b^6*d*e^96 + 7007*(x*e + d)^(9/2)*B*b^6*d^2*
e^96 - 15015*(x*e + d)^(7/2)*B*b^6*d^3*e^96 + 21021*(x*e + d)^(5/2)*B*b^6*d^4*e^96 - 21021*(x*e + d)^(3/2)*B*b
^6*d^5*e^96 + 21021*sqrt(x*e + d)*B*b^6*d^6*e^96 + 1638*(x*e + d)^(11/2)*B*a*b^5*e^97 + 273*(x*e + d)^(11/2)*A
*b^6*e^97 - 12012*(x*e + d)^(9/2)*B*a*b^5*d*e^97 - 2002*(x*e + d)^(9/2)*A*b^6*d*e^97 + 38610*(x*e + d)^(7/2)*B
*a*b^5*d^2*e^97 + 6435*(x*e + d)^(7/2)*A*b^6*d^2*e^97 - 72072*(x*e + d)^(5/2)*B*a*b^5*d^3*e^97 - 12012*(x*e +
d)^(5/2)*A*b^6*d^3*e^97 + 90090*(x*e + d)^(3/2)*B*a*b^5*d^4*e^97 + 15015*(x*e + d)^(3/2)*A*b^6*d^4*e^97 - 1081
08*sqrt(x*e + d)*B*a*b^5*d^5*e^97 - 18018*sqrt(x*e + d)*A*b^6*d^5*e^97 + 5005*(x*e + d)^(9/2)*B*a^2*b^4*e^98 +
2002*(x*e + d)^(9/2)*A*a*b^5*e^98 - 32175*(x*e + d)^(7/2)*B*a^2*b^4*d*e^98 - 12870*(x*e + d)^(7/2)*A*a*b^5*d*
e^98 + 90090*(x*e + d)^(5/2)*B*a^2*b^4*d^2*e^98 + 36036*(x*e + d)^(5/2)*A*a*b^5*d^2*e^98 - 150150*(x*e + d)^(3
/2)*B*a^2*b^4*d^3*e^98 - 60060*(x*e + d)^(3/2)*A*a*b^5*d^3*e^98 + 225225*sqrt(x*e + d)*B*a^2*b^4*d^4*e^98 + 90
090*sqrt(x*e + d)*A*a*b^5*d^4*e^98 + 8580*(x*e + d)^(7/2)*B*a^3*b^3*e^99 + 6435*(x*e + d)^(7/2)*A*a^2*b^4*e^99
- 48048*(x*e + d)^(5/2)*B*a^3*b^3*d*e^99 - 36036*(x*e + d)^(5/2)*A*a^2*b^4*d*e^99 + 120120*(x*e + d)^(3/2)*B*
a^3*b^3*d^2*e^99 + 90090*(x*e + d)^(3/2)*A*a^2*b^4*d^2*e^99 - 240240*sqrt(x*e + d)*B*a^3*b^3*d^3*e^99 - 180180
*sqrt(x*e + d)*A*a^2*b^4*d^3*e^99 + 9009*(x*e + d)^(5/2)*B*a^4*b^2*e^100 + 12012*(x*e + d)^(5/2)*A*a^3*b^3*e^1
00 - 45045*(x*e + d)^(3/2)*B*a^4*b^2*d*e^100 - 60060*(x*e + d)^(3/2)*A*a^3*b^3*d*e^100 + 135135*sqrt(x*e + d)*
B*a^4*b^2*d^2*e^100 + 180180*sqrt(x*e + d)*A*a^3*b^3*d^2*e^100 + 6006*(x*e + d)^(3/2)*B*a^5*b*e^101 + 15015*(x
*e + d)^(3/2)*A*a^4*b^2*e^101 - 36036*sqrt(x*e + d)*B*a^5*b*d*e^101 - 90090*sqrt(x*e + d)*A*a^4*b^2*d*e^101 +
3003*sqrt(x*e + d)*B*a^6*e^102 + 18018*sqrt(x*e + d)*A*a^5*b*e^102)*e^(-104) + 2*(B*b^6*d^7 - 6*B*a*b^5*d^6*e
- A*b^6*d^6*e + 15*B*a^2*b^4*d^5*e^2 + 6*A*a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^3 - 15*A*a^2*b^4*d^4*e^3 + 15*B*
a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 - 6*B*a^5*b*d^2*e^5 - 15*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d*
e^6 - A*a^6*e^7)*e^(-8)/sqrt(x*e + d)