3.1735 \(\int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx\)
Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac{6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac{2 b^3 B (d+e x)^{15/2}}{15 e^5} \]
[Out]
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*
b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e^5)
________________________________________________________________________________________
Rubi [A] time = 0.0762144, antiderivative size = 173, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{77} \[ -\frac{2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac{6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac{2 b^3 B (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*
b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e^5)
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)^3 (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e) (d+e x)^{5/2}}{e^4}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac{b^3 B (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2}}{7 e^5}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2}}{9 e^5}+\frac{6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2}}{13 e^5}+\frac{2 b^3 B (d+e x)^{15/2}}{15 e^5}\\ \end{align*}
Mathematica [A] time = 0.172567, size = 145, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (-3465 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+12285 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-5005 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+6435 (b d-a e)^3 (B d-A e)+3003 b^3 B (d+e x)^4\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]
[Out]
(2*(d + e*x)^(7/2)*(6435*(b*d - a*e)^3*(B*d - A*e) - 5005*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
+ 12285*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 3465*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3
+ 3003*b^3*B*(d + e*x)^4))/(45045*e^5)
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Maple [A] time = 0.006, size = 301, normalized size = 1.7 \begin{align*}{\frac{6006\,{b}^{3}B{x}^{4}{e}^{4}+6930\,A{b}^{3}{e}^{4}{x}^{3}+20790\,Ba{b}^{2}{e}^{4}{x}^{3}-3696\,B{b}^{3}d{e}^{3}{x}^{3}+24570\,Aa{b}^{2}{e}^{4}{x}^{2}-3780\,A{b}^{3}d{e}^{3}{x}^{2}+24570\,B{a}^{2}b{e}^{4}{x}^{2}-11340\,Ba{b}^{2}d{e}^{3}{x}^{2}+2016\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+30030\,A{a}^{2}b{e}^{4}x-10920\,Aa{b}^{2}d{e}^{3}x+1680\,A{b}^{3}{d}^{2}{e}^{2}x+10010\,B{a}^{3}{e}^{4}x-10920\,B{a}^{2}bd{e}^{3}x+5040\,Ba{b}^{2}{d}^{2}{e}^{2}x-896\,B{b}^{3}{d}^{3}ex+12870\,{a}^{3}A{e}^{4}-8580\,A{a}^{2}bd{e}^{3}+3120\,Aa{b}^{2}{d}^{2}{e}^{2}-480\,A{b}^{3}{d}^{3}e-2860\,B{a}^{3}d{e}^{3}+3120\,B{a}^{2}b{d}^{2}{e}^{2}-1440\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x)
[Out]
2/45045*(e*x+d)^(7/2)*(3003*B*b^3*e^4*x^4+3465*A*b^3*e^4*x^3+10395*B*a*b^2*e^4*x^3-1848*B*b^3*d*e^3*x^3+12285*
A*a*b^2*e^4*x^2-1890*A*b^3*d*e^3*x^2+12285*B*a^2*b*e^4*x^2-5670*B*a*b^2*d*e^3*x^2+1008*B*b^3*d^2*e^2*x^2+15015
*A*a^2*b*e^4*x-5460*A*a*b^2*d*e^3*x+840*A*b^3*d^2*e^2*x+5005*B*a^3*e^4*x-5460*B*a^2*b*d*e^3*x+2520*B*a*b^2*d^2
*e^2*x-448*B*b^3*d^3*e*x+6435*A*a^3*e^4-4290*A*a^2*b*d*e^3+1560*A*a*b^2*d^2*e^2-240*A*b^3*d^3*e-1430*B*a^3*d*e
^3+1560*B*a^2*b*d^2*e^2-720*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5
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Maxima [A] time = 1.03972, size = 358, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{3} - 3465 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")
[Out]
2/45045*(3003*(e*x + d)^(15/2)*B*b^3 - 3465*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(13/2) + 12285*(2*B*
b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)*(e*x + d)^(11/2) - 5005*(4*B*b^3*d^3 - 3*(3*B*a*b
^2 + A*b^3)*d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(9/2) + 6435*(B*b^3*d^4 +
A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^
(7/2))/e^5
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Fricas [B] time = 1.34486, size = 1203, normalized size = 6.95 \begin{align*} \frac{2 \,{\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \,{\left (31 \, B b^{3} d e^{6} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \,{\left (71 \, B b^{3} d^{2} e^{5} + 135 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \,{\left (B b^{3} d^{3} e^{4} + 159 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")
[Out]
2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3*B*a*b^2 + A*b^3)*d^6*e + 1560*(B*a^2
*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x
^6 + 63*(71*B*b^3*d^2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A*a*b^2)*e^7)*x^5 + 35*(B*b^3*d^3*e
^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 897*(B*a^2*b + A*a*b^2)*d*e^6 + 143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8
*B*b^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2*b + A*a*b^2)*d^2*e^5 - 2717*(B*
a^3 + 3*A*a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3*d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b^3)*d^4*e^3 + 195*(B
*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5)*x^2 - (64*B*b^3*d^6*e - 19305*A*a^3*d^2*e^5 - 12
0*(3*B*a*b^2 + A*b^3)*d^5*e^2 + 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x
+ d)/e^5
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Sympy [A] time = 41.0612, size = 1564, normalized size = 9.04 \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)**3*(B*x+A)*(e*x+d)**(5/2),x)
[Out]
A*a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*A*a**3*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7
)/e + 6*A*a**2*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*a**2*b*d*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*A*a**2*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*
x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*A*a*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2
*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*A*a*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)*
*(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*A*a*b**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d
+ e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 2*A*b**3*
d**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4
+ 4*A*b**3*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)
**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 2*A*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*
(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*
a**3*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*B*a**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e
*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d
*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 6*B*a**2*b*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/
2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*B*a**2*b*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d
+ e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 6*B*a**2*b*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5
+ 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 6*B*a*b**2*d**2*(-d**3*(d
+ e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 12*B*a*b**2*
d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**4 + 6*B*a*b**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)*
*(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*B*b**3*d**2*
(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d
+ e*x)**(11/2)/11)/e**5 + 4*B*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(
7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*B*b**3*(d**6*(
d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d
**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5
________________________________________________________________________________________
Giac [B] time = 2.06603, size = 1870, normalized size = 10.81 \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)^3*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")
[Out]
2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*d^2*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*A*a^2*b*d^2*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a
^2*b*d^2*e^(-2) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*b^2*d^2*e^(-2)
+ 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*b^
2*d^2*e^(-3) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)
*d^3)*A*b^3*d^2*e^(-3) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(
x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*b^3*d^2*e^(-4) + 15015*(x*e + d)^(3/2)*A*a^3*d^2 + 858*(15*(x
*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*d*e^(-1) + 2574*(15*(x*e + d)^(7/2) - 42*
(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^2*b*d*e^(-1) + 858*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d
+ 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^2*b*d*e^(-2) + 858*(35*(x*e + d)^(9/2) - 135*(x*e +
d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a*b^2*d*e^(-2) + 78*(315*(x*e + d)^(11/2) -
1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a*b
^2*d*e^(-3) + 26*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5
/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*b^3*d*e^(-3) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 1001
0*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*b^3
*d*e^(-4) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^3*d + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(
7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^3*e^(-1) + 429*(35*(x*e + d)^(9/2) - 135*(x*e
+ d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^2*b*e^(-1) + 39*(315*(x*e + d)^(11/2) -
1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^2
*b*e^(-2) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2
)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a*b^2*e^(-2) + 15*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*
(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a*b^2
*e^(-3) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2
)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*b^3*e^(-3) + (3003*(x*e + d)^(15/2) - 20790*(x*
e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*
(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*b^3*e^(-4) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2)*A*a^3)*e^(-1)