3.1734 \(\int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx\)
Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac{6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac{2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac{2 b^3 B (d+e x)^{17/2}}{17 e^5} \]
[Out]
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(11/2))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2))/(13*e^5) - (2*b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*e^5)
________________________________________________________________________________________
Rubi [A] time = 0.105697, 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} \[ -\frac{2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac{6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac{2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac{2 b^3 B (d+e x)^{17/2}}{17 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(11/2))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2))/(13*e^5) - (2*b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*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)^{7/2} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e) (d+e x)^{7/2}}{e^4}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{9/2}}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{13/2}}{e^4}+\frac{b^3 B (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2}}{9 e^5}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2}}{11 e^5}+\frac{6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2}}{13 e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2}}{15 e^5}+\frac{2 b^3 B (d+e x)^{17/2}}{17 e^5}\\ \end{align*}
Mathematica [A] time = 0.217923, size = 145, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (-7293 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+25245 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-9945 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+12155 (b d-a e)^3 (B d-A e)+6435 b^3 B (d+e x)^4\right )}{109395 e^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(2*(d + e*x)^(9/2)*(12155*(b*d - a*e)^3*(B*d - A*e) - 9945*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
+ 25245*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 7293*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^
3 + 6435*b^3*B*(d + e*x)^4))/(109395*e^5)
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Maple [A] time = 0.007, size = 301, normalized size = 1.7 \begin{align*}{\frac{12870\,{b}^{3}B{x}^{4}{e}^{4}+14586\,A{b}^{3}{e}^{4}{x}^{3}+43758\,Ba{b}^{2}{e}^{4}{x}^{3}-6864\,B{b}^{3}d{e}^{3}{x}^{3}+50490\,Aa{b}^{2}{e}^{4}{x}^{2}-6732\,A{b}^{3}d{e}^{3}{x}^{2}+50490\,B{a}^{2}b{e}^{4}{x}^{2}-20196\,Ba{b}^{2}d{e}^{3}{x}^{2}+3168\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+59670\,A{a}^{2}b{e}^{4}x-18360\,Aa{b}^{2}d{e}^{3}x+2448\,A{b}^{3}{d}^{2}{e}^{2}x+19890\,B{a}^{3}{e}^{4}x-18360\,B{a}^{2}bd{e}^{3}x+7344\,Ba{b}^{2}{d}^{2}{e}^{2}x-1152\,B{b}^{3}{d}^{3}ex+24310\,{a}^{3}A{e}^{4}-13260\,A{a}^{2}bd{e}^{3}+4080\,Aa{b}^{2}{d}^{2}{e}^{2}-544\,A{b}^{3}{d}^{3}e-4420\,B{a}^{3}d{e}^{3}+4080\,B{a}^{2}b{d}^{2}{e}^{2}-1632\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{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)^(7/2),x)
[Out]
2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^2*e^4*x^3-3432*B*b^3*d*e^3*x^3+25245
*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+25245*B*a^2*b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+298
35*A*a^2*b*e^4*x-9180*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x-9180*B*a^2*b*d*e^3*x+3672*B*a*b^2*
d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3*e^4-6630*A*a^2*b*d*e^3+2040*A*a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3
*d*e^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5
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Maxima [A] time = 1.84499, size = 358, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} B b^{3} - 7293 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 25245 \,{\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{13}{2}} - 9945 \,{\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{11}{2}} + 12155 \,{\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{9}{2}}\right )}}{109395 \, 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)^(7/2),x, algorithm="maxima")
[Out]
2/109395*(6435*(e*x + d)^(17/2)*B*b^3 - 7293*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(15/2) + 25245*(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)^(13/2) - 9945*(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)^(11/2) + 12155*(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)^(9/2))/e^5
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Fricas [B] time = 1.40294, size = 1431, normalized size = 8.27 \begin{align*} \frac{2 \,{\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \,{\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \,{\left (52 \, B b^{3} d e^{7} + 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \,{\left (802 \, B b^{3} d^{2} e^{6} + 782 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \,{\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \,{\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \,{\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{109395 \, 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)^(7/2),x, algorithm="fricas")
[Out]
2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*(3*B*a*b^2 + A*b^3)*d^7*e + 2040*(B*a
^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)
*x^7 + 33*(802*B*b^3*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*(1212*B*b^
3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2*e^6 + 10200*(B*a^2*b + A*a*b^2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)
*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*
e^6 + 7514*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*B*a*b^2 + A*b^3)*d^4
*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*A*a^2*b)*d^2*e^6)*x^3 + 3*(16*B*b^3*d^6*e^2 + 2431
0*A*a^3*d^2*e^6 - 34*(3*B*a*b^2 + A*b^3)*d^5*e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*
d^3*e^5)*x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d^6*e^2 + 1020*(B*a^2*b + A*a*b
^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4)*x)*sqrt(e*x + d)/e^5
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Sympy [A] time = 15.3319, size = 1523, normalized size = 8.8 \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)**(7/2),x)
[Out]
Piecewise((2*A*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**3*d**3*x*sqrt(d + e*x)/9 + 4*A*a**3*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*A*a**2*b*d**5*sqrt(d +
e*x)/(33*e**2) + 2*A*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*A*a**2*b*d**3*x**2*sqrt(d + e*x)/11 + 92*A*a**2*
b*d**2*e*x**3*sqrt(d + e*x)/33 + 68*A*a**2*b*d*e**2*x**4*sqrt(d + e*x)/33 + 6*A*a**2*b*e**3*x**5*sqrt(d + e*x)
/11 + 16*A*a*b**2*d**6*sqrt(d + e*x)/(429*e**3) - 8*A*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*A*a*b**2*d**4
*x**2*sqrt(d + e*x)/(143*e) + 424*A*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*A*a*b**2*d**2*e*x**4*sqrt(d + e*x
)/429 + 240*A*a*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 6*A*a*b**2*e**3*x**6*sqrt(d + e*x)/13 - 32*A*b**3*d**7*sq
rt(d + e*x)/(6435*e**4) + 16*A*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*A*b**3*d**5*x**2*sqrt(d + e*x)/(2145*
e**2) + 2*A*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*A*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 412*A*b**3*d**2*
e*x**5*sqrt(d + e*x)/715 + 92*A*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 2*A*b**3*e**3*x**7*sqrt(d + e*x)/15 - 4*B
*a**3*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**3*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**3*d**3*x**2*sqrt(d + e*x)/
33 + 92*B*a**3*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**3*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**3*e**3*x**5*sqrt
(d + e*x)/11 + 16*B*a**2*b*d**6*sqrt(d + e*x)/(429*e**3) - 8*B*a**2*b*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*B*a*
*2*b*d**4*x**2*sqrt(d + e*x)/(143*e) + 424*B*a**2*b*d**3*x**3*sqrt(d + e*x)/429 + 916*B*a**2*b*d**2*e*x**4*sqr
t(d + e*x)/429 + 240*B*a**2*b*d*e**2*x**5*sqrt(d + e*x)/143 + 6*B*a**2*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*a*b
**2*d**7*sqrt(d + e*x)/(2145*e**4) + 16*B*a*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3) - 4*B*a*b**2*d**5*x**2*sqrt(
d + e*x)/(715*e**2) + 2*B*a*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 320*B*a*b**2*d**3*x**4*sqrt(d + e*x)/429 +
1236*B*a*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*a*b**2*d*e**2*x**6*sqrt(d + e*x)/65 + 2*B*a*b**2*e**3*x**7*
sqrt(d + e*x)/5 + 256*B*b**3*d**8*sqrt(d + e*x)/(109395*e**5) - 128*B*b**3*d**7*x*sqrt(d + e*x)/(109395*e**4)
+ 32*B*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*B*b**3*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*B*b**3*
d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*B*b**3*d**3*x**5*sqrt(d + e*x)/12155 + 1604*B*b**3*d**2*e*x**6*sqrt(d
+ e*x)/3315 + 104*B*b**3*d*e**2*x**7*sqrt(d + e*x)/255 + 2*B*b**3*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**
(7/2)*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b*
*2*x**4/4 + B*b**3*x**5/5), True))
________________________________________________________________________________________
Giac [B] time = 2.34368, size = 2792, normalized size = 16.14 \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)^(7/2),x, algorithm="giac")
[Out]
2/765765*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*d^3*e^(-1) + 153153*(3*(x*e + d)^(5/2) - 5*(x*
e + d)^(3/2)*d)*A*a^2*b*d^3*e^(-1) + 21879*(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^3*e^(-2) + 21879*(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^3*
e^(-2) + 7293*(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^3*e^(-3) + 2431*(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^3*e^(-3) + 221*(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^3*e^(-4) + 255255*(x*e + d)^(3/2)*A*a^3*d^3 +
21879*(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^2*e^(-1) + 65637*(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^2*e^(-1) + 21879*(35*(x*e + d)^(9/2) - 13
5*(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^2*e^(-2) + 21879*(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^2*e^(-2) + 19
89*(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^2*e^(-3) + 663*(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^2*e^(-3) + 255*(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*b^3*d^2*e^(-4) + 153153*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^3*d^2 + 72
93*(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*d*e^
(-1) + 21879*(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*d*e^(-1) + 1989*(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*d*e^(-2) + 1989*(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*d*e^(-2) + 765*(
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*d*e^(-3) + 255*(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*d*e^(-3) + 51*(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*d*e^(-4) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^3*d + 221*(
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^3*e^(-1) + 663*(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^2*b*e^(-1) + 255*(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^2*b*e^(-2) + 255*(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*a*b^2*e^(-2) + 51*(300
3*(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 + 9652
5*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*a*b^2*e^(-3) + 17*(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)*A*b^3*e^(-3) + 7*(6435*(x*e + d)^(17/2
) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9
/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*b^3*e^(-4) +
2431*(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^3)*e
^(-1)