3.2050 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac{4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac{10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac{2 b^5 (d+e x)^{19/2}}{19 e^6} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^6) - (20*b^2*(b*d - a
*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(b*d - a*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e
*x)^(17/2))/(17*e^6) + (2*b^5*(d + e*x)^(19/2))/(19*e^6)

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Rubi [A]  time = 0.0738924, antiderivative size = 158, 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, 43} \[ -\frac{10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac{4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac{10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac{2 b^5 (d+e x)^{19/2}}{19 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^6) - (20*b^2*(b*d - a
*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(b*d - a*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e
*x)^(17/2))/(17*e^6) + (2*b^5*(d + e*x)^(19/2))/(19*e^6)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^5 (d+e x)^{7/2}}{e^5}+\frac{5 b (b d-a e)^4 (d+e x)^{9/2}}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{11/2}}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{13/2}}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{15/2}}{e^5}+\frac{b^5 (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^5 (d+e x)^{9/2}}{9 e^6}+\frac{10 b (b d-a e)^4 (d+e x)^{11/2}}{11 e^6}-\frac{20 b^2 (b d-a e)^3 (d+e x)^{13/2}}{13 e^6}+\frac{4 b^3 (b d-a e)^2 (d+e x)^{15/2}}{3 e^6}-\frac{10 b^4 (b d-a e) (d+e x)^{17/2}}{17 e^6}+\frac{2 b^5 (d+e x)^{19/2}}{19 e^6}\\ \end{align*}

Mathematica [A]  time = 0.116552, size = 123, normalized size = 0.78 \[ \frac{2 (d+e x)^{9/2} \left (-319770 b^2 (d+e x)^2 (b d-a e)^3+277134 b^3 (d+e x)^3 (b d-a e)^2-122265 b^4 (d+e x)^4 (b d-a e)+188955 b (d+e x) (b d-a e)^4-46189 (b d-a e)^5+21879 b^5 (d+e x)^5\right )}{415701 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(-46189*(b*d - a*e)^5 + 188955*b*(b*d - a*e)^4*(d + e*x) - 319770*b^2*(b*d - a*e)^3*(d + e*
x)^2 + 277134*b^3*(b*d - a*e)^2*(d + e*x)^3 - 122265*b^4*(b*d - a*e)*(d + e*x)^4 + 21879*b^5*(d + e*x)^5))/(41
5701*e^6)

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Maple [B]  time = 0.007, size = 273, normalized size = 1.7 \begin{align*}{\frac{43758\,{x}^{5}{b}^{5}{e}^{5}+244530\,{x}^{4}a{b}^{4}{e}^{5}-25740\,{x}^{4}{b}^{5}d{e}^{4}+554268\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-130416\,{x}^{3}a{b}^{4}d{e}^{4}+13728\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+639540\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-255816\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+60192\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-6336\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+377910\,x{a}^{4}b{e}^{5}-232560\,x{a}^{3}{b}^{2}d{e}^{4}+93024\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-21888\,xa{b}^{4}{d}^{3}{e}^{2}+2304\,x{b}^{5}{d}^{4}e+92378\,{a}^{5}{e}^{5}-83980\,{a}^{4}bd{e}^{4}+51680\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-20672\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+4864\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{415701\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/415701*(e*x+d)^(9/2)*(21879*b^5*e^5*x^5+122265*a*b^4*e^5*x^4-12870*b^5*d*e^4*x^4+277134*a^2*b^3*e^5*x^3-6520
8*a*b^4*d*e^4*x^3+6864*b^5*d^2*e^3*x^3+319770*a^3*b^2*e^5*x^2-127908*a^2*b^3*d*e^4*x^2+30096*a*b^4*d^2*e^3*x^2
-3168*b^5*d^3*e^2*x^2+188955*a^4*b*e^5*x-116280*a^3*b^2*d*e^4*x+46512*a^2*b^3*d^2*e^3*x-10944*a*b^4*d^3*e^2*x+
1152*b^5*d^4*e*x+46189*a^5*e^5-41990*a^4*b*d*e^4+25840*a^3*b^2*d^2*e^3-10336*a^2*b^3*d^3*e^2+2432*a*b^4*d^4*e-
256*b^5*d^5)/e^6

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Maxima [A]  time = 0.963124, size = 350, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (21879 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{5} - 122265 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 277134 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 319770 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 188955 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 46189 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{415701 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/415701*(21879*(e*x + d)^(19/2)*b^5 - 122265*(b^5*d - a*b^4*e)*(e*x + d)^(17/2) + 277134*(b^5*d^2 - 2*a*b^4*d
*e + a^2*b^3*e^2)*(e*x + d)^(15/2) - 319770*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d
)^(13/2) + 188955*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(11/2)
 - 46189*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x +
d)^(9/2))/e^6

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Fricas [B]  time = 1.29616, size = 1357, normalized size = 8.59 \begin{align*} \frac{2 \,{\left (21879 \, b^{5} e^{9} x^{9} - 256 \, b^{5} d^{9} + 2432 \, a b^{4} d^{8} e - 10336 \, a^{2} b^{3} d^{7} e^{2} + 25840 \, a^{3} b^{2} d^{6} e^{3} - 41990 \, a^{4} b d^{5} e^{4} + 46189 \, a^{5} d^{4} e^{5} + 1287 \,{\left (58 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 858 \,{\left (101 \, b^{5} d^{2} e^{7} + 494 \, a b^{4} d e^{8} + 323 \, a^{2} b^{3} e^{9}\right )} x^{7} + 66 \,{\left (524 \, b^{5} d^{3} e^{6} + 7619 \, a b^{4} d^{2} e^{7} + 14858 \, a^{2} b^{3} d e^{8} + 4845 \, a^{3} b^{2} e^{9}\right )} x^{6} + 9 \,{\left (7 \, b^{5} d^{4} e^{5} + 23028 \, a b^{4} d^{3} e^{6} + 133076 \, a^{2} b^{3} d^{2} e^{7} + 129200 \, a^{3} b^{2} d e^{8} + 20995 \, a^{4} b e^{9}\right )} x^{5} -{\left (70 \, b^{5} d^{5} e^{4} - 665 \, a b^{4} d^{4} e^{5} - 516800 \, a^{2} b^{3} d^{3} e^{6} - 1479340 \, a^{3} b^{2} d^{2} e^{7} - 713830 \, a^{4} b d e^{8} - 46189 \, a^{5} e^{9}\right )} x^{4} + 2 \,{\left (40 \, b^{5} d^{6} e^{3} - 380 \, a b^{4} d^{5} e^{4} + 1615 \, a^{2} b^{3} d^{4} e^{5} + 342380 \, a^{3} b^{2} d^{3} e^{6} + 482885 \, a^{4} b d^{2} e^{7} + 92378 \, a^{5} d e^{8}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{7} e^{2} - 152 \, a b^{4} d^{6} e^{3} + 646 \, a^{2} b^{3} d^{5} e^{4} - 1615 \, a^{3} b^{2} d^{4} e^{5} - 83980 \, a^{4} b d^{3} e^{6} - 46189 \, a^{5} d^{2} e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{8} e - 1216 \, a b^{4} d^{7} e^{2} + 5168 \, a^{2} b^{3} d^{6} e^{3} - 12920 \, a^{3} b^{2} d^{5} e^{4} + 20995 \, a^{4} b d^{4} e^{5} + 184756 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt{e x + d}}{415701 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/415701*(21879*b^5*e^9*x^9 - 256*b^5*d^9 + 2432*a*b^4*d^8*e - 10336*a^2*b^3*d^7*e^2 + 25840*a^3*b^2*d^6*e^3 -
 41990*a^4*b*d^5*e^4 + 46189*a^5*d^4*e^5 + 1287*(58*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 858*(101*b^5*d^2*e^7 + 494
*a*b^4*d*e^8 + 323*a^2*b^3*e^9)*x^7 + 66*(524*b^5*d^3*e^6 + 7619*a*b^4*d^2*e^7 + 14858*a^2*b^3*d*e^8 + 4845*a^
3*b^2*e^9)*x^6 + 9*(7*b^5*d^4*e^5 + 23028*a*b^4*d^3*e^6 + 133076*a^2*b^3*d^2*e^7 + 129200*a^3*b^2*d*e^8 + 2099
5*a^4*b*e^9)*x^5 - (70*b^5*d^5*e^4 - 665*a*b^4*d^4*e^5 - 516800*a^2*b^3*d^3*e^6 - 1479340*a^3*b^2*d^2*e^7 - 71
3830*a^4*b*d*e^8 - 46189*a^5*e^9)*x^4 + 2*(40*b^5*d^6*e^3 - 380*a*b^4*d^5*e^4 + 1615*a^2*b^3*d^4*e^5 + 342380*
a^3*b^2*d^3*e^6 + 482885*a^4*b*d^2*e^7 + 92378*a^5*d*e^8)*x^3 - 6*(16*b^5*d^7*e^2 - 152*a*b^4*d^6*e^3 + 646*a^
2*b^3*d^5*e^4 - 1615*a^3*b^2*d^4*e^5 - 83980*a^4*b*d^3*e^6 - 46189*a^5*d^2*e^7)*x^2 + (128*b^5*d^8*e - 1216*a*
b^4*d^7*e^2 + 5168*a^2*b^3*d^6*e^3 - 12920*a^3*b^2*d^5*e^4 + 20995*a^4*b*d^4*e^5 + 184756*a^5*d^3*e^6)*x)*sqrt
(e*x + d)/e^6

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Sympy [A]  time = 19.1999, size = 1187, normalized size = 7.51 \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)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*a**5*d**4*sqrt(d + e*x)/(9*e) + 8*a**5*d**3*x*sqrt(d + e*x)/9 + 4*a**5*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**5*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**5*e**3*x**4*sqrt(d + e*x)/9 - 20*a**4*b*d**5*sqrt(d + e*x)/(99*e
**2) + 10*a**4*b*d**4*x*sqrt(d + e*x)/(99*e) + 80*a**4*b*d**3*x**2*sqrt(d + e*x)/33 + 460*a**4*b*d**2*e*x**3*s
qrt(d + e*x)/99 + 340*a**4*b*d*e**2*x**4*sqrt(d + e*x)/99 + 10*a**4*b*e**3*x**5*sqrt(d + e*x)/11 + 160*a**3*b*
*2*d**6*sqrt(d + e*x)/(1287*e**3) - 80*a**3*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 20*a**3*b**2*d**4*x**2*sqr
t(d + e*x)/(429*e) + 4240*a**3*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 9160*a**3*b**2*d**2*e*x**4*sqrt(d + e*x)/12
87 + 800*a**3*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 20*a**3*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*a**2*b**3*d**7
*sqrt(d + e*x)/(1287*e**4) + 32*a**2*b**3*d**6*x*sqrt(d + e*x)/(1287*e**3) - 8*a**2*b**3*d**5*x**2*sqrt(d + e*
x)/(429*e**2) + 20*a**2*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 3200*a**2*b**3*d**3*x**4*sqrt(d + e*x)/1287 +
824*a**2*b**3*d**2*e*x**5*sqrt(d + e*x)/143 + 184*a**2*b**3*d*e**2*x**6*sqrt(d + e*x)/39 + 4*a**2*b**3*e**3*x*
*7*sqrt(d + e*x)/3 + 256*a*b**4*d**8*sqrt(d + e*x)/(21879*e**5) - 128*a*b**4*d**7*x*sqrt(d + e*x)/(21879*e**4)
 + 32*a*b**4*d**6*x**2*sqrt(d + e*x)/(7293*e**3) - 80*a*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 70*a*b**4*
d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*a*b**4*d**3*x**5*sqrt(d + e*x)/2431 + 1604*a*b**4*d**2*e*x**6*sqrt(d
+ e*x)/663 + 104*a*b**4*d*e**2*x**7*sqrt(d + e*x)/51 + 10*a*b**4*e**3*x**8*sqrt(d + e*x)/17 - 512*b**5*d**9*sq
rt(d + e*x)/(415701*e**6) + 256*b**5*d**8*x*sqrt(d + e*x)/(415701*e**5) - 64*b**5*d**7*x**2*sqrt(d + e*x)/(138
567*e**4) + 160*b**5*d**6*x**3*sqrt(d + e*x)/(415701*e**3) - 140*b**5*d**5*x**4*sqrt(d + e*x)/(415701*e**2) +
14*b**5*d**4*x**5*sqrt(d + e*x)/(46189*e) + 2096*b**5*d**3*x**6*sqrt(d + e*x)/12597 + 404*b**5*d**2*e*x**7*sqr
t(d + e*x)/969 + 116*b**5*d*e**2*x**8*sqrt(d + e*x)/323 + 2*b**5*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7
/2)*(a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6), True))

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Giac [B]  time = 1.25881, size = 2298, normalized size = 14.54 \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)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/14549535*(4849845*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*b*d^3*e^(-1) + 1385670*(15*(x*e + d)^(7/2) -
 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*b^2*d^3*e^(-2) + 461890*(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^2*b^3*d^3*e^(-3) + 20995*(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^4*d^3*e^(-4) + 1615*(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^5*d^3*e^(-5) + 4849845*(x*e + d)^(3/
2)*a^5*d^3 + 2078505*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b*d^2*e^(-1) + 1
385670*(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^3*b^
2*d^2*e^(-2) + 125970*(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^2*b^3*d^2*e^(-3) + 24225*(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^4*d^2*e^(-4) + 969*(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^5*d^2*e^(-5) + 2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*d^2 + 692835*(35*(x*e + d)^(9/2) - 1
35*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^4*b*d*e^(-1) + 125970*(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^3*b^2*d*e^(-2) + 48450*(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^2*b^3*d*e^(-3) + 4845*(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^4*d*e^(-4) + 399*(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^5*d
*e^(-5) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*d + 20995*(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^4*b*e^(-1) + 16150*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12
870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^3*b^2*e^(-2) + 3230*(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^2*b^3*e^(-3) + 665*(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)*a*b^4*e^(
-4) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d
)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 33
25608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*b^5*e^(-5) + 46189*(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^5)*e^(-1)