3.1223 \(\int (A+B x) (d+e x)^{3/2} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac{2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac{2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac{2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac{2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac{2 B c^2 (d+e x)^{15/2}}{15 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(7/2))/(7*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(9/2))/(9*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(
d + e*x)^(11/2))/(11*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(13/2))/(13*e^6) + (2*B*c^2*(d + e*x)^(
15/2))/(15*e^6)

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Rubi [A]  time = 0.158342, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ -\frac{2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac{2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac{2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac{2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac{2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac{2 B c^2 (d+e x)^{15/2}}{15 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(7/2))/(7*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(9/2))/(9*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(
d + e*x)^(11/2))/(11*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(13/2))/(13*e^6) + (2*B*c^2*(d + e*x)^(
15/2))/(15*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{e^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{5/2}}{e^5}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{7/2}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{9/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{11/2}}{e^5}+\frac{B c^2 (d+e x)^{13/2}}{e^5}\right ) \, dx\\ &=-\frac{2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{5/2}}{5 e^6}+\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{7/2}}{7 e^6}+\frac{2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^6}-\frac{2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{13/2}}{13 e^6}+\frac{2 B c^2 (d+e x)^{15/2}}{15 e^6}\\ \end{align*}

Mathematica [A]  time = 0.210188, size = 272, normalized size = 1.02 \[ \frac{2 (d+e x)^{5/2} \left (A e \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )+B \left (39 b^2 e^2 \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )+6 b c e \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )+c^2 \left (-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3+640 d^4 e x-256 d^5-2310 d e^4 x^4+3003 e^5 x^5\right )\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(A*e*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*d^3 + 40*d^2*e*x - 70*d*e
^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)) + B*(3
9*b^2*e^2*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 6*b*c*e*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x
^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + c^2*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*
d*e^4*x^4 + 3003*e^5*x^5))))/(45045*e^6)

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Maple [A]  time = 0.008, size = 341, normalized size = 1.3 \begin{align*}{\frac{6006\,B{c}^{2}{x}^{5}{e}^{5}+6930\,A{c}^{2}{e}^{5}{x}^{4}+13860\,Bbc{e}^{5}{x}^{4}-4620\,B{c}^{2}d{e}^{4}{x}^{4}+16380\,Abc{e}^{5}{x}^{3}-5040\,A{c}^{2}d{e}^{4}{x}^{3}+8190\,B{b}^{2}{e}^{5}{x}^{3}-10080\,Bbcd{e}^{4}{x}^{3}+3360\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+10010\,A{b}^{2}{e}^{5}{x}^{2}-10920\,Abcd{e}^{4}{x}^{2}+3360\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-5460\,B{b}^{2}d{e}^{4}{x}^{2}+6720\,Bbc{d}^{2}{e}^{3}{x}^{2}-2240\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-5720\,A{b}^{2}d{e}^{4}x+6240\,Abc{d}^{2}{e}^{3}x-1920\,A{c}^{2}{d}^{3}{e}^{2}x+3120\,B{b}^{2}{d}^{2}{e}^{3}x-3840\,Bbc{d}^{3}{e}^{2}x+1280\,B{c}^{2}{d}^{4}ex+2288\,A{b}^{2}{d}^{2}{e}^{3}-2496\,Abc{d}^{3}{e}^{2}+768\,A{c}^{2}{d}^{4}e-1248\,B{b}^{2}{d}^{3}{e}^{2}+1536\,Bbc{d}^{4}e-512\,B{c}^{2}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3003*B*c^2*e^5*x^5+3465*A*c^2*e^5*x^4+6930*B*b*c*e^5*x^4-2310*B*c^2*d*e^4*x^4+8190*A*b*
c*e^5*x^3-2520*A*c^2*d*e^4*x^3+4095*B*b^2*e^5*x^3-5040*B*b*c*d*e^4*x^3+1680*B*c^2*d^2*e^3*x^3+5005*A*b^2*e^5*x
^2-5460*A*b*c*d*e^4*x^2+1680*A*c^2*d^2*e^3*x^2-2730*B*b^2*d*e^4*x^2+3360*B*b*c*d^2*e^3*x^2-1120*B*c^2*d^3*e^2*
x^2-2860*A*b^2*d*e^4*x+3120*A*b*c*d^2*e^3*x-960*A*c^2*d^3*e^2*x+1560*B*b^2*d^2*e^3*x-1920*B*b*c*d^3*e^2*x+640*
B*c^2*d^4*e*x+1144*A*b^2*d^2*e^3-1248*A*b*c*d^3*e^2+384*A*c^2*d^4*e-624*B*b^2*d^3*e^2+768*B*b*c*d^4*e-256*B*c^
2*d^5)/e^6

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Maxima [A]  time = 1.07998, size = 393, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B c^{2} - 3465 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 9009 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*c^2 - 3465*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(13/2) + 4095*(10*B*c^
2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(11/2) - 5005*(10*B*c^2*d^3 - A*b^2*e^3 - 6
*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(9/2) + 6435*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*
(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(7/2) - 9009*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*
B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(5/2))/e^6

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Fricas [A]  time = 1.82457, size = 954, normalized size = 3.57 \begin{align*} \frac{2 \,{\left (3003 \, B c^{2} e^{7} x^{7} - 256 \, B c^{2} d^{7} + 1144 \, A b^{2} d^{4} e^{3} + 384 \,{\left (2 \, B b c + A c^{2}\right )} d^{6} e - 624 \,{\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{2} + 231 \,{\left (16 \, B c^{2} d e^{6} + 15 \,{\left (2 \, B b c + A c^{2}\right )} e^{7}\right )} x^{6} + 63 \,{\left (B c^{2} d^{2} e^{5} + 70 \,{\left (2 \, B b c + A c^{2}\right )} d e^{6} + 65 \,{\left (B b^{2} + 2 \, A b c\right )} e^{7}\right )} x^{5} - 35 \,{\left (2 \, B c^{2} d^{3} e^{4} - 143 \, A b^{2} e^{7} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{5} - 156 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{6}\right )} x^{4} + 5 \,{\left (16 \, B c^{2} d^{4} e^{3} + 1430 \, A b^{2} d e^{6} - 24 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{4} + 39 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (32 \, B c^{2} d^{5} e^{2} - 143 \, A b^{2} d^{2} e^{5} - 48 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{3} + 78 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{4}\right )} x^{2} + 4 \,{\left (32 \, B c^{2} d^{6} e - 143 \, A b^{2} d^{3} e^{4} - 48 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e^{2} + 78 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^2*e^7*x^7 - 256*B*c^2*d^7 + 1144*A*b^2*d^4*e^3 + 384*(2*B*b*c + A*c^2)*d^6*e - 624*(B*b^2 +
2*A*b*c)*d^5*e^2 + 231*(16*B*c^2*d*e^6 + 15*(2*B*b*c + A*c^2)*e^7)*x^6 + 63*(B*c^2*d^2*e^5 + 70*(2*B*b*c + A*c
^2)*d*e^6 + 65*(B*b^2 + 2*A*b*c)*e^7)*x^5 - 35*(2*B*c^2*d^3*e^4 - 143*A*b^2*e^7 - 3*(2*B*b*c + A*c^2)*d^2*e^5
- 156*(B*b^2 + 2*A*b*c)*d*e^6)*x^4 + 5*(16*B*c^2*d^4*e^3 + 1430*A*b^2*d*e^6 - 24*(2*B*b*c + A*c^2)*d^3*e^4 + 3
9*(B*b^2 + 2*A*b*c)*d^2*e^5)*x^3 - 3*(32*B*c^2*d^5*e^2 - 143*A*b^2*d^2*e^5 - 48*(2*B*b*c + A*c^2)*d^4*e^3 + 78
*(B*b^2 + 2*A*b*c)*d^3*e^4)*x^2 + 4*(32*B*c^2*d^6*e - 143*A*b^2*d^3*e^4 - 48*(2*B*b*c + A*c^2)*d^5*e^2 + 78*(B
*b^2 + 2*A*b*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^6

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Sympy [B]  time = 38.1458, size = 937, normalized size = 3.51 \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)**(3/2)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*A*b**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**3 + 4*A*b*c*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**4 + 4*A*b
*c*(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*c**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**5 + 2*A*c**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**5 + 2*B*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**4 + 2*B*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**4 + 4*B*b*c*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**5 + 4*B*b
*c*(-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*c**2*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)**(1
3/2)/13)/e**6 + 2*B*c**2*(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 - 2
0*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*
*6

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Giac [B]  time = 1.31391, size = 1126, normalized size = 4.22 \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)^(3/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/45045*(429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b^2*d*e^(-2) + 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*b^2*d*e^(-3) + 286
*(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*c*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)*B*b*c*d*e^(-4) + 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)*A*c^2*d*e^(-4) + 5*(693*(x*e + d)^(13/2) - 4
095*(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 - 30
03*(x*e + d)^(3/2)*d^5)*B*c^2*d*e^(-5) + 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^2*e^(-2) + 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^2*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*
c*e^(-3) + 10*(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*c*e^(-4) + 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*c^2*e^(-4) + (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*c^2*e^(-5))*e^(-1)