3.1225 \(\int \frac{(A+B x) (b x+c x^2)^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=265 \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/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)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(9/2))/(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11
*e^6)

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Rubi [A]  time = 0.158487, antiderivative size = 265, 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} \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/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)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(9/2))/(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11
*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 \frac{(A+B x) \left (b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 \sqrt{d+e x}}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt{d+e x}}{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)^{3/2}}{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)^{5/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{7/2}}{e^5}+\frac{B c^2 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac{2 d^2 (B d-A e) (c d-b e)^2 \sqrt{d+e x}}{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)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{9/2}}{9 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6}\\ \end{align*}

Mathematica [A]  time = 0.207352, size = 273, normalized size = 1.03 \[ \frac{2 \sqrt{d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (99 b^2 e^2 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )-5 c^2 \left (96 d^3 e^2 x^2-80 d^2 e^3 x^3-128 d^4 e x+256 d^5+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(11*A*e*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x
^2 + 5*e^3*x^3) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) + B*(99*b^2*e^2*(-1
6*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 22*b*c*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3
+ 35*e^4*x^4) - 5*c^2*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5))))
/(3465*e^6)

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Maple [A]  time = 0.008, size = 341, normalized size = 1.3 \begin{align*}{\frac{630\,B{c}^{2}{x}^{5}{e}^{5}+770\,A{c}^{2}{e}^{5}{x}^{4}+1540\,Bbc{e}^{5}{x}^{4}-700\,B{c}^{2}d{e}^{4}{x}^{4}+1980\,Abc{e}^{5}{x}^{3}-880\,A{c}^{2}d{e}^{4}{x}^{3}+990\,B{b}^{2}{e}^{5}{x}^{3}-1760\,Bbcd{e}^{4}{x}^{3}+800\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+1386\,A{b}^{2}{e}^{5}{x}^{2}-2376\,Abcd{e}^{4}{x}^{2}+1056\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-1188\,B{b}^{2}d{e}^{4}{x}^{2}+2112\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-1848\,A{b}^{2}d{e}^{4}x+3168\,Abc{d}^{2}{e}^{3}x-1408\,A{c}^{2}{d}^{3}{e}^{2}x+1584\,B{b}^{2}{d}^{2}{e}^{3}x-2816\,Bbc{d}^{3}{e}^{2}x+1280\,B{c}^{2}{d}^{4}ex+3696\,A{b}^{2}{d}^{2}{e}^{3}-6336\,Abc{d}^{3}{e}^{2}+2816\,A{c}^{2}{d}^{4}e-3168\,B{b}^{2}{d}^{3}{e}^{2}+5632\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x)

[Out]

2/3465*(315*B*c^2*e^5*x^5+385*A*c^2*e^5*x^4+770*B*b*c*e^5*x^4-350*B*c^2*d*e^4*x^4+990*A*b*c*e^5*x^3-440*A*c^2*
d*e^4*x^3+495*B*b^2*e^5*x^3-880*B*b*c*d*e^4*x^3+400*B*c^2*d^2*e^3*x^3+693*A*b^2*e^5*x^2-1188*A*b*c*d*e^4*x^2+5
28*A*c^2*d^2*e^3*x^2-594*B*b^2*d*e^4*x^2+1056*B*b*c*d^2*e^3*x^2-480*B*c^2*d^3*e^2*x^2-924*A*b^2*d*e^4*x+1584*A
*b*c*d^2*e^3*x-704*A*c^2*d^3*e^2*x+792*B*b^2*d^2*e^3*x-1408*B*b*c*d^3*e^2*x+640*B*c^2*d^4*e*x+1848*A*b^2*d^2*e
^3-3168*A*b*c*d^3*e^2+1408*A*c^2*d^4*e-1584*B*b^2*d^3*e^2+2816*B*b*c*d^4*e-1280*B*c^2*d^5)*(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 1.02573, size = 393, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{2} - 385 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\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{7}{2}} - 693 \,{\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{5}{2}} + 1155 \,{\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{3}{2}} - 3465 \,{\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 )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(9/2) + 495*(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)^(7/2) - 693*(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)^(5/2) + 1155*(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)^(3/2) - 3465*(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)*sqrt(e*x + d))/e^6

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Fricas [A]  time = 1.65204, size = 670, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \,{\left (10 \, B c^{2} d e^{4} - 11 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{2} e^{3} - 88 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1848*A*b^2*d^2*e^3 + 1408*(2*B*b*c + A*c^2)*d^4*e - 1584*(B*b^2 +
2*A*b*c)*d^3*e^2 - 35*(10*B*c^2*d*e^4 - 11*(2*B*b*c + A*c^2)*e^5)*x^4 + 5*(80*B*c^2*d^2*e^3 - 88*(2*B*b*c + A
*c^2)*d*e^4 + 99*(B*b^2 + 2*A*b*c)*e^5)*x^3 - 3*(160*B*c^2*d^3*e^2 - 231*A*b^2*e^5 - 176*(2*B*b*c + A*c^2)*d^2
*e^3 + 198*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 231*A*b^2*d*e^4 - 176*(2*B*b*c + A*c^2)*d^3*e^2
+ 198*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 114.911, size = 944, normalized size = 3.56 \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)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*A*b**2*(-d**3/s
qrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 4*A*b*c*d*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 4*A*b*c*(d**4/sqrt(d + e*x) +
4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 2*A*c**2*
d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(
7/2)/7)/e**4 + 2*A*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d +
e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*B*b**2*d*(-d**3/sqrt(d + e*x) - 3*d**2*sq
rt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 2*B*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e
*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 4*B*b*c*d*(d**4/sqrt(d + e
*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 4*B
*b*c*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*
(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*B*c**2*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d*
*3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 + 2*B*c**2
*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d +
e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), ((A*b**2*x**3/3 + B*c**2*x
**6/6 + x**5*(A*c**2 + 2*B*b*c)/5 + x**4*(2*A*b*c + B*b**2)/4)/sqrt(d), True))

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Giac [A]  time = 1.30802, size = 510, normalized size = 1.92 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} A b^{2} e^{\left (-2\right )} + 99 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} B b^{2} e^{\left (-3\right )} + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} A b c e^{\left (-3\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} B b c e^{\left (-4\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} A c^{2} e^{\left (-4\right )} + 5 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} B c^{2} e^{\left (-5\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*e^(-2) + 99*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b^2*e^(-3) + 198*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b*c*e^(-3) + 22*(35*(x*e + d)^(
9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b*
c*e^(-4) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3
+ 315*sqrt(x*e + d)*d^4)*A*c^2*e^(-4) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d
^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*c^2*e^(-5))*e^(-1)