3.1798 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{2 b^3 (d+e x)^{5/2} (-4 a B e-A b e+5 b B d)}{5 e^6}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6}-\frac{4 b \sqrt{d+e x} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6}-\frac{2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^4 B (d+e x)^{7/2}}{7 e^6} \]

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(3*e^6*(d + e*x)^(3/2)) - (2*(b*d - a*e)^3*(5*b*B*
d - 4*A*b*e - a*B*e))/(e^6*Sqrt[d + e*x]) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*
e - 2*a*B*e)*Sqrt[d + e*x])/e^6 + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*
e)*(d + e*x)^(3/2))/(3*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(5/2)
)/(5*e^6) + (2*b^4*B*(d + e*x)^(7/2))/(7*e^6)

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Rubi [A]  time = 0.267102, antiderivative size = 214, 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 \[ -\frac{2 b^3 (d+e x)^{5/2} (-4 a B e-A b e+5 b B d)}{5 e^6}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6}-\frac{4 b \sqrt{d+e x} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6}-\frac{2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^{3/2}}+\frac{2 b^4 B (d+e x)^{7/2}}{7 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(3*e^6*(d + e*x)^(3/2)) - (2*(b*d - a*e)^3*(5*b*B*
d - 4*A*b*e - a*B*e))/(e^6*Sqrt[d + e*x]) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*
e - 2*a*B*e)*Sqrt[d + e*x])/e^6 + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*
e)*(d + e*x)^(3/2))/(3*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(5/2)
)/(5*e^6) + (2*b^4*B*(d + e*x)^(7/2))/(7*e^6)

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Rubi in Sympy [A]  time = 100.721, size = 218, normalized size = 1.02 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{5}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{5 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{3 e^{6}} + \frac{4 b \sqrt{d + e x} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6}} - \frac{2 \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{e^{6} \sqrt{d + e x}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{4}}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(5/2),x)

[Out]

2*B*b**4*(d + e*x)**(7/2)/(7*e**6) + 2*b**3*(d + e*x)**(5/2)*(A*b*e + 4*B*a*e -
5*B*b*d)/(5*e**6) + 4*b**2*(d + e*x)**(3/2)*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B
*b*d)/(3*e**6) + 4*b*sqrt(d + e*x)*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)/
e**6 - 2*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b*d)/(e**6*sqrt(d + e*x)) - 2*(A*
e - B*d)*(a*e - b*d)**4/(3*e**6*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.653814, size = 336, normalized size = 1.57 \[ \frac{-70 a^4 e^4 (A e+2 B d+3 B e x)+280 a^3 b e^3 \left (B \left (8 d^2+12 d e x+3 e^2 x^2\right )-A e (2 d+3 e x)\right )+420 a^2 b^2 e^2 \left (A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )+56 a b^3 e \left (5 A e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+2 b^4 \left (7 A e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 B \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{105 e^6 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(5/2),x]

[Out]

(-70*a^4*e^4*(2*B*d + A*e + 3*B*e*x) + 280*a^3*b*e^3*(-(A*e*(2*d + 3*e*x)) + B*(
8*d^2 + 12*d*e*x + 3*e^2*x^2)) + 420*a^2*b^2*e^2*(A*e*(8*d^2 + 12*d*e*x + 3*e^2*
x^2) + B*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3)) + 56*a*b^3*e*(5*A*e*(-1
6*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + B*(128*d^4 + 192*d^3*e*x + 48*d^2*
e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) + 2*b^4*(7*A*e*(128*d^4 + 192*d^3*e*x + 48*d
^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) - 5*B*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*
x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(105*e^6*(d + e*x)^(3/2))

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Maple [B]  time = 0.014, size = 469, normalized size = 2.2 \[ -{\frac{-30\,{b}^{4}B{x}^{5}{e}^{5}-42\,A{b}^{4}{e}^{5}{x}^{4}-168\,Ba{b}^{3}{e}^{5}{x}^{4}+60\,B{b}^{4}d{e}^{4}{x}^{4}-280\,Aa{b}^{3}{e}^{5}{x}^{3}+112\,A{b}^{4}d{e}^{4}{x}^{3}-420\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}+448\,Ba{b}^{3}d{e}^{4}{x}^{3}-160\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}-1260\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}+1680\,Aa{b}^{3}d{e}^{4}{x}^{2}-672\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}-840\,B{a}^{3}b{e}^{5}{x}^{2}+2520\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}-2688\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}+960\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+840\,A{a}^{3}b{e}^{5}x-5040\,A{a}^{2}{b}^{2}d{e}^{4}x+6720\,Aa{b}^{3}{d}^{2}{e}^{3}x-2688\,A{b}^{4}{d}^{3}{e}^{2}x+210\,B{a}^{4}{e}^{5}x-3360\,B{a}^{3}bd{e}^{4}x+10080\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-10752\,Ba{b}^{3}{d}^{3}{e}^{2}x+3840\,B{b}^{4}{d}^{4}ex+70\,A{a}^{4}{e}^{5}+560\,Ad{a}^{3}b{e}^{4}-3360\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}+4480\,Aa{b}^{3}{d}^{3}{e}^{2}-1792\,A{d}^{4}{b}^{4}e+140\,B{a}^{4}d{e}^{4}-2240\,B{a}^{3}b{d}^{2}{e}^{3}+6720\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}-7168\,B{d}^{4}a{b}^{3}e+2560\,{b}^{4}B{d}^{5}}{105\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(5/2),x)

[Out]

-2/105*(-15*B*b^4*e^5*x^5-21*A*b^4*e^5*x^4-84*B*a*b^3*e^5*x^4+30*B*b^4*d*e^4*x^4
-140*A*a*b^3*e^5*x^3+56*A*b^4*d*e^4*x^3-210*B*a^2*b^2*e^5*x^3+224*B*a*b^3*d*e^4*
x^3-80*B*b^4*d^2*e^3*x^3-630*A*a^2*b^2*e^5*x^2+840*A*a*b^3*d*e^4*x^2-336*A*b^4*d
^2*e^3*x^2-420*B*a^3*b*e^5*x^2+1260*B*a^2*b^2*d*e^4*x^2-1344*B*a*b^3*d^2*e^3*x^2
+480*B*b^4*d^3*e^2*x^2+420*A*a^3*b*e^5*x-2520*A*a^2*b^2*d*e^4*x+3360*A*a*b^3*d^2
*e^3*x-1344*A*b^4*d^3*e^2*x+105*B*a^4*e^5*x-1680*B*a^3*b*d*e^4*x+5040*B*a^2*b^2*
d^2*e^3*x-5376*B*a*b^3*d^3*e^2*x+1920*B*b^4*d^4*e*x+35*A*a^4*e^5+280*A*a^3*b*d*e
^4-1680*A*a^2*b^2*d^2*e^3+2240*A*a*b^3*d^3*e^2-896*A*b^4*d^4*e+70*B*a^4*d*e^4-11
20*B*a^3*b*d^2*e^3+3360*B*a^2*b^2*d^3*e^2-3584*B*a*b^3*d^4*e+1280*B*b^4*d^5)/(e*
x+d)^(3/2)/e^6

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Maxima [A]  time = 0.718856, size = 560, normalized size = 2.62 \[ \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{4} - 21 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} \sqrt{e x + d}}{e^{5}} + \frac{35 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{105 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(e*x + d)^(7/2)*B*b^4 - 21*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x +
d)^(5/2) + 70*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b
^3)*e^2)*(e*x + d)^(3/2) - 210*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3
*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*sqrt(e*x + d))/e^
5 + 35*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A
*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^
4 - 3*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d
^2*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d))
/((e*x + d)^(3/2)*e^5))/e

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Fricas [A]  time = 0.299313, size = 566, normalized size = 2.64 \[ \frac{2 \,{\left (15 \, B b^{4} e^{5} x^{5} - 1280 \, B b^{4} d^{5} - 35 \, A a^{4} e^{5} + 896 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 1120 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 560 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 70 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \,{\left (10 \, B b^{4} d e^{4} - 7 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B b^{4} d^{2} e^{3} - 28 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 35 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \,{\left (80 \, B b^{4} d^{3} e^{2} - 56 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 70 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 35 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 3 \,{\left (640 \, B b^{4} d^{4} e - 448 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 560 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 280 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )}}{105 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^4*e^5*x^5 - 1280*B*b^4*d^5 - 35*A*a^4*e^5 + 896*(4*B*a*b^3 + A*b^4
)*d^4*e - 1120*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 560*(2*B*a^3*b + 3*A*a^2*b^2)
*d^2*e^3 - 70*(B*a^4 + 4*A*a^3*b)*d*e^4 - 3*(10*B*b^4*d*e^4 - 7*(4*B*a*b^3 + A*b
^4)*e^5)*x^4 + 2*(40*B*b^4*d^2*e^3 - 28*(4*B*a*b^3 + A*b^4)*d*e^4 + 35*(3*B*a^2*
b^2 + 2*A*a*b^3)*e^5)*x^3 - 6*(80*B*b^4*d^3*e^2 - 56*(4*B*a*b^3 + A*b^4)*d^2*e^3
+ 70*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - 35*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 -
3*(640*B*b^4*d^4*e - 448*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 560*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^2*e^3 - 280*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 35*(B*a^4 + 4*A*a^3*b)*e^5)
*x)/((e^7*x + d*e^6)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{4}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(5/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**4/(d + e*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.288339, size = 765, normalized size = 3.57 \[ \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} e^{36} - 105 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d e^{36} + 350 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{2} e^{36} - 1050 \, \sqrt{x e + d} B b^{4} d^{3} e^{36} + 84 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} e^{37} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} e^{37} - 560 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d e^{37} + 2520 \, \sqrt{x e + d} B a b^{3} d^{2} e^{37} + 630 \, \sqrt{x e + d} A b^{4} d^{2} e^{37} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} e^{38} + 140 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} e^{38} - 1890 \, \sqrt{x e + d} B a^{2} b^{2} d e^{38} - 1260 \, \sqrt{x e + d} A a b^{3} d e^{38} + 420 \, \sqrt{x e + d} B a^{3} b e^{39} + 630 \, \sqrt{x e + d} A a^{2} b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} B b^{4} d^{4} - B b^{4} d^{5} - 48 \,{\left (x e + d\right )} B a b^{3} d^{3} e - 12 \,{\left (x e + d\right )} A b^{4} d^{3} e + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 54 \,{\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{2} + 36 \,{\left (x e + d\right )} A a b^{3} d^{2} e^{2} - 6 \, B a^{2} b^{2} d^{3} e^{2} - 4 \, A a b^{3} d^{3} e^{2} - 24 \,{\left (x e + d\right )} B a^{3} b d e^{3} - 36 \,{\left (x e + d\right )} A a^{2} b^{2} d e^{3} + 4 \, B a^{3} b d^{2} e^{3} + 6 \, A a^{2} b^{2} d^{2} e^{3} + 3 \,{\left (x e + d\right )} B a^{4} e^{4} + 12 \,{\left (x e + d\right )} A a^{3} b e^{4} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} + A a^{4} e^{5}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*b^4*e^36 - 105*(x*e + d)^(5/2)*B*b^4*d*e^36 + 350*(x
*e + d)^(3/2)*B*b^4*d^2*e^36 - 1050*sqrt(x*e + d)*B*b^4*d^3*e^36 + 84*(x*e + d)^
(5/2)*B*a*b^3*e^37 + 21*(x*e + d)^(5/2)*A*b^4*e^37 - 560*(x*e + d)^(3/2)*B*a*b^3
*d*e^37 - 140*(x*e + d)^(3/2)*A*b^4*d*e^37 + 2520*sqrt(x*e + d)*B*a*b^3*d^2*e^37
+ 630*sqrt(x*e + d)*A*b^4*d^2*e^37 + 210*(x*e + d)^(3/2)*B*a^2*b^2*e^38 + 140*(
x*e + d)^(3/2)*A*a*b^3*e^38 - 1890*sqrt(x*e + d)*B*a^2*b^2*d*e^38 - 1260*sqrt(x*
e + d)*A*a*b^3*d*e^38 + 420*sqrt(x*e + d)*B*a^3*b*e^39 + 630*sqrt(x*e + d)*A*a^2
*b^2*e^39)*e^(-42) - 2/3*(15*(x*e + d)*B*b^4*d^4 - B*b^4*d^5 - 48*(x*e + d)*B*a*
b^3*d^3*e - 12*(x*e + d)*A*b^4*d^3*e + 4*B*a*b^3*d^4*e + A*b^4*d^4*e + 54*(x*e +
d)*B*a^2*b^2*d^2*e^2 + 36*(x*e + d)*A*a*b^3*d^2*e^2 - 6*B*a^2*b^2*d^3*e^2 - 4*A
*a*b^3*d^3*e^2 - 24*(x*e + d)*B*a^3*b*d*e^3 - 36*(x*e + d)*A*a^2*b^2*d*e^3 + 4*B
*a^3*b*d^2*e^3 + 6*A*a^2*b^2*d^2*e^3 + 3*(x*e + d)*B*a^4*e^4 + 12*(x*e + d)*A*a^
3*b*e^4 - B*a^4*d*e^4 - 4*A*a^3*b*d*e^4 + A*a^4*e^5)*e^(-6)/(x*e + d)^(3/2)