3.1797 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac{4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac{2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt{d+e x}}+\frac{2 b^4 B (d+e x)^{9/2}}{9 e^6} \]

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(e^6*Sqrt[d + e*x]) + (2*(b*d - a*e)^3*(5*b*B*d -
4*A*b*e - a*B*e)*Sqrt[d + e*x])/e^6 - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*
a*B*e)*(d + e*x)^(3/2))/(3*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*
e)*(d + e*x)^(5/2))/(5*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(7/2)
)/(7*e^6) + (2*b^4*B*(d + e*x)^(9/2))/(9*e^6)

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Rubi [A]  time = 0.263882, 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 \[ -\frac{2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac{4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac{2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt{d+e x}}+\frac{2 b^4 B (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(e^6*Sqrt[d + e*x]) + (2*(b*d - a*e)^3*(5*b*B*d -
4*A*b*e - a*B*e)*Sqrt[d + e*x])/e^6 - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*
a*B*e)*(d + e*x)^(3/2))/(3*e^6) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*
e)*(d + e*x)^(5/2))/(5*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(7/2)
)/(7*e^6) + (2*b^4*B*(d + e*x)^(9/2))/(9*e^6)

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Rubi in Sympy [A]  time = 101.337, size = 218, normalized size = 1.02 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{7}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{7 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{5 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{e^{6}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{4}}{e^{6} \sqrt{d + e x}} \]

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)**(3/2),x)

[Out]

2*B*b**4*(d + e*x)**(9/2)/(9*e**6) + 2*b**3*(d + e*x)**(7/2)*(A*b*e + 4*B*a*e -
5*B*b*d)/(7*e**6) + 4*b**2*(d + e*x)**(5/2)*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B
*b*d)/(5*e**6) + 4*b*(d + e*x)**(3/2)*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*
d)/(3*e**6) + 2*sqrt(d + e*x)*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b*d)/e**6 -
2*(A*e - B*d)*(a*e - b*d)**4/(e**6*sqrt(d + e*x))

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Mathematica [A]  time = 0.429893, size = 335, normalized size = 1.57 \[ \frac{630 a^4 e^4 (-A e+2 B d+B e x)+840 a^3 b e^3 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+252 a^2 b^2 e^2 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-72 a b^3 e \left (B \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+2 b^4 \left (9 A e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 B \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{315 e^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(630*a^4*e^4*(2*B*d - A*e + B*e*x) + 840*a^3*b*e^3*(3*A*e*(2*d + e*x) + B*(-8*d^
2 - 4*d*e*x + e^2*x^2)) + 252*a^2*b^2*e^2*(5*A*e*(-8*d^2 - 4*d*e*x + e^2*x^2) +
3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 72*a*b^3*e*(-7*A*e*(16*d^3 +
8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + B*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 +
8*d*e^3*x^3 - 5*e^4*x^4)) + 2*b^4*(9*A*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^
2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*B*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*
d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(315*e^6*Sqrt[d + e*x])

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Maple [B]  time = 0.015, size = 469, normalized size = 2.2 \[ -{\frac{-70\,{b}^{4}B{x}^{5}{e}^{5}-90\,A{b}^{4}{e}^{5}{x}^{4}-360\,Ba{b}^{3}{e}^{5}{x}^{4}+100\,B{b}^{4}d{e}^{4}{x}^{4}-504\,Aa{b}^{3}{e}^{5}{x}^{3}+144\,A{b}^{4}d{e}^{4}{x}^{3}-756\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}+576\,Ba{b}^{3}d{e}^{4}{x}^{3}-160\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}-1260\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}+1008\,Aa{b}^{3}d{e}^{4}{x}^{2}-288\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}-840\,B{a}^{3}b{e}^{5}{x}^{2}+1512\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}-1152\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}+320\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}-2520\,A{a}^{3}b{e}^{5}x+5040\,A{a}^{2}{b}^{2}d{e}^{4}x-4032\,Aa{b}^{3}{d}^{2}{e}^{3}x+1152\,A{b}^{4}{d}^{3}{e}^{2}x-630\,B{a}^{4}{e}^{5}x+3360\,B{a}^{3}bd{e}^{4}x-6048\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x+4608\,Ba{b}^{3}{d}^{3}{e}^{2}x-1280\,B{b}^{4}{d}^{4}ex+630\,A{a}^{4}{e}^{5}-5040\,Ad{a}^{3}b{e}^{4}+10080\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-8064\,Aa{b}^{3}{d}^{3}{e}^{2}+2304\,A{d}^{4}{b}^{4}e-1260\,B{a}^{4}d{e}^{4}+6720\,B{d}^{2}{a}^{3}b{e}^{3}-12096\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+9216\,B{d}^{4}a{b}^{3}e-2560\,{b}^{4}B{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2),x)

[Out]

-2/315*(-35*B*b^4*e^5*x^5-45*A*b^4*e^5*x^4-180*B*a*b^3*e^5*x^4+50*B*b^4*d*e^4*x^
4-252*A*a*b^3*e^5*x^3+72*A*b^4*d*e^4*x^3-378*B*a^2*b^2*e^5*x^3+288*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+504*A*a*b^3*d*e^4*x^2-144*A*b^4*
d^2*e^3*x^2-420*B*a^3*b*e^5*x^2+756*B*a^2*b^2*d*e^4*x^2-576*B*a*b^3*d^2*e^3*x^2+
160*B*b^4*d^3*e^2*x^2-1260*A*a^3*b*e^5*x+2520*A*a^2*b^2*d*e^4*x-2016*A*a*b^3*d^2
*e^3*x+576*A*b^4*d^3*e^2*x-315*B*a^4*e^5*x+1680*B*a^3*b*d*e^4*x-3024*B*a^2*b^2*d
^2*e^3*x+2304*B*a*b^3*d^3*e^2*x-640*B*b^4*d^4*e*x+315*A*a^4*e^5-2520*A*a^3*b*d*e
^4+5040*A*a^2*b^2*d^2*e^3-4032*A*a*b^3*d^3*e^2+1152*A*b^4*d^4*e-630*B*a^4*d*e^4+
3360*B*a^3*b*d^2*e^3-6048*B*a^2*b^2*d^3*e^2+4608*B*a*b^3*d^4*e-1280*B*b^4*d^5)/(
e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.737314, size = 563, normalized size = 2.63 \[ \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{4} - 45 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 126 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 210 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*B*b^4 - 45*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x +
d)^(7/2) + 126*(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)^(5/2) - 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)*(e*x + d)^(3/2)
+ 315*(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)*sqrt(e*x +
d))/e^5 + 315*(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)/(sqrt(e*x + d)*e^5))/e

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Fricas [A]  time = 0.290443, size = 551, normalized size = 2.57 \[ \frac{2 \,{\left (35 \, B b^{4} e^{5} x^{5} + 1280 \, B b^{4} d^{5} - 315 \, A a^{4} e^{5} - 1152 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2016 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 1680 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 630 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \,{\left (10 \, B b^{4} d e^{4} - 9 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B b^{4} d^{2} e^{3} - 36 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 63 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 2 \,{\left (80 \, B b^{4} d^{3} e^{2} - 72 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 126 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 105 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} +{\left (640 \, B b^{4} d^{4} e - 576 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1008 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 840 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )}}{315 \, \sqrt{e x + d} e^{6}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^4*e^5*x^5 + 1280*B*b^4*d^5 - 315*A*a^4*e^5 - 1152*(4*B*a*b^3 + A*b
^4)*d^4*e + 2016*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 1680*(2*B*a^3*b + 3*A*a^2*b
^2)*d^2*e^3 + 630*(B*a^4 + 4*A*a^3*b)*d*e^4 - 5*(10*B*b^4*d*e^4 - 9*(4*B*a*b^3 +
A*b^4)*e^5)*x^4 + 2*(40*B*b^4*d^2*e^3 - 36*(4*B*a*b^3 + A*b^4)*d*e^4 + 63*(3*B*
a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 - 2*(80*B*b^4*d^3*e^2 - 72*(4*B*a*b^3 + A*b^4)*d^2
*e^3 + 126*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - 105*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*
x^2 + (640*B*b^4*d^4*e - 576*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 1008*(3*B*a^2*b^2 + 2
*A*a*b^3)*d^2*e^3 - 840*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 315*(B*a^4 + 4*A*a^3*b
)*e^5)*x)/(sqrt(e*x + d)*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{4}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**4/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.288707, size = 792, normalized size = 3.7 \[ \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B b^{4} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B b^{4} d^{4} e^{48} + 180 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{3} e^{49} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{4} e^{49} - 1008 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{49} + 2520 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{49} - 5040 \, \sqrt{x e + d} B a b^{3} d^{3} e^{49} - 1260 \, \sqrt{x e + d} A b^{4} d^{3} e^{49} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{50} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{50} - 1890 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{50} + 5670 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{50} + 3780 \, \sqrt{x e + d} A a b^{3} d^{2} e^{50} + 420 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{51} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{51} - 2520 \, \sqrt{x e + d} B a^{3} b d e^{51} - 3780 \, \sqrt{x e + d} A a^{2} b^{2} d e^{51} + 315 \, \sqrt{x e + d} B a^{4} e^{52} + 1260 \, \sqrt{x e + d} A a^{3} b e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B b^{4} d^{5} - 4 \, B a b^{3} d^{4} e - A b^{4} d^{4} e + 6 \, B a^{2} b^{2} d^{3} e^{2} + 4 \, A a b^{3} d^{3} e^{2} - 4 \, B a^{3} b d^{2} e^{3} - 6 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} - A a^{4} e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \]

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)^(3/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^4*e^48 - 225*(x*e + d)^(7/2)*B*b^4*d*e^48 + 630*(x
*e + d)^(5/2)*B*b^4*d^2*e^48 - 1050*(x*e + d)^(3/2)*B*b^4*d^3*e^48 + 1575*sqrt(x
*e + d)*B*b^4*d^4*e^48 + 180*(x*e + d)^(7/2)*B*a*b^3*e^49 + 45*(x*e + d)^(7/2)*A
*b^4*e^49 - 1008*(x*e + d)^(5/2)*B*a*b^3*d*e^49 - 252*(x*e + d)^(5/2)*A*b^4*d*e^
49 + 2520*(x*e + d)^(3/2)*B*a*b^3*d^2*e^49 + 630*(x*e + d)^(3/2)*A*b^4*d^2*e^49
- 5040*sqrt(x*e + d)*B*a*b^3*d^3*e^49 - 1260*sqrt(x*e + d)*A*b^4*d^3*e^49 + 378*
(x*e + d)^(5/2)*B*a^2*b^2*e^50 + 252*(x*e + d)^(5/2)*A*a*b^3*e^50 - 1890*(x*e +
d)^(3/2)*B*a^2*b^2*d*e^50 - 1260*(x*e + d)^(3/2)*A*a*b^3*d*e^50 + 5670*sqrt(x*e
+ d)*B*a^2*b^2*d^2*e^50 + 3780*sqrt(x*e + d)*A*a*b^3*d^2*e^50 + 420*(x*e + d)^(3
/2)*B*a^3*b*e^51 + 630*(x*e + d)^(3/2)*A*a^2*b^2*e^51 - 2520*sqrt(x*e + d)*B*a^3
*b*d*e^51 - 3780*sqrt(x*e + d)*A*a^2*b^2*d*e^51 + 315*sqrt(x*e + d)*B*a^4*e^52 +
1260*sqrt(x*e + d)*A*a^3*b*e^52)*e^(-54) + 2*(B*b^4*d^5 - 4*B*a*b^3*d^4*e - A*b
^4*d^4*e + 6*B*a^2*b^2*d^3*e^2 + 4*A*a*b^3*d^3*e^2 - 4*B*a^3*b*d^2*e^3 - 6*A*a^2
*b^2*d^2*e^3 + B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 - A*a^4*e^5)*e^(-6)/sqrt(x*e + d)