3.1876 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=424 \[ -\frac{(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}+\frac{35 e^3 (a+b x) \sqrt{d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(35*e^3*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^5*(b*d - a*e)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^
(3/2))/(192*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(8*b*B*d + A*b
*e - 9*a*B*e)*(d + e*x)^(5/2))/(96*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - ((8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(24*b^2*(b*d - a*e)*
(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(9/2))/(4*b*
(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d + A*b*
e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(
11/2)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.835142, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{9/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{35 e^3 (a+b x) (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}+\frac{35 e^3 (a+b x) \sqrt{d+e x} (-9 a B e+A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{35 e^2 (d+e x)^{3/2} (-9 a B e+A b e+8 b B d)}{192 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-9 a B e+A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(35*e^3*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^5*(b*d - a*e)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^
(3/2))/(192*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(8*b*B*d + A*b
*e - 9*a*B*e)*(d + e*x)^(5/2))/(96*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - ((8*b*B*d + A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(24*b^2*(b*d - a*e)*
(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(9/2))/(4*b*
(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d + A*b*
e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(
11/2)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.19061, size = 233, normalized size = 0.55 \[ \frac{(a+b x) \left (\frac{\sqrt{d+e x} \left (-\frac{3 e^2 (-325 a B e+93 A b e+232 b B d)}{a+b x}+\frac{2 e (a e-b d) (-315 a B e+163 A b e+152 b B d)}{(a+b x)^2}-\frac{8 (b d-a e)^2 (-33 a B e+25 A b e+8 b B d)}{(a+b x)^3}-\frac{48 (A b-a B) (b d-a e)^3}{(a+b x)^4}+384 B e^3\right )}{3 b^5}-\frac{35 e^3 (-9 a B e+A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2} \sqrt{b d-a e}}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)*((Sqrt[d + e*x]*(384*B*e^3 - (48*(A*b - a*B)*(b*d - a*e)^3)/(a + b*x)
^4 - (8*(b*d - a*e)^2*(8*b*B*d + 25*A*b*e - 33*a*B*e))/(a + b*x)^3 + (2*e*(-(b*d
) + a*e)*(152*b*B*d + 163*A*b*e - 315*a*B*e))/(a + b*x)^2 - (3*e^2*(232*b*B*d +
93*A*b*e - 325*a*B*e))/(a + b*x)))/(3*b^5) - (35*e^3*(8*b*B*d + A*b*e - 9*a*B*e)
*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(11/2)*Sqrt[b*d - a*e])))/
(64*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.039, size = 1390, normalized size = 3.3 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-1/192*(5670*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^5+385*A
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3+385*A*(b*(a*e-b*d))^(1/2)*(e*x+d)
^(3/2)*b^4*d^2*e+105*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4-105*A*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e-630*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*x^2*a^2*b^3*e^5-1929*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3-840*B*
arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*d*e^4-3360*B*arctan((e*x+d)^(1
/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*d*e^4-1536*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*x^3*a*b^3*e^4+4079*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*d*e+945*B*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^4*e^5-840*B*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4-420*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*x^3*a*b^4*e^5-384*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^4*b^4*e^4+3780*B*
arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^2*b^3*e^5-975*B*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(7/2)*a*b^3*e+511*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-51
1*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d*e-5017*B*(b*(a*e-b*d))^(1/2)*(e*x+d)
^(3/2)*a*b^3*d^2*e-1536*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b*e^4-2304*B*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^2*e^4-3360*B*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4-770*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^
3*d*e^2-315*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3+315*A*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2+2139*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3
*b*d*e^3-3051*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2+1929*B*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+5402*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*
a^2*b^2*d*e^2-5040*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*d*e
^4-105*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*e^5+279*A*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(7/2)*b^4*e+696*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^4*d-178
4*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2-105*A*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*a^4*b*e^5+1544*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^4*d^3-945*B
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*e^4-456*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*b^4*d^4+945*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5-420*A*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5-2295*B*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(5/2)*a^2*b^2*e^2+3780*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a
^4*b*e^5)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^5/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300159, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(2*(384*B*b^4*e^3*x^4 - 16*(B*a*b^3 + 3*A*b^4)*d^3 - 56*(B*a^2*b^2 + A*a*
b^3)*d^2*e - 70*(3*B*a^3*b + A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - A*a^3*b)*e^3 - 3*
(232*B*b^4*d*e^2 - 93*(9*B*a*b^3 - A*b^4)*e^3)*x^3 - (304*B*b^4*d^2*e + 2*(577*B
*a*b^3 + 163*A*b^4)*d*e^2 - 511*(9*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 - (64*B*b^4*d^3
+ 8*(27*B*a*b^3 + 25*A*b^4)*d^2*e + 28*(29*B*a^2*b^2 + 9*A*a*b^3)*d*e^2 - 385*(
9*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 105*(8*B*a^4*
b*d*e^3 - (9*B*a^5 - A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (9*B*a*b^4 - A*b^5)*e^4)*x^
4 + 4*(8*B*a*b^4*d*e^3 - (9*B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3
- (9*B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (9*B*a^4*b - A*a^
3*b^2)*e^4)*x)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e
)*sqrt(e*x + d))/(b*x + a)))/((b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6
*x + a^4*b^5)*sqrt(b^2*d - a*b*e)), 1/192*((384*B*b^4*e^3*x^4 - 16*(B*a*b^3 + 3*
A*b^4)*d^3 - 56*(B*a^2*b^2 + A*a*b^3)*d^2*e - 70*(3*B*a^3*b + A*a^2*b^2)*d*e^2 +
105*(9*B*a^4 - A*a^3*b)*e^3 - 3*(232*B*b^4*d*e^2 - 93*(9*B*a*b^3 - A*b^4)*e^3)*
x^3 - (304*B*b^4*d^2*e + 2*(577*B*a*b^3 + 163*A*b^4)*d*e^2 - 511*(9*B*a^2*b^2 -
A*a*b^3)*e^3)*x^2 - (64*B*b^4*d^3 + 8*(27*B*a*b^3 + 25*A*b^4)*d^2*e + 28*(29*B*a
^2*b^2 + 9*A*a*b^3)*d*e^2 - 385*(9*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(-b^2*d + a*
b*e)*sqrt(e*x + d) - 105*(8*B*a^4*b*d*e^3 - (9*B*a^5 - A*a^4*b)*e^4 + (8*B*b^5*d
*e^3 - (9*B*a*b^4 - A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (9*B*a^2*b^3 - A*a*b^
4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (9*B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 4*(8*B
*a^3*b^2*d*e^3 - (9*B*a^4*b - A*a^3*b^2)*e^4)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*
d + a*b*e)*sqrt(e*x + d))))/((b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*
x + a^4*b^5)*sqrt(-b^2*d + a*b*e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.33704, size = 841, normalized size = 1.98 \[ -\frac{2 \, \sqrt{x e + d} B e^{3}}{b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{35 \,{\left (8 \, B b d e^{3} - 9 \, B a e^{4} + A b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \sqrt{-b^{2} d + a b e} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{696 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} d e^{3} - 1784 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{3} + 1544 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{3} - 456 \, \sqrt{x e + d} B b^{4} d^{4} e^{3} - 975 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{3} e^{4} + 279 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{4} e^{4} + 4079 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{4} - 511 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{4} - 5017 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{4} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{4} + 1929 \, \sqrt{x e + d} B a b^{3} d^{3} e^{4} - 105 \, \sqrt{x e + d} A b^{4} d^{3} e^{4} - 2295 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{5} + 511 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{5} + 5402 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{5} - 770 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{5} - 3051 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{5} + 315 \, \sqrt{x e + d} A a b^{3} d^{2} e^{5} - 1929 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{6} + 385 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{6} + 2139 \, \sqrt{x e + d} B a^{3} b d e^{6} - 315 \, \sqrt{x e + d} A a^{2} b^{2} d e^{6} - 561 \, \sqrt{x e + d} B a^{4} e^{7} + 105 \, \sqrt{x e + d} A a^{3} b e^{7}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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)^(5/2),x, algorithm="giac")

[Out]

-2*sqrt(x*e + d)*B*e^3/(b^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 35/64*(8*B*b
*d*e^3 - 9*B*a*e^4 + A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt
(-b^2*d + a*b*e)*b^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/192*(696*(x*e + d
)^(7/2)*B*b^4*d*e^3 - 1784*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 1544*(x*e + d)^(3/2)*
B*b^4*d^3*e^3 - 456*sqrt(x*e + d)*B*b^4*d^4*e^3 - 975*(x*e + d)^(7/2)*B*a*b^3*e^
4 + 279*(x*e + d)^(7/2)*A*b^4*e^4 + 4079*(x*e + d)^(5/2)*B*a*b^3*d*e^4 - 511*(x*
e + d)^(5/2)*A*b^4*d*e^4 - 5017*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 + 385*(x*e + d)^
(3/2)*A*b^4*d^2*e^4 + 1929*sqrt(x*e + d)*B*a*b^3*d^3*e^4 - 105*sqrt(x*e + d)*A*b
^4*d^3*e^4 - 2295*(x*e + d)^(5/2)*B*a^2*b^2*e^5 + 511*(x*e + d)^(5/2)*A*a*b^3*e^
5 + 5402*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 - 770*(x*e + d)^(3/2)*A*a*b^3*d*e^5 - 3
051*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 + 315*sqrt(x*e + d)*A*a*b^3*d^2*e^5 - 1929*(
x*e + d)^(3/2)*B*a^3*b*e^6 + 385*(x*e + d)^(3/2)*A*a^2*b^2*e^6 + 2139*sqrt(x*e +
d)*B*a^3*b*d*e^6 - 315*sqrt(x*e + d)*A*a^2*b^2*d*e^6 - 561*sqrt(x*e + d)*B*a^4*
e^7 + 105*sqrt(x*e + d)*A*a^3*b*e^7)/(((x*e + d)*b - b*d + a*e)^4*b^5*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2))