3.1748 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=281 \[ \frac{7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}}-\frac{7 b e (5 a B e-9 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^5}-\frac{7 e (5 a B e-9 A b e+4 b B d)}{12 (d+e x)^{3/2} (b d-a e)^4}-\frac{7 e (5 a B e-9 A b e+4 b B d)}{20 b (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-9 A b e+4 b B d}{4 b (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-7*e*(4*b*B*d - 9*A*b*e + 5*a*B*e))/(20*b*(b*d - a*e)^3*(d + e*x)^(5/2)) - (A*b
- a*B)/(2*b*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(5/2)) - (4*b*B*d - 9*A*b*e + 5*a
*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(5/2)) - (7*e*(4*b*B*d - 9*A*b*e +
5*a*B*e))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)) - (7*b*e*(4*b*B*d - 9*A*b*e + 5*a*B
*e))/(4*(b*d - a*e)^5*Sqrt[d + e*x]) + (7*b^(3/2)*e*(4*b*B*d - 9*A*b*e + 5*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(11/2))

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Rubi [A]  time = 0.664483, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}}-\frac{7 b e (5 a B e-9 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^5}-\frac{7 e (5 a B e-9 A b e+4 b B d)}{12 (d+e x)^{3/2} (b d-a e)^4}-\frac{7 e (5 a B e-9 A b e+4 b B d)}{20 b (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-9 A b e+4 b B d}{4 b (a+b x) (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^3*(d + e*x)^(7/2)),x]

[Out]

(-7*e*(4*b*B*d - 9*A*b*e + 5*a*B*e))/(20*b*(b*d - a*e)^3*(d + e*x)^(5/2)) - (A*b
- a*B)/(2*b*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(5/2)) - (4*b*B*d - 9*A*b*e + 5*a
*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(5/2)) - (7*e*(4*b*B*d - 9*A*b*e +
5*a*B*e))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)) - (7*b*e*(4*b*B*d - 9*A*b*e + 5*a*B
*e))/(4*(b*d - a*e)^5*Sqrt[d + e*x]) + (7*b^(3/2)*e*(4*b*B*d - 9*A*b*e + 5*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(11/2))

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Rubi in Sympy [A]  time = 68.6842, size = 274, normalized size = 0.98 \[ - \frac{7 b^{\frac{3}{2}} e \left (9 A b e - 5 B a e - 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{11}{2}}} - \frac{7 b e \left (9 A b e - 5 B a e - 4 B b d\right )}{4 \sqrt{d + e x} \left (a e - b d\right )^{5}} + \frac{7 e \left (9 A b e - 5 B a e - 4 B b d\right )}{12 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} - \frac{7 e \left (9 A b e - 5 B a e - 4 B b d\right )}{20 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{9 A b e - 5 B a e - 4 B b d}{4 b \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{2 b \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d)**(7/2),x)

[Out]

-7*b**(3/2)*e*(9*A*b*e - 5*B*a*e - 4*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e
- b*d))/(4*(a*e - b*d)**(11/2)) - 7*b*e*(9*A*b*e - 5*B*a*e - 4*B*b*d)/(4*sqrt(d
+ e*x)*(a*e - b*d)**5) + 7*e*(9*A*b*e - 5*B*a*e - 4*B*b*d)/(12*(d + e*x)**(3/2)*
(a*e - b*d)**4) - 7*e*(9*A*b*e - 5*B*a*e - 4*B*b*d)/(20*b*(d + e*x)**(5/2)*(a*e
- b*d)**3) + (9*A*b*e - 5*B*a*e - 4*B*b*d)/(4*b*(a + b*x)*(d + e*x)**(5/2)*(a*e
- b*d)**2) + (A*b - B*a)/(2*b*(a + b*x)**2*(d + e*x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 1.21268, size = 236, normalized size = 0.84 \[ \frac{7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2}}+\frac{\sqrt{d+e x} \left (-\frac{15 b^2 (11 a B e-15 A b e+4 b B d)}{a+b x}-\frac{30 b^2 (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{360 b e (-a B e+2 A b e-b B d)}{d+e x}+\frac{40 e (b d-a e) (-a B e+3 A b e-2 b B d)}{(d+e x)^2}+\frac{24 e (b d-a e)^2 (A e-B d)}{(d+e x)^3}\right )}{60 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^(7/2)),x]

[Out]

(Sqrt[d + e*x]*((-30*b^2*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2 - (15*b^2*(4*b*B*d
- 15*A*b*e + 11*a*B*e))/(a + b*x) + (24*e*(b*d - a*e)^2*(-(B*d) + A*e))/(d + e*
x)^3 + (40*e*(b*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)^2 + (360*b*e*(-
(b*B*d) + 2*A*b*e - a*B*e))/(d + e*x)))/(60*(b*d - a*e)^5) + (7*b^(3/2)*e*(4*b*B
*d - 9*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*
d - a*e)^(11/2))

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Maple [B]  time = 0.039, size = 648, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^3/(e*x+d)^(7/2),x)

[Out]

-2/5/(a*e-b*d)^3/(e*x+d)^(5/2)*A*e^2+2/5*e/(a*e-b*d)^3/(e*x+d)^(5/2)*B*d+2/(a*e-
b*d)^4/(e*x+d)^(3/2)*A*b*e^2-2/3/(a*e-b*d)^4/(e*x+d)^(3/2)*B*a*e^2-4/3*e/(a*e-b*
d)^4/(e*x+d)^(3/2)*B*b*d-12*b^2/(a*e-b*d)^5/(e*x+d)^(1/2)*A*e^2+6*b/(a*e-b*d)^5/
(e*x+d)^(1/2)*B*a*e^2+6*e*b^2/(a*e-b*d)^5/(e*x+d)^(1/2)*B*d-15/4/(a*e-b*d)^5*b^4
/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*e^2+11/4/(a*e-b*d)^5*b^3/(b*e*x+a*e)^2*(e*x+d)^(3
/2)*B*a*e^2+e/(a*e-b*d)^5*b^4/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*d-17/4/(a*e-b*d)^5*b
^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a*e^3+17/4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^2*(e*x+d
)^(1/2)*A*d*e^2+13/4/(a*e-b*d)^5*b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*e^3-9/4/(
a*e-b*d)^5*b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*d*e^2-e/(a*e-b*d)^5*b^4/(b*e*x+a*
e)^2*(e*x+d)^(1/2)*B*d^2-63/4/(a*e-b*d)^5*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)
^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*e^2+35/4/(a*e-b*d)^5*b^2/((a*e-b*d)*b)^(1/2)*arc
tan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*e^2+7*e/(a*e-b*d)^5*b^3/((a*e-b*d)*
b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*d

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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)/((b*x + a)^3*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244841, 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)/((b*x + a)^3*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/120*(48*A*a^4*e^4 - 60*(B*a*b^3 + A*b^4)*d^4 - 2*(659*B*a^2*b^2 - 255*A*a*b^3
)*d^3*e - 32*(17*B*a^3*b - 54*A*a^2*b^2)*d^2*e^2 + 16*(2*B*a^4 - 21*A*a^3*b)*d*e
^3 - 210*(4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 - 70*(28*B*b^4*d^2*e^2
+ (55*B*a*b^3 - 63*A*b^4)*d*e^3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 14*(92*
B*b^4*d^3*e + 9*(39*B*a*b^3 - 23*A*b^4)*d^2*e^2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3
)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 + 105*(4*B*a^2*b^2*d^3*e + (5*B*a
^3*b - 9*A*a^2*b^2)*d^2*e^2 + (4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 +
2*(4*B*b^4*d^2*e^2 + 9*(B*a*b^3 - A*b^4)*d*e^3 + (5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*
x^3 + (4*B*b^4*d^3*e + 3*(7*B*a*b^3 - 3*A*b^4)*d^2*e^2 + 12*(2*B*a^2*b^2 - 3*A*a
*b^3)*d*e^3 + (5*B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 + 2*(4*B*a*b^3*d^3*e + 9*(B*a^2
*b^2 - A*a*b^3)*d^2*e^2 + (5*B*a^3*b - 9*A*a^2*b^2)*d*e^3)*x)*sqrt(e*x + d)*sqrt
(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b
*d - a*e)))/(b*x + a)) - 2*(60*B*b^4*d^4 + (1183*B*a*b^3 - 135*A*b^4)*d^3*e + 3*
(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 72*(9*B*a^3*b - 17*A*a^2*b^2)*d*e^3 - 8*
(5*B*a^4 - 9*A*a^3*b)*e^4)*x)/((a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e
^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6
*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e
^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5
+ 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2
+ 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a
^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3
*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)*sqrt(e*x + d)), 1/60*(24*A*a^4*e^4 - 30*(
B*a*b^3 + A*b^4)*d^4 - (659*B*a^2*b^2 - 255*A*a*b^3)*d^3*e - 16*(17*B*a^3*b - 54
*A*a^2*b^2)*d^2*e^2 + 8*(2*B*a^4 - 21*A*a^3*b)*d*e^3 - 105*(4*B*b^4*d*e^3 + (5*B
*a*b^3 - 9*A*b^4)*e^4)*x^4 - 35*(28*B*b^4*d^2*e^2 + (55*B*a*b^3 - 63*A*b^4)*d*e^
3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 7*(92*B*b^4*d^3*e + 9*(39*B*a*b^3 - 2
3*A*b^4)*d^2*e^2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^
2*b^2)*e^4)*x^2 + 105*(4*B*a^2*b^2*d^3*e + (5*B*a^3*b - 9*A*a^2*b^2)*d^2*e^2 + (
4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 + 2*(4*B*b^4*d^2*e^2 + 9*(B*a*b^3
- A*b^4)*d*e^3 + (5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 + (4*B*b^4*d^3*e + 3*(7*B*a
*b^3 - 3*A*b^4)*d^2*e^2 + 12*(2*B*a^2*b^2 - 3*A*a*b^3)*d*e^3 + (5*B*a^3*b - 9*A*
a^2*b^2)*e^4)*x^2 + 2*(4*B*a*b^3*d^3*e + 9*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (5*B*
a^3*b - 9*A*a^2*b^2)*d*e^3)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d -
a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) - (60*B*b^4*d^4 + (1183*B*a*b^3 -
135*A*b^4)*d^3*e + 3*(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 72*(9*B*a^3*b - 17*
A*a^2*b^2)*d*e^3 - 8*(5*B*a^4 - 9*A*a^3*b)*e^4)*x)/((a^2*b^5*d^7 - 5*a^3*b^4*d^6
*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (
b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*
b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^
3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*
e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*
e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^
5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)*sqrt(e*x + d))]

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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)/(b*x+a)**3/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233492, size = 822, normalized size = 2.93 \[ -\frac{7 \,{\left (4 \, B b^{3} d e + 5 \, B a b^{2} e^{2} - 9 \, A b^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d e - 4 \, \sqrt{x e + d} B b^{4} d^{2} e + 11 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} e^{2} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} e^{2} - 9 \, \sqrt{x e + d} B a b^{3} d e^{2} + 17 \, \sqrt{x e + d} A b^{4} d e^{2} + 13 \, \sqrt{x e + d} B a^{2} b^{2} e^{3} - 17 \, \sqrt{x e + d} A a b^{3} e^{3}}{4 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} - \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} B b^{2} d e + 10 \,{\left (x e + d\right )} B b^{2} d^{2} e + 3 \, B b^{2} d^{3} e + 45 \,{\left (x e + d\right )}^{2} B a b e^{2} - 90 \,{\left (x e + d\right )}^{2} A b^{2} e^{2} - 5 \,{\left (x e + d\right )} B a b d e^{2} - 15 \,{\left (x e + d\right )} A b^{2} d e^{2} - 6 \, B a b d^{2} e^{2} - 3 \, A b^{2} d^{2} e^{2} - 5 \,{\left (x e + d\right )} B a^{2} e^{3} + 15 \,{\left (x e + d\right )} A a b e^{3} + 3 \, B a^{2} d e^{3} + 6 \, A a b d e^{3} - 3 \, A a^{2} e^{4}\right )}}{15 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

-7/4*(4*B*b^3*d*e + 5*B*a*b^2*e^2 - 9*A*b^3*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^
2*d + a*b*e))/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^
3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(x*e + d)^(3/2)*B*b^
4*d*e - 4*sqrt(x*e + d)*B*b^4*d^2*e + 11*(x*e + d)^(3/2)*B*a*b^3*e^2 - 15*(x*e +
d)^(3/2)*A*b^4*e^2 - 9*sqrt(x*e + d)*B*a*b^3*d*e^2 + 17*sqrt(x*e + d)*A*b^4*d*e
^2 + 13*sqrt(x*e + d)*B*a^2*b^2*e^3 - 17*sqrt(x*e + d)*A*a*b^3*e^3)/((b^5*d^5 -
5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^
5)*((x*e + d)*b - b*d + a*e)^2) - 2/15*(45*(x*e + d)^2*B*b^2*d*e + 10*(x*e + d)*
B*b^2*d^2*e + 3*B*b^2*d^3*e + 45*(x*e + d)^2*B*a*b*e^2 - 90*(x*e + d)^2*A*b^2*e^
2 - 5*(x*e + d)*B*a*b*d*e^2 - 15*(x*e + d)*A*b^2*d*e^2 - 6*B*a*b*d^2*e^2 - 3*A*b
^2*d^2*e^2 - 5*(x*e + d)*B*a^2*e^3 + 15*(x*e + d)*A*a*b*e^3 + 3*B*a^2*d*e^3 + 6*
A*a*b*d*e^3 - 3*A*a^2*e^4)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a
^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(x*e + d)^(5/2))