3.1823 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)
Optimal. Leaf size=291 \[ \frac{35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac{35 \sqrt{b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac{7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]
[Out]
(35*e^2*(2*b*B*d - 3*A*b*e + a*B*e))/(24*b*(b*d - a*e)^4*(d + e*x)^(3/2)) - (A*b
- a*B)/(3*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)) - (2*b*B*d - 3*A*b*e + a*B
*e)/(4*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) + (7*e*(2*b*B*d - 3*A*b*e +
a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) + (35*e^2*(2*b*B*d - 3*A*b
*e + a*B*e))/(8*(b*d - a*e)^5*Sqrt[d + e*x]) - (35*Sqrt[b]*e^2*(2*b*B*d - 3*A*b*
e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(11/
2))
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Rubi [A] time = 0.725216, antiderivative size = 291, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ \frac{35 e^2 (a B e-3 A b e+2 b B d)}{8 \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^2 (a B e-3 A b e+2 b B d)}{24 b (d+e x)^{3/2} (b d-a e)^4}-\frac{35 \sqrt{b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}+\frac{7 e (a B e-3 A b e+2 b B d)}{8 b (a+b x) (d+e x)^{3/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{4 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
(35*e^2*(2*b*B*d - 3*A*b*e + a*B*e))/(24*b*(b*d - a*e)^4*(d + e*x)^(3/2)) - (A*b
- a*B)/(3*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)) - (2*b*B*d - 3*A*b*e + a*B
*e)/(4*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) + (7*e*(2*b*B*d - 3*A*b*e +
a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) + (35*e^2*(2*b*B*d - 3*A*b
*e + a*B*e))/(8*(b*d - a*e)^5*Sqrt[d + e*x]) - (35*Sqrt[b]*e^2*(2*b*B*d - 3*A*b*
e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(11/
2))
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Rubi in Sympy [A] time = 136.938, size = 277, normalized size = 0.95 \[ \frac{35 \sqrt{b} e^{2} \left (3 A b e - B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{11}{2}}} + \frac{35 e^{2} \left (3 A b e - B a e - 2 B b d\right )}{8 \sqrt{d + e x} \left (a e - b d\right )^{5}} - \frac{35 e^{2} \left (3 A b e - B a e - 2 B b d\right )}{24 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}} + \frac{7 e \left (3 A b e - B a e - 2 B b d\right )}{8 b \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{3 A b e - B a e - 2 B b d}{4 b \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{3 b \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
35*sqrt(b)*e**2*(3*A*b*e - B*a*e - 2*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e
- b*d))/(8*(a*e - b*d)**(11/2)) + 35*e**2*(3*A*b*e - B*a*e - 2*B*b*d)/(8*sqrt(d
+ e*x)*(a*e - b*d)**5) - 35*e**2*(3*A*b*e - B*a*e - 2*B*b*d)/(24*b*(d + e*x)**(3
/2)*(a*e - b*d)**4) + 7*e*(3*A*b*e - B*a*e - 2*B*b*d)/(8*b*(a + b*x)*(d + e*x)**
(3/2)*(a*e - b*d)**3) + (3*A*b*e - B*a*e - 2*B*b*d)/(4*b*(a + b*x)**2*(d + e*x)*
*(3/2)*(a*e - b*d)**2) + (A*b - B*a)/(3*b*(a + b*x)**3*(d + e*x)**(3/2)*(a*e - b
*d))
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Mathematica [A] time = 1.29252, size = 237, normalized size = 0.81 \[ -\frac{\sqrt{d+e x} \left (\frac{48 e^2 (-a B e+4 A b e-3 b B d)}{d+e x}-\frac{16 e^2 (a e-b d) (A e-B d)}{(d+e x)^2}+\frac{3 b e (-19 a B e+41 A b e-22 b B d)}{a+b x}+\frac{2 b (b d-a e) (11 a B e-17 A b e+6 b B d)}{(a+b x)^2}+\frac{8 b (A b-a B) (b d-a e)^2}{(a+b x)^3}\right )}{24 (b d-a e)^5}-\frac{35 \sqrt{b} e^2 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
-(Sqrt[d + e*x]*((8*b*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^3 + (2*b*(b*d - a*e)*
(6*b*B*d - 17*A*b*e + 11*a*B*e))/(a + b*x)^2 + (3*b*e*(-22*b*B*d + 41*A*b*e - 19
*a*B*e))/(a + b*x) - (16*e^2*(-(b*d) + a*e)*(-(B*d) + A*e))/(d + e*x)^2 + (48*e^
2*(-3*b*B*d + 4*A*b*e - a*B*e))/(d + e*x)))/(24*(b*d - a*e)^5) - (35*Sqrt[b]*e^2
*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
8*(b*d - a*e)^(11/2))
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Maple [B] time = 0.047, size = 853, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
-2/3*e^3/(a*e-b*d)^4/(e*x+d)^(3/2)*A+2/3*e^2/(a*e-b*d)^4/(e*x+d)^(3/2)*B*d+8*e^3
/(a*e-b*d)^5/(e*x+d)^(1/2)*A*b-2*e^3/(a*e-b*d)^5/(e*x+d)^(1/2)*a*B-6*e^2/(a*e-b*
d)^5/(e*x+d)^(1/2)*B*b*d+41/8*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A-
19/8*e^3/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a-11/4*e^2/(a*e-b*d)^5*b^
4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d+35/3*e^4/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*A*(e*x+
d)^(3/2)*a-35/3*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d-17/3*e^4/(a*
e-b*d)^5*b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2-1/3*e^3/(a*e-b*d)^5*b^3/(b*e*x+a*
e)^3*B*(e*x+d)^(3/2)*a*d+6*e^2/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2
+55/8*e^5/(a*e-b*d)^5*b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2-55/4*e^4/(a*e-b*d)^5
*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d+55/8*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e
*x+d)^(1/2)*A*d^2-29/8*e^5/(a*e-b*d)^5*b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3+23/8*
e^3/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^2-13/4*e^2/(a*e-b*d)^5*b^4
/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^3+4*e^4/(a*e-b*d)^5*b^2/(b*e*x+a*e)^3*(e*x+d)^(
1/2)*B*a^2*d+105/8*e^3/(a*e-b*d)^5*b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*A-35/8*e^3/(a*e-b*d)^5*b/(b*(a*e-b*d))^(1/2)*arctan((e*x+
d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B-35/4*e^2/(a*e-b*d)^5*b^2/(b*(a*e-b*d))^(1/2)
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(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^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.341182, 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^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2)),x, algorithm="fricas")
[Out]
[1/48*(32*A*a^4*e^4 - 8*(B*a*b^3 + 2*A*b^4)*d^4 + 20*(4*B*a^2*b^2 + 5*A*a*b^3)*d
^3*e + 2*(247*B*a^3*b - 165*A*a^2*b^2)*d^2*e^2 + 32*(2*B*a^4 - 13*A*a^3*b)*d*e^3
+ 210*(2*B*b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 + 280*(2*B*b^4*d^2*e^2 + (5
*B*a*b^3 - 3*A*b^4)*d*e^3 + 2*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 42*(2*B*b^4*d^3
*e + (37*B*a*b^3 - 3*A*b^4)*d^2*e^2 + 2*(20*B*a^2*b^2 - 27*A*a*b^3)*d*e^3 + 11*(
B*a^3*b - 3*A*a^2*b^2)*e^4)*x^2 + 105*(2*B*a^3*b*d^2*e^2 + (B*a^4 - 3*A*a^3*b)*d
*e^3 + (2*B*b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 + (2*B*b^4*d^2*e^2 + (7*B*a
*b^3 - 3*A*b^4)*d*e^3 + 3*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 3*(2*B*a*b^3*d^2*e^
2 + 3*(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 3*A*a^2*b^2)*e^4)*x^2 + (6*B*a^2*
b^2*d^2*e^2 + (5*B*a^3*b - 9*A*a^2*b^2)*d*e^3 + (B*a^4 - 3*A*a^3*b)*e^4)*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)) - 12*(2*B*b^4*d^4 - (19*B*a*b^3 + 3*A*b^4)*
d^3*e - 2*(58*B*a^2*b^2 - 15*A*a*b^3)*d^2*e^2 - 3*(23*B*a^3*b - 53*A*a^2*b^2)*d*
e^3 - 8*(B*a^4 - 3*A*a^3*b)*e^4)*x)/((a^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3
*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b
^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3
*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 -
25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a*b^7*d^6 - 4*a^
2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^
6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3
*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*x)*sqrt(e*x + d)), 1/24*(16*
A*a^4*e^4 - 4*(B*a*b^3 + 2*A*b^4)*d^4 + 10*(4*B*a^2*b^2 + 5*A*a*b^3)*d^3*e + (24
7*B*a^3*b - 165*A*a^2*b^2)*d^2*e^2 + 16*(2*B*a^4 - 13*A*a^3*b)*d*e^3 + 105*(2*B*
b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 + 140*(2*B*b^4*d^2*e^2 + (5*B*a*b^3 - 3
*A*b^4)*d*e^3 + 2*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 21*(2*B*b^4*d^3*e + (37*B*a
*b^3 - 3*A*b^4)*d^2*e^2 + 2*(20*B*a^2*b^2 - 27*A*a*b^3)*d*e^3 + 11*(B*a^3*b - 3*
A*a^2*b^2)*e^4)*x^2 - 105*(2*B*a^3*b*d^2*e^2 + (B*a^4 - 3*A*a^3*b)*d*e^3 + (2*B*
b^4*d*e^3 + (B*a*b^3 - 3*A*b^4)*e^4)*x^4 + (2*B*b^4*d^2*e^2 + (7*B*a*b^3 - 3*A*b
^4)*d*e^3 + 3*(B*a^2*b^2 - 3*A*a*b^3)*e^4)*x^3 + 3*(2*B*a*b^3*d^2*e^2 + 3*(B*a^2
*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 3*A*a^2*b^2)*e^4)*x^2 + (6*B*a^2*b^2*d^2*e^2
+ (5*B*a^3*b - 9*A*a^2*b^2)*d*e^3 + (B*a^4 - 3*A*a^3*b)*e^4)*x)*sqrt(e*x + d)*sq
rt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) -
6*(2*B*b^4*d^4 - (19*B*a*b^3 + 3*A*b^4)*d^3*e - 2*(58*B*a^2*b^2 - 15*A*a*b^3)*d
^2*e^2 - 3*(23*B*a^3*b - 53*A*a^2*b^2)*d*e^3 - 8*(B*a^4 - 3*A*a^3*b)*e^4)*x)/((a
^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b
*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^
3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e -
5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 -
3*a^6*b^2*e^6)*x^3 + 3*(a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5
*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^
5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^
5 - a^8*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)/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.300468, size = 1013, normalized size = 3.48 \[ \text{result too large to display} \]
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, algorithm="giac")
[Out]
35/8*(2*B*b^2*d*e^2 + B*a*b*e^3 - 3*A*b^2*e^3)*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)) + 1/24*(210*(x*e + d)^4*B*b^4*d
*e^2 - 560*(x*e + d)^3*B*b^4*d^2*e^2 + 462*(x*e + d)^2*B*b^4*d^3*e^2 - 96*(x*e +
d)*B*b^4*d^4*e^2 - 16*B*b^4*d^5*e^2 + 105*(x*e + d)^4*B*a*b^3*e^3 - 315*(x*e +
d)^4*A*b^4*e^3 + 280*(x*e + d)^3*B*a*b^3*d*e^3 + 840*(x*e + d)^3*A*b^4*d*e^3 - 6
93*(x*e + d)^2*B*a*b^3*d^2*e^3 - 693*(x*e + d)^2*A*b^4*d^2*e^3 + 240*(x*e + d)*B
*a*b^3*d^3*e^3 + 144*(x*e + d)*A*b^4*d^3*e^3 + 64*B*a*b^3*d^4*e^3 + 16*A*b^4*d^4
*e^3 + 280*(x*e + d)^3*B*a^2*b^2*e^4 - 840*(x*e + d)^3*A*a*b^3*e^4 + 1386*(x*e +
d)^2*A*a*b^3*d*e^4 - 144*(x*e + d)*B*a^2*b^2*d^2*e^4 - 432*(x*e + d)*A*a*b^3*d^
2*e^4 - 96*B*a^2*b^2*d^3*e^4 - 64*A*a*b^3*d^3*e^4 + 231*(x*e + d)^2*B*a^3*b*e^5
- 693*(x*e + d)^2*A*a^2*b^2*e^5 - 48*(x*e + d)*B*a^3*b*d*e^5 + 432*(x*e + d)*A*a
^2*b^2*d*e^5 + 64*B*a^3*b*d^2*e^5 + 96*A*a^2*b^2*d^2*e^5 + 48*(x*e + d)*B*a^4*e^
6 - 144*(x*e + d)*A*a^3*b*e^6 - 16*B*a^4*d*e^6 - 64*A*a^3*b*d*e^6 + 16*A*a^4*e^7
)/((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)^(3/2)*b - sqrt(x*e + d)*b*d + sqrt(x*e + d)*a*e)^3)