3.379 \(\int \frac{(d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=305 \[ \frac{d (d+e x)^{3/2} (8 c d-9 b e)}{4 b^2 x (b+c x)^2}-\frac{d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{3/2}}+\frac{\sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 c (b+c x)}+\frac{\sqrt{d+e x} (c d-b e) \left (2 b^2 e^2-15 b c d e+12 c^2 d^2\right )}{4 b^3 c (b+c x)^2}-\frac{d (d+e x)^{5/2}}{2 b x^2 (b+c x)^2} \]

[Out]

((c*d - b*e)*(12*c^2*d^2 - 15*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*c*(b +
c*x)^2) + ((2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4*b
^4*c*(b + c*x)) + (d*(8*c*d - 9*b*e)*(d + e*x)^(3/2))/(4*b^2*x*(b + c*x)^2) - (d
*(d + e*x)^(5/2))/(2*b*x^2*(b + c*x)^2) - (d^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35
*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*e)^(3/2)*(48*c^2*d
^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*
b^5*c^(3/2))

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Rubi [A]  time = 1.20974, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{d (d+e x)^{3/2} (8 c d-9 b e)}{4 b^2 x (b+c x)^2}-\frac{d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{3/2}}+\frac{\sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 c (b+c x)}+\frac{\sqrt{d+e x} (c d-b e) \left (2 b^2 e^2-15 b c d e+12 c^2 d^2\right )}{4 b^3 c (b+c x)^2}-\frac{d (d+e x)^{5/2}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

((c*d - b*e)*(12*c^2*d^2 - 15*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*c*(b +
c*x)^2) + ((2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4*b
^4*c*(b + c*x)) + (d*(8*c*d - 9*b*e)*(d + e*x)^(3/2))/(4*b^2*x*(b + c*x)^2) - (d
*(d + e*x)^(5/2))/(2*b*x^2*(b + c*x)^2) - (d^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35
*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*e)^(3/2)*(48*c^2*d
^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*
b^5*c^(3/2))

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Rubi in Sympy [A]  time = 152.855, size = 287, normalized size = 0.94 \[ - \frac{d \left (d + e x\right )^{\frac{5}{2}}}{2 b x^{2} \left (b + c x\right )^{2}} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{2 b^{2} c x \left (b + c x\right )^{2}} - \frac{\sqrt{d + e x} \left (b e - c d\right ) \left (b^{2} e^{2} + 9 b c d e - 12 c^{2} d^{2}\right )}{4 b^{3} c^{2} x \left (b + c x\right )} + \frac{\sqrt{d + e x} \left (b e - 2 c d\right ) \left (b^{2} e^{2} + 12 b c d e - 12 c^{2} d^{2}\right )}{4 b^{4} c^{2} x} - \frac{d^{\frac{3}{2}} \left (35 b^{2} e^{2} - 84 b c d e + 48 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{4 b^{5}} + \frac{\left (b e - c d\right )^{\frac{3}{2}} \left (b^{2} e^{2} + 12 b c d e - 48 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{4 b^{5} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*(d + e*x)**(5/2)/(2*b*x**2*(b + c*x)**2) - (d + e*x)**(3/2)*(b*e - 2*c*d)*(b*
e - c*d)/(2*b**2*c*x*(b + c*x)**2) - sqrt(d + e*x)*(b*e - c*d)*(b**2*e**2 + 9*b*
c*d*e - 12*c**2*d**2)/(4*b**3*c**2*x*(b + c*x)) + sqrt(d + e*x)*(b*e - 2*c*d)*(b
**2*e**2 + 12*b*c*d*e - 12*c**2*d**2)/(4*b**4*c**2*x) - d**(3/2)*(35*b**2*e**2 -
84*b*c*d*e + 48*c**2*d**2)*atanh(sqrt(d + e*x)/sqrt(d))/(4*b**5) + (b*e - c*d)*
*(3/2)*(b**2*e**2 + 12*b*c*d*e - 48*c**2*d**2)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b
*e - c*d))/(4*b**5*c**(3/2))

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Mathematica [A]  time = 0.60823, size = 212, normalized size = 0.7 \[ \frac{d^{3/2} \left (-\left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{3/2}}+b \sqrt{d+e x} \left (\frac{d^2 (12 c d-13 b e)}{x}+\frac{(c d-b e)^2 (b e+12 c d)}{c (b+c x)}-\frac{2 b (b e-c d)^3}{c (b+c x)^2}-\frac{2 b d^3}{x^2}\right )}{4 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[d + e*x]*((-2*b*d^3)/x^2 + (d^2*(12*c*d - 13*b*e))/x - (2*b*(-(c*d) + b*
e)^3)/(c*(b + c*x)^2) + ((c*d - b*e)^2*(12*c*d + b*e))/(c*(b + c*x))) - d^(3/2)*
(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + ((c*d -
b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/S
qrt[c*d - b*e]])/c^(3/2))/(4*b^5)

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Maple [B]  time = 0.031, size = 627, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)+5/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*c*
d-23/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*c^2*d^2+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)
^(3/2)*c^3*d^3-1/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)+15/4*e^4/b/(c*e*x+b*e)^2*(e
*x+d)^(1/2)*d-39/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*d^2+37/4*e^2/b^3/(c*e*x
+b*e)^2*(e*x+d)^(1/2)*d^3*c^2-3*e/b^4/(c*e*x+b*e)^2*c^3*(e*x+d)^(1/2)*d^4+1/4*e^
4/b/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+5/2*e^3/b^
2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d-71/4*e^2/b^3
*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^2+27*e/b^4/
((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^3*c^2-12/b^5*c
^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^4-13/4*d^2/
b^3/x^2*(e*x+d)^(3/2)+3/e*d^3/b^4/x^2*(e*x+d)^(3/2)*c+11/4*d^3/b^3/x^2*(e*x+d)^(
1/2)-3/e*d^4/b^4/x^2*c*(e*x+d)^(1/2)-35/4*e^2*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/
d^(1/2))+21*e*d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(7/2)/b^5*arctan
h((e*x+d)^(1/2)/d^(1/2))*c^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((e*x + d)^(7/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.650269, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(4
8*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3
*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt((c*d - b*e)/c)*log
((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((48
*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3
*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d
*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c*d^3
- (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^
2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 -
13*b^4*c*d^2*e)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8
*(2*((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*
b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d
^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*arct
an(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + ((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*
c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (4
8*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqr
t(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e
+ 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b
^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt(e*x + d))/(
b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), -1/8*(2*((48*c^5*d^3 - 84*b*c^4*d^2*e
+ 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)
*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(-d)*arctan
(sqrt(e*x + d)/sqrt(-d)) - ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^
3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e
^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sq
rt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/
c))/(c*x + b)) + 2*(2*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*
d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b
^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 +
2*b^6*c^2*x^3 + b^7*c*x^2), -1/4*(((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e
^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c
^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)/s
qrt(-d)) - ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 +
2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^
2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-(c*d - b*e)/
c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (2*b^4*c*d^3 - (24*b*c^4*d^3 - 3
6*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c
^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*s
qrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238784, size = 745, normalized size = 2.44 \[ \frac{{\left (48 \, c^{2} d^{4} - 84 \, b c d^{3} e + 35 \, b^{2} d^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} - \frac{{\left (48 \, c^{4} d^{4} - 108 \, b c^{3} d^{3} e + 71 \, b^{2} c^{2} d^{2} e^{2} - 10 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5} c} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{4} d^{3} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{4} d^{4} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{5} e - 24 \, \sqrt{x e + d} c^{4} d^{6} e - 36 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{3} d^{2} e^{2} + 144 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{3} d^{3} e^{2} - 180 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{4} e^{2} + 72 \, \sqrt{x e + d} b c^{3} d^{5} e^{2} + 10 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{2} d e^{3} - 85 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} e^{3} + 148 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{3} e^{3} - 73 \, \sqrt{x e + d} b^{2} c^{2} d^{4} e^{3} +{\left (x e + d\right )}^{\frac{7}{2}} b^{3} c e^{4} + 13 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c d e^{4} - 42 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c d^{2} e^{4} + 26 \, \sqrt{x e + d} b^{3} c d^{3} e^{4} -{\left (x e + d\right )}^{\frac{5}{2}} b^{4} e^{5} + 2 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d e^{5} - \sqrt{x e + d} b^{4} d^{2} e^{5}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(48*c^2*d^4 - 84*b*c*d^3*e + 35*b^2*d^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/
(b^5*sqrt(-d)) - 1/4*(48*c^4*d^4 - 108*b*c^3*d^3*e + 71*b^2*c^2*d^2*e^2 - 10*b^3
*c*d*e^3 - b^4*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d +
b*c*e)*b^5*c) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^3*e - 72*(x*e + d)^(5/2)*c^4*d^4*e
+ 72*(x*e + d)^(3/2)*c^4*d^5*e - 24*sqrt(x*e + d)*c^4*d^6*e - 36*(x*e + d)^(7/2
)*b*c^3*d^2*e^2 + 144*(x*e + d)^(5/2)*b*c^3*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^3*
d^4*e^2 + 72*sqrt(x*e + d)*b*c^3*d^5*e^2 + 10*(x*e + d)^(7/2)*b^2*c^2*d*e^3 - 85
*(x*e + d)^(5/2)*b^2*c^2*d^2*e^3 + 148*(x*e + d)^(3/2)*b^2*c^2*d^3*e^3 - 73*sqrt
(x*e + d)*b^2*c^2*d^4*e^3 + (x*e + d)^(7/2)*b^3*c*e^4 + 13*(x*e + d)^(5/2)*b^3*c
*d*e^4 - 42*(x*e + d)^(3/2)*b^3*c*d^2*e^4 + 26*sqrt(x*e + d)*b^3*c*d^3*e^4 - (x*
e + d)^(5/2)*b^4*e^5 + 2*(x*e + d)^(3/2)*b^4*d*e^5 - sqrt(x*e + d)*b^4*d^2*e^5)/
(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c)