3.380 \(\int \frac{(d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=280 \[ \frac{d \sqrt{d+e x} (8 c d-7 b e)}{4 b^2 x (b+c x)^2}-\frac{3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 \sqrt{c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c}}+\frac{3 \sqrt{d+e x} \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 (b+c x)}+\frac{\sqrt{d+e x} \left (2 b^2 e^2-13 b c d e+12 c^2 d^2\right )}{4 b^3 (b+c x)^2}-\frac{d (d+e x)^{3/2}}{2 b x^2 (b+c x)^2} \]

[Out]

((12*c^2*d^2 - 13*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*(b + c*x)^2) + (d*(
8*c*d - 7*b*e)*Sqrt[d + e*x])/(4*b^2*x*(b + c*x)^2) + (3*(8*c^2*d^2 - 8*b*c*d*e
+ b^2*e^2)*Sqrt[d + e*x])/(4*b^4*(b + c*x)) - (d*(d + e*x)^(3/2))/(2*b*x^2*(b +
c*x)^2) - (3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/(4*b^5) + (3*Sqrt[c*d - b*e]*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c])

_______________________________________________________________________________________

Rubi [A]  time = 1.148, antiderivative size = 280, 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 \[ \frac{d \sqrt{d+e x} (8 c d-7 b e)}{4 b^2 x (b+c x)^2}-\frac{3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 \sqrt{c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c}}+\frac{3 \sqrt{d+e x} \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 (b+c x)}+\frac{\sqrt{d+e x} \left (2 b^2 e^2-13 b c d e+12 c^2 d^2\right )}{4 b^3 (b+c x)^2}-\frac{d (d+e x)^{3/2}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(b*x + c*x^2)^3,x]

[Out]

((12*c^2*d^2 - 13*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*(b + c*x)^2) + (d*(
8*c*d - 7*b*e)*Sqrt[d + e*x])/(4*b^2*x*(b + c*x)^2) + (3*(8*c^2*d^2 - 8*b*c*d*e
+ b^2*e^2)*Sqrt[d + e*x])/(4*b^4*(b + c*x)) - (d*(d + e*x)^(3/2))/(2*b*x^2*(b +
c*x)^2) - (3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]
/Sqrt[d]])/(4*b^5) + (3*Sqrt[c*d - b*e]*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 171.174, size = 277, normalized size = 0.99 \[ - \frac{d \left (d + e x\right )^{\frac{3}{2}}}{2 b x^{2} \left (b + c x\right )^{2}} - \frac{\sqrt{d + e x} \left (b e - 2 c d\right ) \left (b e - c d\right )}{2 b^{2} c x \left (b + c x\right )^{2}} + \frac{\sqrt{d + e x} \left (2 b^{2} e^{2} - 13 b c d e + 12 c^{2} d^{2}\right )}{4 b^{3} c x \left (b + c x\right )} + \frac{3 \sqrt{d + e x} \left (b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{4 b^{4} \left (b + c x\right )} - \frac{3 \sqrt{d} \left (5 b^{2} e^{2} - 20 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{4 b^{5}} + \frac{3 \sqrt{b e - c d} \left (b^{2} e^{2} - 12 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{4 b^{5} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

-d*(d + e*x)**(3/2)/(2*b*x**2*(b + c*x)**2) - sqrt(d + e*x)*(b*e - 2*c*d)*(b*e -
c*d)/(2*b**2*c*x*(b + c*x)**2) + sqrt(d + e*x)*(2*b**2*e**2 - 13*b*c*d*e + 12*c
**2*d**2)/(4*b**3*c*x*(b + c*x)) + 3*sqrt(d + e*x)*(b**2*e**2 - 8*b*c*d*e + 8*c*
*2*d**2)/(4*b**4*(b + c*x)) - 3*sqrt(d)*(5*b**2*e**2 - 20*b*c*d*e + 16*c**2*d**2
)*atanh(sqrt(d + e*x)/sqrt(d))/(4*b**5) + 3*sqrt(b*e - c*d)*(b**2*e**2 - 12*b*c*
d*e + 16*c**2*d**2)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(4*b**5*sqrt(c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.419975, size = 237, normalized size = 0.85 \[ \frac{-3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{b \sqrt{d+e x} \left (b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )+b^2 c x \left (8 d^2-37 d e x+3 e^2 x^2\right )+12 b c^2 d x^2 (3 d-2 e x)+24 c^3 d^2 x^3\right )}{x^2 (b+c x)^2}+\frac{3 \left (-b^3 e^3+13 b^2 c d e^2-28 b c^2 d^2 e+16 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{c d-b e}}}{4 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^3,x]

[Out]

((b*Sqrt[d + e*x]*(24*c^3*d^2*x^3 + 12*b*c^2*d*x^2*(3*d - 2*e*x) + b^2*c*x*(8*d^
2 - 37*d*e*x + 3*e^2*x^2) + b^3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2)))/(x^2*(b + c*x)^
2) - 3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[
d]] + (3*(16*c^3*d^3 - 28*b*c^2*d^2*e + 13*b^2*c*d*e^2 - b^3*e^3)*ArcTanh[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(Sqrt[c]*Sqrt[c*d - b*e]))/(4*b^5)

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 521, normalized size = 1.9 \[{\frac{3\,{e}^{3}c}{4\,{b}^{2} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{e}^{2}{c}^{2}d}{4\,{b}^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e \left ( ex+d \right ) ^{3/2}{d}^{2}{c}^{3}}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{5\,{e}^{4}}{4\,b \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-{\frac{11\,{e}^{3}cd}{2\,{b}^{2} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{29\,{e}^{2}{c}^{2}{d}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e\sqrt{ex+d}{c}^{3}{d}^{3}}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{3\,{e}^{3}}{4\,{b}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-{\frac{39\,{e}^{2}cd}{4\,{b}^{3}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+21\,{\frac{e{c}^{2}{d}^{2}}{{b}^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{3}{d}^{3}}{{b}^{5}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{9\,d}{4\,{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{{d}^{2} \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}}}+{\frac{7\,{d}^{2}}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{{d}^{3}\sqrt{ex+d}c}{e{b}^{4}{x}^{2}}}-{\frac{15\,{e}^{2}}{4\,{b}^{3}}\sqrt{d}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ) }+15\,{\frac{e{d}^{3/2}c}{{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{d}^{5/2}{c}^{2}}{{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

3/4*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*c-15/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/
2)*c^2*d+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d^2*c^3+5/4*e^4/b/(c*e*x+b*e)^2*(e*
x+d)^(1/2)-11/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(1/2)*c*d+29/4*e^2/b^3/(c*e*x+b*e)
^2*(e*x+d)^(1/2)*c^2*d^2-3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*c^3*d^3+3/4*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))-39/4*e^2/b^3/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*c*d+21*e/b^4/((b*e
-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*c^2*d^2-12/b^5/((b*e-
c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*c^3*d^3-9/4*d/b^3/x^2*
(e*x+d)^(3/2)+3/e*d^2/b^4/x^2*(e*x+d)^(3/2)*c+7/4*d^2/b^3/x^2*(e*x+d)^(1/2)-3/e*
d^3/b^4/x^2*(e*x+d)^(1/2)*c-15/4*e^2*d^(1/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))+
15*e*d^(3/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(5/2)/b^5*arctanh((e*x+d)
^(1/2)/d^(1/2))*c^2

_______________________________________________________________________________________

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)^(5/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.336302, 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)^(5/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^
2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*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)) + 3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*
d^2 - 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e + 5*b^4
*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*d^2 -
3*(8*b*c^3*d^2 - 8*b^2*c^2*d*e + b^3*c*e^2)*x^3 - (36*b^2*c^2*d^2 - 37*b^3*c*d*
e + 5*b^4*e^2)*x^2 - (8*b^3*c*d^2 - 9*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*x^4 +
2*b^6*c*x^3 + b^7*x^2), 1/8*(6*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 +
2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d
*e + b^4*e^2)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c
)) + 3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 20*b
^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e + 5*b^4*e^2)*x^2)
*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*d^2 - 3*(8*b*c^
3*d^2 - 8*b^2*c^2*d*e + b^3*c*e^2)*x^3 - (36*b^2*c^2*d^2 - 37*b^3*c*d*e + 5*b^4*
e^2)*x^2 - (8*b^3*c*d^2 - 9*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*x^4 + 2*b^6*c*x^
3 + b^7*x^2), -1/8*(6*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b
*c^3*d^2 - 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e +
5*b^4*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - 3*((16*c^4*d^2 - 12*b*
c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 +
(16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*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)) + 2*(2*b^4*d^2
- 3*(8*b*c^3*d^2 - 8*b^2*c^2*d*e + b^3*c*e^2)*x^3 - (36*b^2*c^2*d^2 - 37*b^3*c*d
*e + 5*b^4*e^2)*x^2 - (8*b^3*c*d^2 - 9*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*x^4 +
2*b^6*c*x^3 + b^7*x^2), -1/4*(3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^
4 + 2*(16*b*c^3*d^2 - 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b
^3*c*d*e + 5*b^4*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - 3*((16*c^4*
d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c
*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*sqrt(-(c*d - b*e)/c)*
arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (2*b^4*d^2 - 3*(8*b*c^3*d^2 - 8*b^2
*c^2*d*e + b^3*c*e^2)*x^3 - (36*b^2*c^2*d^2 - 37*b^3*c*d*e + 5*b^4*e^2)*x^2 - (8
*b^3*c*d^2 - 9*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)]

_______________________________________________________________________________________

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)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234766, size = 605, normalized size = 2.16 \[ -\frac{3 \,{\left (16 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5}} + \frac{3 \,{\left (16 \, c^{2} d^{3} - 20 \, b c d^{2} e + 5 \, b^{2} d e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{2} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{3} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{4} e - 24 \, \sqrt{x e + d} c^{3} d^{5} e - 24 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} d e^{2} + 108 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d^{2} e^{2} - 144 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{3} e^{2} + 60 \, \sqrt{x e + d} b c^{2} d^{4} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c e^{3} - 46 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c d e^{3} + 91 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d^{2} e^{3} - 48 \, \sqrt{x e + d} b^{2} c d^{3} e^{3} + 5 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{4} - 19 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{4} + 12 \, \sqrt{x e + d} b^{3} d^{2} e^{4}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^3*d^3 - 28*b*c^2*d^2*e + 13*b^2*c*d*e^2 - b^3*e^3)*arctan(sqrt(x*e +
d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5) + 3/4*(16*c^2*d^3 - 20*b*c
*d^2*e + 5*b^2*d*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*(x
*e + d)^(7/2)*c^3*d^2*e - 72*(x*e + d)^(5/2)*c^3*d^3*e + 72*(x*e + d)^(3/2)*c^3*
d^4*e - 24*sqrt(x*e + d)*c^3*d^5*e - 24*(x*e + d)^(7/2)*b*c^2*d*e^2 + 108*(x*e +
d)^(5/2)*b*c^2*d^2*e^2 - 144*(x*e + d)^(3/2)*b*c^2*d^3*e^2 + 60*sqrt(x*e + d)*b
*c^2*d^4*e^2 + 3*(x*e + d)^(7/2)*b^2*c*e^3 - 46*(x*e + d)^(5/2)*b^2*c*d*e^3 + 91
*(x*e + d)^(3/2)*b^2*c*d^2*e^3 - 48*sqrt(x*e + d)*b^2*c*d^3*e^3 + 5*(x*e + d)^(5
/2)*b^3*e^4 - 19*(x*e + d)^(3/2)*b^3*d*e^4 + 12*sqrt(x*e + d)*b^3*d^2*e^4)/(((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)