3.1955 \(\int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=172 \[ \frac{e^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{5/2} d^{5/2}}-\frac{2 e (d+e x)}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^3)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (2*e*(d
+ e*x))/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (e^(3/2)*ArcTan
h[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2])])/(c^(5/2)*d^(5/2))

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Rubi [A]  time = 0.285278, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{e^{3/2} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{5/2} d^{5/2}}-\frac{2 e (d+e x)}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^3)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (2*e*(d
+ e*x))/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (e^(3/2)*ArcTan
h[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2])])/(c^(5/2)*d^(5/2))

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Rubi in Sympy [A]  time = 51.3038, size = 167, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{3}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{2 e \left (d + e x\right )}{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{c^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-2*(d + e*x)**3/(3*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 2*e*
(d + e*x)/(c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + e**(3/2)*
atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*
e*x**2 + x*(a*e**2 + c*d**2))))/(c**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.446062, size = 153, normalized size = 0.89 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{3 e^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}-\frac{2 \sqrt{c} \sqrt{d} \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}\right )}{3 c^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((-2*Sqrt[c]*Sqrt[d]*(3*a*e^2 + c*d*(d + 4*e*x)))
/(a*e + c*d*x)^2 + (3*e^(3/2)*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c
*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(3*c
^(5/2)*d^(5/2))

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Maple [B]  time = 0.019, size = 2538, normalized size = 14.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

e^2/d^2/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-1/6*e*d^7*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)
^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/48*e^6/d^4/c^4/(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*a^3-1/3*e^3*x^3/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-
31/8*e*d/c*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+7/16*e^4/d^2/c^3/(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-7/24*e^6/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/
(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-7/3*e^5*d^3/(-a^2*e^4+2*a*c*d^2*e^2-
c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+5/16*e^2*d^4/(-a^2*e^4+2*
a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-e^2/d^2/c^2*x/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+e^2*d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a
*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/2*e^3/d^3/c^3/(a*e*d+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*a-5/4*e^5*d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*x*a^2+16/3*e^4*d^4*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-3/4*e^3/d/c^2*x/(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(3/2)*a+5/2*e^3*d^5*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1/2*e^7/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)
/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+3/2*e^5/d/c^2/(-a^2*e^4+2*a*c*d^2*e
^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+3/2*e^3*d/c/(-a^2*e^4+2*
a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+2*e^4/c/(-a^2*e^4
+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-10*e^6*d^2/(
-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+
16/3*e^8/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x*a^3+2/3*e^3*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(3/2)*x*a-7/24*e^4*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-1/6*e^11/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^
4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+5/2*e^9/d/c^2/(-a^2*e^4+2*a*c*d
^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-1/3*e^2*d^6*c^2/(-
a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-1/24*
e*d^5*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*x-1/48*e^10/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*a^5+5/16*e^8/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-7/3*e^7*d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+1/2*e^4/d^2/c^2*x^2/(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*a+1/8*e^5/d^3/c^3*x/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*a^2-1/3*e^10/d^2/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*x*a^4-1/24*e^9/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+e^6/d^2/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2
*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+2/3*e^7/d/c^2/(-a^2*e^4+2*a*
c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3-11/16*d^2/c/(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*e*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/
(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/48*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d
^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-25/16*e^2/c^2/(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(3/2)*a-7/2*e^2/c*x^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*e/
d/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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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)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.654946, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{6 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}, \frac{3 \,{\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{3 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(c^2*d^2*e*x^2 + 2*a*c*d*e^2*x + a^2*e^3)*sqrt(e/(c*d))*log(8*c^2*d^2*e^
2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x + 4*(2*c
^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(e/(c*d))) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + c*d^2
+ 3*a*e^2))/(c^4*d^4*x^2 + 2*a*c^3*d^3*e*x + a^2*c^2*d^2*e^2), 1/3*(3*(c^2*d^2*e
*x^2 + 2*a*c*d*e^2*x + a^2*e^3)*sqrt(-e/(c*d))*arctan(1/2*(2*c*d*e*x + c*d^2 + a
*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*sqrt(-e/(c*d)))) - 2*sqrt
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + c*d^2 + 3*a*e^2))/(c^4*d^4*
x^2 + 2*a*c^3*d^3*e*x + a^2*c^2*d^2*e^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)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267387, size = 837, normalized size = 4.87 \[ -\frac{2 \,{\left ({\left ({\left (\frac{4 \,{\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}} + \frac{3 \,{\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )} x + \frac{6 \,{\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )} x + \frac{c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}}{c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}}\right )}}{3 \,{\left (c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}} - \frac{\sqrt{c d} e^{\frac{3}{2}}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="giac")

[Out]

-2/3*(((4*(c^5*d^9*e^3 - 4*a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e^9
+ a^4*c*d*e^11)*x/(c^6*d^10 - 4*a*c^5*d^8*e^2 + 6*a^2*c^4*d^6*e^4 - 4*a^3*c^3*d
^4*e^6 + a^4*c^2*d^2*e^8) + 3*(3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 + 14*a^2*c^3*d^
6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)/(c^6*d^10 - 4*a*c^5*d^8*e
^2 + 6*a^2*c^4*d^6*e^4 - 4*a^3*c^3*d^4*e^6 + a^4*c^2*d^2*e^8))*x + 6*(c^5*d^11*e
- 3*a*c^4*d^9*e^3 + 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3*a^4*c*d^3*e^9 + a
^5*d*e^11)/(c^6*d^10 - 4*a*c^5*d^8*e^2 + 6*a^2*c^4*d^6*e^4 - 4*a^3*c^3*d^4*e^6 +
a^4*c^2*d^2*e^8))*x + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*c^3*d^8*e^4 + 14*a^3*c
^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)/(c^6*d^10 - 4*a*c^5*d^8*e^2 + 6*
a^2*c^4*d^6*e^4 - 4*a^3*c^3*d^4*e^6 + a^4*c^2*d^2*e^8))/(c*d*x^2*e + a*d*e + (c*
d^2 + a*e^2)*x)^(3/2) - sqrt(c*d)*e^(3/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c
*d)*a*e^(5/2)))/(c^3*d^3)