3.1909 \(\int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=341 \[ \frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}-\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \]

[Out]

(-7*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^4*d^4*e^2) + (7*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^3*d^3*e) + (7*(c*d^2 -
a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*c^2*d^2) + ((d + e*x)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6*c*d) + (7*(c*d^2 - a*e^2)^6*A
rcTanh[(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])])/(1024*c^(9/2)*d^(9/2)*e^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.630446, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}-\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^4*d^4*e^2) + (7*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^3*d^3*e) + (7*(c*d^2 -
a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*c^2*d^2) + ((d + e*x)
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6*c*d) + (7*(c*d^2 - a*e^2)^6*A
rcTanh[(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])])/(1024*c^(9/2)*d^(9/2)*e^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 85.0151, size = 328, normalized size = 0.96 \[ \frac{\left (d + e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{6 c d} - \frac{7 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{60 c^{2} d^{2}} + \frac{7 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{192 c^{3} d^{3} e} - \frac{7 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 c^{4} d^{4} e^{2}} + \frac{7 \left (a e^{2} - c d^{2}\right )^{6} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 c^{\frac{9}{2}} d^{\frac{9}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(6*c*d) - 7*(a*e**2
- c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(60*c**2*d**2) + 7*(
a*e**2 - c*d**2)**2*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**
2 + c*d**2))**(3/2)/(192*c**3*d**3*e) - 7*(a*e**2 - c*d**2)**4*(a*e**2 + c*d**2
+ 2*c*d*e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(512*c**4*d**4*e**2)
+ 7*(a*e**2 - c*d**2)**6*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))))/(1024*c**(9/2)*d**(9/2
)*e**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.948371, size = 374, normalized size = 1.1 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \left (-105 a^5 e^{10}+35 a^4 c d e^8 (17 d+2 e x)-14 a^3 c^2 d^2 e^6 \left (99 d^2+28 d e x+4 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (281 d^3+150 d^2 e x+52 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (595 d^4+5752 d^3 e x+9528 d^2 e^2 x^2+6560 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x+3016 d^3 e^2 x^2+6192 d^2 e^3 x^3+4736 d e^4 x^4+1280 e^5 x^5\right )\right )}{15 c^4 d^4 e^2 (d+e x) (a e+c d x)}+\frac{7 \left (c d^2-a e^2\right )^6 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right )}{1024} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((2*(-105*a^5*e^10 + 35*a^4*c*d*e^8*(17*d + 2*e
*x) - 14*a^3*c^2*d^2*e^6*(99*d^2 + 28*d*e*x + 4*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(28
1*d^3 + 150*d^2*e*x + 52*d*e^2*x^2 + 8*e^3*x^3) + a*c^4*d^4*e^2*(595*d^4 + 5752*
d^3*e*x + 9528*d^2*e^2*x^2 + 6560*d*e^3*x^3 + 1664*e^4*x^4) + c^5*d^5*(-105*d^5
+ 70*d^4*e*x + 3016*d^3*e^2*x^2 + 6192*d^2*e^3*x^3 + 4736*d*e^4*x^4 + 1280*e^5*x
^5)))/(15*c^4*d^4*e^2*(a*e + c*d*x)*(d + e*x)) + (7*(c*d^2 - a*e^2)^6*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)])/
(c^(9/2)*d^(9/2)*e^(5/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/1024

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 1302, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/256*e^7/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+7/64*e^5*a^3/c^
2/d*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-21/128*d*e^3/c*(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x*a^2-35/256*d^2*e^4/c*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)*a^3-7/48*e^2/c
*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+7/96*d^2*(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*x+7/192*d^3/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+21/512*d^4*(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+7/96*e^4/d^2/c^2*(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(3/2)*x*a^2+7/1024*e^10/d^4/c^4*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)*a^6-21/512*e^8
/d^2/c^3*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)*a^5+21/512*e^6/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*a^4-7/256*d^2*e^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+10
5/1024*e^6/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)*a^4+105/1024*d^4*e^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)*
a^2-7/256*d^5/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-7/192*d*e/c*(a*e*d+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+7/64*d^3*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*x*a-21/512*d^6*c*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)*a+7/1024*d^8/e^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)-7/256*e^4/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-7/512*d^6/e^2*c
*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/6*e*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)/d/c-7/512*e^8/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5-7/60*
e^2/d^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+7/192*e^5/d^3/c^3*(a*e*d+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-7/192*e^3/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)*a^2+17/60/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.291701, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*c^5*d^5*e^5*x^5 - 105*c^5*d^10 + 595*a*c^4*d^8*e^2 + 1686*a^2*
c^3*d^6*e^4 - 1386*a^3*c^2*d^4*e^6 + 595*a^4*c*d^2*e^8 - 105*a^5*e^10 + 128*(37*
c^5*d^6*e^4 + 13*a*c^4*d^4*e^6)*x^4 + 16*(387*c^5*d^7*e^3 + 410*a*c^4*d^5*e^5 +
3*a^2*c^3*d^3*e^7)*x^3 + 8*(377*c^5*d^8*e^2 + 1191*a*c^4*d^6*e^4 + 39*a^2*c^3*d^
4*e^6 - 7*a^3*c^2*d^2*e^8)*x^2 + 2*(35*c^5*d^9*e + 2876*a*c^4*d^7*e^3 + 450*a^2*
c^3*d^5*e^5 - 196*a^3*c^2*d^3*e^7 + 35*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 105*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d
^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*
log(4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x) + (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)*sqrt(c*d*e)))/(sqrt(c*d*e)*c^4*d^4*e^2), 1/15360*(2*(1280*c^
5*d^5*e^5*x^5 - 105*c^5*d^10 + 595*a*c^4*d^8*e^2 + 1686*a^2*c^3*d^6*e^4 - 1386*a
^3*c^2*d^4*e^6 + 595*a^4*c*d^2*e^8 - 105*a^5*e^10 + 128*(37*c^5*d^6*e^4 + 13*a*c
^4*d^4*e^6)*x^4 + 16*(387*c^5*d^7*e^3 + 410*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x
^3 + 8*(377*c^5*d^8*e^2 + 1191*a*c^4*d^6*e^4 + 39*a^2*c^3*d^4*e^6 - 7*a^3*c^2*d^
2*e^8)*x^2 + 2*(35*c^5*d^9*e + 2876*a*c^4*d^7*e^3 + 450*a^2*c^3*d^5*e^5 - 196*a^
3*c^2*d^3*e^7 + 35*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*s
qrt(-c*d*e) + 105*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3
*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*arctan(1/2*(2*c*d*e
*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*
d*e)))/(sqrt(-c*d*e)*c^4*d^4*e^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)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255906, size = 671, normalized size = 1.97 \[ \frac{1}{7680} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c d x e^{3} + \frac{{\left (37 \, c^{6} d^{7} e^{7} + 13 \, a c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (387 \, c^{6} d^{8} e^{6} + 410 \, a c^{5} d^{6} e^{8} + 3 \, a^{2} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (377 \, c^{6} d^{9} e^{5} + 1191 \, a c^{5} d^{7} e^{7} + 39 \, a^{2} c^{4} d^{5} e^{9} - 7 \, a^{3} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (35 \, c^{6} d^{10} e^{4} + 2876 \, a c^{5} d^{8} e^{6} + 450 \, a^{2} c^{4} d^{6} e^{8} - 196 \, a^{3} c^{3} d^{4} e^{10} + 35 \, a^{4} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac{{\left (105 \, c^{6} d^{11} e^{3} - 595 \, a c^{5} d^{9} e^{5} - 1686 \, a^{2} c^{4} d^{7} e^{7} + 1386 \, a^{3} c^{3} d^{5} e^{9} - 595 \, a^{4} c^{2} d^{3} e^{11} + 105 \, a^{5} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} - \frac{7 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*c*d*x*e^3 + (
37*c^6*d^7*e^7 + 13*a*c^5*d^5*e^9)*e^(-5)/(c^5*d^5))*x + (387*c^6*d^8*e^6 + 410*
a*c^5*d^6*e^8 + 3*a^2*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x + (377*c^6*d^9*e^5 + 119
1*a*c^5*d^7*e^7 + 39*a^2*c^4*d^5*e^9 - 7*a^3*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x +
(35*c^6*d^10*e^4 + 2876*a*c^5*d^8*e^6 + 450*a^2*c^4*d^6*e^8 - 196*a^3*c^3*d^4*e
^10 + 35*a^4*c^2*d^2*e^12)*e^(-5)/(c^5*d^5))*x - (105*c^6*d^11*e^3 - 595*a*c^5*d
^9*e^5 - 1686*a^2*c^4*d^7*e^7 + 1386*a^3*c^3*d^5*e^9 - 595*a^4*c^2*d^3*e^11 + 10
5*a^5*c*d*e^13)*e^(-5)/(c^5*d^5)) - 7/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2
*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*
e^12)*sqrt(c*d)*e^(-5/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2
) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(
c^5*d^5)