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

Optimal. Leaf size=416 \[ -\frac{\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (a^3 e^2+3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac{3 c x^2 \left (a c+b^2\right )}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e} \]

[Out]

-(((c^3*d - b^3*e - 6*a*b*c*e)*x)/e^2) + (3*c*(b^2 + a*c)*x^2)/(2*e) + (b*c^2*x^
3)/e + (c^3*x^4)/(4*e) - ((c^3*d^2 - 3*b^2*c*d^(4/3)*e^(2/3) - 3*a*c^2*d^(4/3)*e
^(2/3) - b^3*d*e - 6*a*b*c*d*e + 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*ArcTan[(d^(1
/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(7/3)) + ((c^3*d^2 - 6
*a*b*c*d*e - e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e)
)*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(7/3)) - ((c^3*d^2 - 6*a*b*c*d*e - e*(b
^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e))*Log[d^(2/3) - d
^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) - ((b*c^2*d - a*b^2*e - a^2
*c*e)*Log[d + e*x^3])/e^2

_______________________________________________________________________________________

Rubi [A]  time = 1.31076, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (a^3 e^2+3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac{3 c x^2 \left (a c+b^2\right )}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

-(((c^3*d - b^3*e - 6*a*b*c*e)*x)/e^2) + (3*c*(b^2 + a*c)*x^2)/(2*e) + (b*c^2*x^
3)/e + (c^3*x^4)/(4*e) - ((c^3*d^2 - 3*b^2*c*d^(4/3)*e^(2/3) - 3*a*c^2*d^(4/3)*e
^(2/3) - b^3*d*e - 6*a*b*c*d*e + 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*ArcTan[(d^(1
/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(7/3)) + ((c^3*d^2 - 6
*a*b*c*d*e - e*(b^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e)
)*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(7/3)) - ((c^3*d^2 - 6*a*b*c*d*e - e*(b
^3*d - a^3*e) + 3*d^(1/3)*e^(2/3)*(b^2*c*d + a*c^2*d - a^2*b*e))*Log[d^(2/3) - d
^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(7/3)) - ((b*c^2*d - a*b^2*e - a^2
*c*e)*Log[d + e*x^3])/e^2

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.866991, size = 439, normalized size = 1.06 \[ \frac{12 \sqrt [3]{e} \log \left (d+e x^3\right ) \left (a^2 c e+a b^2 e-b c^2 d\right )-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e \left (a^3 e+3 a^2 b \sqrt [3]{d} e^{2/3}-b^3 d\right )-3 c \left (2 a b d e+b^2 d^{4/3} e^{2/3}\right )-3 a c^2 d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}-\frac{2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^3 e^2-3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}+\frac{4 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a^3 e^2-3 a^2 b \sqrt [3]{d} e^{5/3}-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}-b^3 d e+3 b^2 c d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}+12 \sqrt [3]{e} x \left (6 a b c e+b^3 e-c^3 d\right )+18 c e^{4/3} x^2 \left (a c+b^2\right )+12 b c^2 e^{4/3} x^3+3 c^3 e^{4/3} x^4}{12 e^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(12*e^(1/3)*(-(c^3*d) + b^3*e + 6*a*b*c*e)*x + 18*c*(b^2 + a*c)*e^(4/3)*x^2 + 12
*b*c^2*e^(4/3)*x^3 + 3*c^3*e^(4/3)*x^4 - (4*Sqrt[3]*(c^3*d^2 - 3*a*c^2*d^(4/3)*e
^(2/3) + e*(-(b^3*d) + 3*a^2*b*d^(1/3)*e^(2/3) + a^3*e) - 3*c*(b^2*d^(4/3)*e^(2/
3) + 2*a*b*d*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (4*(c^3*
d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e
- 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (2*(c^3
*d^2 + 3*b^2*c*d^(4/3)*e^(2/3) + 3*a*c^2*d^(4/3)*e^(2/3) - b^3*d*e - 6*a*b*c*d*e
- 3*a^2*b*d^(1/3)*e^(5/3) + a^3*e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*
x^2])/d^(2/3) + 12*e^(1/3)*(-(b*c^2*d) + a*b^2*e + a^2*c*e)*Log[d + e*x^3])/(12*
e^(7/3))

_______________________________________________________________________________________

Maple [B]  time = 0.008, size = 837, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*b*c*d-1/3/e
^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^3*d+1/3/e^3/(d/
e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*c^3*d^2+1/e^2/(d/e)^(1/
3)*ln(x+(d/e)^(1/3))*a*c^2*d+1/e^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*b^2*c*d-1/2/e^2
/(d/e)^(1/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a*c^2*d-1/2/e^2/(d/e)^(1/3)*ln(x^
2-x*(d/e)^(1/3)+(d/e)^(2/3))*b^2*c*d+1/e*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*
(2/(d/e)^(1/3)*x-1))*a^2*b+1/3/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a^3+1/2/e/(d/e)^(
1/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a^2*b-1/e^2*ln(e*x^3+d)*b*c^2*d-1/6/e^3/(
d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*c^3*d^2+1/3/e/(d/e)^(2/3)*3^(1/2)*a
rctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a^3+1/6/e^2/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1
/3)+(d/e)^(2/3))*b^3*d-2/e^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a*b*c*d-1/e^2*3^(1/2)
/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a*c^2*d+1/3/e^3/(d/e)^(2/3)
*ln(x+(d/e)^(1/3))*c^3*d^2+6/e*a*b*c*x-1/6/e/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d
/e)^(2/3))*a^3+1/e*ln(e*x^3+d)*a^2*c+1/e*ln(e*x^3+d)*a*b^2+1/4*c^3*x^4/e+b*c^2*x
^3/e+1/e*b^3*x+3/2/e*x^2*a*c^2+3/2/e*b^2*c*x^2-1/e^2*c^3*d*x-1/3/e^2/(d/e)^(2/3)
*ln(x+(d/e)^(1/3))*b^3*d-1/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*a^2*b-1/e^2*3^(1/2)/(
d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b^2*c*d+1/e^2/(d/e)^(2/3)*ln(
x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a*b*c*d

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 124.454, size = 1314, normalized size = 3.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*c**2*x**3/e + c**3*x**4/(4*e) + RootSum(27*_t**3*d**2*e**7 + _t**2*(-81*a**2*c
*d**2*e**6 - 81*a*b**2*d**2*e**6 + 81*b*c**2*d**3*e**5) + _t*(27*a**5*b*d*e**6 +
54*a**4*c**2*d**2*e**5 - 27*a**3*b**2*c*d**2*e**5 + 54*a**2*b**4*d**2*e**5 + 27
*a**2*b*c**3*d**3*e**4 + 27*a*b**3*c**2*d**3*e**4 - 27*a*c**5*d**4*e**3 + 27*b**
5*c*d**3*e**4 + 54*b**2*c**4*d**4*e**3) - a**9*e**6 - 9*a**7*b*c*d*e**5 + 3*a**6
*b**3*d*e**5 - 3*a**6*c**3*d**2*e**4 - 27*a**5*b**2*c**2*d**2*e**4 + 18*a**4*b**
4*c*d**2*e**4 - 18*a**4*b*c**4*d**3*e**3 - 3*a**3*b**6*d**2*e**4 - 21*a**3*b**3*
c**3*d**3*e**3 - 3*a**3*c**6*d**4*e**2 + 27*a**2*b**5*c**2*d**3*e**3 - 27*a**2*b
**2*c**5*d**4*e**2 - 9*a*b**7*c*d**3*e**3 + 18*a*b**4*c**4*d**4*e**2 - 9*a*b*c**
7*d**5*e + b**9*d**3*e**3 - 3*b**6*c**3*d**4*e**2 + 3*b**3*c**6*d**5*e - c**9*d*
*6, Lambda(_t, _t*log(x + (27*_t**2*a**2*b*d**2*e**6 - 27*_t**2*a*c**2*d**3*e**5
- 27*_t**2*b**2*c*d**3*e**5 + 3*_t*a**6*d*e**6 - 90*_t*a**4*b*c*d**2*e**5 - 60*
_t*a**3*b**3*d**2*e**5 + 60*_t*a**3*c**3*d**3*e**4 + 270*_t*a**2*b**2*c**2*d**3*
e**4 + 90*_t*a*b**4*c*d**3*e**4 - 90*_t*a*b*c**4*d**4*e**3 + 3*_t*b**6*d**3*e**4
- 60*_t*b**3*c**3*d**4*e**3 + 3*_t*c**6*d**5*e**2 - 3*a**8*c*d*e**5 + 15*a**7*b
**2*d*e**5 + 30*a**6*b*c**2*d**2*e**4 - 48*a**5*b**3*c*d**2*e**4 - 15*a**5*c**4*
d**3*e**3 + 15*a**4*b**5*d**2*e**4 - 15*a**4*b**2*c**3*d**3*e**3 - 15*a**3*b**4*
c**2*d**3*e**3 - 48*a**3*b*c**5*d**4*e**2 - 30*a**2*b**6*c*d**3*e**3 + 15*a**2*b
**3*c**4*d**4*e**2 + 15*a**2*c**7*d**5*e - 3*a*b**8*d**3*e**3 - 48*a*b**5*c**3*d
**4*e**2 - 30*a*b**2*c**6*d**5*e - 15*b**7*c**2*d**4*e**2 - 15*b**4*c**5*d**5*e
+ 3*b*c**8*d**6)/(a**9*e**6 - 18*a**7*b*c*d*e**5 + 24*a**6*b**3*d*e**5 + 3*a**6*
c**3*d**2*e**4 + 27*a**5*b**2*c**2*d**2*e**4 - 45*a**4*b**4*c*d**2*e**4 + 45*a**
4*b*c**4*d**3*e**3 + 3*a**3*b**6*d**2*e**4 - 60*a**3*b**3*c**3*d**3*e**3 - 24*a*
*3*c**6*d**4*e**2 - 27*a**2*b**5*c**2*d**3*e**3 + 27*a**2*b**2*c**5*d**4*e**2 -
18*a*b**7*c*d**3*e**3 - 45*a*b**4*c**4*d**4*e**2 - 18*a*b*c**7*d**5*e - b**9*d**
3*e**3 - 24*b**6*c**3*d**4*e**2 - 3*b**3*c**6*d**5*e + c**9*d**6)))) + x**2*(3*a
*c**2 + 3*b**2*c)/(2*e) + x*(6*a*b*c*e + b**3*e - c**3*d)/e**2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224708, size = 648, normalized size = 1.56 \[ -{\left (b c^{2} d - a b^{2} e - a^{2} c e\right )} e^{\left (-2\right )}{\rm ln}\left ({\left | x^{3} e + d \right |}\right ) + \frac{\sqrt{3}{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c^{3} d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b^{3} d e - 6 \, \left (-d e^{2}\right )^{\frac{1}{3}} a b c d e + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} b^{2} c d + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a c^{2} d - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a^{2} b e + \left (-d e^{2}\right )^{\frac{1}{3}} a^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{3 \, d} - \frac{{\left (c^{3} d^{2} e^{7} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} b^{2} c d e^{8} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a c^{2} d e^{8} - b^{3} d e^{8} - 6 \, a b c d e^{8} + 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a^{2} b e^{9} + a^{3} e^{9}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-9\right )}{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{1}{4} \,{\left (c^{3} x^{4} e^{3} + 4 \, b c^{2} x^{3} e^{3} + 6 \, b^{2} c x^{2} e^{3} + 6 \, a c^{2} x^{2} e^{3} - 4 \, c^{3} d x e^{2} + 4 \, b^{3} x e^{3} + 24 \, a b c x e^{3}\right )} e^{\left (-4\right )} + \frac{{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c^{3} d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b^{3} d e - 6 \, \left (-d e^{2}\right )^{\frac{1}{3}} a b c d e - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} b^{2} c d - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a c^{2} d + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a^{2} b e + \left (-d e^{2}\right )^{\frac{1}{3}} a^{3} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*c^2*d - a*b^2*e - a^2*c*e)*e^(-2)*ln(abs(x^3*e + d)) + 1/3*sqrt(3)*((-d*e^2)
^(1/3)*c^3*d^2 - (-d*e^2)^(1/3)*b^3*d*e - 6*(-d*e^2)^(1/3)*a*b*c*d*e + 3*(-d*e^2
)^(2/3)*b^2*c*d + 3*(-d*e^2)^(2/3)*a*c^2*d - 3*(-d*e^2)^(2/3)*a^2*b*e + (-d*e^2)
^(1/3)*a^3*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d*e^(-1))^(1/3))/(-d*e^(-1))^(1/3))*
e^(-3)/d - 1/3*(c^3*d^2*e^7 - 3*(-d*e^(-1))^(1/3)*b^2*c*d*e^8 - 3*(-d*e^(-1))^(1
/3)*a*c^2*d*e^8 - b^3*d*e^8 - 6*a*b*c*d*e^8 + 3*(-d*e^(-1))^(1/3)*a^2*b*e^9 + a^
3*e^9)*(-d*e^(-1))^(1/3)*e^(-9)*ln(abs(x - (-d*e^(-1))^(1/3)))/d + 1/4*(c^3*x^4*
e^3 + 4*b*c^2*x^3*e^3 + 6*b^2*c*x^2*e^3 + 6*a*c^2*x^2*e^3 - 4*c^3*d*x*e^2 + 4*b^
3*x*e^3 + 24*a*b*c*x*e^3)*e^(-4) + 1/6*((-d*e^2)^(1/3)*c^3*d^2 - (-d*e^2)^(1/3)*
b^3*d*e - 6*(-d*e^2)^(1/3)*a*b*c*d*e - 3*(-d*e^2)^(2/3)*b^2*c*d - 3*(-d*e^2)^(2/
3)*a*c^2*d + 3*(-d*e^2)^(2/3)*a^2*b*e + (-d*e^2)^(1/3)*a^3*e^2)*e^(-3)*ln(x^2 +
(-d*e^(-1))^(1/3)*x + (-d*e^(-1))^(2/3))/d