3.660 \(\int \frac{1}{(d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx\)

Optimal. Leaf size=343 \[ \frac{3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3}-\frac{3 b \log (d+e x)}{a^4 e f^3}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2}+\frac{20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2-20 a b^2 c+3 b^4}{4 a^2 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 e f^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2) + (b^
2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*
x)^2 + c*(d + e*x)^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c
)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(
d + e*x)^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b +
2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) - (3*b*Lo
g[d + e*x])/(a^4*e*f^3) + (3*b*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(4*a^4*e*
f^3)

_______________________________________________________________________________________

Rubi [A]  time = 1.22462, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3}-\frac{3 b \log (d+e x)}{a^4 e f^3}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2}+\frac{20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2-20 a b^2 c+3 b^4}{4 a^2 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 e f^3 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2) + (b^
2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*
x)^2 + c*(d + e*x)^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c
)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(
d + e*x)^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b +
2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) - (3*b*Lo
g[d + e*x])/(a^4*e*f^3) + (3*b*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(4*a^4*e*
f^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 176.663, size = 333, normalized size = 0.97 \[ \frac{- 2 a c + b^{2} + b c \left (d + e x\right )^{2}}{4 a e f^{3} \left (d + e x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )^{2}} + \frac{20 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4} + 3 b c \left (d + e x\right )^{2} \left (- 6 a c + b^{2}\right )}{4 a^{2} e f^{3} \left (d + e x\right )^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} - \frac{3 \left (- 5 a c + b^{2}\right ) \left (- 2 a c + b^{2}\right )}{2 a^{3} e f^{3} \left (d + e x\right )^{2} \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b \log{\left (\left (d + e x\right )^{2} \right )}}{2 a^{4} e f^{3}} + \frac{3 b \log{\left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4} \right )}}{4 a^{4} e f^{3}} - \frac{3 \left (- 20 a^{3} c^{3} + 30 a^{2} b^{2} c^{2} - 10 a b^{4} c + b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{4} e f^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*f*x+d*f)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

(-2*a*c + b**2 + b*c*(d + e*x)**2)/(4*a*e*f**3*(d + e*x)**2*(-4*a*c + b**2)*(a +
b*(d + e*x)**2 + c*(d + e*x)**4)**2) + (20*a**2*c**2 - 20*a*b**2*c + 3*b**4 + 3
*b*c*(d + e*x)**2*(-6*a*c + b**2))/(4*a**2*e*f**3*(d + e*x)**2*(-4*a*c + b**2)**
2*(a + b*(d + e*x)**2 + c*(d + e*x)**4)) - 3*(-5*a*c + b**2)*(-2*a*c + b**2)/(2*
a**3*e*f**3*(d + e*x)**2*(-4*a*c + b**2)**2) - 3*b*log((d + e*x)**2)/(2*a**4*e*f
**3) + 3*b*log(a + b*(d + e*x)**2 + c*(d + e*x)**4)/(4*a**4*e*f**3) - 3*(-20*a**
3*c**3 + 30*a**2*b**2*c**2 - 10*a*b**4*c + b**6)*atanh((b + 2*c*(d + e*x)**2)/sq
rt(-4*a*c + b**2))/(2*a**4*e*f**3*(-4*a*c + b**2)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 6.24747, size = 509, normalized size = 1.48 \[ -\frac{3 b \log (d+e x)}{a^4 e f^3}-\frac{1}{2 a^3 e f^3 (d+e x)^2}+\frac{-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2}{4 a^2 e f^3 \left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac{-46 a^2 b c^2-28 a^2 c^3 (d+e x)^2+29 a b^3 c+26 a b^2 c^2 (d+e x)^2-4 b^5-4 b^4 c (d+e x)^2}{4 a^3 e f^3 \left (4 a c-b^2\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a^4 e f^3 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a^4 e f^3 \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

-1/(2*a^3*e*f^3*(d + e*x)^2) + (b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d +
e*x)^2)/(4*a^2*(-b^2 + 4*a*c)*e*f^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-
4*b^5 + 29*a*b^3*c - 46*a^2*b*c^2 - 4*b^4*c*(d + e*x)^2 + 26*a*b^2*c^2*(d + e*x)
^2 - 28*a^2*c^3*(d + e*x)^2)/(4*a^3*(-b^2 + 4*a*c)^2*e*f^3*(a + b*(d + e*x)^2 +
c*(d + e*x)^4)) - (3*b*Log[d + e*x])/(a^4*e*f^3) + (3*(b^6 - 10*a*b^4*c + 30*a^2
*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16
*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a
^4*(b^2 - 4*a*c)^(5/2)*e*f^3) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*
c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^
2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(
5/2)*e*f^3)

_______________________________________________________________________________________

Maple [C]  time = 0.027, size = 5737, normalized size = 16.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*f*x+d*f)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^3*(e*f*x + d*f)^3),x, algorithm="maxima")

[Out]

-1/4*(6*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*e^8*x^8 + 48*(b^4*c^2 - 7*a*b^2*c^3
+ 10*a^2*c^4)*d*e^7*x^7 + 3*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3 + 56*(b^4*c^
2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^2)*e^6*x^6 + 6*(56*(b^4*c^2 - 7*a*b^2*c^3 + 10*a
^2*c^4)*d^3 + 3*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3)*d)*e^5*x^5 + 6*(b^4*c^2
- 7*a*b^2*c^3 + 10*a^2*c^4)*d^8 + (6*b^6 - 36*a*b^4*c + 14*a^2*b^2*c^2 + 100*a^3
*c^3 + 420*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^4 + 45*(4*b^5*c - 29*a*b^3*c^2
+ 46*a^2*b*c^3)*d^2)*e^4*x^4 + 3*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3)*d^6 +
4*(84*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^5 + 15*(4*b^5*c - 29*a*b^3*c^2 + 46
*a^2*b*c^3)*d^3 + 2*(3*b^6 - 18*a*b^4*c + 7*a^2*b^2*c^2 + 50*a^3*c^3)*d)*e^3*x^3
+ 2*a^2*b^4 - 16*a^3*b^2*c + 32*a^4*c^2 + 2*(3*b^6 - 18*a*b^4*c + 7*a^2*b^2*c^2
+ 50*a^3*c^3)*d^4 + (168*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^6 + 9*a*b^5 - 6
8*a^2*b^3*c + 122*a^3*b*c^2 + 45*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3)*d^4 + 1
2*(3*b^6 - 18*a*b^4*c + 7*a^2*b^2*c^2 + 50*a^3*c^3)*d^2)*e^2*x^2 + (9*a*b^5 - 68
*a^2*b^3*c + 122*a^3*b*c^2)*d^2 + 2*(24*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^7
+ 9*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3)*d^5 + 4*(3*b^6 - 18*a*b^4*c + 7*a^2
*b^2*c^2 + 50*a^3*c^3)*d^3 + (9*a*b^5 - 68*a^2*b^3*c + 122*a^3*b*c^2)*d)*e*x)/((
a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^11*f^3*x^10 + 10*(a^3*b^4*c^2 - 8*a^
4*b^2*c^3 + 16*a^5*c^4)*d*e^10*f^3*x^9 + (2*a^3*b^5*c - 16*a^4*b^3*c^2 + 32*a^5*
b*c^3 + 45*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^2)*e^9*f^3*x^8 + 8*(15*(
a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^3 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 1
6*a^5*b*c^3)*d)*e^8*f^3*x^7 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3 + 210*(a^3*b^4
*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^4 + 56*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*
b*c^3)*d^2)*e^7*f^3*x^6 + 2*(126*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5
+ 56*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^3 + 3*(a^3*b^6 - 6*a^4*b^4*c +
32*a^6*c^3)*d)*e^6*f^3*x^5 + (2*a^4*b^5 - 16*a^5*b^3*c + 32*a^6*b*c^2 + 210*(a^
3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^6 + 140*(a^3*b^5*c - 8*a^4*b^3*c^2 + 1
6*a^5*b*c^3)*d^4 + 15*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^2)*e^5*f^3*x^4 + 4*
(30*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^7 + 28*(a^3*b^5*c - 8*a^4*b^3*c
^2 + 16*a^5*b*c^3)*d^5 + 5*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^3 + 2*(a^4*b^5
- 8*a^5*b^3*c + 16*a^6*b*c^2)*d)*e^4*f^3*x^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*
c^2 + 45*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^8 + 56*(a^3*b^5*c - 8*a^4*
b^3*c^2 + 16*a^5*b*c^3)*d^6 + 15*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^4 + 12*(
a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d^2)*e^3*f^3*x^2 + 2*(5*(a^3*b^4*c^2 - 8*a
^4*b^2*c^3 + 16*a^5*c^4)*d^9 + 8*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^7
+ 3*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^5 + 4*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6
*b*c^2)*d^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*d)*e^2*f^3*x + ((a^3*b^4*c^2
- 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^10 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3
)*d^8 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^6 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16
*a^6*b*c^2)*d^4 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*d^2)*e*f^3) + 3*integrate
(((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^3*x^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*d*e^2*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 9*a*b^4*c
+ 23*a^2*b^2*c^2 - 10*a^3*c^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e*x
+ (b^6 - 9*a*b^4*c + 23*a^2*b^2*c^2 - 10*a^3*c^3)*d)/(c*e^4*x^4 + 4*c*d*e^3*x^3
+ c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a), x)/((a^4*b
^4 - 8*a^5*b^2*c + 16*a^6*c^2)*f^3) - 3*b*log(e*x + d)/(a^4*e*f^3)

_______________________________________________________________________________________

Fricas [A]  time = 6.10038, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^3*(e*f*x + d*f)^3),x, algorithm="fricas")

[Out]

[-1/4*(3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*e^10*x^10 + 10*
(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d*e^9*x^9 + (2*b^7*c - 20
*a*b^5*c^2 + 60*a^2*b^3*c^3 - 40*a^3*b*c^4 + 45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2
*b^2*c^4 - 20*a^3*c^5)*d^2)*e^8*x^8 + 8*(15*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2
*c^4 - 20*a^3*c^5)*d^3 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4
)*d)*e^7*x^7 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4 +
210*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^4 + 56*(b^7*c - 10
*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^2)*e^6*x^6 + (b^6*c^2 - 10*a*b^4*c
^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^10 + 2*(126*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2
*b^2*c^4 - 20*a^3*c^5)*d^5 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*
b*c^4)*d^3 + 3*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*
d)*e^5*x^5 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^8 + (2*a
*b^7 - 20*a^2*b^5*c + 60*a^3*b^3*c^2 - 40*a^4*b*c^3 + 210*(b^6*c^2 - 10*a*b^4*c^
3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^6 + 140*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^
3 - 20*a^3*b*c^4)*d^4 + 15*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 -
40*a^4*c^4)*d^2)*e^4*x^4 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 -
40*a^4*c^4)*d^6 + 4*(30*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d
^7 + 28*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^5 + 5*(b^8 - 8*
a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^3 + 2*(a*b^7 - 10*a^2*
b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d)*e^3*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30
*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^4 + (45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4
- 20*a^3*c^5)*d^8 + a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + 56*(b
^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^6 + 15*(b^8 - 8*a*b^6*c +
10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^4 + 12*(a*b^7 - 10*a^2*b^5*c +
30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^2)*e^2*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b
^2*c^2 - 20*a^5*c^3)*d^2 + 2*(5*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^
3*c^5)*d^9 + 8*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^7 + 3*(b
^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^5 + 4*(a*b^7 -
10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^3 + (a^2*b^6 - 10*a^3*b^4*c + 30
*a^4*b^2*c^2 - 20*a^5*c^3)*d)*e*x)*log((2*(b^2*c - 4*a*c^2)*e^2*x^2 + 4*(b^2*c -
4*a*c^2)*d*e*x + b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*d^2 + (2*c^2*e^4*x^4 + 8*c
^2*d*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^
3 + b*c*d)*e*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*
d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) + (6*(a*b^4*c^
2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*e^8*x^8 + 48*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3
*c^4)*d*e^7*x^7 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3 + 56*(a*b^4*c^2 -
7*a^2*b^2*c^3 + 10*a^3*c^4)*d^2)*e^6*x^6 + 6*(56*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 1
0*a^3*c^4)*d^3 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d)*e^5*x^5 + 6*(a
*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^8 + (6*a*b^6 - 36*a^2*b^4*c + 14*a^3*b^
2*c^2 + 100*a^4*c^3 + 420*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^4 + 45*(4*a
*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^2)*e^4*x^4 + 2*a^3*b^4 - 16*a^4*b^2*c
+ 32*a^5*c^2 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^6 + 4*(84*(a*b^4*
c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^5 + 15*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*
b*c^3)*d^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*d)*e^3*x^3
+ 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*d^4 + (9*a^2*b^5 - 68*
a^3*b^3*c + 122*a^4*b*c^2 + 168*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^6 + 4
5*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^4 + 12*(3*a*b^6 - 18*a^2*b^4*c +
7*a^3*b^2*c^2 + 50*a^4*c^3)*d^2)*e^2*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*
b*c^2)*d^2 + 2*(24*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^7 + 9*(4*a*b^5*c -
29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^5 + 4*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2
+ 50*a^4*c^3)*d^3 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2)*d)*e*x - 3*((b^5*
c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^10*x^10 + 10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2
*b*c^4)*d*e^9*x^9 + (2*b^6*c - 16*a*b^4*c^2 + 32*a^2*b^2*c^3 + 45*(b^5*c^2 - 8*a
*b^3*c^3 + 16*a^2*b*c^4)*d^2)*e^8*x^8 + 8*(15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*
c^4)*d^3 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d)*e^7*x^7 + (b^7 - 6*a*b^5*
c + 32*a^3*b*c^3 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^4 + 56*(b^6*c -
8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^2)*e^6*x^6 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c
^4)*d^10 + 2*(126*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^5 + 56*(b^6*c - 8*a*b
^4*c^2 + 16*a^2*b^2*c^3)*d^3 + 3*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d)*e^5*x^5 + 2
*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^8 + (2*a*b^6 - 16*a^2*b^4*c + 32*a^3*b
^2*c^2 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^6 + 140*(b^6*c - 8*a*b^4*c
^2 + 16*a^2*b^2*c^3)*d^4 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^2)*e^4*x^4 + (b
^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^6 + 4*(30*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4
)*d^7 + 28*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^5 + 5*(b^7 - 6*a*b^5*c + 32*
a^3*b*c^3)*d^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d)*e^3*x^3 + 2*(a*b^6
- 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 + (45*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)
*d^8 + a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b
^2*c^3)*d^6 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^4 + 12*(a*b^6 - 8*a^2*b^4*c
+ 16*a^3*b^2*c^2)*d^2)*e^2*x^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2 + 2*
(5*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^9 + 8*(b^6*c - 8*a*b^4*c^2 + 16*a^2*
b^2*c^3)*d^7 + 3*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^5 + 4*(a*b^6 - 8*a^2*b^4*c +
16*a^3*b^2*c^2)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*e*x)*log(c*e^4*
x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*
e*x + a) + 12*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^10*x^10 + 10*(b^5*c^2 -
8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^9*x^9 + (2*b^6*c - 16*a*b^4*c^2 + 32*a^2*b^2*c^3
+ 45*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2)*e^8*x^8 + 8*(15*(b^5*c^2 - 8*a
*b^3*c^3 + 16*a^2*b*c^4)*d^3 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d)*e^7*x
^7 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4
)*d^4 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^2)*e^6*x^6 + (b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*d^10 + 2*(126*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^5
+ 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3 + 3*(b^7 - 6*a*b^5*c + 32*a^3*b
*c^3)*d)*e^5*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^8 + (2*a*b^6 - 16*
a^2*b^4*c + 32*a^3*b^2*c^2 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^6 + 14
0*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^4 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*c^
3)*d^2)*e^4*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^6 + 4*(30*(b^5*c^2 - 8*a*b^
3*c^3 + 16*a^2*b*c^4)*d^7 + 28*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^5 + 5*(b
^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d)
*e^3*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 + (45*(b^5*c^2 - 8*a*b^3
*c^3 + 16*a^2*b*c^4)*d^8 + a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 56*(b^6*c - 8*
a*b^4*c^2 + 16*a^2*b^2*c^3)*d^6 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^4 + 12*(
a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^2)*e^2*x^2 + (a^2*b^5 - 8*a^3*b^3*c + 16
*a^4*b*c^2)*d^2 + 2*(5*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^9 + 8*(b^6*c - 8
*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^7 + 3*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^5 + 4*(a
*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2
)*d)*e*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a
^6*c^4)*e^11*f^3*x^10 + 10*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d*e^10*f^3
*x^9 + (2*a^4*b^5*c - 16*a^5*b^3*c^2 + 32*a^6*b*c^3 + 45*(a^4*b^4*c^2 - 8*a^5*b^
2*c^3 + 16*a^6*c^4)*d^2)*e^9*f^3*x^8 + 8*(15*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a
^6*c^4)*d^3 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d)*e^8*f^3*x^7 + (a^4
*b^6 - 6*a^5*b^4*c + 32*a^7*c^3 + 210*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)
*d^4 + 56*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^2)*e^7*f^3*x^6 + 2*(126*(
a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^5 + 56*(a^4*b^5*c - 8*a^5*b^3*c^2 +
16*a^6*b*c^3)*d^3 + 3*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d)*e^6*f^3*x^5 + (2*a
^5*b^5 - 16*a^6*b^3*c + 32*a^7*b*c^2 + 210*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6
*c^4)*d^6 + 140*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^4 + 15*(a^4*b^6 - 6
*a^5*b^4*c + 32*a^7*c^3)*d^2)*e^5*f^3*x^4 + 4*(30*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 +
16*a^6*c^4)*d^7 + 28*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^5 + 5*(a^4*b^
6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^3 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d)*
e^4*f^3*x^3 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2 + 45*(a^4*b^4*c^2 - 8*a^5*b^2*
c^3 + 16*a^6*c^4)*d^8 + 56*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^6 + 15*(
a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^4 + 12*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c
^2)*d^2)*e^3*f^3*x^2 + 2*(5*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^9 + 8*(
a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^7 + 3*(a^4*b^6 - 6*a^5*b^4*c + 32*a^
7*c^3)*d^5 + 4*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^3 + (a^6*b^4 - 8*a^7*b^2
*c + 16*a^8*c^2)*d)*e^2*f^3*x + ((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^10
+ 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^8 + (a^4*b^6 - 6*a^5*b^4*c + 3
2*a^7*c^3)*d^6 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^4 + (a^6*b^4 - 8*a^7
*b^2*c + 16*a^8*c^2)*d^2)*e*f^3)*sqrt(b^2 - 4*a*c)), 1/4*(6*((b^6*c^2 - 10*a*b^4
*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*e^10*x^10 + 10*(b^6*c^2 - 10*a*b^4*c^3 + 30*
a^2*b^2*c^4 - 20*a^3*c^5)*d*e^9*x^9 + (2*b^7*c - 20*a*b^5*c^2 + 60*a^2*b^3*c^3 -
40*a^3*b*c^4 + 45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^2)*e
^8*x^8 + 8*(15*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^3 + 2*(b
^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d)*e^7*x^7 + (b^8 - 8*a*b^6
*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4 + 210*(b^6*c^2 - 10*a*b^4*c^3
+ 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^4 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 -
20*a^3*b*c^4)*d^2)*e^6*x^6 + (b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*
c^5)*d^10 + 2*(126*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^5 +
56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^3 + 3*(b^8 - 8*a*b^6
*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d)*e^5*x^5 + 2*(b^7*c - 10*a*
b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^8 + (2*a*b^7 - 20*a^2*b^5*c + 60*a^3*
b^3*c^2 - 40*a^4*b*c^3 + 210*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c
^5)*d^6 + 140*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^4 + 15*(b
^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^2)*e^4*x^4 + (b
^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^6 + 4*(30*(b^6*
c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^7 + 28*(b^7*c - 10*a*b^5*c^2
+ 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^5 + 5*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40
*a^3*b^2*c^3 - 40*a^4*c^4)*d^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a
^4*b*c^3)*d)*e^3*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*
d^4 + (45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^8 + a^2*b^6 -
10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*
b^3*c^3 - 20*a^3*b*c^4)*d^6 + 15*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*
c^3 - 40*a^4*c^4)*d^4 + 12*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3
)*d^2)*e^2*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*d^2 + 2*
(5*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^9 + 8*(b^7*c - 10*a*
b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^7 + 3*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c
^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^5 + 4*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2
- 20*a^4*b*c^3)*d^3 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*d)
*e*x)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 -
4*a*c)) - (6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*e^8*x^8 + 48*(a*b^4*c^2 -
7*a^2*b^2*c^3 + 10*a^3*c^4)*d*e^7*x^7 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b
*c^3 + 56*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^2)*e^6*x^6 + 6*(56*(a*b^4*c
^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^3 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*
c^3)*d)*e^5*x^5 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^8 + (6*a*b^6 - 36
*a^2*b^4*c + 14*a^3*b^2*c^2 + 100*a^4*c^3 + 420*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*
a^3*c^4)*d^4 + 45*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^2)*e^4*x^4 + 2*a
^3*b^4 - 16*a^4*b^2*c + 32*a^5*c^2 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^
3)*d^6 + 4*(84*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*d^5 + 15*(4*a*b^5*c - 29
*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 5
0*a^4*c^3)*d)*e^3*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*
d^4 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2 + 168*(a*b^4*c^2 - 7*a^2*b^2*c^3
+ 10*a^3*c^4)*d^6 + 45*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^4 + 12*(3*
a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*d^2)*e^2*x^2 + (9*a^2*b^5 - 6
8*a^3*b^3*c + 122*a^4*b*c^2)*d^2 + 2*(24*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4
)*d^7 + 9*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*d^5 + 4*(3*a*b^6 - 18*a^2*
b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*d^3 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*
c^2)*d)*e*x - 3*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^10*x^10 + 10*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^9*x^9 + (2*b^6*c - 16*a*b^4*c^2 + 32*a^2*b^2*c
^3 + 45*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2)*e^8*x^8 + 8*(15*(b^5*c^2 - 8
*a*b^3*c^3 + 16*a^2*b*c^4)*d^3 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d)*e^7
*x^7 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c
^4)*d^4 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^2)*e^6*x^6 + (b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*d^10 + 2*(126*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d
^5 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3 + 3*(b^7 - 6*a*b^5*c + 32*a^3
*b*c^3)*d)*e^5*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^8 + (2*a*b^6 - 1
6*a^2*b^4*c + 32*a^3*b^2*c^2 + 210*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^6 +
140*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^4 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*
c^3)*d^2)*e^4*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^6 + 4*(30*(b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*d^7 + 28*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^5 + 5*
(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*
d)*e^3*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 + (45*(b^5*c^2 - 8*a*b
^3*c^3 + 16*a^2*b*c^4)*d^8 + a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 56*(b^6*c -
8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^6 + 15*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^4 + 12
*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^2)*e^2*x^2 + (a^2*b^5 - 8*a^3*b^3*c +
16*a^4*b*c^2)*d^2 + 2*(5*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^9 + 8*(b^6*c -
8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^7 + 3*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^5 + 4*
(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c
^2)*d)*e*x)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^
2 + 2*(2*c*d^3 + b*d)*e*x + a) + 12*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^10
*x^10 + 10*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^9*x^9 + (2*b^6*c - 16*a*b^
4*c^2 + 32*a^2*b^2*c^3 + 45*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2)*e^8*x^8
+ 8*(15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3 + 2*(b^6*c - 8*a*b^4*c^2 + 16
*a^2*b^2*c^3)*d)*e^7*x^7 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3 + 210*(b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*d^4 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^2)*e^6
*x^6 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^10 + 2*(126*(b^5*c^2 - 8*a*b^3*c
^3 + 16*a^2*b*c^4)*d^5 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3 + 3*(b^7
- 6*a*b^5*c + 32*a^3*b*c^3)*d)*e^5*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3
)*d^8 + (2*a*b^6 - 16*a^2*b^4*c + 32*a^3*b^2*c^2 + 210*(b^5*c^2 - 8*a*b^3*c^3 +
16*a^2*b*c^4)*d^6 + 140*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^4 + 15*(b^7 - 6
*a*b^5*c + 32*a^3*b*c^3)*d^2)*e^4*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^6 + 4
*(30*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^7 + 28*(b^6*c - 8*a*b^4*c^2 + 16*a
^2*b^2*c^3)*d^5 + 5*(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d^3 + 2*(a*b^6 - 8*a^2*b^4*
c + 16*a^3*b^2*c^2)*d)*e^3*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^4 +
(45*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^8 + a^2*b^5 - 8*a^3*b^3*c + 16*a^4*
b*c^2 + 56*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^6 + 15*(b^7 - 6*a*b^5*c + 32
*a^3*b*c^3)*d^4 + 12*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^2)*e^2*x^2 + (a^2*
b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2 + 2*(5*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c
^4)*d^9 + 8*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^7 + 3*(b^7 - 6*a*b^5*c + 32
*a^3*b*c^3)*d^5 + 4*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3 + (a^2*b^5 - 8*a^
3*b^3*c + 16*a^4*b*c^2)*d)*e*x)*log(e*x + d))*sqrt(-b^2 + 4*a*c))/(((a^4*b^4*c^2
- 8*a^5*b^2*c^3 + 16*a^6*c^4)*e^11*f^3*x^10 + 10*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 +
16*a^6*c^4)*d*e^10*f^3*x^9 + (2*a^4*b^5*c - 16*a^5*b^3*c^2 + 32*a^6*b*c^3 + 45*
(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^2)*e^9*f^3*x^8 + 8*(15*(a^4*b^4*c^2
- 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^3 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3
)*d)*e^8*f^3*x^7 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3 + 210*(a^4*b^4*c^2 - 8*a^
5*b^2*c^3 + 16*a^6*c^4)*d^4 + 56*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^2)
*e^7*f^3*x^6 + 2*(126*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^5 + 56*(a^4*b
^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^3 + 3*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3
)*d)*e^6*f^3*x^5 + (2*a^5*b^5 - 16*a^6*b^3*c + 32*a^7*b*c^2 + 210*(a^4*b^4*c^2 -
8*a^5*b^2*c^3 + 16*a^6*c^4)*d^6 + 140*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3
)*d^4 + 15*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^2)*e^5*f^3*x^4 + 4*(30*(a^4*b^
4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^7 + 28*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6
*b*c^3)*d^5 + 5*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^3 + 2*(a^5*b^5 - 8*a^6*b^
3*c + 16*a^7*b*c^2)*d)*e^4*f^3*x^3 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2 + 45*(a
^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*d^8 + 56*(a^4*b^5*c - 8*a^5*b^3*c^2 + 1
6*a^6*b*c^3)*d^6 + 15*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^4 + 12*(a^5*b^5 - 8
*a^6*b^3*c + 16*a^7*b*c^2)*d^2)*e^3*f^3*x^2 + 2*(5*(a^4*b^4*c^2 - 8*a^5*b^2*c^3
+ 16*a^6*c^4)*d^9 + 8*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^7 + 3*(a^4*b^
6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^5 + 4*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^3
+ (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d)*e^2*f^3*x + ((a^4*b^4*c^2 - 8*a^5*b^2
*c^3 + 16*a^6*c^4)*d^10 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d^8 + (a^
4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^6 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)
*d^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d^2)*e*f^3)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

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(1/(e*f*x+d*f)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.335031, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((e*x + d)^4*c + (e*x + d)^2*b + a)^3*(e*f*x + d*f)^3),x, algorithm="giac")

[Out]

Done