\(\int \frac {1}{(a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3)^2} \, dx\) [19]

Optimal result

Integrand size = 46, antiderivative size = 234 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \]

[Out]

-b^3/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)-d^3/(-a*d+b*c)^2/(-c*f+d*e)^2/(d*x+c)-f^3/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*
x+e)-2*b^3*(-2*a*d*f+b*c*f+b*d*e)*ln(b*x+a)/(-a*d+b*c)^3/(-a*f+b*e)^3+2*d^3*(a*d*f-2*b*c*f+b*d*e)*ln(d*x+c)/(-
a*d+b*c)^3/(-c*f+d*e)^3+2*f^3*(-a*d*f-b*c*f+2*b*d*e)*ln(f*x+e)/(-a*f+b*e)^3/(-c*f+d*e)^3

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2083} \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=-\frac {b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac {d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac {2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac {f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac {2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \]

[In]

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

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) +
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(d*e - c*f)^3) + (2*f^3*(2*b*d*e - b*c*f - a*d*f
)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

Rule 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^4}{(b c-a d)^2 (b e-a f)^2 (a+b x)^2}-\frac {2 b^4 (b d e+b c f-2 a d f)}{(b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {d^4}{(b c-a d)^2 (-d e+c f)^2 (c+d x)^2}-\frac {2 d^4 (b d e-2 b c f+a d f)}{(b c-a d)^3 (-d e+c f)^3 (c+d x)}+\frac {f^4}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 f^4 (-2 b d e+b c f+a d f)}{(b e-a f)^3 (d e-c f)^3 (e+f x)}\right ) \, dx \\ & = -\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}-\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (-d e+c f)^3}-\frac {2 f^3 (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \]

[In]

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

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) -
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(-(d*e) + c*f)^3) - (2*f^3*(-2*b*d*e + b*c*f + a
*d*f)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {f^{3}}{\left (c f -e d \right )^{2} \left (a f -b e \right )^{2} \left (f x +e \right )}-\frac {2 f^{3} \left (a d f +f b c -2 b d e \right ) \ln \left (f x +e \right )}{\left (c f -e d \right )^{3} \left (a f -b e \right )^{3}}-\frac {d^{3}}{\left (c f -e d \right )^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 d^{3} \left (a d f -2 f b c +b d e \right ) \ln \left (d x +c \right )}{\left (c f -e d \right )^{3} \left (d a -b c \right )^{3}}-\frac {b^{3}}{\left (a f -b e \right )^{2} \left (d a -b c \right )^{2} \left (b x +a \right )}+\frac {2 b^{3} \left (2 a d f -f b c -b d e \right ) \ln \left (b x +a \right )}{\left (a f -b e \right )^{3} \left (d a -b c \right )^{3}}\)	\(235\)
norman	\(\text {Expression too large to display}\)	\(1232\)
risch	\(\text {Expression too large to display}\)	\(3349\)
parallelrisch	\(\text {Expression too large to display}\)	\(4300\)

[In]

int(1/(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^2,x,method=_RETURNVERBOSE)

[Out]

-f^3/(c*f-d*e)^2/(a*f-b*e)^2/(f*x+e)-2*f^3*(a*d*f+b*c*f-2*b*d*e)/(c*f-d*e)^3/(a*f-b*e)^3*ln(f*x+e)-d^3/(c*f-d*
e)^2/(a*d-b*c)^2/(d*x+c)+2*d^3*(a*d*f-2*b*c*f+b*d*e)/(c*f-d*e)^3/(a*d-b*c)^3*ln(d*x+c)-b^3/(a*f-b*e)^2/(a*d-b*
c)^2/(b*x+a)+2*b^3*(2*a*d*f-b*c*f-b*d*e)/(a*f-b*e)^3/(a*d-b*c)^3*ln(b*x+a)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (234) = 468\).

Time = 0.32 (sec) , antiderivative size = 2096, normalized size of antiderivative = 8.96 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2*(b^4*d*e + (b^4*c - 2*a*b^3*d)*f)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*e
^3 - 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*e^2*f + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d
+ 3*a^4*b^2*c*d^2 - a^5*b*d^3)*e*f^2 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*f^3) + 2*(b*d
^4*e - (2*b*c*d^3 - a*d^4)*f)*log(d*x + c)/((b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6)*e^3 - 3*
(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*e^2*f + 3*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b
*c^3*d^3 - a^3*c^2*d^4)*e*f^2 - (b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*f^3) + 2*(2*b*d*e*f^
3 - (b*c + a*d)*f^4)*log(f*x + e)/(b^3*d^3*e^6 + a^3*c^3*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f + 3*(b^3*c^2*d
+ 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^3 + 3*(a*b^2*
c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^5) - ((b^3*c*d^2 + a*b^2*d^3)*e^3 - 2
*(b^3*c^2*d + a^2*b*d^3)*e^2*f + (b^3*c^3 + a^3*d^3)*e*f^2 + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*f^3 + 2*(
b^3*d^3*e^2*f - (b^3*c*d^2 + a*b^2*d^3)*e*f^2 + (b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d^3)*f^3)*x^2 + (2*b^3*d^3*e^
3 - (b^3*c*d^2 + a*b^2*d^3)*e^2*f - (b^3*c^2*d + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 2
*a^3*d^3)*f^3)*x)/((a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*e^5 - 2*(a*b^4*c^4*d - a^2*b^3*c^3*d^2
- a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*e^4*f + (a*b^4*c^5 + 2*a^2*b^3*c^4*d - 6*a^3*b^2*c^3*d^2 + 2*a^4*b*c^2*d^3 +
a^5*c*d^4)*e^3*f^2 - 2*(a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*e^2*f^3 + (a^3*b^2*c^5 - 2*
a^4*b*c^4*d + a^5*c^3*d^2)*e*f^4 + ((b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*e^4*f - 2*(b^5*c^3*d^2 - a*b^4
*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^3*f^2 + (b^5*c^4*d + 2*a*b^4*c^3*d^2 - 6*a^2*b^3*c^2*d^3 + 2*a^3*b^2
*c*d^4 + a^4*b*d^5)*e^2*f^3 - 2*(a*b^4*c^4*d - a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*e*f^4 + (a^2*b
^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*f^5)*x^3 + ((b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*e^5 - (b
^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^4*f - (b^5*c^4*d - 2*a*b^4*c^3*d^2 + 2*a^2*b^3*c^2
*d^3 - 2*a^3*b^2*c*d^4 + a^4*b*d^5)*e^3*f^2 + (b^5*c^5 + a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 +
a^4*b*c*d^4 + a^5*d^5)*e^2*f^3 - (2*a*b^4*c^5 - a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - a^4*b*c^2*d^3 + 2*a^5*c*d
^4)*e*f^4 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*f^5)*x^2 + ((b^5*c^3*d^2 - a*b^4*c^2*d
^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*e^5 - (2*b^5*c^4*d - a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - a^3*b^2*c*d^4 + 2*a
^4*b*d^5)*e^4*f + (b^5*c^5 + a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + a^5*d^5)*e^3*
f^2 - (a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^3*b^2*c^3*d^2 - 2*a^4*b*c^2*d^3 + a^5*c*d^4)*e^2*f^3 - (a^2*b^3*c^5 -
a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*e*f^4 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*f^5)*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1435 vs. \(2 (234) = 468\).

Time = 0.28 (sec) , antiderivative size = 1435, normalized size of antiderivative = 6.13 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2*(b^5*d*e + b^5*c*f - 2*a*b^4*d*f)*log(abs(b*x + a))/(b^7*c^3*e^3 - 3*a*b^6*c^2*d*e^3 + 3*a^2*b^5*c*d^2*e^3
- a^3*b^4*d^3*e^3 - 3*a*b^6*c^3*e^2*f + 9*a^2*b^5*c^2*d*e^2*f - 9*a^3*b^4*c*d^2*e^2*f + 3*a^4*b^3*d^3*e^2*f +
3*a^2*b^5*c^3*e*f^2 - 9*a^3*b^4*c^2*d*e*f^2 + 9*a^4*b^3*c*d^2*e*f^2 - 3*a^5*b^2*d^3*e*f^2 - a^3*b^4*c^3*f^3 +
3*a^4*b^3*c^2*d*f^3 - 3*a^5*b^2*c*d^2*f^3 + a^6*b*d^3*f^3) + 2*(b*d^5*e - 2*b*c*d^4*f + a*d^5*f)*log(abs(d*x +
c))/(b^3*c^3*d^4*e^3 - 3*a*b^2*c^2*d^5*e^3 + 3*a^2*b*c*d^6*e^3 - a^3*d^7*e^3 - 3*b^3*c^4*d^3*e^2*f + 9*a*b^2*
c^3*d^4*e^2*f - 9*a^2*b*c^2*d^5*e^2*f + 3*a^3*c*d^6*e^2*f + 3*b^3*c^5*d^2*e*f^2 - 9*a*b^2*c^4*d^3*e*f^2 + 9*a^
2*b*c^3*d^4*e*f^2 - 3*a^3*c^2*d^5*e*f^2 - b^3*c^6*d*f^3 + 3*a*b^2*c^5*d^2*f^3 - 3*a^2*b*c^4*d^3*f^3 + a^3*c^3*
d^4*f^3) + 2*(2*b*d*e*f^4 - b*c*f^5 - a*d*f^5)*log(abs(f*x + e))/(b^3*d^3*e^6*f - 3*b^3*c*d^2*e^5*f^2 - 3*a*b^
2*d^3*e^5*f^2 + 3*b^3*c^2*d*e^4*f^3 + 9*a*b^2*c*d^2*e^4*f^3 + 3*a^2*b*d^3*e^4*f^3 - b^3*c^3*e^3*f^4 - 9*a*b^2*
c^2*d*e^3*f^4 - 9*a^2*b*c*d^2*e^3*f^4 - a^3*d^3*e^3*f^4 + 3*a*b^2*c^3*e^2*f^5 + 9*a^2*b*c^2*d*e^2*f^5 + 3*a^3*
c*d^2*e^2*f^5 - 3*a^2*b*c^3*e*f^6 - 3*a^3*c^2*d*e*f^6 + a^3*c^3*f^7) - (2*b^3*d^3*e^2*f*x^2 - 2*b^3*c*d^2*e*f^
2*x^2 - 2*a*b^2*d^3*e*f^2*x^2 + 2*b^3*c^2*d*f^3*x^2 - 2*a*b^2*c*d^2*f^3*x^2 + 2*a^2*b*d^3*f^3*x^2 + 2*b^3*d^3*
e^3*x - b^3*c*d^2*e^2*f*x - a*b^2*d^3*e^2*f*x - b^3*c^2*d*e*f^2*x - a^2*b*d^3*e*f^2*x + 2*b^3*c^3*f^3*x - a*b^
2*c^2*d*f^3*x - a^2*b*c*d^2*f^3*x + 2*a^3*d^3*f^3*x + b^3*c*d^2*e^3 + a*b^2*d^3*e^3 - 2*b^3*c^2*d*e^2*f - 2*a^
2*b*d^3*e^2*f + b^3*c^3*e*f^2 + a^3*d^3*e*f^2 + a*b^2*c^3*f^3 - 2*a^2*b*c^2*d*f^3 + a^3*c*d^2*f^3)/((b^4*c^2*d
^2*e^4 - 2*a*b^3*c*d^3*e^4 + a^2*b^2*d^4*e^4 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a^2*b^2*c*d^3*e^3
*f - 2*a^3*b*d^4*e^3*f + b^4*c^4*e^2*f^2 + 2*a*b^3*c^3*d*e^2*f^2 - 6*a^2*b^2*c^2*d^2*e^2*f^2 + 2*a^3*b*c*d^3*e
^2*f^2 + a^4*d^4*e^2*f^2 - 2*a*b^3*c^4*e*f^3 + 2*a^2*b^2*c^3*d*e*f^3 + 2*a^3*b*c^2*d^2*e*f^3 - 2*a^4*c*d^3*e*f
^3 + a^2*b^2*c^4*f^4 - 2*a^3*b*c^3*d*f^4 + a^4*c^2*d^2*f^4)*(b*d*f*x^3 + b*d*e*x^2 + b*c*f*x^2 + a*d*f*x^2 + b
*c*e*x + a*d*e*x + a*c*f*x + a*c*e))

Mupad [B] (verification not implemented)

Time = 13.89 (sec) , antiderivative size = 1940, normalized size of antiderivative = 8.29 \[ \int \frac {1}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((a*b^2*c^3*f^3 + a*b^2*d^3*e^3 + a^3*c*d^2*f^3 + b^3*c*d^2*e^3 + a^3*d^3*e*f^2 + b^3*c^3*e*f^2 - 2*a^2*b*c^
2*d*f^3 - 2*a^2*b*d^3*e^2*f - 2*b^3*c^2*d*e^2*f)/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^
2*d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*
a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^
2*b^2*c*d^3*e^3*f + 2*a^2*b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*
e^2*f^2) + (2*x^2*(a^2*b*d^3*f^3 + b^3*c^2*d*f^3 + b^3*d^3*e^2*f - a*b^2*c*d^2*f^3 - a*b^2*d^3*e*f^2 - b^3*c*d
^2*e*f^2))/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4*
e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 -
2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^2*c*d^3*e^3*f + 2*a^2*b^2*c^3*d*e*
f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2) - (x*(a*b^2*c^2*d*f^3 - 2*b^3
*c^3*f^3 - 2*b^3*d^3*e^3 - 2*a^3*d^3*f^3 + a^2*b*c*d^2*f^3 + a*b^2*d^3*e^2*f + a^2*b*d^3*e*f^2 + b^3*c*d^2*e^2
*f + b^3*c^2*d*e*f^2))/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^4 + a^4*d^4*e^2*f^
2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*
d^3*e*f^3 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^2*c*d^3*e^3*f + 2*a^2*
b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2))/(x^2*(a*d*f + b*
c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d*f*x^3) - (log(a + b*x)*(b^4*(2*c*f + 2*d*e) - 4*a*b^3*d
*f))/(b^6*c^3*e^3 + a^6*d^3*f^3 - a^3*b^3*c^3*f^3 - a^3*b^3*d^3*e^3 - 3*a*b^5*c^2*d*e^3 - 3*a^5*b*c*d^2*f^3 -
3*a*b^5*c^3*e^2*f - 3*a^5*b*d^3*e*f^2 + 3*a^2*b^4*c*d^2*e^3 + 3*a^4*b^2*c^2*d*f^3 + 3*a^2*b^4*c^3*e*f^2 + 3*a^
4*b^2*d^3*e^2*f + 9*a^2*b^4*c^2*d*e^2*f - 9*a^3*b^3*c*d^2*e^2*f - 9*a^3*b^3*c^2*d*e*f^2 + 9*a^4*b^2*c*d^2*e*f^
2) - (log(c + d*x)*(d^4*(2*a*f + 2*b*e) - 4*b*c*d^3*f))/(a^3*d^6*e^3 + b^3*c^6*f^3 - a^3*c^3*d^3*f^3 - b^3*c^3
*d^3*e^3 - 3*a^2*b*c*d^5*e^3 - 3*a*b^2*c^5*d*f^3 - 3*a^3*c*d^5*e^2*f - 3*b^3*c^5*d*e*f^2 + 3*a*b^2*c^2*d^4*e^3
+ 3*a^2*b*c^4*d^2*f^3 + 3*a^3*c^2*d^4*e*f^2 + 3*b^3*c^4*d^2*e^2*f - 9*a*b^2*c^3*d^3*e^2*f + 9*a*b^2*c^4*d^2*e
*f^2 + 9*a^2*b*c^2*d^4*e^2*f - 9*a^2*b*c^3*d^3*e*f^2) - (log(e + f*x)*(f^4*(2*a*d + 2*b*c) - 4*b*d*e*f^3))/(a^
3*c^3*f^6 + b^3*d^3*e^6 - a^3*d^3*e^3*f^3 - b^3*c^3*e^3*f^3 - 3*a^2*b*c^3*e*f^5 - 3*a*b^2*d^3*e^5*f - 3*a^3*c^
2*d*e*f^5 - 3*b^3*c*d^2*e^5*f + 3*a*b^2*c^3*e^2*f^4 + 3*a^2*b*d^3*e^4*f^2 + 3*a^3*c*d^2*e^2*f^4 + 3*b^3*c^2*d*
e^4*f^2 + 9*a*b^2*c*d^2*e^4*f^2 - 9*a*b^2*c^2*d*e^3*f^3 - 9*a^2*b*c*d^2*e^3*f^3 + 9*a^2*b*c^2*d*e^2*f^4)